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Comment:(experimental) Upgrade to libtommath 1.0 (actually by merging all changes between libtommath 0.42.0 and 1.0). Still to be tested thourougly, before doing anything with it.
Downloads: Tarball | ZIP archive
Timelines: family | ancestors | descendants | both | libtommath-1.0
Files: files | file ages | folders
SHA1: d1210bac0ba7064f80757bb3b3ba0d2984e1113e
User & Date: jan.nijtmans 2016-11-16 15:22:26.588
Context
2016-11-17
10:46
Merge trunk. Re-generate tclTomMath.h. Use faster exponentiation-function from libtommath 1.0 (in tc... check-in: 1e2d716ec7 user: jan.nijtmans tags: libtommath-1.0
2016-11-16
15:22
(experimental) Upgrade to libtommath 1.0 (actually by merging all changes between libtommath 0.42.0 ... check-in: d1210bac0b user: jan.nijtmans tags: libtommath-1.0
13:04
import libtommath 1.0 check-in: dfaa44e279 user: jan.nijtmans tags: libtommath
10:55
Use more "size_t" in stead of "int" internall. Also eliminate a lot of type-casts which are not nece... check-in: 521d320b7b user: jan.nijtmans tags: trunk
Changes
Unified Diff Ignore Whitespace Patch
Changes to generic/tclStubInit.c.
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    TclBN_s_mp_sub, /* 60 */
    TclBN_mp_init_set_int, /* 61 */
    TclBN_mp_set_int, /* 62 */
    TclBN_mp_cnt_lsb, /* 63 */
    TclBNInitBignumFromLong, /* 64 */
    TclBNInitBignumFromWideInt, /* 65 */
    TclBNInitBignumFromWideUInt, /* 66 */

};

static const TclStubHooks tclStubHooks = {
    &tclPlatStubs,
    &tclIntStubs,
    &tclIntPlatStubs
};







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    TclBN_s_mp_sub, /* 60 */
    TclBN_mp_init_set_int, /* 61 */
    TclBN_mp_set_int, /* 62 */
    TclBN_mp_cnt_lsb, /* 63 */
    TclBNInitBignumFromLong, /* 64 */
    TclBNInitBignumFromWideInt, /* 65 */
    TclBNInitBignumFromWideUInt, /* 66 */
    TclBN_mp_expt_d_ex, /* 67 */
};

static const TclStubHooks tclStubHooks = {
    &tclPlatStubs,
    &tclIntStubs,
    &tclIntPlatStubs
};
Changes to generic/tclTomMath.decls.
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}
declare 65 {
    void TclBNInitBignumFromWideInt(mp_int *bignum, Tcl_WideInt initVal)
}
declare 66 {
    void TclBNInitBignumFromWideUInt(mp_int *bignum, Tcl_WideUInt initVal)
}






# Local Variables:
# mode: tcl
# End:







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}
declare 65 {
    void TclBNInitBignumFromWideInt(mp_int *bignum, Tcl_WideInt initVal)
}
declare 66 {
    void TclBNInitBignumFromWideUInt(mp_int *bignum, Tcl_WideUInt initVal)
}

# Added in libtommath 1.0
declare 67 {
    int TclBN_mp_expt_d_ex(mp_int *a, mp_digit b, mp_int *c, int fast)
}

# Local Variables:
# mode: tcl
# End:
Changes to generic/tclTomMathDecls.h.
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#define mp_div TclBN_mp_div
#define mp_div_2 TclBN_mp_div_2
#define mp_div_2d TclBN_mp_div_2d
#define mp_div_3 TclBN_mp_div_3
#define mp_div_d TclBN_mp_div_d
#define mp_exch TclBN_mp_exch
#define mp_expt_d TclBN_mp_expt_d

#define mp_grow TclBN_mp_grow
#define mp_init TclBN_mp_init
#define mp_init_copy TclBN_mp_init_copy
#define mp_init_multi TclBN_mp_init_multi
#define mp_init_set TclBN_mp_init_set
#define mp_init_set_int TclBN_mp_init_set_int
#define mp_init_size TclBN_mp_init_size







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#define mp_div TclBN_mp_div
#define mp_div_2 TclBN_mp_div_2
#define mp_div_2d TclBN_mp_div_2d
#define mp_div_3 TclBN_mp_div_3
#define mp_div_d TclBN_mp_div_d
#define mp_exch TclBN_mp_exch
#define mp_expt_d TclBN_mp_expt_d
#define mp_expt_d_ex TclBN_mp_expt_d_ex
#define mp_grow TclBN_mp_grow
#define mp_init TclBN_mp_init
#define mp_init_copy TclBN_mp_init_copy
#define mp_init_multi TclBN_mp_init_multi
#define mp_init_set TclBN_mp_init_set
#define mp_init_set_int TclBN_mp_init_set_int
#define mp_init_size TclBN_mp_init_size
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EXTERN void		TclBNInitBignumFromLong(mp_int *bignum, long initVal);
/* 65 */
EXTERN void		TclBNInitBignumFromWideInt(mp_int *bignum,
				Tcl_WideInt initVal);
/* 66 */
EXTERN void		TclBNInitBignumFromWideUInt(mp_int *bignum,
				Tcl_WideUInt initVal);




typedef struct TclTomMathStubs {
    int magic;
    void *hooks;

    int (*tclBN_epoch) (void); /* 0 */
    int (*tclBN_revision) (void); /* 1 */







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EXTERN void		TclBNInitBignumFromLong(mp_int *bignum, long initVal);
/* 65 */
EXTERN void		TclBNInitBignumFromWideInt(mp_int *bignum,
				Tcl_WideInt initVal);
/* 66 */
EXTERN void		TclBNInitBignumFromWideUInt(mp_int *bignum,
				Tcl_WideUInt initVal);
/* 67 */
EXTERN int		TclBN_mp_expt_d_ex(mp_int *a, mp_digit b, mp_int *c,
				int fast);

typedef struct TclTomMathStubs {
    int magic;
    void *hooks;

    int (*tclBN_epoch) (void); /* 0 */
    int (*tclBN_revision) (void); /* 1 */
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    int (*tclBN_s_mp_sub) (mp_int *a, mp_int *b, mp_int *c); /* 60 */
    int (*tclBN_mp_init_set_int) (mp_int *a, unsigned long i); /* 61 */
    int (*tclBN_mp_set_int) (mp_int *a, unsigned long i); /* 62 */
    int (*tclBN_mp_cnt_lsb) (const mp_int *a); /* 63 */
    void (*tclBNInitBignumFromLong) (mp_int *bignum, long initVal); /* 64 */
    void (*tclBNInitBignumFromWideInt) (mp_int *bignum, Tcl_WideInt initVal); /* 65 */
    void (*tclBNInitBignumFromWideUInt) (mp_int *bignum, Tcl_WideUInt initVal); /* 66 */

} TclTomMathStubs;

extern const TclTomMathStubs *tclTomMathStubsPtr;

#ifdef __cplusplus
}
#endif







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    int (*tclBN_s_mp_sub) (mp_int *a, mp_int *b, mp_int *c); /* 60 */
    int (*tclBN_mp_init_set_int) (mp_int *a, unsigned long i); /* 61 */
    int (*tclBN_mp_set_int) (mp_int *a, unsigned long i); /* 62 */
    int (*tclBN_mp_cnt_lsb) (const mp_int *a); /* 63 */
    void (*tclBNInitBignumFromLong) (mp_int *bignum, long initVal); /* 64 */
    void (*tclBNInitBignumFromWideInt) (mp_int *bignum, Tcl_WideInt initVal); /* 65 */
    void (*tclBNInitBignumFromWideUInt) (mp_int *bignum, Tcl_WideUInt initVal); /* 66 */
    int (*tclBN_mp_expt_d_ex) (mp_int *a, mp_digit b, mp_int *c, int fast); /* 67 */
} TclTomMathStubs;

extern const TclTomMathStubs *tclTomMathStubsPtr;

#ifdef __cplusplus
}
#endif
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	(tclTomMathStubsPtr->tclBN_mp_cnt_lsb) /* 63 */
#define TclBNInitBignumFromLong \
	(tclTomMathStubsPtr->tclBNInitBignumFromLong) /* 64 */
#define TclBNInitBignumFromWideInt \
	(tclTomMathStubsPtr->tclBNInitBignumFromWideInt) /* 65 */
#define TclBNInitBignumFromWideUInt \
	(tclTomMathStubsPtr->tclBNInitBignumFromWideUInt) /* 66 */



#endif /* defined(USE_TCL_STUBS) */

/* !END!: Do not edit above this line. */

#undef TCL_STORAGE_CLASS
#define TCL_STORAGE_CLASS DLLIMPORT

#endif /* _TCLINTDECLS */







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	(tclTomMathStubsPtr->tclBN_mp_cnt_lsb) /* 63 */
#define TclBNInitBignumFromLong \
	(tclTomMathStubsPtr->tclBNInitBignumFromLong) /* 64 */
#define TclBNInitBignumFromWideInt \
	(tclTomMathStubsPtr->tclBNInitBignumFromWideInt) /* 65 */
#define TclBNInitBignumFromWideUInt \
	(tclTomMathStubsPtr->tclBNInitBignumFromWideUInt) /* 66 */
#define TclBN_mp_expt_d_ex \
	(tclTomMathStubsPtr->tclBN_mp_expt_d_ex) /* 67 */

#endif /* defined(USE_TCL_STUBS) */

/* !END!: Do not edit above this line. */

#undef TCL_STORAGE_CLASS
#define TCL_STORAGE_CLASS DLLIMPORT

#endif /* _TCLINTDECLS */
Changes to libtommath/LICENSE.

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LibTomMath is hereby released into the Public Domain.  






-- Tom St Denis




















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LibTomMath is licensed under DUAL licensing terms.

Choose and use the license of your needs.

[LICENSE #1]

LibTomMath is public domain.  As should all quality software be.

Tom St Denis

[/LICENSE #1]

[LICENSE #2]

            DO WHAT THE FUCK YOU WANT TO PUBLIC LICENSE
                    Version 2, December 2004

 Copyright (C) 2004 Sam Hocevar <[email protected]>

 Everyone is permitted to copy and distribute verbatim or modified
 copies of this license document, and changing it is allowed as long
 as the name is changed.

            DO WHAT THE FUCK YOU WANT TO PUBLIC LICENSE
   TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION

  0. You just DO WHAT THE FUCK YOU WANT TO. 

[/LICENSE #2]
Changes to libtommath/bn.ilg.
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This is makeindex, version 2.14 [02-Oct-2002] (kpathsea + Thai support).
Scanning input file bn.idx....done (79 entries accepted, 0 rejected).
Sorting entries....done (511 comparisons).
Generating output file bn.ind....done (82 lines written, 0 warnings).
Output written in bn.ind.
Transcript written in bn.ilg.
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This is makeindex, version 2.15 [TeX Live 2013] (kpathsea + Thai support).
Scanning input file bn.idx....done (85 entries accepted, 0 rejected).
Sorting entries....done (554 comparisons).
Generating output file bn.ind....done (88 lines written, 0 warnings).
Output written in bn.ind.
Transcript written in bn.ilg.
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\begin{theindex}

  \item mp\_add, \hyperpage{29}
  \item mp\_add\_d, \hyperpage{52}
  \item mp\_and, \hyperpage{29}
  \item mp\_clear, \hyperpage{11}
  \item mp\_clear\_multi, \hyperpage{12}
  \item mp\_cmp, \hyperpage{24}
  \item mp\_cmp\_d, \hyperpage{25}
  \item mp\_cmp\_mag, \hyperpage{23}
  \item mp\_div, \hyperpage{30}
  \item mp\_div\_2, \hyperpage{26}
  \item mp\_div\_2d, \hyperpage{28}
  \item mp\_div\_d, \hyperpage{52}
  \item mp\_dr\_reduce, \hyperpage{40}
  \item mp\_dr\_setup, \hyperpage{40}
  \item MP\_EQ, \hyperpage{22}
  \item mp\_error\_to\_string, \hyperpage{10}
  \item mp\_expt\_d, \hyperpage{43}

  \item mp\_exptmod, \hyperpage{43}
  \item mp\_exteuclid, \hyperpage{51}
  \item mp\_gcd, \hyperpage{51}
  \item mp\_get\_int, \hyperpage{20}


  \item mp\_grow, \hyperpage{16}
  \item MP\_GT, \hyperpage{22}
  \item mp\_init, \hyperpage{11}
  \item mp\_init\_copy, \hyperpage{13}
  \item mp\_init\_multi, \hyperpage{12}
  \item mp\_init\_set, \hyperpage{21}
  \item mp\_init\_set\_int, \hyperpage{21}
  \item mp\_init\_size, \hyperpage{14}
  \item mp\_int, \hyperpage{10}
  \item mp\_invmod, \hyperpage{52}
  \item mp\_jacobi, \hyperpage{52}
  \item mp\_lcm, \hyperpage{51}
  \item mp\_lshd, \hyperpage{28}
  \item MP\_LT, \hyperpage{22}
  \item MP\_MEM, \hyperpage{9}
  \item mp\_mod, \hyperpage{35}
  \item mp\_mod\_d, \hyperpage{52}
  \item mp\_montgomery\_calc\_normalization, \hyperpage{38}
  \item mp\_montgomery\_reduce, \hyperpage{37}
  \item mp\_montgomery\_setup, \hyperpage{37}
  \item mp\_mul, \hyperpage{31}
  \item mp\_mul\_2, \hyperpage{26}
  \item mp\_mul\_2d, \hyperpage{28}
  \item mp\_mul\_d, \hyperpage{52}
  \item mp\_n\_root, \hyperpage{44}
  \item mp\_neg, \hyperpage{29}
  \item MP\_NO, \hyperpage{9}
  \item MP\_OKAY, \hyperpage{9}
  \item mp\_or, \hyperpage{29}
  \item mp\_prime\_fermat, \hyperpage{45}
  \item mp\_prime\_is\_divisible, \hyperpage{45}
  \item mp\_prime\_is\_prime, \hyperpage{46}
  \item mp\_prime\_miller\_rabin, \hyperpage{45}
  \item mp\_prime\_next\_prime, \hyperpage{46}
  \item mp\_prime\_rabin\_miller\_trials, \hyperpage{46}
  \item mp\_prime\_random, \hyperpage{47}
  \item mp\_prime\_random\_ex, \hyperpage{47}
  \item mp\_radix\_size, \hyperpage{49}
  \item mp\_read\_radix, \hyperpage{49}
  \item mp\_read\_unsigned\_bin, \hyperpage{50}
  \item mp\_reduce, \hyperpage{36}
  \item mp\_reduce\_2k, \hyperpage{41}
  \item mp\_reduce\_2k\_setup, \hyperpage{41}
  \item mp\_reduce\_setup, \hyperpage{36}
  \item mp\_rshd, \hyperpage{28}
  \item mp\_set, \hyperpage{19}
  \item mp\_set\_int, \hyperpage{20}


  \item mp\_shrink, \hyperpage{15}
  \item mp\_sqr, \hyperpage{33}

  \item mp\_sub, \hyperpage{29}
  \item mp\_sub\_d, \hyperpage{52}
  \item mp\_to\_unsigned\_bin, \hyperpage{50}
  \item mp\_toradix, \hyperpage{49}
  \item mp\_unsigned\_bin\_size, \hyperpage{50}
  \item MP\_VAL, \hyperpage{9}
  \item mp\_xor, \hyperpage{29}
  \item MP\_YES, \hyperpage{9}

\end{theindex}


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\begin{theindex}

  \item mp\_add, \hyperpage{24}
  \item mp\_add\_d, \hyperpage{44}
  \item mp\_and, \hyperpage{24}
  \item mp\_clear, \hyperpage{9}
  \item mp\_clear\_multi, \hyperpage{10}
  \item mp\_cmp, \hyperpage{19}
  \item mp\_cmp\_d, \hyperpage{20}
  \item mp\_cmp\_mag, \hyperpage{18}
  \item mp\_div, \hyperpage{24}
  \item mp\_div\_2, \hyperpage{22}
  \item mp\_div\_2d, \hyperpage{23}
  \item mp\_div\_d, \hyperpage{44}
  \item mp\_dr\_reduce, \hyperpage{33}
  \item mp\_dr\_setup, \hyperpage{33}
  \item MP\_EQ, \hyperpage{18}
  \item mp\_error\_to\_string, \hyperpage{7}
  \item mp\_expt\_d, \hyperpage{35}
  \item mp\_expt\_d\_ex, \hyperpage{35}
  \item mp\_exptmod, \hyperpage{35}
  \item mp\_exteuclid, \hyperpage{43}
  \item mp\_gcd, \hyperpage{43}
  \item mp\_get\_int, \hyperpage{16}
  \item mp\_get\_long, \hyperpage{17}
  \item mp\_get\_long\_long, \hyperpage{17}
  \item mp\_grow, \hyperpage{13}
  \item MP\_GT, \hyperpage{18}
  \item mp\_init, \hyperpage{8}
  \item mp\_init\_copy, \hyperpage{10}
  \item mp\_init\_multi, \hyperpage{10}
  \item mp\_init\_set, \hyperpage{17}
  \item mp\_init\_set\_int, \hyperpage{17}
  \item mp\_init\_size, \hyperpage{11}
  \item mp\_int, \hyperpage{8}
  \item mp\_invmod, \hyperpage{44}
  \item mp\_jacobi, \hyperpage{43}
  \item mp\_lcm, \hyperpage{43}
  \item mp\_lshd, \hyperpage{23}
  \item MP\_LT, \hyperpage{18}
  \item MP\_MEM, \hyperpage{7}
  \item mp\_mod, \hyperpage{29}
  \item mp\_mod\_d, \hyperpage{44}
  \item mp\_montgomery\_calc\_normalization, \hyperpage{31}
  \item mp\_montgomery\_reduce, \hyperpage{31}
  \item mp\_montgomery\_setup, \hyperpage{31}
  \item mp\_mul, \hyperpage{25}
  \item mp\_mul\_2, \hyperpage{22}
  \item mp\_mul\_2d, \hyperpage{23}
  \item mp\_mul\_d, \hyperpage{44}
  \item mp\_n\_root, \hyperpage{36}
  \item mp\_neg, \hyperpage{24}
  \item MP\_NO, \hyperpage{7}
  \item MP\_OKAY, \hyperpage{7}
  \item mp\_or, \hyperpage{24}
  \item mp\_prime\_fermat, \hyperpage{37}
  \item mp\_prime\_is\_divisible, \hyperpage{37}
  \item mp\_prime\_is\_prime, \hyperpage{38}
  \item mp\_prime\_miller\_rabin, \hyperpage{37}
  \item mp\_prime\_next\_prime, \hyperpage{38}
  \item mp\_prime\_rabin\_miller\_trials, \hyperpage{38}
  \item mp\_prime\_random, \hyperpage{38}
  \item mp\_prime\_random\_ex, \hyperpage{39}
  \item mp\_radix\_size, \hyperpage{41}
  \item mp\_read\_radix, \hyperpage{41}
  \item mp\_read\_unsigned\_bin, \hyperpage{42}
  \item mp\_reduce, \hyperpage{30}
  \item mp\_reduce\_2k, \hyperpage{34}
  \item mp\_reduce\_2k\_setup, \hyperpage{34}
  \item mp\_reduce\_setup, \hyperpage{29}
  \item mp\_rshd, \hyperpage{23}
  \item mp\_set, \hyperpage{15}
  \item mp\_set\_int, \hyperpage{16}
  \item mp\_set\_long, \hyperpage{17}
  \item mp\_set\_long\_long, \hyperpage{17}
  \item mp\_shrink, \hyperpage{12}
  \item mp\_sqr, \hyperpage{26}
  \item mp\_sqrtmod\_prime, \hyperpage{44}
  \item mp\_sub, \hyperpage{24}
  \item mp\_sub\_d, \hyperpage{44}
  \item mp\_to\_unsigned\_bin, \hyperpage{42}
  \item mp\_toradix, \hyperpage{41}
  \item mp\_unsigned\_bin\_size, \hyperpage{41}
  \item MP\_VAL, \hyperpage{7}
  \item mp\_xor, \hyperpage{24}
  \item MP\_YES, \hyperpage{7}

\end{theindex}
Changes to libtommath/bn.tex.
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\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}}
\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}}
\def\gap{\vspace{0.5ex}}
\makeindex
\begin{document}
\frontmatter
\pagestyle{empty}
\title{LibTomMath User Manual \\ v0.39}
\author{Tom St Denis \\ tomstdenis@iahu.ca}
\maketitle
This text, the library and the accompanying textbook are all hereby placed in the public domain.  This book has been 
formatted for B5 [176x250] paper using the \LaTeX{} {\em book} macro package.

\vspace{10cm}

\begin{flushright}Open Source.  Open Academia.  Open Minds.

\mbox{ }







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\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}}
\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}}
\def\gap{\vspace{0.5ex}}
\makeindex
\begin{document}
\frontmatter
\pagestyle{empty}
\title{LibTomMath User Manual \\ v1.0.0}
\author{Tom St Denis \\ tstdenis82@gmail.com}
\maketitle
This text, the library and the accompanying textbook are all hereby placed in the public domain.  This book has been
formatted for B5 [176x250] paper using the \LaTeX{} {\em book} macro package.

\vspace{10cm}

\begin{flushright}Open Source.  Open Academia.  Open Minds.

\mbox{ }
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\listoffigures
\mainmatter
\pagestyle{headings}
\chapter{Introduction}
\section{What is LibTomMath?}
LibTomMath is a library of source code which provides a series of efficient and carefully written functions for manipulating
large integer numbers.  It was written in portable ISO C source code so that it will build on any platform with a conforming
C compiler.  

In a nutshell the library was written from scratch with verbose comments to help instruct computer science students how
to implement ``bignum'' math.  However, the resulting code has proven to be very useful.  It has been used by numerous 
universities, commercial and open source software developers.  It has been used on a variety of platforms ranging from
Linux and Windows based x86 to ARM based Gameboys and PPC based MacOS machines.  

\section{License}
As of the v0.25 the library source code has been placed in the public domain with every new release.  As of the v0.28
release the textbook ``Implementing Multiple Precision Arithmetic'' has been placed in the public domain with every new
release as well.  This textbook is meant to compliment the project by providing a more solid walkthrough of the development
algorithms used in the library.

Since both\footnote{Note that the MPI files under mtest/ are copyrighted by Michael Fromberger.  They are not required to use LibTomMath.} are in the 
public domain everyone is entitled to do with them as they see fit.

\section{Building LibTomMath}

LibTomMath is meant to be very ``GCC friendly'' as it comes with a makefile well suited for GCC.  However, the library will
also build in MSVC, Borland C out of the box.  For any other ISO C compiler a makefile will have to be made by the end
developer.  

\subsection{Static Libraries}
To build as a static library for GCC issue the following
\begin{alltt}
make
\end{alltt}

command.  This will build the library and archive the object files in ``libtommath.a''.  Now you link against 
that and include ``tommath.h'' within your programs.  Alternatively to build with MSVC issue the following
\begin{alltt}
nmake -f makefile.msvc
\end{alltt}

This will build the library and archive the object files in ``tommath.lib''.  This has been tested with MSVC 
version 6.00 with service pack 5.  

\subsection{Shared Libraries}
To build as a shared library for GCC issue the following
\begin{alltt}
make -f makefile.shared
\end{alltt}
This requires the ``libtool'' package (common on most Linux/BSD systems).  It will build LibTomMath as both shared
and static then install (by default) into /usr/lib as well as install the header files in /usr/include.  The shared 
library (resource) will be called ``libtommath.la'' while the static library called ``libtommath.a''.  Generally 
you use libtool to link your application against the shared object.  

There is limited support for making a ``DLL'' in windows via the ``makefile.cygwin\_dll'' makefile.  It requires 
Cygwin to work with since it requires the auto-export/import functionality.  The resulting DLL and import library 
``libtommath.dll.a'' can be used to link LibTomMath dynamically to any Windows program using Cygwin.

\subsection{Testing}
To build the library and the test harness type

\begin{alltt}
make test
\end{alltt}

This will build the library, ``test'' and ``mtest/mtest''.  The ``test'' program will accept test vectors and verify the
results.  ``mtest/mtest'' will generate test vectors using the MPI library by Michael Fromberger\footnote{A copy of MPI
is included in the package}.  Simply pipe mtest into test using

\begin{alltt}
mtest/mtest | test
\end{alltt}

If you do not have a ``/dev/urandom'' style RNG source you will have to write your own PRNG and simply pipe that into 
mtest.  For example, if your PRNG program is called ``myprng'' simply invoke

\begin{alltt}
myprng | mtest/mtest | test
\end{alltt}

This will output a row of numbers that are increasing.  Each column is a different test (such as addition, multiplication, etc)
that is being performed.  The numbers represent how many times the test was invoked.  If an error is detected the program
will exit with a dump of the relevent numbers it was working with.

\section{Build Configuration}
LibTomMath can configured at build time in three phases we shall call ``depends'', ``tweaks'' and ``trims''.  
Each phase changes how the library is built and they are applied one after another respectively.  

To make the system more powerful you can tweak the build process.  Classes are defined in the file
``tommath\_superclass.h''.  By default, the symbol ``LTM\_ALL'' shall be defined which simply 
instructs the system to build all of the functions.  This is how LibTomMath used to be packaged.  This will give you 
access to every function LibTomMath offers.

However, there are cases where such a build is not optional.  For instance, you want to perform RSA operations.  You 
don't need the vast majority of the library to perform these operations.  Aside from LTM\_ALL there is 
another pre--defined class ``SC\_RSA\_1'' which works in conjunction with the RSA from LibTomCrypt.  Additional 
classes can be defined base on the need of the user.

\subsection{Build Depends}
In the file tommath\_class.h you will see a large list of C ``defines'' followed by a series of ``ifdefs''
which further define symbols.  All of the symbols (technically they're macros $\ldots$) represent a given C source
file.  For instance, BN\_MP\_ADD\_C represents the file ``bn\_mp\_add.c''.  When a define has been enabled the
function in the respective file will be compiled and linked into the library.  Accordingly when the define
is absent the file will not be compiled and not contribute any size to the library.

You will also note that the header tommath\_class.h is actually recursively included (it includes itself twice).  
This is to help resolve as many dependencies as possible.  In the last pass the symbol LTM\_LAST will be defined.  
This is useful for ``trims''.

\subsection{Build Tweaks}
A tweak is an algorithm ``alternative''.  For example, to provide tradeoffs (usually between size and space).
They can be enabled at any pass of the configuration phase.

\begin{small}
\begin{center}
\begin{tabular}{|l|l|}
\hline \textbf{Define} & \textbf{Purpose} \\
\hline BN\_MP\_DIV\_SMALL & Enables a slower, smaller and equally \\
                          & functional mp\_div() function \\
\hline
\end{tabular}
\end{center}
\end{small}

\subsection{Build Trims}
A trim is a manner of removing functionality from a function that is not required.  For instance, to perform
RSA cryptography you only require exponentiation with odd moduli so even moduli support can be safely removed.  
Build trims are meant to be defined on the last pass of the configuration which means they are to be defined
only if LTM\_LAST has been defined.

\subsubsection{Moduli Related}
\begin{small}
\begin{center}
\begin{tabular}{|l|l|}







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\listoffigures
\mainmatter
\pagestyle{headings}
\chapter{Introduction}
\section{What is LibTomMath?}
LibTomMath is a library of source code which provides a series of efficient and carefully written functions for manipulating
large integer numbers.  It was written in portable ISO C source code so that it will build on any platform with a conforming
C compiler.

In a nutshell the library was written from scratch with verbose comments to help instruct computer science students how
to implement ``bignum'' math.  However, the resulting code has proven to be very useful.  It has been used by numerous
universities, commercial and open source software developers.  It has been used on a variety of platforms ranging from
Linux and Windows based x86 to ARM based Gameboys and PPC based MacOS machines.

\section{License}
As of the v0.25 the library source code has been placed in the public domain with every new release.  As of the v0.28
release the textbook ``Implementing Multiple Precision Arithmetic'' has been placed in the public domain with every new
release as well.  This textbook is meant to compliment the project by providing a more solid walkthrough of the development
algorithms used in the library.

Since both\footnote{Note that the MPI files under mtest/ are copyrighted by Michael Fromberger.  They are not required to use LibTomMath.} are in the
public domain everyone is entitled to do with them as they see fit.

\section{Building LibTomMath}

LibTomMath is meant to be very ``GCC friendly'' as it comes with a makefile well suited for GCC.  However, the library will
also build in MSVC, Borland C out of the box.  For any other ISO C compiler a makefile will have to be made by the end
developer.

\subsection{Static Libraries}
To build as a static library for GCC issue the following
\begin{alltt}
make
\end{alltt}

command.  This will build the library and archive the object files in ``libtommath.a''.  Now you link against
that and include ``tommath.h'' within your programs.  Alternatively to build with MSVC issue the following
\begin{alltt}
nmake -f makefile.msvc
\end{alltt}

This will build the library and archive the object files in ``tommath.lib''.  This has been tested with MSVC
version 6.00 with service pack 5.

\subsection{Shared Libraries}
To build as a shared library for GCC issue the following
\begin{alltt}
make -f makefile.shared
\end{alltt}
This requires the ``libtool'' package (common on most Linux/BSD systems).  It will build LibTomMath as both shared
and static then install (by default) into /usr/lib as well as install the header files in /usr/include.  The shared
library (resource) will be called ``libtommath.la'' while the static library called ``libtommath.a''.  Generally
you use libtool to link your application against the shared object.

There is limited support for making a ``DLL'' in windows via the ``makefile.cygwin\_dll'' makefile.  It requires
Cygwin to work with since it requires the auto-export/import functionality.  The resulting DLL and import library
``libtommath.dll.a'' can be used to link LibTomMath dynamically to any Windows program using Cygwin.

\subsection{Testing}
To build the library and the test harness type

\begin{alltt}
make test
\end{alltt}

This will build the library, ``test'' and ``mtest/mtest''.  The ``test'' program will accept test vectors and verify the
results.  ``mtest/mtest'' will generate test vectors using the MPI library by Michael Fromberger\footnote{A copy of MPI
is included in the package}.  Simply pipe mtest into test using

\begin{alltt}
mtest/mtest | test
\end{alltt}

If you do not have a ``/dev/urandom'' style RNG source you will have to write your own PRNG and simply pipe that into
mtest.  For example, if your PRNG program is called ``myprng'' simply invoke

\begin{alltt}
myprng | mtest/mtest | test
\end{alltt}

This will output a row of numbers that are increasing.  Each column is a different test (such as addition, multiplication, etc)
that is being performed.  The numbers represent how many times the test was invoked.  If an error is detected the program
will exit with a dump of the relevent numbers it was working with.

\section{Build Configuration}
LibTomMath can configured at build time in three phases we shall call ``depends'', ``tweaks'' and ``trims''.
Each phase changes how the library is built and they are applied one after another respectively.

To make the system more powerful you can tweak the build process.  Classes are defined in the file
``tommath\_superclass.h''.  By default, the symbol ``LTM\_ALL'' shall be defined which simply
instructs the system to build all of the functions.  This is how LibTomMath used to be packaged.  This will give you
access to every function LibTomMath offers.

However, there are cases where such a build is not optional.  For instance, you want to perform RSA operations.  You
don't need the vast majority of the library to perform these operations.  Aside from LTM\_ALL there is
another pre--defined class ``SC\_RSA\_1'' which works in conjunction with the RSA from LibTomCrypt.  Additional
classes can be defined base on the need of the user.

\subsection{Build Depends}
In the file tommath\_class.h you will see a large list of C ``defines'' followed by a series of ``ifdefs''
which further define symbols.  All of the symbols (technically they're macros $\ldots$) represent a given C source
file.  For instance, BN\_MP\_ADD\_C represents the file ``bn\_mp\_add.c''.  When a define has been enabled the
function in the respective file will be compiled and linked into the library.  Accordingly when the define
is absent the file will not be compiled and not contribute any size to the library.

You will also note that the header tommath\_class.h is actually recursively included (it includes itself twice).
This is to help resolve as many dependencies as possible.  In the last pass the symbol LTM\_LAST will be defined.
This is useful for ``trims''.

\subsection{Build Tweaks}
A tweak is an algorithm ``alternative''.  For example, to provide tradeoffs (usually between size and space).
They can be enabled at any pass of the configuration phase.

\begin{small}
\begin{center}
\begin{tabular}{|l|l|}
\hline \textbf{Define} & \textbf{Purpose} \\
\hline BN\_MP\_DIV\_SMALL & Enables a slower, smaller and equally \\
                          & functional mp\_div() function \\
\hline
\end{tabular}
\end{center}
\end{small}

\subsection{Build Trims}
A trim is a manner of removing functionality from a function that is not required.  For instance, to perform
RSA cryptography you only require exponentiation with odd moduli so even moduli support can be safely removed.
Build trims are meant to be defined on the last pass of the configuration which means they are to be defined
only if LTM\_LAST has been defined.

\subsubsection{Moduli Related}
\begin{small}
\begin{center}
\begin{tabular}{|l|l|}
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\hline \textbf{Restriction} & \textbf{Undefine} \\
\hline Moduli $\le 2560$ bits              & BN\_MP\_MONTGOMERY\_REDUCE\_C \\
                                           & BN\_S\_MP\_MUL\_DIGS\_C \\
                                           & BN\_S\_MP\_MUL\_HIGH\_DIGS\_C \\
                                           & BN\_S\_MP\_SQR\_C \\
\hline Polynomial Schmolynomial            & BN\_MP\_KARATSUBA\_MUL\_C \\
                                           & BN\_MP\_KARATSUBA\_SQR\_C \\
                                           & BN\_MP\_TOOM\_MUL\_C \\ 
                                           & BN\_MP\_TOOM\_SQR\_C \\

\hline
\end{tabular}
\end{center}
\end{small}


\section{Purpose of LibTomMath}
Unlike  GNU MP (GMP) Library, LIP, OpenSSL or various other commercial kits (Miracl), LibTomMath was not written with 
bleeding edge performance in mind.  First and foremost LibTomMath was written to be entirely open.  Not only is the 
source code public domain (unlike various other GPL/etc licensed code), not only is the code freely downloadable but the
source code is also accessible for computer science students attempting to learn ``BigNum'' or multiple precision
arithmetic techniques. 

LibTomMath was written to be an instructive collection of source code.  This is why there are many comments, only one
function per source file and often I use a ``middle-road'' approach where I don't cut corners for an extra 2\% speed
increase.

Source code alone cannot really teach how the algorithms work which is why I also wrote a textbook that accompanies
the library (beat that!).







|









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\hline \textbf{Restriction} & \textbf{Undefine} \\
\hline Moduli $\le 2560$ bits              & BN\_MP\_MONTGOMERY\_REDUCE\_C \\
                                           & BN\_S\_MP\_MUL\_DIGS\_C \\
                                           & BN\_S\_MP\_MUL\_HIGH\_DIGS\_C \\
                                           & BN\_S\_MP\_SQR\_C \\
\hline Polynomial Schmolynomial            & BN\_MP\_KARATSUBA\_MUL\_C \\
                                           & BN\_MP\_KARATSUBA\_SQR\_C \\
                                           & BN\_MP\_TOOM\_MUL\_C \\
                                           & BN\_MP\_TOOM\_SQR\_C \\

\hline
\end{tabular}
\end{center}
\end{small}


\section{Purpose of LibTomMath}
Unlike  GNU MP (GMP) Library, LIP, OpenSSL or various other commercial kits (Miracl), LibTomMath was not written with
bleeding edge performance in mind.  First and foremost LibTomMath was written to be entirely open.  Not only is the
source code public domain (unlike various other GPL/etc licensed code), not only is the code freely downloadable but the
source code is also accessible for computer science students attempting to learn ``BigNum'' or multiple precision
arithmetic techniques.

LibTomMath was written to be an instructive collection of source code.  This is why there are many comments, only one
function per source file and often I use a ``middle-road'' approach where I don't cut corners for an extra 2\% speed
increase.

Source code alone cannot really teach how the algorithms work which is why I also wrote a textbook that accompanies
the library (beat that!).
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\hline
\end{tabular}
\end{center}
\end{small}
\caption{LibTomMath Valuation}
\end{figure}

It may seem odd to compare LibTomMath to GnuPG since the math in GnuPG is only a small portion of the entire application. 
However, LibTomMath was written with cryptography in mind.  It provides essentially all of the functions a cryptosystem
would require when working with large integers.  

So it may feel tempting to just rip the math code out of GnuPG (or GnuMP where it was taken from originally) in your
own application but I think there are reasons not to.  While LibTomMath is slower than libraries such as GnuMP it is
not normally significantly slower.  On x86 machines the difference is normally a factor of two when performing modular
exponentiations.  It depends largely on the processor, compiler and the moduli being used.

Essentially the only time you wouldn't use LibTomMath is when blazing speed is the primary concern.  However,
on the other side of the coin LibTomMath offers you a totally free (public domain) well structured math library
that is very flexible, complete and performs well in resource contrained environments.  Fast RSA for example can
be performed with as little as 8KB of ram for data (again depending on build options).  

\chapter{Getting Started with LibTomMath}
\section{Building Programs}
In order to use LibTomMath you must include ``tommath.h'' and link against the appropriate library file (typically 
libtommath.a).  There is no library initialization required and the entire library is thread safe.

\section{Return Codes}
There are three possible return codes a function may return.

\index{MP\_OKAY}\index{MP\_YES}\index{MP\_NO}\index{MP\_VAL}\index{MP\_MEM}
\begin{figure}[here!]







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\hline
\end{tabular}
\end{center}
\end{small}
\caption{LibTomMath Valuation}
\end{figure}

It may seem odd to compare LibTomMath to GnuPG since the math in GnuPG is only a small portion of the entire application.
However, LibTomMath was written with cryptography in mind.  It provides essentially all of the functions a cryptosystem
would require when working with large integers.

So it may feel tempting to just rip the math code out of GnuPG (or GnuMP where it was taken from originally) in your
own application but I think there are reasons not to.  While LibTomMath is slower than libraries such as GnuMP it is
not normally significantly slower.  On x86 machines the difference is normally a factor of two when performing modular
exponentiations.  It depends largely on the processor, compiler and the moduli being used.

Essentially the only time you wouldn't use LibTomMath is when blazing speed is the primary concern.  However,
on the other side of the coin LibTomMath offers you a totally free (public domain) well structured math library
that is very flexible, complete and performs well in resource contrained environments.  Fast RSA for example can
be performed with as little as 8KB of ram for data (again depending on build options).

\chapter{Getting Started with LibTomMath}
\section{Building Programs}
In order to use LibTomMath you must include ``tommath.h'' and link against the appropriate library file (typically
libtommath.a).  There is no library initialization required and the entire library is thread safe.

\section{Return Codes}
There are three possible return codes a function may return.

\index{MP\_OKAY}\index{MP\_YES}\index{MP\_NO}\index{MP\_VAL}\index{MP\_MEM}
\begin{figure}[here!]
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to a string use the following function.

\index{mp\_error\_to\_string}
\begin{alltt}
char *mp_error_to_string(int code);
\end{alltt}

This will return a pointer to a string which describes the given error code.  It will not work for the return codes 
MP\_YES and MP\_NO.  

\section{Data Types}
The basic ``multiple precision integer'' type is known as the ``mp\_int'' within LibTomMath.  This data type is used to
organize all of the data required to manipulate the integer it represents.  Within LibTomMath it has been prototyped
as the following.

\index{mp\_int}
\begin{alltt}
typedef struct  \{
    int used, alloc, sign;
    mp_digit *dp;
\} mp_int;
\end{alltt}

Where ``mp\_digit'' is a data type that represents individual digits of the integer.  By default, an mp\_digit is the
ISO C ``unsigned long'' data type and each digit is $28-$bits long.  The mp\_digit type can be configured to suit other
platforms by defining the appropriate macros.  

All LTM functions that use the mp\_int type will expect a pointer to mp\_int structure.  You must allocate memory to
hold the structure itself by yourself (whether off stack or heap it doesn't matter).  The very first thing that must be
done to use an mp\_int is that it must be initialized.

\section{Function Organization}








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to a string use the following function.

\index{mp\_error\_to\_string}
\begin{alltt}
char *mp_error_to_string(int code);
\end{alltt}

This will return a pointer to a string which describes the given error code.  It will not work for the return codes
MP\_YES and MP\_NO.

\section{Data Types}
The basic ``multiple precision integer'' type is known as the ``mp\_int'' within LibTomMath.  This data type is used to
organize all of the data required to manipulate the integer it represents.  Within LibTomMath it has been prototyped
as the following.

\index{mp\_int}
\begin{alltt}
typedef struct  \{
    int used, alloc, sign;
    mp_digit *dp;
\} mp_int;
\end{alltt}

Where ``mp\_digit'' is a data type that represents individual digits of the integer.  By default, an mp\_digit is the
ISO C ``unsigned long'' data type and each digit is $28-$bits long.  The mp\_digit type can be configured to suit other
platforms by defining the appropriate macros.

All LTM functions that use the mp\_int type will expect a pointer to mp\_int structure.  You must allocate memory to
hold the structure itself by yourself (whether off stack or heap it doesn't matter).  The very first thing that must be
done to use an mp\_int is that it must be initialized.

\section{Function Organization}

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mp_div(&a, &b, &a, &c);   /* a = [a/b], c = a mod b */
\end{alltt}

This allows operands to be re-used which can make programming simpler.

\section{Initialization}
\subsection{Single Initialization}
A single mp\_int can be initialized with the ``mp\_init'' function. 

\index{mp\_init}
\begin{alltt}
int mp_init (mp_int * a);
\end{alltt}

This function expects a pointer to an mp\_int structure and will initialize the members of the structure so the mp\_int
represents the default integer which is zero.  If the functions returns MP\_OKAY then the mp\_int is ready to be used
by the other LibTomMath functions.

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
 
   /* use the number */

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

\subsection{Single Free}
When you are finished with an mp\_int it is ideal to return the heap it used back to the system.  The following function 
provides this functionality.

\index{mp\_clear}
\begin{alltt}
void mp_clear (mp_int * a);
\end{alltt}

The function expects a pointer to a previously initialized mp\_int structure and frees the heap it uses.  It sets the 
pointer\footnote{The ``dp'' member.} within the mp\_int to \textbf{NULL} which is used to prevent double free situations. 
Is is legal to call mp\_clear() twice on the same mp\_int in a row.  

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
 
   /* use the number */

   /* We're done with it. */
   mp_clear(&number);

   return EXIT_SUCCESS;
\}







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mp_div(&a, &b, &a, &c);   /* a = [a/b], c = a mod b */
\end{alltt}

This allows operands to be re-used which can make programming simpler.

\section{Initialization}
\subsection{Single Initialization}
A single mp\_int can be initialized with the ``mp\_init'' function.

\index{mp\_init}
\begin{alltt}
int mp_init (mp_int * a);
\end{alltt}

This function expects a pointer to an mp\_int structure and will initialize the members of the structure so the mp\_int
represents the default integer which is zero.  If the functions returns MP\_OKAY then the mp\_int is ready to be used
by the other LibTomMath functions.

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* use the number */

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

\subsection{Single Free}
When you are finished with an mp\_int it is ideal to return the heap it used back to the system.  The following function
provides this functionality.

\index{mp\_clear}
\begin{alltt}
void mp_clear (mp_int * a);
\end{alltt}

The function expects a pointer to a previously initialized mp\_int structure and frees the heap it uses.  It sets the
pointer\footnote{The ``dp'' member.} within the mp\_int to \textbf{NULL} which is used to prevent double free situations.
Is is legal to call mp\_clear() twice on the same mp\_int in a row.

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* use the number */

   /* We're done with it. */
   mp_clear(&number);

   return EXIT_SUCCESS;
\}
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\index{mp\_init\_multi} \index{mp\_clear\_multi}
\begin{alltt}
int mp_init_multi(mp_int *mp, ...);
\end{alltt}

It accepts a \textbf{NULL} terminated list of pointers to mp\_int structures.  It will attempt to initialize them all
at once.  If the function returns MP\_OKAY then all of the mp\_int variables are ready to use, otherwise none of them
are available for use.  A complementary mp\_clear\_multi() function allows multiple mp\_int variables to be free'd 
from the heap at the same time.  

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int num1, num2, num3;
   int result;

   if ((result = mp_init_multi(&num1, 
                               &num2,
                               &num3, NULL)) != MP\_OKAY) \{      
      printf("Error initializing the numbers.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
 
   /* use the numbers */

   /* We're done with them. */
   mp_clear_multi(&num1, &num2, &num3, NULL);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

\subsection{Other Initializers}
To initialized and make a copy of an mp\_int the mp\_init\_copy() function has been provided.  

\index{mp\_init\_copy}
\begin{alltt}
int mp_init_copy (mp_int * a, mp_int * b);
\end{alltt}

This function will initialize $a$ and make it a copy of $b$ if all goes well.

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int num1, num2;
   int result;

   /* initialize and do work on num1 ... */

   /* We want a copy of num1 in num2 now */
   if ((result = mp_init_copy(&num2, &num1)) != MP_OKAY) \{
     printf("Error initializing the copy.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
 
   /* now num2 is ready and contains a copy of num1 */

   /* We're done with them. */
   mp_clear_multi(&num1, &num2, NULL);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

Another less common initializer is mp\_init\_size() which allows the user to initialize an mp\_int with a given
default number of digits.  By default, all initializers allocate \textbf{MP\_PREC} digits.  This function lets
you override this behaviour.

\index{mp\_init\_size}
\begin{alltt}
int mp_init_size (mp_int * a, int size);
\end{alltt}

The $size$ parameter must be greater than zero.  If the function succeeds the mp\_int $a$ will be initialized
to have $size$ digits (which are all initially zero).  

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   /* we need a 60-digit number */
   if ((result = mp_init_size(&number, 60)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
 
   /* use the number */

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

\section{Maintenance Functions}

\subsection{Reducing Memory Usage}
When an mp\_int is in a state where it won't be changed again\footnote{A Diffie-Hellman modulus for instance.} excess
digits can be removed to return memory to the heap with the mp\_shrink() function.

\index{mp\_shrink}
\begin{alltt}
int mp_shrink (mp_int * a);
\end{alltt}

This will remove excess digits of the mp\_int $a$.  If the operation fails the mp\_int should be intact without the
excess digits being removed.  Note that you can use a shrunk mp\_int in further computations, however, such operations
will require heap operations which can be slow.  It is not ideal to shrink mp\_int variables that you will further
modify in the system (unless you are seriously low on memory).  

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
 
   /* use the number [e.g. pre-computation]  */

   /* We're done with it for now. */
   if ((result = mp_shrink(&number)) != MP_OKAY) \{
      printf("Error shrinking the number.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* use it .... */


   /* we're done with it. */ 
   mp_clear(&number);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

\subsection{Adding additional digits}

Within the mp\_int structure are two parameters which control the limitations of the array of digits that represent
the integer the mp\_int is meant to equal.   The \textit{used} parameter dictates how many digits are significant, that is,
contribute to the value of the mp\_int.  The \textit{alloc} parameter dictates how many digits are currently available in
the array.  If you need to perform an operation that requires more digits you will have to mp\_grow() the mp\_int to
your desired size.  

\index{mp\_grow}
\begin{alltt}
int mp_grow (mp_int * a, int size);
\end{alltt}

This will grow the array of digits of $a$ to $size$.  If the \textit{alloc} parameter is already bigger than
$size$ the function will not do anything.

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
 
   /* use the number */

   /* We need to add 20 digits to the number  */
   if ((result = mp_grow(&number, number.alloc + 20)) != MP_OKAY) \{
      printf("Error growing the number.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}


   /* use the number */

   /* we're done with it. */ 
   mp_clear(&number);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

\chapter{Basic Operations}
\section{Small Constants}
Setting mp\_ints to small constants is a relatively common operation.  To accomodate these instances there are two
small constant assignment functions.  The first function is used to set a single digit constant while the second sets
an ISO C style ``unsigned long'' constant.  The reason for both functions is efficiency.  Setting a single digit is quick but the
domain of a digit can change (it's always at least $0 \ldots 127$).  

\subsection{Single Digit}

Setting a single digit can be accomplished with the following function.

\index{mp\_set}
\begin{alltt}







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\index{mp\_init\_multi} \index{mp\_clear\_multi}
\begin{alltt}
int mp_init_multi(mp_int *mp, ...);
\end{alltt}

It accepts a \textbf{NULL} terminated list of pointers to mp\_int structures.  It will attempt to initialize them all
at once.  If the function returns MP\_OKAY then all of the mp\_int variables are ready to use, otherwise none of them
are available for use.  A complementary mp\_clear\_multi() function allows multiple mp\_int variables to be free'd
from the heap at the same time.

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int num1, num2, num3;
   int result;

   if ((result = mp_init_multi(&num1,
                               &num2,
                               &num3, NULL)) != MP\_OKAY) \{
      printf("Error initializing the numbers.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* use the numbers */

   /* We're done with them. */
   mp_clear_multi(&num1, &num2, &num3, NULL);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

\subsection{Other Initializers}
To initialized and make a copy of an mp\_int the mp\_init\_copy() function has been provided.

\index{mp\_init\_copy}
\begin{alltt}
int mp_init_copy (mp_int * a, mp_int * b);
\end{alltt}

This function will initialize $a$ and make it a copy of $b$ if all goes well.

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int num1, num2;
   int result;

   /* initialize and do work on num1 ... */

   /* We want a copy of num1 in num2 now */
   if ((result = mp_init_copy(&num2, &num1)) != MP_OKAY) \{
     printf("Error initializing the copy.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* now num2 is ready and contains a copy of num1 */

   /* We're done with them. */
   mp_clear_multi(&num1, &num2, NULL);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

Another less common initializer is mp\_init\_size() which allows the user to initialize an mp\_int with a given
default number of digits.  By default, all initializers allocate \textbf{MP\_PREC} digits.  This function lets
you override this behaviour.

\index{mp\_init\_size}
\begin{alltt}
int mp_init_size (mp_int * a, int size);
\end{alltt}

The $size$ parameter must be greater than zero.  If the function succeeds the mp\_int $a$ will be initialized
to have $size$ digits (which are all initially zero).

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   /* we need a 60-digit number */
   if ((result = mp_init_size(&number, 60)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* use the number */

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

\section{Maintenance Functions}

\subsection{Reducing Memory Usage}
When an mp\_int is in a state where it won't be changed again\footnote{A Diffie-Hellman modulus for instance.} excess
digits can be removed to return memory to the heap with the mp\_shrink() function.

\index{mp\_shrink}
\begin{alltt}
int mp_shrink (mp_int * a);
\end{alltt}

This will remove excess digits of the mp\_int $a$.  If the operation fails the mp\_int should be intact without the
excess digits being removed.  Note that you can use a shrunk mp\_int in further computations, however, such operations
will require heap operations which can be slow.  It is not ideal to shrink mp\_int variables that you will further
modify in the system (unless you are seriously low on memory).

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* use the number [e.g. pre-computation]  */

   /* We're done with it for now. */
   if ((result = mp_shrink(&number)) != MP_OKAY) \{
      printf("Error shrinking the number.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* use it .... */


   /* we're done with it. */
   mp_clear(&number);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

\subsection{Adding additional digits}

Within the mp\_int structure are two parameters which control the limitations of the array of digits that represent
the integer the mp\_int is meant to equal.   The \textit{used} parameter dictates how many digits are significant, that is,
contribute to the value of the mp\_int.  The \textit{alloc} parameter dictates how many digits are currently available in
the array.  If you need to perform an operation that requires more digits you will have to mp\_grow() the mp\_int to
your desired size.

\index{mp\_grow}
\begin{alltt}
int mp_grow (mp_int * a, int size);
\end{alltt}

This will grow the array of digits of $a$ to $size$.  If the \textit{alloc} parameter is already bigger than
$size$ the function will not do anything.

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* use the number */

   /* We need to add 20 digits to the number  */
   if ((result = mp_grow(&number, number.alloc + 20)) != MP_OKAY) \{
      printf("Error growing the number.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}


   /* use the number */

   /* we're done with it. */
   mp_clear(&number);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

\chapter{Basic Operations}
\section{Small Constants}
Setting mp\_ints to small constants is a relatively common operation.  To accomodate these instances there are two
small constant assignment functions.  The first function is used to set a single digit constant while the second sets
an ISO C style ``unsigned long'' constant.  The reason for both functions is efficiency.  Setting a single digit is quick but the
domain of a digit can change (it's always at least $0 \ldots 127$).

\subsection{Single Digit}

Setting a single digit can be accomplished with the following function.

\index{mp\_set}
\begin{alltt}
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\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
 
   /* set the number to 5 */
   mp_set(&number, 5);

   /* we're done with it. */ 
   mp_clear(&number);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

\subsection{Long Constants}

To set a constant that is the size of an ISO C ``unsigned long'' and larger than a single digit the following function 
can be used.

\index{mp\_set\_int}
\begin{alltt}
int mp_set_int (mp_int * a, unsigned long b);
\end{alltt}

This will assign the value of the 32-bit variable $b$ to the mp\_int $a$.  Unlike mp\_set() this function will always
accept a 32-bit input regardless of the size of a single digit.  However, since the value may span several digits 
this function can fail if it runs out of heap memory.

To get the ``unsigned long'' copy of an mp\_int the following function can be used.

\index{mp\_get\_int}
\begin{alltt}
unsigned long mp_get_int (mp_int * a);
\end{alltt}

This will return the 32 least significant bits of the mp\_int $a$.  

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
 
   /* set the number to 654321 (note this is bigger than 127) */
   if ((result = mp_set_int(&number, 654321)) != MP_OKAY) \{
      printf("Error setting the value of the number.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   printf("number == \%lu", mp_get_int(&number));

   /* we're done with it. */ 
   mp_clear(&number);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

This should output the following if the program succeeds.

\begin{alltt}
number == 654321
\end{alltt}





































\subsection{Initialize and Setting Constants}
To both initialize and set small constants the following two functions are available.
\index{mp\_init\_set} \index{mp\_init\_set\_int}
\begin{alltt}
int mp_init_set (mp_int * a, mp_digit b);
int mp_init_set_int (mp_int * a, unsigned long b);
\end{alltt}

Both functions work like the previous counterparts except they first mp\_init $a$ before setting the values.  

\begin{alltt}
int main(void)
\{
   mp_int number1, number2;
   int    result;

   /* initialize and set a single digit */
   if ((result = mp_init_set(&number1, 100)) != MP_OKAY) \{
      printf("Error setting number1: \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}             

   /* initialize and set a long */
   if ((result = mp_init_set_int(&number2, 1023)) != MP_OKAY) \{
      printf("Error setting number2: \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* display */
   printf("Number1, Number2 == \%lu, \%lu",
          mp_get_int(&number1), mp_get_int(&number2));







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\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* set the number to 5 */
   mp_set(&number, 5);

   /* we're done with it. */
   mp_clear(&number);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

\subsection{Long Constants}

To set a constant that is the size of an ISO C ``unsigned long'' and larger than a single digit the following function
can be used.

\index{mp\_set\_int}
\begin{alltt}
int mp_set_int (mp_int * a, unsigned long b);
\end{alltt}

This will assign the value of the 32-bit variable $b$ to the mp\_int $a$.  Unlike mp\_set() this function will always
accept a 32-bit input regardless of the size of a single digit.  However, since the value may span several digits
this function can fail if it runs out of heap memory.

To get the ``unsigned long'' copy of an mp\_int the following function can be used.

\index{mp\_get\_int}
\begin{alltt}
unsigned long mp_get_int (mp_int * a);
\end{alltt}

This will return the 32 least significant bits of the mp\_int $a$.

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* set the number to 654321 (note this is bigger than 127) */
   if ((result = mp_set_int(&number, 654321)) != MP_OKAY) \{
      printf("Error setting the value of the number.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   printf("number == \%lu", mp_get_int(&number));

   /* we're done with it. */
   mp_clear(&number);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

This should output the following if the program succeeds.

\begin{alltt}
number == 654321
\end{alltt}

\subsection{Long Constants - platform dependant}

\index{mp\_set\_long}
\begin{alltt}
int mp_set_long (mp_int * a, unsigned long b);
\end{alltt}

This will assign the value of the platform-dependant sized variable $b$ to the mp\_int $a$.

To get the ``unsigned long'' copy of an mp\_int the following function can be used.

\index{mp\_get\_long}
\begin{alltt}
unsigned long mp_get_long (mp_int * a);
\end{alltt}

This will return the least significant bits of the mp\_int $a$ that fit into an ``unsigned long''.

\subsection{Long Long Constants}

\index{mp\_set\_long\_long}
\begin{alltt}
int mp_set_long_long (mp_int * a, unsigned long long b);
\end{alltt}

This will assign the value of the 64-bit variable $b$ to the mp\_int $a$.

To get the ``unsigned long long'' copy of an mp\_int the following function can be used.

\index{mp\_get\_long\_long}
\begin{alltt}
unsigned long long mp_get_long_long (mp_int * a);
\end{alltt}

This will return the 64 least significant bits of the mp\_int $a$.

\subsection{Initialize and Setting Constants}
To both initialize and set small constants the following two functions are available.
\index{mp\_init\_set} \index{mp\_init\_set\_int}
\begin{alltt}
int mp_init_set (mp_int * a, mp_digit b);
int mp_init_set_int (mp_int * a, unsigned long b);
\end{alltt}

Both functions work like the previous counterparts except they first mp\_init $a$ before setting the values.

\begin{alltt}
int main(void)
\{
   mp_int number1, number2;
   int    result;

   /* initialize and set a single digit */
   if ((result = mp_init_set(&number1, 100)) != MP_OKAY) \{
      printf("Error setting number1: \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* initialize and set a long */
   if ((result = mp_init_set_int(&number2, 1023)) != MP_OKAY) \{
      printf("Error setting number2: \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* display */
   printf("Number1, Number2 == \%lu, \%lu",
          mp_get_int(&number1), mp_get_int(&number2));
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\hline
\end{tabular}
\end{center}
\caption{Comparison Codes for $a, b$}
\label{fig:CMP}
\end{figure}

In figure \ref{fig:CMP} two integers $a$ and $b$ are being compared.  In this case $a$ is said to be ``to the left'' of 
$b$.  

\subsection{Unsigned comparison}

An unsigned comparison considers only the digits themselves and not the associated \textit{sign} flag of the 
mp\_int structures.  This is analogous to an absolute comparison.  The function mp\_cmp\_mag() will compare two
mp\_int variables based on their digits only. 

\index{mp\_cmp\_mag}
\begin{alltt}
int mp_cmp_mag(mp_int * a, mp_int * b);
\end{alltt}
This will compare $a$ to $b$ placing $a$ to the left of $b$.  This function cannot fail and will return one of the
three compare codes listed in figure \ref{fig:CMP}.

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number1, number2;
   int result;

   if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{
      printf("Error initializing the numbers.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
 
   /* set the number1 to 5 */
   mp_set(&number1, 5);
  
   /* set the number2 to -6 */
   mp_set(&number2, 6);
   if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{
      printf("Error negating number2.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   switch(mp_cmp_mag(&number1, &number2)) \{
       case MP_GT:  printf("|number1| > |number2|"); break;
       case MP_EQ:  printf("|number1| = |number2|"); break;
       case MP_LT:  printf("|number1| < |number2|"); break;
   \}

   /* we're done with it. */ 
   mp_clear_multi(&number1, &number2, NULL);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes 
successfully it should print the following.

\begin{alltt}
|number1| < |number2|
\end{alltt}

This is because $\vert -6 \vert = 6$ and obviously $5 < 6$.







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\hline
\end{tabular}
\end{center}
\caption{Comparison Codes for $a, b$}
\label{fig:CMP}
\end{figure}

In figure \ref{fig:CMP} two integers $a$ and $b$ are being compared.  In this case $a$ is said to be ``to the left'' of
$b$.

\subsection{Unsigned comparison}

An unsigned comparison considers only the digits themselves and not the associated \textit{sign} flag of the
mp\_int structures.  This is analogous to an absolute comparison.  The function mp\_cmp\_mag() will compare two
mp\_int variables based on their digits only.

\index{mp\_cmp\_mag}
\begin{alltt}
int mp_cmp_mag(mp_int * a, mp_int * b);
\end{alltt}
This will compare $a$ to $b$ placing $a$ to the left of $b$.  This function cannot fail and will return one of the
three compare codes listed in figure \ref{fig:CMP}.

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number1, number2;
   int result;

   if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{
      printf("Error initializing the numbers.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* set the number1 to 5 */
   mp_set(&number1, 5);

   /* set the number2 to -6 */
   mp_set(&number2, 6);
   if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{
      printf("Error negating number2.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   switch(mp_cmp_mag(&number1, &number2)) \{
       case MP_GT:  printf("|number1| > |number2|"); break;
       case MP_EQ:  printf("|number1| = |number2|"); break;
       case MP_LT:  printf("|number1| < |number2|"); break;
   \}

   /* we're done with it. */
   mp_clear_multi(&number1, &number2, NULL);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes
successfully it should print the following.

\begin{alltt}
|number1| < |number2|
\end{alltt}

This is because $\vert -6 \vert = 6$ and obviously $5 < 6$.
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\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number1, number2;
   int result;

   if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{
      printf("Error initializing the numbers.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
 
   /* set the number1 to 5 */
   mp_set(&number1, 5);
  
   /* set the number2 to -6 */
   mp_set(&number2, 6);
   if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{
      printf("Error negating number2.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   switch(mp_cmp(&number1, &number2)) \{
       case MP_GT:  printf("number1 > number2"); break;
       case MP_EQ:  printf("number1 = number2"); break;
       case MP_LT:  printf("number1 < number2"); break;
   \}

   /* we're done with it. */ 
   mp_clear_multi(&number1, &number2, NULL);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes 
successfully it should print the following.

\begin{alltt}
number1 > number2
\end{alltt}

\subsection{Single Digit}

To compare a single digit against an mp\_int the following function has been provided.

\index{mp\_cmp\_d}
\begin{alltt}
int mp_cmp_d(mp_int * a, mp_digit b);
\end{alltt}

This will compare $a$ to the left of $b$ using a signed comparison.  Note that it will always treat $b$ as 
positive.  This function is rather handy when you have to compare against small values such as $1$ (which often
comes up in cryptography).  The function cannot fail and will return one of the tree compare condition codes
listed in figure \ref{fig:CMP}.


\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
 
   /* set the number to 5 */
   mp_set(&number, 5);

   switch(mp_cmp_d(&number, 7)) \{
       case MP_GT:  printf("number > 7"); break;
       case MP_EQ:  printf("number = 7"); break;
       case MP_LT:  printf("number < 7"); break;
   \}

   /* we're done with it. */ 
   mp_clear(&number);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

If this program functions properly it will print out the following.

\begin{alltt}
number < 7
\end{alltt}

\section{Logical Operations}

Logical operations are operations that can be performed either with simple shifts or boolean operators such as
AND, XOR and OR directly.  These operations are very quick.

\subsection{Multiplication by two}

Multiplications and divisions by any power of two can be performed with quick logical shifts either left or
right depending on the operation.  

When multiplying or dividing by two a special case routine can be used which are as follows.
\index{mp\_mul\_2} \index{mp\_div\_2}
\begin{alltt}
int mp_mul_2(mp_int * a, mp_int * b);
int mp_div_2(mp_int * a, mp_int * b);
\end{alltt}

The former will assign twice $a$ to $b$ while the latter will assign half $a$ to $b$.  These functions are fast
since the shift counts and maskes are hardcoded into the routines.

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
 
   /* set the number to 5 */
   mp_set(&number, 5);

   /* multiply by two */
   if ((result = mp\_mul\_2(&number, &number)) != MP_OKAY) \{
      printf("Error multiplying the number.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
   switch(mp_cmp_d(&number, 7)) \{
       case MP_GT:  printf("2*number > 7"); break;
       case MP_EQ:  printf("2*number = 7"); break;
       case MP_LT:  printf("2*number < 7"); break;
   \}

   /* now divide by two */
   if ((result = mp\_div\_2(&number, &number)) != MP_OKAY) \{
      printf("Error dividing the number.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
   switch(mp_cmp_d(&number, 7)) \{
       case MP_GT:  printf("2*number/2 > 7"); break;
       case MP_EQ:  printf("2*number/2 = 7"); break;
       case MP_LT:  printf("2*number/2 < 7"); break;
   \}

   /* we're done with it. */ 
   mp_clear(&number);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

If this program is successful it will print out the following text.

\begin{alltt}
2*number > 7
2*number/2 < 7
\end{alltt}



Since $10 > 7$ and $5 < 7$.  To multiply by a power of two the following function can be used.

\index{mp\_mul\_2d}
\begin{alltt}
int mp_mul_2d(mp_int * a, int b, mp_int * c);
\end{alltt}

This will multiply $a$ by $2^b$ and store the result in ``c''.  If the value of $b$ is less than or equal to 
zero the function will copy $a$ to ``c'' without performing any further actions.  


To divide by a power of two use the following.

\index{mp\_div\_2d}
\begin{alltt}
int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d);
\end{alltt}
Which will divide $a$ by $2^b$, store the quotient in ``c'' and the remainder in ``d'.  If $b \le 0$ then the
function simply copies $a$ over to ``c'' and zeroes $d$.  The variable $d$ may be passed as a \textbf{NULL}
value to signal that the remainder is not desired.


\subsection{Polynomial Basis Operations}

Strictly speaking the organization of the integers within the mp\_int structures is what is known as a 
``polynomial basis''.  This simply means a field element is stored by divisions of a radix.  For example, if
$f(x) = \sum_{i=0}^{k} y_ix^k$ for any vector $\vec y$ then the array of digits in $\vec y$ are said to be 
the polynomial basis representation of $z$ if $f(\beta) = z$ for a given radix $\beta$.  

To multiply by the polynomial $g(x) = x$ all you have todo is shift the digits of the basis left one place.  The
following function provides this operation.

\index{mp\_lshd}
\begin{alltt}
int mp_lshd (mp_int * a, int b);







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\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number1, number2;
   int result;

   if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{
      printf("Error initializing the numbers.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* set the number1 to 5 */
   mp_set(&number1, 5);

   /* set the number2 to -6 */
   mp_set(&number2, 6);
   if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{
      printf("Error negating number2.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   switch(mp_cmp(&number1, &number2)) \{
       case MP_GT:  printf("number1 > number2"); break;
       case MP_EQ:  printf("number1 = number2"); break;
       case MP_LT:  printf("number1 < number2"); break;
   \}

   /* we're done with it. */
   mp_clear_multi(&number1, &number2, NULL);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes
successfully it should print the following.

\begin{alltt}
number1 > number2
\end{alltt}

\subsection{Single Digit}

To compare a single digit against an mp\_int the following function has been provided.

\index{mp\_cmp\_d}
\begin{alltt}
int mp_cmp_d(mp_int * a, mp_digit b);
\end{alltt}

This will compare $a$ to the left of $b$ using a signed comparison.  Note that it will always treat $b$ as
positive.  This function is rather handy when you have to compare against small values such as $1$ (which often
comes up in cryptography).  The function cannot fail and will return one of the tree compare condition codes
listed in figure \ref{fig:CMP}.


\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* set the number to 5 */
   mp_set(&number, 5);

   switch(mp_cmp_d(&number, 7)) \{
       case MP_GT:  printf("number > 7"); break;
       case MP_EQ:  printf("number = 7"); break;
       case MP_LT:  printf("number < 7"); break;
   \}

   /* we're done with it. */
   mp_clear(&number);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

If this program functions properly it will print out the following.

\begin{alltt}
number < 7
\end{alltt}

\section{Logical Operations}

Logical operations are operations that can be performed either with simple shifts or boolean operators such as
AND, XOR and OR directly.  These operations are very quick.

\subsection{Multiplication by two}

Multiplications and divisions by any power of two can be performed with quick logical shifts either left or
right depending on the operation.

When multiplying or dividing by two a special case routine can be used which are as follows.
\index{mp\_mul\_2} \index{mp\_div\_2}
\begin{alltt}
int mp_mul_2(mp_int * a, mp_int * b);
int mp_div_2(mp_int * a, mp_int * b);
\end{alltt}

The former will assign twice $a$ to $b$ while the latter will assign half $a$ to $b$.  These functions are fast
since the shift counts and maskes are hardcoded into the routines.

\begin{small} \begin{alltt}
int main(void)
\{
   mp_int number;
   int result;

   if ((result = mp_init(&number)) != MP_OKAY) \{
      printf("Error initializing the number.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* set the number to 5 */
   mp_set(&number, 5);

   /* multiply by two */
   if ((result = mp\_mul\_2(&number, &number)) != MP_OKAY) \{
      printf("Error multiplying the number.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
   switch(mp_cmp_d(&number, 7)) \{
       case MP_GT:  printf("2*number > 7"); break;
       case MP_EQ:  printf("2*number = 7"); break;
       case MP_LT:  printf("2*number < 7"); break;
   \}

   /* now divide by two */
   if ((result = mp\_div\_2(&number, &number)) != MP_OKAY) \{
      printf("Error dividing the number.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
   switch(mp_cmp_d(&number, 7)) \{
       case MP_GT:  printf("2*number/2 > 7"); break;
       case MP_EQ:  printf("2*number/2 = 7"); break;
       case MP_LT:  printf("2*number/2 < 7"); break;
   \}

   /* we're done with it. */
   mp_clear(&number);

   return EXIT_SUCCESS;
\}
\end{alltt} \end{small}

If this program is successful it will print out the following text.

\begin{alltt}
2*number > 7
2*number/2 < 7
\end{alltt}

Since $10 > 7$ and $5 < 7$.

To multiply by a power of two the following function can be used.

\index{mp\_mul\_2d}
\begin{alltt}
int mp_mul_2d(mp_int * a, int b, mp_int * c);
\end{alltt}

This will multiply $a$ by $2^b$ and store the result in ``c''.  If the value of $b$ is less than or equal to
zero the function will copy $a$ to ``c'' without performing any further actions.  The multiplication itself
is implemented as a right-shift operation of $a$ by $b$ bits.

To divide by a power of two use the following.

\index{mp\_div\_2d}
\begin{alltt}
int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d);
\end{alltt}
Which will divide $a$ by $2^b$, store the quotient in ``c'' and the remainder in ``d'.  If $b \le 0$ then the
function simply copies $a$ over to ``c'' and zeroes $d$.  The variable $d$ may be passed as a \textbf{NULL}
value to signal that the remainder is not desired.  The division itself is implemented as a left-shift
operation of $a$ by $b$ bits.

\subsection{Polynomial Basis Operations}

Strictly speaking the organization of the integers within the mp\_int structures is what is known as a
``polynomial basis''.  This simply means a field element is stored by divisions of a radix.  For example, if
$f(x) = \sum_{i=0}^{k} y_ix^k$ for any vector $\vec y$ then the array of digits in $\vec y$ are said to be
the polynomial basis representation of $z$ if $f(\beta) = z$ for a given radix $\beta$.

To multiply by the polynomial $g(x) = x$ all you have todo is shift the digits of the basis left one place.  The
following function provides this operation.

\index{mp\_lshd}
\begin{alltt}
int mp_lshd (mp_int * a, int b);
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\index{mp\_or} \index{mp\_and} \index{mp\_xor}
\begin{alltt}
int mp_or  (mp_int * a, mp_int * b, mp_int * c);
int mp_and (mp_int * a, mp_int * b, mp_int * c);
int mp_xor (mp_int * a, mp_int * b, mp_int * c);
\end{alltt}

Which compute $c = a \odot b$ where $\odot$ is one of OR, AND or XOR.  

\section{Addition and Subtraction}

To compute an addition or subtraction the following two functions can be used.

\index{mp\_add} \index{mp\_sub}
\begin{alltt}







|







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\index{mp\_or} \index{mp\_and} \index{mp\_xor}
\begin{alltt}
int mp_or  (mp_int * a, mp_int * b, mp_int * c);
int mp_and (mp_int * a, mp_int * b, mp_int * c);
int mp_xor (mp_int * a, mp_int * b, mp_int * c);
\end{alltt}

Which compute $c = a \odot b$ where $\odot$ is one of OR, AND or XOR.

\section{Addition and Subtraction}

To compute an addition or subtraction the following two functions can be used.

\index{mp\_add} \index{mp\_sub}
\begin{alltt}
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Simple integer negation can be performed with the following.

\index{mp\_neg}
\begin{alltt}
int mp_neg (mp_int * a, mp_int * b);
\end{alltt}

Which assigns $-a$ to $b$.  

\subsection{Absolute}
Simple integer absolutes can be performed with the following.

\index{mp\_neg}
\begin{alltt}
int mp_abs (mp_int * a, mp_int * b);
\end{alltt}

Which assigns $\vert a \vert$ to $b$.  

\section{Integer Division and Remainder}
To perform a complete and general integer division with remainder use the following function.

\index{mp\_div}
\begin{alltt}
int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d);
\end{alltt}
                                                        
This divides $a$ by $b$ and stores the quotient in $c$ and $d$.  The signed quotient is computed such that 
$bc + d = a$.  Note that either of $c$ or $d$ can be set to \textbf{NULL} if their value is not required.  If 
$b$ is zero the function returns \textbf{MP\_VAL}.  


\chapter{Multiplication and Squaring}
\section{Multiplication}
A full signed integer multiplication can be performed with the following.
\index{mp\_mul}
\begin{alltt}
int mp_mul (mp_int * a, mp_int * b, mp_int * c);
\end{alltt}
Which assigns the full signed product $ab$ to $c$.  This function actually breaks into one of four cases which are 
specific multiplication routines optimized for given parameters.  First there are the Toom-Cook multiplications which
should only be used with very large inputs.  This is followed by the Karatsuba multiplications which are for moderate
sized inputs.  Then followed by the Comba and baseline multipliers.

Fortunately for the developer you don't really need to know this unless you really want to fine tune the system.  mp\_mul()
will determine on its own\footnote{Some tweaking may be required.} what routine to use automatically when it is called.

\begin{alltt}
int main(void)
\{
   mp_int number1, number2;
   int result;

   /* Initialize the numbers */
   if ((result = mp_init_multi(&number1, 
                               &number2, NULL)) != MP_OKAY) \{
      printf("Error initializing the numbers.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* set the terms */
   if ((result = mp_set_int(&number, 257)) != MP_OKAY) \{
      printf("Error setting number1.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
 
   if ((result = mp_set_int(&number2, 1023)) != MP_OKAY) \{
      printf("Error setting number2.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* multiply them */
   if ((result = mp_mul(&number1, &number2,
                        &number1)) != MP_OKAY) \{
      printf("Error multiplying terms.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* display */
   printf("number1 * number2 == \%lu", mp_get_int(&number1));

   /* free terms and return */
   mp_clear_multi(&number1, &number2, NULL);

   return EXIT_SUCCESS;
\}
\end{alltt}   

If this program succeeds it shall output the following.

\begin{alltt}
number1 * number2 == 262911
\end{alltt}

\section{Squaring}
Since squaring can be performed faster than multiplication it is performed it's own function instead of just using
mp\_mul().

\index{mp\_sqr}
\begin{alltt}
int mp_sqr (mp_int * a, mp_int * b);
\end{alltt}

Will square $a$ and store it in $b$.  Like the case of multiplication there are four different squaring
algorithms all which can be called from mp\_sqr().  It is ideal to use mp\_sqr over mp\_mul when squaring terms because
of the speed difference.  

\section{Tuning Polynomial Basis Routines}

Both of the Toom-Cook and Karatsuba multiplication algorithms are faster than the traditional $O(n^2)$ approach that
the Comba and baseline algorithms use.  At $O(n^{1.464973})$ and $O(n^{1.584962})$ running times respectively they require 
considerably less work.  For example, a 10000-digit multiplication would take roughly 724,000 single precision
multiplications with Toom-Cook or 100,000,000 single precision multiplications with the standard Comba (a factor
of 138).

So why not always use Karatsuba or Toom-Cook?   The simple answer is that they have so much overhead that they're not
actually faster than Comba until you hit distinct  ``cutoff'' points.  For Karatsuba with the default configuration, 
GCC 3.3.1 and an Athlon XP processor the cutoff point is roughly 110 digits (about 70 for the Intel P4).  That is, at 
110 digits Karatsuba and Comba multiplications just about break even and for 110+ digits Karatsuba is faster.

Toom-Cook has incredible overhead and is probably only useful for very large inputs.  So far no known cutoff points 
exist and for the most part I just set the cutoff points very high to make sure they're not called.

A demo program in the ``etc/'' directory of the project called ``tune.c'' can be used to find the cutoff points.  This
can be built with GCC as follows

\begin{alltt}
make XXX







|









|








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Simple integer negation can be performed with the following.

\index{mp\_neg}
\begin{alltt}
int mp_neg (mp_int * a, mp_int * b);
\end{alltt}

Which assigns $-a$ to $b$.

\subsection{Absolute}
Simple integer absolutes can be performed with the following.

\index{mp\_neg}
\begin{alltt}
int mp_abs (mp_int * a, mp_int * b);
\end{alltt}

Which assigns $\vert a \vert$ to $b$.

\section{Integer Division and Remainder}
To perform a complete and general integer division with remainder use the following function.

\index{mp\_div}
\begin{alltt}
int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d);
\end{alltt}

This divides $a$ by $b$ and stores the quotient in $c$ and $d$.  The signed quotient is computed such that
$bc + d = a$.  Note that either of $c$ or $d$ can be set to \textbf{NULL} if their value is not required.  If
$b$ is zero the function returns \textbf{MP\_VAL}.


\chapter{Multiplication and Squaring}
\section{Multiplication}
A full signed integer multiplication can be performed with the following.
\index{mp\_mul}
\begin{alltt}
int mp_mul (mp_int * a, mp_int * b, mp_int * c);
\end{alltt}
Which assigns the full signed product $ab$ to $c$.  This function actually breaks into one of four cases which are
specific multiplication routines optimized for given parameters.  First there are the Toom-Cook multiplications which
should only be used with very large inputs.  This is followed by the Karatsuba multiplications which are for moderate
sized inputs.  Then followed by the Comba and baseline multipliers.

Fortunately for the developer you don't really need to know this unless you really want to fine tune the system.  mp\_mul()
will determine on its own\footnote{Some tweaking may be required.} what routine to use automatically when it is called.

\begin{alltt}
int main(void)
\{
   mp_int number1, number2;
   int result;

   /* Initialize the numbers */
   if ((result = mp_init_multi(&number1,
                               &number2, NULL)) != MP_OKAY) \{
      printf("Error initializing the numbers.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* set the terms */
   if ((result = mp_set_int(&number, 257)) != MP_OKAY) \{
      printf("Error setting number1.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   if ((result = mp_set_int(&number2, 1023)) != MP_OKAY) \{
      printf("Error setting number2.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* multiply them */
   if ((result = mp_mul(&number1, &number2,
                        &number1)) != MP_OKAY) \{
      printf("Error multiplying terms.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* display */
   printf("number1 * number2 == \%lu", mp_get_int(&number1));

   /* free terms and return */
   mp_clear_multi(&number1, &number2, NULL);

   return EXIT_SUCCESS;
\}
\end{alltt}

If this program succeeds it shall output the following.

\begin{alltt}
number1 * number2 == 262911
\end{alltt}

\section{Squaring}
Since squaring can be performed faster than multiplication it is performed it's own function instead of just using
mp\_mul().

\index{mp\_sqr}
\begin{alltt}
int mp_sqr (mp_int * a, mp_int * b);
\end{alltt}

Will square $a$ and store it in $b$.  Like the case of multiplication there are four different squaring
algorithms all which can be called from mp\_sqr().  It is ideal to use mp\_sqr over mp\_mul when squaring terms because
of the speed difference.

\section{Tuning Polynomial Basis Routines}

Both of the Toom-Cook and Karatsuba multiplication algorithms are faster than the traditional $O(n^2)$ approach that
the Comba and baseline algorithms use.  At $O(n^{1.464973})$ and $O(n^{1.584962})$ running times respectively they require
considerably less work.  For example, a 10000-digit multiplication would take roughly 724,000 single precision
multiplications with Toom-Cook or 100,000,000 single precision multiplications with the standard Comba (a factor
of 138).

So why not always use Karatsuba or Toom-Cook?   The simple answer is that they have so much overhead that they're not
actually faster than Comba until you hit distinct  ``cutoff'' points.  For Karatsuba with the default configuration,
GCC 3.3.1 and an Athlon XP processor the cutoff point is roughly 110 digits (about 70 for the Intel P4).  That is, at
110 digits Karatsuba and Comba multiplications just about break even and for 110+ digits Karatsuba is faster.

Toom-Cook has incredible overhead and is probably only useful for very large inputs.  So far no known cutoff points
exist and for the most part I just set the cutoff points very high to make sure they're not called.

A demo program in the ``etc/'' directory of the project called ``tune.c'' can be used to find the cutoff points.  This
can be built with GCC as follows

\begin{alltt}
make XXX
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1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
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1305

When the program is running it will output a series of measurements for different cutoff points.  It will first find
good Karatsuba squaring and multiplication points.  Then it proceeds to find Toom-Cook points.  Note that the Toom-Cook
tuning takes a very long time as the cutoff points are likely to be very high.

\chapter{Modular Reduction}

Modular reduction is process of taking the remainder of one quantity divided by another.  Expressed 
as (\ref{eqn:mod}) the modular reduction is equivalent to the remainder of $b$ divided by $c$.  

\begin{equation}
a \equiv b \mbox{ (mod }c\mbox{)}
\label{eqn:mod}
\end{equation}

Of particular interest to cryptography are reductions where $b$ is limited to the range $0 \le b < c^2$ since particularly 
fast reduction algorithms can be written for the limited range.  

Note that one of the four optimized reduction algorithms are automatically chosen in the modular exponentiation
algorithm mp\_exptmod when an appropriate modulus is detected.  

\section{Straight Division}
In order to effect an arbitrary modular reduction the following algorithm is provided.

\index{mp\_mod}
\begin{alltt}
int mp_mod(mp_int *a, mp_int *b, mp_int *c);
\end{alltt}

This reduces $a$ modulo $b$ and stores the result in $c$.  The sign of $c$ shall agree with the sign 
of $b$.  This algorithm accepts an input $a$ of any range and is not limited by $0 \le a < b^2$.

\section{Barrett Reduction}

Barrett reduction is a generic optimized reduction algorithm that requires pre--computation to achieve
a decent speedup over straight division.  First a $\mu$ value must be precomputed with the following function.








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|







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When the program is running it will output a series of measurements for different cutoff points.  It will first find
good Karatsuba squaring and multiplication points.  Then it proceeds to find Toom-Cook points.  Note that the Toom-Cook
tuning takes a very long time as the cutoff points are likely to be very high.

\chapter{Modular Reduction}

Modular reduction is process of taking the remainder of one quantity divided by another.  Expressed
as (\ref{eqn:mod}) the modular reduction is equivalent to the remainder of $b$ divided by $c$.

\begin{equation}
a \equiv b \mbox{ (mod }c\mbox{)}
\label{eqn:mod}
\end{equation}

Of particular interest to cryptography are reductions where $b$ is limited to the range $0 \le b < c^2$ since particularly
fast reduction algorithms can be written for the limited range.

Note that one of the four optimized reduction algorithms are automatically chosen in the modular exponentiation
algorithm mp\_exptmod when an appropriate modulus is detected.

\section{Straight Division}
In order to effect an arbitrary modular reduction the following algorithm is provided.

\index{mp\_mod}
\begin{alltt}
int mp_mod(mp_int *a, mp_int *b, mp_int *c);
\end{alltt}

This reduces $a$ modulo $b$ and stores the result in $c$.  The sign of $c$ shall agree with the sign
of $b$.  This algorithm accepts an input $a$ of any range and is not limited by $0 \le a < b^2$.

\section{Barrett Reduction}

Barrett reduction is a generic optimized reduction algorithm that requires pre--computation to achieve
a decent speedup over straight division.  First a $\mu$ value must be precomputed with the following function.

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\begin{alltt}
int main(void)
\{
   mp_int   a, b, c, mu;
   int      result;

   /* initialize a,b to desired values, mp_init mu, 
    * c and set c to 1...we want to compute a^3 mod b 
    */

   /* get mu value */
   if ((result = mp_reduce_setup(&mu, b)) != MP_OKAY) \{
      printf("Error getting mu.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* square a to get c = a^2 */
   if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{
      printf("Error squaring.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* now reduce `c' modulo b */
   if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{
      printf("Error reducing.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
   
   /* multiply a to get c = a^3 */
   if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{
      printf("Error reducing.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* now reduce `c' modulo b  */
   if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{
      printf("Error reducing.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
  
   /* c now equals a^3 mod b */

   return EXIT_SUCCESS;
\}
\end{alltt} 

This program will calculate $a^3 \mbox{ mod }b$ if all the functions succeed.  

\section{Montgomery Reduction}

Montgomery is a specialized reduction algorithm for any odd moduli.  Like Barrett reduction a pre--computation
step is required.  This is accomplished with the following.

\index{mp\_montgomery\_setup}
\begin{alltt}
int mp_montgomery_setup(mp_int *a, mp_digit *mp);
\end{alltt}

For the given odd moduli $a$ the precomputation value is placed in $mp$.  The reduction is computed with the 
following.

\index{mp\_montgomery\_reduce}
\begin{alltt}
int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
\end{alltt}
This reduces $a$ in place modulo $m$ with the pre--computed value $mp$.   $a$ must be in the range
$0 \le a < b^2$.

Montgomery reduction is faster than Barrett reduction for moduli smaller than the ``comba'' limit.  With the default
setup for instance, the limit is $127$ digits ($3556$--bits).   Note that this function is not limited to
$127$ digits just that it falls back to a baseline algorithm after that point.  

An important observation is that this reduction does not return $a \mbox{ mod }m$ but $aR^{-1} \mbox{ mod }m$ 
where $R = \beta^n$, $n$ is the n number of digits in $m$ and $\beta$ is radix used (default is $2^{28}$).  

To quickly calculate $R$ the following function was provided.

\index{mp\_montgomery\_calc\_normalization}
\begin{alltt}
int mp_montgomery_calc_normalization(mp_int *a, mp_int *b);
\end{alltt}
Which calculates $a = R$ for the odd moduli $b$ without using multiplication or division.  

The normal modus operandi for Montgomery reductions is to normalize the integers before entering the system.  For
example, to calculate $a^3 \mbox { mod }b$ using Montgomery reduction the value of $a$ can be normalized by
multiplying it by $R$.  Consider the following code snippet.

\begin{alltt}
int main(void)
\{
   mp_int   a, b, c, R;
   mp_digit mp;
   int      result;

   /* initialize a,b to desired values, 
    * mp_init R, c and set c to 1.... 
    */

   /* get normalization */
   if ((result = mp_montgomery_calc_normalization(&R, b)) != MP_OKAY) \{
      printf("Error getting norm.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* get mp value */
   if ((result = mp_montgomery_setup(&c, &mp)) != MP_OKAY) \{
      printf("Error setting up montgomery.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* normalize `a' so now a is equal to aR */
   if ((result = mp_mulmod(&a, &R, &b, &a)) != MP_OKAY) \{
      printf("Error computing aR.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* square a to get c = a^2R^2 */
   if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{
      printf("Error squaring.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* now reduce `c' back down to c = a^2R^2 * R^-1 == a^2R */
   if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
      printf("Error reducing.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
   
   /* multiply a to get c = a^3R^2 */
   if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{
      printf("Error reducing.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* now reduce `c' back down to c = a^3R^2 * R^-1 == a^3R */
   if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
      printf("Error reducing.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}
   
   /* now reduce (again) `c' back down to c = a^3R * R^-1 == a^3 */
   if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
      printf("Error reducing.  \%s", 
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* c now equals a^3 mod b */

   return EXIT_SUCCESS;
\}
\end{alltt} 

This particular example does not look too efficient but it demonstrates the point of the algorithm.  By 
normalizing the inputs the reduced results are always of the form $aR$ for some variable $a$.  This allows
a single final reduction to correct for the normalization and the fast reduction used within the algorithm.

For more details consider examining the file \textit{bn\_mp\_exptmod\_fast.c}.

\section{Restricted Dimminished Radix}

``Dimminished Radix'' reduction refers to reduction with respect to moduli that are ameniable to simple
digit shifting and small multiplications.  In this case the ``restricted'' variant refers to moduli of the
form $\beta^k - p$ for some $k \ge 0$ and $0 < p < \beta$ where $\beta$ is the radix (default to $2^{28}$).  

As in the case of Montgomery reduction there is a pre--computation phase required for a given modulus.

\index{mp\_dr\_setup}
\begin{alltt}
void mp_dr_setup(mp_int *a, mp_digit *d);
\end{alltt}

This computes the value required for the modulus $a$ and stores it in $d$.  This function cannot fail
and does not return any error codes.  After the pre--computation a reduction can be performed with the
following.

\index{mp\_dr\_reduce}
\begin{alltt}
int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp);
\end{alltt}

This reduces $a$ in place modulo $b$ with the pre--computed value $mp$.  $b$ must be of a restricted
dimminished radix form and $a$ must be in the range $0 \le a < b^2$.  Dimminished radix reductions are 
much faster than both Barrett and Montgomery reductions as they have a much lower asymtotic running time.  

Since the moduli are restricted this algorithm is not particularly useful for something like Rabin, RSA or
BBS cryptographic purposes.  This reduction algorithm is useful for Diffie-Hellman and ECC where fixed
primes are acceptable.  

Note that unlike Montgomery reduction there is no normalization process.  The result of this function is
equal to the correct residue.

\section{Unrestricted Dimminshed Radix}

Unrestricted reductions work much like the restricted counterparts except in this case the moduli is of the 
form $2^k - p$ for $0 < p < \beta$.  In this sense the unrestricted reductions are more flexible as they 
can be applied to a wider range of numbers.  

\index{mp\_reduce\_2k\_setup}
\begin{alltt}
int mp_reduce_2k_setup(mp_int *a, mp_digit *d);
\end{alltt}

This will compute the required $d$ value for the given moduli $a$.  

\index{mp\_reduce\_2k}
\begin{alltt}
int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d);
\end{alltt}

This will reduce $a$ in place modulo $n$ with the pre--computed value $d$.  From my experience this routine is 
slower than mp\_dr\_reduce but faster for most moduli sizes than the Montgomery reduction.  

\chapter{Exponentiation}
\section{Single Digit Exponentiation}


















\index{mp\_expt\_d}
\begin{alltt}
int mp_expt_d (mp_int * a, mp_digit b, mp_int * c)
\end{alltt}
This computes $c = a^b$ using a simple binary left-to-right algorithm.  It is faster than repeated multiplications by 
$a$ for all values of $b$ greater than three.  

\section{Modular Exponentiation}
\index{mp\_exptmod}
\begin{alltt}
int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
\end{alltt}
This computes $Y \equiv G^X \mbox{ (mod }P\mbox{)}$ using a variable width sliding window algorithm.  This function
will automatically detect the fastest modular reduction technique to use during the operation.  For negative values of 
$X$ the operation is performed as $Y \equiv (G^{-1} \mbox{ mod }P)^{\vert X \vert} \mbox{ (mod }P\mbox{)}$ provided that 
$gcd(G, P) = 1$.

This function is actually a shell around the two internal exponentiation functions.  This routine will automatically
detect when Barrett, Montgomery, Restricted and Unrestricted Dimminished Radix based exponentiation can be used.  Generally
moduli of the a ``restricted dimminished radix'' form lead to the fastest modular exponentiations.  Followed by Montgomery
and the other two algorithms.

\section{Root Finding}
\index{mp\_n\_root}
\begin{alltt}
int mp_n_root (mp_int * a, mp_digit b, mp_int * c)
\end{alltt}
This computes $c = a^{1/b}$ such that $c^b \le a$ and $(c+1)^b > a$.  The implementation of this function is not 
ideal for values of $b$ greater than three.  It will work but become very slow.  So unless you are working with very small
numbers (less than 1000 bits) I'd avoid $b > 3$ situations.  Will return a positive root only for even roots and return
a root with the sign of the input for odd roots.  For example, performing $4^{1/2}$ will return $2$ whereas $(-8)^{1/3}$ 
will return $-2$.  

This algorithm uses the ``Newton Approximation'' method and will converge on the correct root fairly quickly.  Since
the algorithm requires raising $a$ to the power of $b$ it is not ideal to attempt to find roots for large
values of $b$.  If particularly large roots are required then a factor method could be used instead.  For example,
$a^{1/16}$ is equivalent to $\left (a^{1/4} \right)^{1/4}$ or simply 
$\left ( \left ( \left ( a^{1/2} \right )^{1/2} \right )^{1/2} \right )^{1/2}$

\chapter{Prime Numbers}
\section{Trial Division}
\index{mp\_prime\_is\_divisible}
\begin{alltt}
int mp_prime_is_divisible (mp_int * a, int *result)
\end{alltt}
This will attempt to evenly divide $a$ by a list of primes\footnote{Default is the first 256 primes.} and store the 
outcome in ``result''.  That is if $result = 0$ then $a$ is not divisible by the primes, otherwise it is.  Note that 
if the function does not return \textbf{MP\_OKAY} the value in ``result'' should be considered undefined\footnote{Currently
the default is to set it to zero first.}.

\section{Fermat Test}
\index{mp\_prime\_fermat}
\begin{alltt}
int mp_prime_fermat (mp_int * a, mp_int * b, int *result)
\end{alltt}
Performs a Fermat primality test to the base $b$.  That is it computes $b^a \mbox{ mod }a$ and tests whether the value is
equal to $b$ or not.  If the values are equal then $a$ is probably prime and $result$ is set to one.  Otherwise $result$
is set to zero.

\section{Miller-Rabin Test}
\index{mp\_prime\_miller\_rabin}
\begin{alltt}
int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result)
\end{alltt}
Performs a Miller-Rabin test to the base $b$ of $a$.  This test is much stronger than the Fermat test and is very hard to
fool (besides with Carmichael numbers).  If $a$ passes the test (therefore is probably prime) $result$ is set to one.  
Otherwise $result$ is set to zero.  

Note that is suggested that you use the Miller-Rabin test instead of the Fermat test since all of the failures of 
Miller-Rabin are a subset of the failures of the Fermat test.

\subsection{Required Number of Tests}
Generally to ensure a number is very likely to be prime you have to perform the Miller-Rabin with at least a half-dozen
or so unique bases.  However, it has been proven that the probability of failure goes down as the size of the input goes up.
This is why a simple function has been provided to help out.

\index{mp\_prime\_rabin\_miller\_trials}
\begin{alltt}
int mp_prime_rabin_miller_trials(int size)
\end{alltt}
This returns the number of trials required for a $2^{-96}$ (or lower) probability of failure for a given ``size'' expressed
in bits.  This comes in handy specially since larger numbers are slower to test.  For example, a 512-bit number would
require ten tests whereas a 1024-bit number would only require four tests. 

You should always still perform a trial division before a Miller-Rabin test though.

\section{Primality Testing}
\index{mp\_prime\_is\_prime}
\begin{alltt}
int mp_prime_is_prime (mp_int * a, int t, int *result)
\end{alltt}
This will perform a trial division followed by $t$ rounds of Miller-Rabin tests on $a$ and store the result in $result$.  
If $a$ passes all of the tests $result$ is set to one, otherwise it is set to zero.  Note that $t$ is bounded by 
$1 \le t < PRIME\_SIZE$ where $PRIME\_SIZE$ is the number of primes in the prime number table (by default this is $256$).

\section{Next Prime}
\index{mp\_prime\_next\_prime}
\begin{alltt}
int mp_prime_next_prime(mp_int *a, int t, int bbs_style)
\end{alltt}
This finds the next prime after $a$ that passes mp\_prime\_is\_prime() with $t$ tests.  Set $bbs\_style$ to one if you 
want only the next prime congruent to $3 \mbox{ mod } 4$, otherwise set it to zero to find any next prime.  

\section{Random Primes}
\index{mp\_prime\_random}
\begin{alltt}
int mp_prime_random(mp_int *a, int t, int size, int bbs, 
                    ltm_prime_callback cb, void *dat)
\end{alltt}
This will find a prime greater than $256^{size}$ which can be ``bbs\_style'' or not depending on $bbs$ and must pass
$t$ rounds of tests.  The ``ltm\_prime\_callback'' is a typedef for 

\begin{alltt}
typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat);
\end{alltt}

Which is a function that must read $len$ bytes (and return the amount stored) into $dst$.  The $dat$ variable is simply
copied from the original input.  It can be used to pass RNG context data to the callback.  The function 
mp\_prime\_random() is more suitable for generating primes which must be secret (as in the case of RSA) since there 
is no skew on the least significant bits.

\textit{Note:}  As of v0.30 of the LibTomMath library this function has been deprecated.  It is still available
but users are encouraged to use the new mp\_prime\_random\_ex() function instead.

\subsection{Extended Generation}
\index{mp\_prime\_random\_ex}
\begin{alltt}
int mp_prime_random_ex(mp_int *a,    int t, 
                       int     size, int flags, 
                       ltm_prime_callback cb, void *dat);
\end{alltt}
This will generate a prime in $a$ using $t$ tests of the primality testing algorithms.  The variable $size$
specifies the bit length of the prime desired.  The variable $flags$ specifies one of several options available
(see fig. \ref{fig:primeopts}) which can be OR'ed together.  The callback parameters are used as in 
mp\_prime\_random().

\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|r|l|}
\hline \textbf{Flag}         & \textbf{Meaning} \\







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\begin{alltt}
int main(void)
\{
   mp_int   a, b, c, mu;
   int      result;

   /* initialize a,b to desired values, mp_init mu,
    * c and set c to 1...we want to compute a^3 mod b
    */

   /* get mu value */
   if ((result = mp_reduce_setup(&mu, b)) != MP_OKAY) \{
      printf("Error getting mu.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* square a to get c = a^2 */
   if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{
      printf("Error squaring.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* now reduce `c' modulo b */
   if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{
      printf("Error reducing.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* multiply a to get c = a^3 */
   if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{
      printf("Error reducing.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* now reduce `c' modulo b  */
   if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{
      printf("Error reducing.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* c now equals a^3 mod b */

   return EXIT_SUCCESS;
\}
\end{alltt}

This program will calculate $a^3 \mbox{ mod }b$ if all the functions succeed.

\section{Montgomery Reduction}

Montgomery is a specialized reduction algorithm for any odd moduli.  Like Barrett reduction a pre--computation
step is required.  This is accomplished with the following.

\index{mp\_montgomery\_setup}
\begin{alltt}
int mp_montgomery_setup(mp_int *a, mp_digit *mp);
\end{alltt}

For the given odd moduli $a$ the precomputation value is placed in $mp$.  The reduction is computed with the
following.

\index{mp\_montgomery\_reduce}
\begin{alltt}
int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
\end{alltt}
This reduces $a$ in place modulo $m$ with the pre--computed value $mp$.   $a$ must be in the range
$0 \le a < b^2$.

Montgomery reduction is faster than Barrett reduction for moduli smaller than the ``comba'' limit.  With the default
setup for instance, the limit is $127$ digits ($3556$--bits).   Note that this function is not limited to
$127$ digits just that it falls back to a baseline algorithm after that point.

An important observation is that this reduction does not return $a \mbox{ mod }m$ but $aR^{-1} \mbox{ mod }m$
where $R = \beta^n$, $n$ is the n number of digits in $m$ and $\beta$ is radix used (default is $2^{28}$).

To quickly calculate $R$ the following function was provided.

\index{mp\_montgomery\_calc\_normalization}
\begin{alltt}
int mp_montgomery_calc_normalization(mp_int *a, mp_int *b);
\end{alltt}
Which calculates $a = R$ for the odd moduli $b$ without using multiplication or division.

The normal modus operandi for Montgomery reductions is to normalize the integers before entering the system.  For
example, to calculate $a^3 \mbox { mod }b$ using Montgomery reduction the value of $a$ can be normalized by
multiplying it by $R$.  Consider the following code snippet.

\begin{alltt}
int main(void)
\{
   mp_int   a, b, c, R;
   mp_digit mp;
   int      result;

   /* initialize a,b to desired values,
    * mp_init R, c and set c to 1....
    */

   /* get normalization */
   if ((result = mp_montgomery_calc_normalization(&R, b)) != MP_OKAY) \{
      printf("Error getting norm.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* get mp value */
   if ((result = mp_montgomery_setup(&c, &mp)) != MP_OKAY) \{
      printf("Error setting up montgomery.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* normalize `a' so now a is equal to aR */
   if ((result = mp_mulmod(&a, &R, &b, &a)) != MP_OKAY) \{
      printf("Error computing aR.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* square a to get c = a^2R^2 */
   if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{
      printf("Error squaring.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* now reduce `c' back down to c = a^2R^2 * R^-1 == a^2R */
   if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
      printf("Error reducing.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* multiply a to get c = a^3R^2 */
   if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{
      printf("Error reducing.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* now reduce `c' back down to c = a^3R^2 * R^-1 == a^3R */
   if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
      printf("Error reducing.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* now reduce (again) `c' back down to c = a^3R * R^-1 == a^3 */
   if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
      printf("Error reducing.  \%s",
             mp_error_to_string(result));
      return EXIT_FAILURE;
   \}

   /* c now equals a^3 mod b */

   return EXIT_SUCCESS;
\}
\end{alltt}

This particular example does not look too efficient but it demonstrates the point of the algorithm.  By
normalizing the inputs the reduced results are always of the form $aR$ for some variable $a$.  This allows
a single final reduction to correct for the normalization and the fast reduction used within the algorithm.

For more details consider examining the file \textit{bn\_mp\_exptmod\_fast.c}.

\section{Restricted Dimminished Radix}

``Dimminished Radix'' reduction refers to reduction with respect to moduli that are ameniable to simple
digit shifting and small multiplications.  In this case the ``restricted'' variant refers to moduli of the
form $\beta^k - p$ for some $k \ge 0$ and $0 < p < \beta$ where $\beta$ is the radix (default to $2^{28}$).

As in the case of Montgomery reduction there is a pre--computation phase required for a given modulus.

\index{mp\_dr\_setup}
\begin{alltt}
void mp_dr_setup(mp_int *a, mp_digit *d);
\end{alltt}

This computes the value required for the modulus $a$ and stores it in $d$.  This function cannot fail
and does not return any error codes.  After the pre--computation a reduction can be performed with the
following.

\index{mp\_dr\_reduce}
\begin{alltt}
int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp);
\end{alltt}

This reduces $a$ in place modulo $b$ with the pre--computed value $mp$.  $b$ must be of a restricted
dimminished radix form and $a$ must be in the range $0 \le a < b^2$.  Dimminished radix reductions are
much faster than both Barrett and Montgomery reductions as they have a much lower asymtotic running time.

Since the moduli are restricted this algorithm is not particularly useful for something like Rabin, RSA or
BBS cryptographic purposes.  This reduction algorithm is useful for Diffie-Hellman and ECC where fixed
primes are acceptable.

Note that unlike Montgomery reduction there is no normalization process.  The result of this function is
equal to the correct residue.

\section{Unrestricted Dimminshed Radix}

Unrestricted reductions work much like the restricted counterparts except in this case the moduli is of the
form $2^k - p$ for $0 < p < \beta$.  In this sense the unrestricted reductions are more flexible as they
can be applied to a wider range of numbers.

\index{mp\_reduce\_2k\_setup}
\begin{alltt}
int mp_reduce_2k_setup(mp_int *a, mp_digit *d);
\end{alltt}

This will compute the required $d$ value for the given moduli $a$.

\index{mp\_reduce\_2k}
\begin{alltt}
int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d);
\end{alltt}

This will reduce $a$ in place modulo $n$ with the pre--computed value $d$.  From my experience this routine is
slower than mp\_dr\_reduce but faster for most moduli sizes than the Montgomery reduction.

\chapter{Exponentiation}
\section{Single Digit Exponentiation}
\index{mp\_expt\_d\_ex}
\begin{alltt}
int mp_expt_d_ex (mp_int * a, mp_digit b, mp_int * c, int fast)
\end{alltt}
This function computes $c = a^b$.

With parameter \textit{fast} set to $0$ the old version of the algorithm is used,
when \textit{fast} is $1$, a faster but not statically timed version of the algorithm is used.

The old version uses a simple binary left-to-right algorithm.
It is faster than repeated multiplications by $a$ for all values of $b$ greater than three.

The new version uses a binary right-to-left algorithm.

The difference between the old and the new version is that the old version always
executes $DIGIT\_BIT$ iterations. The new algorithm executes only $n$ iterations
where $n$ is equal to the position of the highest bit that is set in $b$.

\index{mp\_expt\_d}
\begin{alltt}
int mp_expt_d (mp_int * a, mp_digit b, mp_int * c)
\end{alltt}
mp\_expt\_d(a, b, c) is a wrapper function to mp\_expt\_d\_ex(a, b, c, 0).


\section{Modular Exponentiation}
\index{mp\_exptmod}
\begin{alltt}
int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
\end{alltt}
This computes $Y \equiv G^X \mbox{ (mod }P\mbox{)}$ using a variable width sliding window algorithm.  This function
will automatically detect the fastest modular reduction technique to use during the operation.  For negative values of
$X$ the operation is performed as $Y \equiv (G^{-1} \mbox{ mod }P)^{\vert X \vert} \mbox{ (mod }P\mbox{)}$ provided that
$gcd(G, P) = 1$.

This function is actually a shell around the two internal exponentiation functions.  This routine will automatically
detect when Barrett, Montgomery, Restricted and Unrestricted Dimminished Radix based exponentiation can be used.  Generally
moduli of the a ``restricted dimminished radix'' form lead to the fastest modular exponentiations.  Followed by Montgomery
and the other two algorithms.

\section{Root Finding}
\index{mp\_n\_root}
\begin{alltt}
int mp_n_root (mp_int * a, mp_digit b, mp_int * c)
\end{alltt}
This computes $c = a^{1/b}$ such that $c^b \le a$ and $(c+1)^b > a$.  The implementation of this function is not
ideal for values of $b$ greater than three.  It will work but become very slow.  So unless you are working with very small
numbers (less than 1000 bits) I'd avoid $b > 3$ situations.  Will return a positive root only for even roots and return
a root with the sign of the input for odd roots.  For example, performing $4^{1/2}$ will return $2$ whereas $(-8)^{1/3}$
will return $-2$.

This algorithm uses the ``Newton Approximation'' method and will converge on the correct root fairly quickly.  Since
the algorithm requires raising $a$ to the power of $b$ it is not ideal to attempt to find roots for large
values of $b$.  If particularly large roots are required then a factor method could be used instead.  For example,
$a^{1/16}$ is equivalent to $\left (a^{1/4} \right)^{1/4}$ or simply
$\left ( \left ( \left ( a^{1/2} \right )^{1/2} \right )^{1/2} \right )^{1/2}$

\chapter{Prime Numbers}
\section{Trial Division}
\index{mp\_prime\_is\_divisible}
\begin{alltt}
int mp_prime_is_divisible (mp_int * a, int *result)
\end{alltt}
This will attempt to evenly divide $a$ by a list of primes\footnote{Default is the first 256 primes.} and store the
outcome in ``result''.  That is if $result = 0$ then $a$ is not divisible by the primes, otherwise it is.  Note that
if the function does not return \textbf{MP\_OKAY} the value in ``result'' should be considered undefined\footnote{Currently
the default is to set it to zero first.}.

\section{Fermat Test}
\index{mp\_prime\_fermat}
\begin{alltt}
int mp_prime_fermat (mp_int * a, mp_int * b, int *result)
\end{alltt}
Performs a Fermat primality test to the base $b$.  That is it computes $b^a \mbox{ mod }a$ and tests whether the value is
equal to $b$ or not.  If the values are equal then $a$ is probably prime and $result$ is set to one.  Otherwise $result$
is set to zero.

\section{Miller-Rabin Test}
\index{mp\_prime\_miller\_rabin}
\begin{alltt}
int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result)
\end{alltt}
Performs a Miller-Rabin test to the base $b$ of $a$.  This test is much stronger than the Fermat test and is very hard to
fool (besides with Carmichael numbers).  If $a$ passes the test (therefore is probably prime) $result$ is set to one.
Otherwise $result$ is set to zero.

Note that is suggested that you use the Miller-Rabin test instead of the Fermat test since all of the failures of
Miller-Rabin are a subset of the failures of the Fermat test.

\subsection{Required Number of Tests}
Generally to ensure a number is very likely to be prime you have to perform the Miller-Rabin with at least a half-dozen
or so unique bases.  However, it has been proven that the probability of failure goes down as the size of the input goes up.
This is why a simple function has been provided to help out.

\index{mp\_prime\_rabin\_miller\_trials}
\begin{alltt}
int mp_prime_rabin_miller_trials(int size)
\end{alltt}
This returns the number of trials required for a $2^{-96}$ (or lower) probability of failure for a given ``size'' expressed
in bits.  This comes in handy specially since larger numbers are slower to test.  For example, a 512-bit number would
require ten tests whereas a 1024-bit number would only require four tests.

You should always still perform a trial division before a Miller-Rabin test though.

\section{Primality Testing}
\index{mp\_prime\_is\_prime}
\begin{alltt}
int mp_prime_is_prime (mp_int * a, int t, int *result)
\end{alltt}
This will perform a trial division followed by $t$ rounds of Miller-Rabin tests on $a$ and store the result in $result$.
If $a$ passes all of the tests $result$ is set to one, otherwise it is set to zero.  Note that $t$ is bounded by
$1 \le t < PRIME\_SIZE$ where $PRIME\_SIZE$ is the number of primes in the prime number table (by default this is $256$).

\section{Next Prime}
\index{mp\_prime\_next\_prime}
\begin{alltt}
int mp_prime_next_prime(mp_int *a, int t, int bbs_style)
\end{alltt}
This finds the next prime after $a$ that passes mp\_prime\_is\_prime() with $t$ tests.  Set $bbs\_style$ to one if you
want only the next prime congruent to $3 \mbox{ mod } 4$, otherwise set it to zero to find any next prime.

\section{Random Primes}
\index{mp\_prime\_random}
\begin{alltt}
int mp_prime_random(mp_int *a, int t, int size, int bbs,
                    ltm_prime_callback cb, void *dat)
\end{alltt}
This will find a prime greater than $256^{size}$ which can be ``bbs\_style'' or not depending on $bbs$ and must pass
$t$ rounds of tests.  The ``ltm\_prime\_callback'' is a typedef for

\begin{alltt}
typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat);
\end{alltt}

Which is a function that must read $len$ bytes (and return the amount stored) into $dst$.  The $dat$ variable is simply
copied from the original input.  It can be used to pass RNG context data to the callback.  The function
mp\_prime\_random() is more suitable for generating primes which must be secret (as in the case of RSA) since there
is no skew on the least significant bits.

\textit{Note:}  As of v0.30 of the LibTomMath library this function has been deprecated.  It is still available
but users are encouraged to use the new mp\_prime\_random\_ex() function instead.

\subsection{Extended Generation}
\index{mp\_prime\_random\_ex}
\begin{alltt}
int mp_prime_random_ex(mp_int *a,    int t,
                       int     size, int flags,
                       ltm_prime_callback cb, void *dat);
\end{alltt}
This will generate a prime in $a$ using $t$ tests of the primality testing algorithms.  The variable $size$
specifies the bit length of the prime desired.  The variable $flags$ specifies one of several options available
(see fig. \ref{fig:primeopts}) which can be OR'ed together.  The callback parameters are used as in
mp\_prime\_random().

\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|r|l|}
\hline \textbf{Flag}         & \textbf{Meaning} \\
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to terminate the string.  Valid values of ``radix'' line in the range $[2, 64]$.  To determine the size (exact) required
by the conversion before storing any data use the following function.

\index{mp\_radix\_size}
\begin{alltt}
int mp_radix_size (mp_int * a, int radix, int *size)
\end{alltt}
This stores in ``size'' the number of characters (including space for the NUL terminator) required.  Upon error this 
function returns an error code and ``size'' will be zero.  

\subsection{From ASCII}
\index{mp\_read\_radix}
\begin{alltt}
int mp_read_radix (mp_int * a, char *str, int radix);
\end{alltt}
This will read the base-``radix'' NUL terminated string from ``str'' into $a$.  It will stop reading when it reads a







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to terminate the string.  Valid values of ``radix'' line in the range $[2, 64]$.  To determine the size (exact) required
by the conversion before storing any data use the following function.

\index{mp\_radix\_size}
\begin{alltt}
int mp_radix_size (mp_int * a, int radix, int *size)
\end{alltt}
This stores in ``size'' the number of characters (including space for the NUL terminator) required.  Upon error this
function returns an error code and ``size'' will be zero.

\subsection{From ASCII}
\index{mp\_read\_radix}
\begin{alltt}
int mp_read_radix (mp_int * a, char *str, int radix);
\end{alltt}
This will read the base-``radix'' NUL terminated string from ``str'' into $a$.  It will stop reading when it reads a
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\begin{alltt}
int mp_signed_bin_size(mp_int *a);
int mp_read_signed_bin(mp_int *a, unsigned char *b, int c);
int mp_to_signed_bin(mp_int *a, unsigned char *b);
\end{alltt}
They operate essentially the same as the unsigned copies except they prefix the data with zero or non--zero
byte depending on the sign.  If the sign is zpos (e.g. not negative) the prefix is zero, otherwise the prefix
is non--zero.  

\chapter{Algebraic Functions}
\section{Extended Euclidean Algorithm}
\index{mp\_exteuclid}
\begin{alltt}
int mp_exteuclid(mp_int *a, mp_int *b, 
                 mp_int *U1, mp_int *U2, mp_int *U3);
\end{alltt}

This finds the triple U1/U2/U3 using the Extended Euclidean algorithm such that the following equation holds.

\begin{equation}
a \cdot U1 + b \cdot U2 = U3
\end{equation}

Any of the U1/U2/U3 paramters can be set to \textbf{NULL} if they are not desired.  

\section{Greatest Common Divisor}
\index{mp\_gcd}
\begin{alltt}
int mp_gcd (mp_int * a, mp_int * b, mp_int * c)
\end{alltt}
This will compute the greatest common divisor of $a$ and $b$ and store it in $c$.







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\begin{alltt}
int mp_signed_bin_size(mp_int *a);
int mp_read_signed_bin(mp_int *a, unsigned char *b, int c);
int mp_to_signed_bin(mp_int *a, unsigned char *b);
\end{alltt}
They operate essentially the same as the unsigned copies except they prefix the data with zero or non--zero
byte depending on the sign.  If the sign is zpos (e.g. not negative) the prefix is zero, otherwise the prefix
is non--zero.

\chapter{Algebraic Functions}
\section{Extended Euclidean Algorithm}
\index{mp\_exteuclid}
\begin{alltt}
int mp_exteuclid(mp_int *a, mp_int *b,
                 mp_int *U1, mp_int *U2, mp_int *U3);
\end{alltt}

This finds the triple U1/U2/U3 using the Extended Euclidean algorithm such that the following equation holds.

\begin{equation}
a \cdot U1 + b \cdot U2 = U3
\end{equation}

Any of the U1/U2/U3 paramters can be set to \textbf{NULL} if they are not desired.

\section{Greatest Common Divisor}
\index{mp\_gcd}
\begin{alltt}
int mp_gcd (mp_int * a, mp_int * b, mp_int * c)
\end{alltt}
This will compute the greatest common divisor of $a$ and $b$ and store it in $c$.
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\index{mp\_jacobi}
\begin{alltt}
int mp_jacobi (mp_int * a, mp_int * p, int *c)
\end{alltt}
This will compute the Jacobi symbol for $a$ with respect to $p$.  If $p$ is prime this essentially computes the Legendre
symbol.  The result is stored in $c$ and can take on one of three values $\lbrace -1, 0, 1 \rbrace$.  If $p$ is prime
then the result will be $-1$ when $a$ is not a quadratic residue modulo $p$.  The result will be $0$ if $a$ divides $p$
and the result will be $1$ if $a$ is a quadratic residue modulo $p$.  






















\section{Modular Inverse}
\index{mp\_invmod}
\begin{alltt}
int mp_invmod (mp_int * a, mp_int * b, mp_int * c)
\end{alltt}
Computes the multiplicative inverse of $a$ modulo $b$ and stores the result in $c$ such that $ac \equiv 1 \mbox{ (mod }b\mbox{)}$.







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\index{mp\_jacobi}
\begin{alltt}
int mp_jacobi (mp_int * a, mp_int * p, int *c)
\end{alltt}
This will compute the Jacobi symbol for $a$ with respect to $p$.  If $p$ is prime this essentially computes the Legendre
symbol.  The result is stored in $c$ and can take on one of three values $\lbrace -1, 0, 1 \rbrace$.  If $p$ is prime
then the result will be $-1$ when $a$ is not a quadratic residue modulo $p$.  The result will be $0$ if $a$ divides $p$
and the result will be $1$ if $a$ is a quadratic residue modulo $p$.

\section{Modular square root}
\index{mp\_sqrtmod\_prime}
\begin{alltt}
int mp_sqrtmod_prime(mp_int *n, mp_int *p, mp_int *r)
\end{alltt}

This will solve the modular equatioon $r^2 = n \mod p$ where $p$ is a prime number greater than 2 (odd prime).
The result is returned in the third argument $r$, the function returns \textbf{MP\_OKAY} on success,
other return values indicate failure.

The implementation is split for two different cases:

1. if $p \mod 4 == 3$ we apply \href{http://cacr.uwaterloo.ca/hac/}{Handbook of Applied Cryptography algorithm 3.36} and compute $r$ directly as
$r = n^{(p+1)/4} \mod p$

2. otherwise we use \href{https://en.wikipedia.org/wiki/Tonelli-Shanks_algorithm}{Tonelli-Shanks algorithm}

The function does not check the primality of parameter $p$ thus it is up to the caller to assure that this parameter
is a prime number. When $p$ is a composite the function behaviour is undefined, it may even return a false-positive
\textbf{MP\_OKAY}.

\section{Modular Inverse}
\index{mp\_invmod}
\begin{alltt}
int mp_invmod (mp_int * a, mp_int * b, mp_int * c)
\end{alltt}
Computes the multiplicative inverse of $a$ modulo $b$ and stores the result in $c$ such that $ac \equiv 1 \mbox{ (mod }b\mbox{)}$.
Changes to libtommath/bn_error.c.
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#include <tommath.h>
#ifdef BN_ERROR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

static const struct {
     int code;
     char *msg;
} msgs[] = {
     { MP_OKAY, "Successful" },
     { MP_MEM,  "Out of heap" },
     { MP_VAL,  "Value out of range" }
};

/* return a char * string for a given code */
char *mp_error_to_string(int code)
{
   int x;

   /* scan the lookup table for the given message */
   for (x = 0; x < (int)(sizeof(msgs) / sizeof(msgs[0])); x++) {
       if (msgs[x].code == code) {
          return msgs[x].msg;
       }
   }

   /* generic reply for invalid code */
   return "Invalid error code";
}

#endif




|













|




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>
>
>
>
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#include <tommath_private.h>
#ifdef BN_ERROR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

static const struct {
     int code;
     const char *msg;
} msgs[] = {
     { MP_OKAY, "Successful" },
     { MP_MEM,  "Out of heap" },
     { MP_VAL,  "Value out of range" }
};

/* return a char * string for a given code */
const char *mp_error_to_string(int code)
{
   int x;

   /* scan the lookup table for the given message */
   for (x = 0; x < (int)(sizeof(msgs) / sizeof(msgs[0])); x++) {
       if (msgs[x].code == code) {
          return msgs[x].msg;
       }
   }

   /* generic reply for invalid code */
   return "Invalid error code";
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_fast_mp_invmod.c.
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#include <tommath.h>
#ifdef BN_FAST_MP_INVMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* computes the modular inverse via binary extended euclidean algorithm, 
 * that is c = 1/a mod b 
 *
 * Based on slow invmod except this is optimized for the case where b is 
 * odd as per HAC Note 14.64 on pp. 610
 */
int fast_mp_invmod (mp_int * a, mp_int * b, mp_int * c)
{
  mp_int  x, y, u, v, B, D;
  int     res, neg;

  /* 2. [modified] b must be odd   */
  if (mp_iseven (b) == 1) {
    return MP_VAL;
  }

  /* init all our temps */
  if ((res = mp_init_multi(&x, &y, &u, &v, &B, &D, NULL)) != MP_OKAY) {
     return res;
  }
|













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#include <tommath_private.h>
#ifdef BN_FAST_MP_INVMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* computes the modular inverse via binary extended euclidean algorithm, 
 * that is c = 1/a mod b 
 *
 * Based on slow invmod except this is optimized for the case where b is 
 * odd as per HAC Note 14.64 on pp. 610
 */
int fast_mp_invmod (mp_int * a, mp_int * b, mp_int * c)
{
  mp_int  x, y, u, v, B, D;
  int     res, neg;

  /* 2. [modified] b must be odd   */
  if (mp_iseven (b) == MP_YES) {
    return MP_VAL;
  }

  /* init all our temps */
  if ((res = mp_init_multi(&x, &y, &u, &v, &B, &D, NULL)) != MP_OKAY) {
     return res;
  }
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  if ((res = mp_copy (&y, &v)) != MP_OKAY) {
    goto LBL_ERR;
  }
  mp_set (&D, 1);

top:
  /* 4.  while u is even do */
  while (mp_iseven (&u) == 1) {
    /* 4.1 u = u/2 */
    if ((res = mp_div_2 (&u, &u)) != MP_OKAY) {
      goto LBL_ERR;
    }
    /* 4.2 if B is odd then */
    if (mp_isodd (&B) == 1) {
      if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) {
        goto LBL_ERR;
      }
    }
    /* B = B/2 */
    if ((res = mp_div_2 (&B, &B)) != MP_OKAY) {
      goto LBL_ERR;
    }
  }

  /* 5.  while v is even do */
  while (mp_iseven (&v) == 1) {
    /* 5.1 v = v/2 */
    if ((res = mp_div_2 (&v, &v)) != MP_OKAY) {
      goto LBL_ERR;
    }
    /* 5.2 if D is odd then */
    if (mp_isodd (&D) == 1) {
      /* D = (D-x)/2 */
      if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) {
        goto LBL_ERR;
      }
    }
    /* D = D/2 */
    if ((res = mp_div_2 (&D, &D)) != MP_OKAY) {







|





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  if ((res = mp_copy (&y, &v)) != MP_OKAY) {
    goto LBL_ERR;
  }
  mp_set (&D, 1);

top:
  /* 4.  while u is even do */
  while (mp_iseven (&u) == MP_YES) {
    /* 4.1 u = u/2 */
    if ((res = mp_div_2 (&u, &u)) != MP_OKAY) {
      goto LBL_ERR;
    }
    /* 4.2 if B is odd then */
    if (mp_isodd (&B) == MP_YES) {
      if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) {
        goto LBL_ERR;
      }
    }
    /* B = B/2 */
    if ((res = mp_div_2 (&B, &B)) != MP_OKAY) {
      goto LBL_ERR;
    }
  }

  /* 5.  while v is even do */
  while (mp_iseven (&v) == MP_YES) {
    /* 5.1 v = v/2 */
    if ((res = mp_div_2 (&v, &v)) != MP_OKAY) {
      goto LBL_ERR;
    }
    /* 5.2 if D is odd then */
    if (mp_isodd (&D) == MP_YES) {
      /* D = (D-x)/2 */
      if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) {
        goto LBL_ERR;
      }
    }
    /* D = D/2 */
    if ((res = mp_div_2 (&D, &D)) != MP_OKAY) {
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    if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) {
      goto LBL_ERR;
    }
  }

  /* if not zero goto step 4 */
  if (mp_iszero (&u) == 0) {
    goto top;
  }

  /* now a = C, b = D, gcd == g*v */

  /* if v != 1 then there is no inverse */
  if (mp_cmp_d (&v, 1) != MP_EQ) {







|







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    if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) {
      goto LBL_ERR;
    }
  }

  /* if not zero goto step 4 */
  if (mp_iszero (&u) == MP_NO) {
    goto top;
  }

  /* now a = C, b = D, gcd == g*v */

  /* if v != 1 then there is no inverse */
  if (mp_cmp_d (&v, 1) != MP_EQ) {
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  c->sign = neg;
  res = MP_OKAY;

LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &B, &D, NULL);
  return res;
}
#endif











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  c->sign = neg;
  res = MP_OKAY;

LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &B, &D, NULL);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_fast_mp_montgomery_reduce.c.
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#include <tommath.h>
#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* computes xR**-1 == x (mod N) via Montgomery Reduction
 *
 * This is an optimized implementation of montgomery_reduce
 * which uses the comba method to quickly calculate the columns of the
 * reduction.
 *
 * Based on Algorithm 14.32 on pp.601 of HAC.
*/
int fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
{
  int     ix, res, olduse;
  mp_word W[MP_WARRAY];

  /* get old used count */
  olduse = x->used;

  /* grow a as required */
  if (x->alloc < n->used + 1) {
    if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) {
      return res;
    }
  }

  /* first we have to get the digits of the input into
   * an array of double precision words W[...]
   */
  {
    register mp_word *_W;
    register mp_digit *tmpx;

    /* alias for the W[] array */
    _W   = W;

    /* alias for the digits of  x*/
    tmpx = x->dp;

    /* copy the digits of a into W[0..a->used-1] */
    for (ix = 0; ix < x->used; ix++) {
      *_W++ = *tmpx++;
    }

    /* zero the high words of W[a->used..m->used*2] */
    for (; ix < n->used * 2 + 1; ix++) {
      *_W++ = 0;
    }
  }

  /* now we proceed to zero successive digits
   * from the least significant upwards
   */
  for (ix = 0; ix < n->used; ix++) {
    /* mu = ai * m' mod b
     *
     * We avoid a double precision multiplication (which isn't required)
     * by casting the value down to a mp_digit.  Note this requires
     * that W[ix-1] have  the carry cleared (see after the inner loop)
     */
    register mp_digit mu;
    mu = (mp_digit) (((W[ix] & MP_MASK) * rho) & MP_MASK);

    /* a = a + mu * m * b**i
     *
     * This is computed in place and on the fly.  The multiplication
     * by b**i is handled by offseting which columns the results
     * are added to.
     *
     * Note the comba method normally doesn't handle carries in the
     * inner loop In this case we fix the carry from the previous
     * column since the Montgomery reduction requires digits of the
     * result (so far) [see above] to work.  This is
     * handled by fixing up one carry after the inner loop.  The
     * carry fixups are done in order so after these loops the
     * first m->used words of W[] have the carries fixed
     */
    {
      register int iy;
      register mp_digit *tmpn;
      register mp_word *_W;

      /* alias for the digits of the modulus */
      tmpn = n->dp;

      /* Alias for the columns set by an offset of ix */
      _W = W + ix;

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#include <tommath_private.h>
#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* computes xR**-1 == x (mod N) via Montgomery Reduction
 *
 * This is an optimized implementation of montgomery_reduce
 * which uses the comba method to quickly calculate the columns of the
 * reduction.
 *
 * Based on Algorithm 14.32 on pp.601 of HAC.
*/
int fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
{
  int     ix, res, olduse;
  mp_word W[MP_WARRAY];

  /* get old used count */
  olduse = x->used;

  /* grow a as required */
  if (x->alloc < (n->used + 1)) {
    if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) {
      return res;
    }
  }

  /* first we have to get the digits of the input into
   * an array of double precision words W[...]
   */
  {
    mp_word *_W;
    mp_digit *tmpx;

    /* alias for the W[] array */
    _W   = W;

    /* alias for the digits of  x*/
    tmpx = x->dp;

    /* copy the digits of a into W[0..a->used-1] */
    for (ix = 0; ix < x->used; ix++) {
      *_W++ = *tmpx++;
    }

    /* zero the high words of W[a->used..m->used*2] */
    for (; ix < ((n->used * 2) + 1); ix++) {
      *_W++ = 0;
    }
  }

  /* now we proceed to zero successive digits
   * from the least significant upwards
   */
  for (ix = 0; ix < n->used; ix++) {
    /* mu = ai * m' mod b
     *
     * We avoid a double precision multiplication (which isn't required)
     * by casting the value down to a mp_digit.  Note this requires
     * that W[ix-1] have  the carry cleared (see after the inner loop)
     */
    mp_digit mu;
    mu = (mp_digit) (((W[ix] & MP_MASK) * rho) & MP_MASK);

    /* a = a + mu * m * b**i
     *
     * This is computed in place and on the fly.  The multiplication
     * by b**i is handled by offseting which columns the results
     * are added to.
     *
     * Note the comba method normally doesn't handle carries in the
     * inner loop In this case we fix the carry from the previous
     * column since the Montgomery reduction requires digits of the
     * result (so far) [see above] to work.  This is
     * handled by fixing up one carry after the inner loop.  The
     * carry fixups are done in order so after these loops the
     * first m->used words of W[] have the carries fixed
     */
    {
      int iy;
      mp_digit *tmpn;
      mp_word *_W;

      /* alias for the digits of the modulus */
      tmpn = n->dp;

      /* Alias for the columns set by an offset of ix */
      _W = W + ix;

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  }

  /* now we have to propagate the carries and
   * shift the words downward [all those least
   * significant digits we zeroed].
   */
  {
    register mp_digit *tmpx;
    register mp_word *_W, *_W1;

    /* nox fix rest of carries */

    /* alias for current word */
    _W1 = W + ix;

    /* alias for next word, where the carry goes */
    _W = W + ++ix;

    for (; ix <= n->used * 2 + 1; ix++) {
      *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT);
    }

    /* copy out, A = A/b**n
     *
     * The result is A/b**n but instead of converting from an
     * array of mp_word to mp_digit than calling mp_rshd
     * we just copy them in the right order
     */

    /* alias for destination word */
    tmpx = x->dp;

    /* alias for shifted double precision result */
    _W = W + n->used;

    for (ix = 0; ix < n->used + 1; ix++) {
      *tmpx++ = (mp_digit)(*_W++ & ((mp_word) MP_MASK));
    }

    /* zero oldused digits, if the input a was larger than
     * m->used+1 we'll have to clear the digits
     */
    for (; ix < olduse; ix++) {







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|







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  }

  /* now we have to propagate the carries and
   * shift the words downward [all those least
   * significant digits we zeroed].
   */
  {
    mp_digit *tmpx;
    mp_word *_W, *_W1;

    /* nox fix rest of carries */

    /* alias for current word */
    _W1 = W + ix;

    /* alias for next word, where the carry goes */
    _W = W + ++ix;

    for (; ix <= ((n->used * 2) + 1); ix++) {
      *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT);
    }

    /* copy out, A = A/b**n
     *
     * The result is A/b**n but instead of converting from an
     * array of mp_word to mp_digit than calling mp_rshd
     * we just copy them in the right order
     */

    /* alias for destination word */
    tmpx = x->dp;

    /* alias for shifted double precision result */
    _W = W + n->used;

    for (ix = 0; ix < (n->used + 1); ix++) {
      *tmpx++ = (mp_digit)(*_W++ & ((mp_word) MP_MASK));
    }

    /* zero oldused digits, if the input a was larger than
     * m->used+1 we'll have to clear the digits
     */
    for (; ix < olduse; ix++) {
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168




  /* if A >= m then A = A - m */
  if (mp_cmp_mag (x, n) != MP_LT) {
    return s_mp_sub (x, n, x);
  }
  return MP_OKAY;
}
#endif











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  /* if A >= m then A = A - m */
  if (mp_cmp_mag (x, n) != MP_LT) {
    return s_mp_sub (x, n, x);
  }
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_fast_s_mp_mul_digs.c.
1
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#include <tommath.h>
#ifdef BN_FAST_S_MP_MUL_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* Fast (comba) multiplier
 *
 * This is the fast column-array [comba] multiplier.  It is 
 * designed to compute the columns of the product first 
 * then handle the carries afterwards.  This has the effect 
|













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#include <tommath_private.h>
#ifdef BN_FAST_S_MP_MUL_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* Fast (comba) multiplier
 *
 * This is the fast column-array [comba] multiplier.  It is 
 * designed to compute the columns of the product first 
 * then handle the carries afterwards.  This has the effect 
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 * Based on Algorithm 14.12 on pp.595 of HAC.
 *
 */
int fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
{
  int     olduse, res, pa, ix, iz;
  mp_digit W[MP_WARRAY];
  register mp_word  _W;

  /* grow the destination as required */
  if (c->alloc < digs) {
    if ((res = mp_grow (c, digs)) != MP_OKAY) {
      return res;
    }
  }







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 * Based on Algorithm 14.12 on pp.595 of HAC.
 *
 */
int fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
{
  int     olduse, res, pa, ix, iz;
  mp_digit W[MP_WARRAY];
  mp_word  _W;

  /* grow the destination as required */
  if (c->alloc < digs) {
    if ((res = mp_grow (c, digs)) != MP_OKAY) {
      return res;
    }
  }
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      }

      /* store term */
      W[ix] = ((mp_digit)_W) & MP_MASK;

      /* make next carry */
      _W = _W >> ((mp_word)DIGIT_BIT);
 }

  /* setup dest */
  olduse  = c->used;
  c->used = pa;

  {
    register mp_digit *tmpc;
    tmpc = c->dp;
    for (ix = 0; ix < pa+1; ix++) {
      /* now extract the previous digit [below the carry] */
      *tmpc++ = W[ix];
    }

    /* clear unused digits [that existed in the old copy of c] */
    for (; ix < olduse; ix++) {
      *tmpc++ = 0;
    }
  }
  mp_clamp (c);
  return MP_OKAY;
}
#endif











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      }

      /* store term */
      W[ix] = ((mp_digit)_W) & MP_MASK;

      /* make next carry */
      _W = _W >> ((mp_word)DIGIT_BIT);
  }

  /* setup dest */
  olduse  = c->used;
  c->used = pa;

  {
    mp_digit *tmpc;
    tmpc = c->dp;
    for (ix = 0; ix < (pa + 1); ix++) {
      /* now extract the previous digit [below the carry] */
      *tmpc++ = W[ix];
    }

    /* clear unused digits [that existed in the old copy of c] */
    for (; ix < olduse; ix++) {
      *tmpc++ = 0;
    }
  }
  mp_clamp (c);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_fast_s_mp_mul_high_digs.c.
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#include <tommath.h>
#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* this is a modified version of fast_s_mul_digs that only produces
 * output digits *above* digs.  See the comments for fast_s_mul_digs
 * to see how it works.
 *
 * This is used in the Barrett reduction since for one of the multiplications
|













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#include <tommath_private.h>
#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* this is a modified version of fast_s_mul_digs that only produces
 * output digits *above* digs.  See the comments for fast_s_mul_digs
 * to see how it works.
 *
 * This is used in the Barrett reduction since for one of the multiplications
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  }
  
  /* setup dest */
  olduse  = c->used;
  c->used = pa;

  {
    register mp_digit *tmpc;

    tmpc = c->dp + digs;
    for (ix = digs; ix < pa; ix++) {
      /* now extract the previous digit [below the carry] */
      *tmpc++ = W[ix];
    }

    /* clear unused digits [that existed in the old copy of c] */
    for (; ix < olduse; ix++) {
      *tmpc++ = 0;
    }
  }
  mp_clamp (c);
  return MP_OKAY;
}
#endif











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  }
  
  /* setup dest */
  olduse  = c->used;
  c->used = pa;

  {
    mp_digit *tmpc;

    tmpc = c->dp + digs;
    for (ix = digs; ix < pa; ix++) {
      /* now extract the previous digit [below the carry] */
      *tmpc++ = W[ix];
    }

    /* clear unused digits [that existed in the old copy of c] */
    for (; ix < olduse; ix++) {
      *tmpc++ = 0;
    }
  }
  mp_clamp (c);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_fast_s_mp_sqr.c.
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#include <tommath.h>
#ifdef BN_FAST_S_MP_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* the jist of squaring...
 * you do like mult except the offset of the tmpx [one that 
 * starts closer to zero] can't equal the offset of tmpy.  
 * So basically you set up iy like before then you min it with
 * (ty-tx) so that it never happens.  You double all those 
|













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#include <tommath_private.h>
#ifdef BN_FAST_S_MP_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* the jist of squaring...
 * you do like mult except the offset of the tmpx [one that 
 * starts closer to zero] can't equal the offset of tmpy.  
 * So basically you set up iy like before then you min it with
 * (ty-tx) so that it never happens.  You double all those 
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       */
      iy = MIN(a->used-tx, ty+1);

      /* now for squaring tx can never equal ty 
       * we halve the distance since they approach at a rate of 2x
       * and we have to round because odd cases need to be executed
       */
      iy = MIN(iy, (ty-tx+1)>>1);

      /* execute loop */
      for (iz = 0; iz < iy; iz++) {
         _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--);
      }

      /* double the inner product and add carry */







|







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       */
      iy = MIN(a->used-tx, ty+1);

      /* now for squaring tx can never equal ty 
       * we halve the distance since they approach at a rate of 2x
       * and we have to round because odd cases need to be executed
       */
      iy = MIN(iy, ((ty-tx)+1)>>1);

      /* execute loop */
      for (iz = 0; iz < iy; iz++) {
         _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--);
      }

      /* double the inner product and add carry */
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      *tmpb++ = 0;
    }
  }
  mp_clamp (b);
  return MP_OKAY;
}
#endif











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      *tmpb++ = 0;
    }
  }
  mp_clamp (b);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_2expt.c.
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#include <tommath.h>
#ifdef BN_MP_2EXPT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* computes a = 2**b 
 *
 * Simple algorithm which zeroes the int, grows it then just sets one bit
 * as required.
 */
int
mp_2expt (mp_int * a, int b)
{
  int     res;

  /* zero a as per default */
  mp_zero (a);

  /* grow a to accomodate the single bit */
  if ((res = mp_grow (a, b / DIGIT_BIT + 1)) != MP_OKAY) {
    return res;
  }

  /* set the used count of where the bit will go */
  a->used = b / DIGIT_BIT + 1;

  /* put the single bit in its place */
  a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT);

  return MP_OKAY;
}
#endif




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#include <tommath_private.h>
#ifdef BN_MP_2EXPT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* computes a = 2**b 
 *
 * Simple algorithm which zeroes the int, grows it then just sets one bit
 * as required.
 */
int
mp_2expt (mp_int * a, int b)
{
  int     res;

  /* zero a as per default */
  mp_zero (a);

  /* grow a to accomodate the single bit */
  if ((res = mp_grow (a, (b / DIGIT_BIT) + 1)) != MP_OKAY) {
    return res;
  }

  /* set the used count of where the bit will go */
  a->used = (b / DIGIT_BIT) + 1;

  /* put the single bit in its place */
  a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT);

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_abs.c.
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#include <tommath.h>
#ifdef BN_MP_ABS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* b = |a| 
 *
 * Simple function copies the input and fixes the sign to positive
 */
int
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#include <tommath_private.h>
#ifdef BN_MP_ABS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* b = |a| 
 *
 * Simple function copies the input and fixes the sign to positive
 */
int
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  /* force the sign of b to positive */
  b->sign = MP_ZPOS;

  return MP_OKAY;
}
#endif











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  /* force the sign of b to positive */
  b->sign = MP_ZPOS;

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_add.c.
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#include <tommath.h>
#ifdef BN_MP_ADD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* high level addition (handles signs) */
int mp_add (mp_int * a, mp_int * b, mp_int * c)
{
  int     sa, sb, res;

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#include <tommath_private.h>
#ifdef BN_MP_ADD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* high level addition (handles signs) */
int mp_add (mp_int * a, mp_int * b, mp_int * c)
{
  int     sa, sb, res;

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49




      res = s_mp_sub (a, b, c);
    }
  }
  return res;
}

#endif











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      res = s_mp_sub (a, b, c);
    }
  }
  return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_add_d.c.
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#include <tommath.h>
#ifdef BN_MP_ADD_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* single digit addition */
int
mp_add_d (mp_int * a, mp_digit b, mp_int * c)
{
  int     res, ix, oldused;
  mp_digit *tmpa, *tmpc, mu;

  /* grow c as required */
  if (c->alloc < a->used + 1) {
     if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) {
        return res;
     }
  }

  /* if a is negative and |a| >= b, call c = |a| - b */
  if (a->sign == MP_NEG && (a->used > 1 || a->dp[0] >= b)) {
     /* temporarily fix sign of a */
     a->sign = MP_ZPOS;

     /* c = |a| - b */
     res = mp_sub_d(a, b, c);

     /* fix signs  */
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#include <tommath_private.h>
#ifdef BN_MP_ADD_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* single digit addition */
int
mp_add_d (mp_int * a, mp_digit b, mp_int * c)
{
  int     res, ix, oldused;
  mp_digit *tmpa, *tmpc, mu;

  /* grow c as required */
  if (c->alloc < (a->used + 1)) {
     if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) {
        return res;
     }
  }

  /* if a is negative and |a| >= b, call c = |a| - b */
  if ((a->sign == MP_NEG) && ((a->used > 1) || (a->dp[0] >= b))) {
     /* temporarily fix sign of a */
     a->sign = MP_ZPOS;

     /* c = |a| - b */
     res = mp_sub_d(a, b, c);

     /* fix signs  */
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  }
  mp_clamp(c);

  return MP_OKAY;
}

#endif











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  }
  mp_clamp(c);

  return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_addmod.c.
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#include <tommath.h>
#ifdef BN_MP_ADDMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* d = a + b (mod c) */
int
mp_addmod (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
{
  int     res;
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#include <tommath_private.h>
#ifdef BN_MP_ADDMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* d = a + b (mod c) */
int
mp_addmod (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
{
  int     res;
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    return res;
  }
  res = mp_mod (&t, c, d);
  mp_clear (&t);
  return res;
}
#endif











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    return res;
  }
  res = mp_mod (&t, c, d);
  mp_clear (&t);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_and.c.
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#include <tommath.h>
#ifdef BN_MP_AND_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* AND two ints together */
int
mp_and (mp_int * a, mp_int * b, mp_int * c)
{
  int     res, ix, px;
|













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#include <tommath_private.h>
#ifdef BN_MP_AND_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* AND two ints together */
int
mp_and (mp_int * a, mp_int * b, mp_int * c)
{
  int     res, ix, px;
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  mp_clamp (&t);
  mp_exch (c, &t);
  mp_clear (&t);
  return MP_OKAY;
}
#endif











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  mp_clamp (&t);
  mp_exch (c, &t);
  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_clamp.c.
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#include <tommath.h>
#ifdef BN_MP_CLAMP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* trim unused digits 
 *
 * This is used to ensure that leading zero digits are
 * trimed and the leading "used" digit will be non-zero
 * Typically very fast.  Also fixes the sign if there
 * are no more leading digits
 */
void
mp_clamp (mp_int * a)
{
  /* decrease used while the most significant digit is
   * zero.
   */
  while (a->used > 0 && a->dp[a->used - 1] == 0) {
    --(a->used);
  }

  /* reset the sign flag if used == 0 */
  if (a->used == 0) {
    a->sign = MP_ZPOS;
  }
}
#endif




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#include <tommath_private.h>
#ifdef BN_MP_CLAMP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* trim unused digits 
 *
 * This is used to ensure that leading zero digits are
 * trimed and the leading "used" digit will be non-zero
 * Typically very fast.  Also fixes the sign if there
 * are no more leading digits
 */
void
mp_clamp (mp_int * a)
{
  /* decrease used while the most significant digit is
   * zero.
   */
  while ((a->used > 0) && (a->dp[a->used - 1] == 0)) {
    --(a->used);
  }

  /* reset the sign flag if used == 0 */
  if (a->used == 0) {
    a->sign = MP_ZPOS;
  }
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_clear.c.
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#include <tommath.h>
#ifdef BN_MP_CLEAR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* clear one (frees)  */
void
mp_clear (mp_int * a)
{
  int i;
|













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#include <tommath_private.h>
#ifdef BN_MP_CLEAR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* clear one (frees)  */
void
mp_clear (mp_int * a)
{
  int i;
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    /* reset members to make debugging easier */
    a->dp    = NULL;
    a->alloc = a->used = 0;
    a->sign  = MP_ZPOS;
  }
}
#endif











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    /* reset members to make debugging easier */
    a->dp    = NULL;
    a->alloc = a->used = 0;
    a->sign  = MP_ZPOS;
  }
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_clear_multi.c.
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#include <tommath.h>
#ifdef BN_MP_CLEAR_MULTI_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */
#include <stdarg.h>

void mp_clear_multi(mp_int *mp, ...) 
{
    mp_int* next_mp = mp;
    va_list args;
    va_start(args, mp);
    while (next_mp != NULL) {
        mp_clear(next_mp);
        next_mp = va_arg(args, mp_int*);
    }
    va_end(args);
}
#endif




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#include <tommath_private.h>
#ifdef BN_MP_CLEAR_MULTI_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */
#include <stdarg.h>

void mp_clear_multi(mp_int *mp, ...) 
{
    mp_int* next_mp = mp;
    va_list args;
    va_start(args, mp);
    while (next_mp != NULL) {
        mp_clear(next_mp);
        next_mp = va_arg(args, mp_int*);
    }
    va_end(args);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_cmp.c.
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#include <tommath.h>
#ifdef BN_MP_CMP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* compare two ints (signed)*/
int
mp_cmp (const mp_int * a, const mp_int * b)
{
  /* compare based on sign */
|













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#include <tommath_private.h>
#ifdef BN_MP_CMP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* compare two ints (signed)*/
int
mp_cmp (const mp_int * a, const mp_int * b)
{
  /* compare based on sign */
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     /* if negative compare opposite direction */
     return mp_cmp_mag(b, a);
  } else {
     return mp_cmp_mag(a, b);
  }
}
#endif











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     /* if negative compare opposite direction */
     return mp_cmp_mag(b, a);
  } else {
     return mp_cmp_mag(a, b);
  }
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_cmp_d.c.
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#include <tommath.h>
#ifdef BN_MP_CMP_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* compare a digit */
int mp_cmp_d(const mp_int * a, mp_digit b)
{
  /* compare based on sign */
  if (a->sign == MP_NEG) {
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#include <tommath_private.h>
#ifdef BN_MP_CMP_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* compare a digit */
int mp_cmp_d(const mp_int * a, mp_digit b)
{
  /* compare based on sign */
  if (a->sign == MP_NEG) {
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  } else if (a->dp[0] < b) {
    return MP_LT;
  } else {
    return MP_EQ;
  }
}
#endif











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  } else if (a->dp[0] < b) {
    return MP_LT;
  } else {
    return MP_EQ;
  }
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_cmp_mag.c.
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#include <tommath.h>
#ifdef BN_MP_CMP_MAG_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* compare maginitude of two ints (unsigned) */
int mp_cmp_mag (const mp_int * a, const mp_int * b)
{
  int     n;
  mp_digit *tmpa, *tmpb;
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#include <tommath_private.h>
#ifdef BN_MP_CMP_MAG_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* compare maginitude of two ints (unsigned) */
int mp_cmp_mag (const mp_int * a, const mp_int * b)
{
  int     n;
  mp_digit *tmpa, *tmpb;
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    if (*tmpa < *tmpb) {
      return MP_LT;
    }
  }
  return MP_EQ;
}
#endif











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    if (*tmpa < *tmpb) {
      return MP_LT;
    }
  }
  return MP_EQ;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_cnt_lsb.c.
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#include <tommath.h>
#ifdef BN_MP_CNT_LSB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

static const int lnz[16] = { 
   4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0
};

/* Counts the number of lsbs which are zero before the first zero bit */
int mp_cnt_lsb(const mp_int *a)
{
   int x;
   mp_digit q, qq;

   /* easy out */
   if (mp_iszero(a) == 1) {
      return 0;
   }

   /* scan lower digits until non-zero */
   for (x = 0; x < a->used && a->dp[x] == 0; x++);
   q = a->dp[x];
   x *= DIGIT_BIT;

   /* now scan this digit until a 1 is found */
   if ((q & 1) == 0) {
      do {
         qq  = q & 15;
         x  += lnz[qq];
         q >>= 4;
      } while (qq == 0);
   }
   return x;
}

#endif




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#include <tommath_private.h>
#ifdef BN_MP_CNT_LSB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

static const int lnz[16] = { 
   4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0
};

/* Counts the number of lsbs which are zero before the first zero bit */
int mp_cnt_lsb(const mp_int *a)
{
   int x;
   mp_digit q, qq;

   /* easy out */
   if (mp_iszero(a) == MP_YES) {
      return 0;
   }

   /* scan lower digits until non-zero */
   for (x = 0; (x < a->used) && (a->dp[x] == 0); x++) {}
   q = a->dp[x];
   x *= DIGIT_BIT;

   /* now scan this digit until a 1 is found */
   if ((q & 1) == 0) {
      do {
         qq  = q & 15;
         x  += lnz[qq];
         q >>= 4;
      } while (qq == 0);
   }
   return x;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_copy.c.
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#include <tommath.h>
#ifdef BN_MP_COPY_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* copy, b = a */
int
mp_copy (const mp_int * a, mp_int * b)
{
  int     res, n;
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#include <tommath_private.h>
#ifdef BN_MP_COPY_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* copy, b = a */
int
mp_copy (const mp_int * a, mp_int * b)
{
  int     res, n;
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     if ((res = mp_grow (b, a->used)) != MP_OKAY) {
        return res;
     }
  }

  /* zero b and copy the parameters over */
  {
    register mp_digit *tmpa, *tmpb;

    /* pointer aliases */

    /* source */
    tmpa = a->dp;

    /* destination */







|







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     if ((res = mp_grow (b, a->used)) != MP_OKAY) {
        return res;
     }
  }

  /* zero b and copy the parameters over */
  {
    mp_digit *tmpa, *tmpb;

    /* pointer aliases */

    /* source */
    tmpa = a->dp;

    /* destination */
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  /* copy used count and sign */
  b->used = a->used;
  b->sign = a->sign;
  return MP_OKAY;
}
#endif











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  /* copy used count and sign */
  b->used = a->used;
  b->sign = a->sign;
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_count_bits.c.
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#include <tommath.h>
#ifdef BN_MP_COUNT_BITS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* returns the number of bits in an int */
int
mp_count_bits (const mp_int * a)
{
  int     r;
|













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#include <tommath_private.h>
#ifdef BN_MP_COUNT_BITS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* returns the number of bits in an int */
int
mp_count_bits (const mp_int * a)
{
  int     r;
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  while (q > ((mp_digit) 0)) {
    ++r;
    q >>= ((mp_digit) 1);
  }
  return r;
}
#endif











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  while (q > ((mp_digit) 0)) {
    ++r;
    q >>= ((mp_digit) 1);
  }
  return r;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_div.c.
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#include <tommath.h>
#ifdef BN_MP_DIV_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

#ifdef BN_MP_DIV_SMALL

/* slower bit-bang division... also smaller */
int mp_div(mp_int * a, mp_int * b, mp_int * c, mp_int * d)
{
   mp_int ta, tb, tq, q;
   int    res, n, n2;

  /* is divisor zero ? */
  if (mp_iszero (b) == 1) {
    return MP_VAL;
  }

  /* if a < b then q=0, r = a */
  if (mp_cmp_mag (a, b) == MP_LT) {
    if (d != NULL) {
      res = mp_copy (a, d);
    } else {
      res = MP_OKAY;
    }
    if (c != NULL) {
      mp_zero (c);
    }
    return res;
  }
	
  /* init our temps */
  if ((res = mp_init_multi(&ta, &tb, &tq, &q, NULL) != MP_OKAY)) {
     return res;
  }


  mp_set(&tq, 1);
  n = mp_count_bits(a) - mp_count_bits(b);
  if (((res = mp_abs(a, &ta)) != MP_OKAY) ||
      ((res = mp_abs(b, &tb)) != MP_OKAY) || 
      ((res = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) ||
      ((res = mp_mul_2d(&tq, n, &tq)) != MP_OKAY)) {
      goto LBL_ERR;
  }

  while (n-- >= 0) {
     if (mp_cmp(&tb, &ta) != MP_GT) {
        if (((res = mp_sub(&ta, &tb, &ta)) != MP_OKAY) ||
            ((res = mp_add(&q, &tq, &q)) != MP_OKAY)) {
           goto LBL_ERR;
        }
     }
     if (((res = mp_div_2d(&tb, 1, &tb, NULL)) != MP_OKAY) ||
         ((res = mp_div_2d(&tq, 1, &tq, NULL)) != MP_OKAY)) {
           goto LBL_ERR;
     }
  }

  /* now q == quotient and ta == remainder */
  n  = a->sign;
  n2 = (a->sign == b->sign ? MP_ZPOS : MP_NEG);
  if (c != NULL) {
     mp_exch(c, &q);
     c->sign  = (mp_iszero(c) == MP_YES) ? MP_ZPOS : n2;
  }
  if (d != NULL) {
     mp_exch(d, &ta);
     d->sign = (mp_iszero(d) == MP_YES) ? MP_ZPOS : n;
  }
LBL_ERR:
   mp_clear_multi(&ta, &tb, &tq, &q, NULL);
   return res;
}

#else

/* integer signed division. 
 * c*b + d == a [e.g. a/b, c=quotient, d=remainder]
 * HAC pp.598 Algorithm 14.20
 *
 * Note that the description in HAC is horribly 
 * incomplete.  For example, it doesn't consider 
 * the case where digits are removed from 'x' in 
 * the inner loop.  It also doesn't consider the 
 * case that y has fewer than three digits, etc..
 *
 * The overall algorithm is as described as 
 * 14.20 from HAC but fixed to treat these cases.
*/
int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
{
  mp_int  q, x, y, t1, t2;
  int     res, n, t, i, norm, neg;

  /* is divisor zero ? */
  if (mp_iszero (b) == 1) {
    return MP_VAL;
  }

  /* if a < b then q=0, r = a */
  if (mp_cmp_mag (a, b) == MP_LT) {
    if (d != NULL) {
      res = mp_copy (a, d);
|













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#include <tommath_private.h>
#ifdef BN_MP_DIV_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

#ifdef BN_MP_DIV_SMALL

/* slower bit-bang division... also smaller */
int mp_div(mp_int * a, mp_int * b, mp_int * c, mp_int * d)
{
   mp_int ta, tb, tq, q;
   int    res, n, n2;

  /* is divisor zero ? */
  if (mp_iszero (b) == MP_YES) {
    return MP_VAL;
  }

  /* if a < b then q=0, r = a */
  if (mp_cmp_mag (a, b) == MP_LT) {
    if (d != NULL) {
      res = mp_copy (a, d);
    } else {
      res = MP_OKAY;
    }
    if (c != NULL) {
      mp_zero (c);
    }
    return res;
  }

  /* init our temps */
  if ((res = mp_init_multi(&ta, &tb, &tq, &q, NULL)) != MP_OKAY) {
     return res;
  }


  mp_set(&tq, 1);
  n = mp_count_bits(a) - mp_count_bits(b);
  if (((res = mp_abs(a, &ta)) != MP_OKAY) ||
      ((res = mp_abs(b, &tb)) != MP_OKAY) ||
      ((res = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) ||
      ((res = mp_mul_2d(&tq, n, &tq)) != MP_OKAY)) {
      goto LBL_ERR;
  }

  while (n-- >= 0) {
     if (mp_cmp(&tb, &ta) != MP_GT) {
        if (((res = mp_sub(&ta, &tb, &ta)) != MP_OKAY) ||
            ((res = mp_add(&q, &tq, &q)) != MP_OKAY)) {
           goto LBL_ERR;
        }
     }
     if (((res = mp_div_2d(&tb, 1, &tb, NULL)) != MP_OKAY) ||
         ((res = mp_div_2d(&tq, 1, &tq, NULL)) != MP_OKAY)) {
           goto LBL_ERR;
     }
  }

  /* now q == quotient and ta == remainder */
  n  = a->sign;
  n2 = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;
  if (c != NULL) {
     mp_exch(c, &q);
     c->sign  = (mp_iszero(c) == MP_YES) ? MP_ZPOS : n2;
  }
  if (d != NULL) {
     mp_exch(d, &ta);
     d->sign = (mp_iszero(d) == MP_YES) ? MP_ZPOS : n;
  }
LBL_ERR:
   mp_clear_multi(&ta, &tb, &tq, &q, NULL);
   return res;
}

#else

/* integer signed division.
 * c*b + d == a [e.g. a/b, c=quotient, d=remainder]
 * HAC pp.598 Algorithm 14.20
 *
 * Note that the description in HAC is horribly
 * incomplete.  For example, it doesn't consider
 * the case where digits are removed from 'x' in
 * the inner loop.  It also doesn't consider the
 * case that y has fewer than three digits, etc..
 *
 * The overall algorithm is as described as
 * 14.20 from HAC but fixed to treat these cases.
*/
int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
{
  mp_int  q, x, y, t1, t2;
  int     res, n, t, i, norm, neg;

  /* is divisor zero ? */
  if (mp_iszero (b) == MP_YES) {
    return MP_VAL;
  }

  /* if a < b then q=0, r = a */
  if (mp_cmp_mag (a, b) == MP_LT) {
    if (d != NULL) {
      res = mp_copy (a, d);
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  /* step 3. for i from n down to (t + 1) */
  for (i = n; i >= (t + 1); i--) {
    if (i > x.used) {
      continue;
    }

    /* step 3.1 if xi == yt then set q{i-t-1} to b-1, 
     * otherwise set q{i-t-1} to (xi*b + x{i-1})/yt */
    if (x.dp[i] == y.dp[t]) {
      q.dp[i - t - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1);
    } else {
      mp_word tmp;
      tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT);
      tmp |= ((mp_word) x.dp[i - 1]);
      tmp /= ((mp_word) y.dp[t]);
      if (tmp > (mp_word) MP_MASK)
        tmp = MP_MASK;

      q.dp[i - t - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK));
    }

    /* while (q{i-t-1} * (yt * b + y{t-1})) > 
             xi * b**2 + xi-1 * b + xi-2 
     
       do q{i-t-1} -= 1; 
    */
    q.dp[i - t - 1] = (q.dp[i - t - 1] + 1) & MP_MASK;
    do {
      q.dp[i - t - 1] = (q.dp[i - t - 1] - 1) & MP_MASK;

      /* find left hand */
      mp_zero (&t1);
      t1.dp[0] = (t - 1 < 0) ? 0 : y.dp[t - 1];
      t1.dp[1] = y.dp[t];
      t1.used = 2;
      if ((res = mp_mul_d (&t1, q.dp[i - t - 1], &t1)) != MP_OKAY) {
        goto LBL_Y;
      }

      /* find right hand */
      t2.dp[0] = (i - 2 < 0) ? 0 : x.dp[i - 2];
      t2.dp[1] = (i - 1 < 0) ? 0 : x.dp[i - 1];
      t2.dp[2] = x.dp[i];
      t2.used = 3;
    } while (mp_cmp_mag(&t1, &t2) == MP_GT);

    /* step 3.3 x = x - q{i-t-1} * y * b**{i-t-1} */
    if ((res = mp_mul_d (&y, q.dp[i - t - 1], &t1)) != MP_OKAY) {
      goto LBL_Y;
    }

    if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) {
      goto LBL_Y;
    }

    if ((res = mp_sub (&x, &t1, &x)) != MP_OKAY) {
      goto LBL_Y;
    }

    /* if x < 0 then { x = x + y*b**{i-t-1}; q{i-t-1} -= 1; } */
    if (x.sign == MP_NEG) {
      if ((res = mp_copy (&y, &t1)) != MP_OKAY) {
        goto LBL_Y;
      }
      if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) {
        goto LBL_Y;
      }
      if ((res = mp_add (&x, &t1, &x)) != MP_OKAY) {
        goto LBL_Y;
      }

      q.dp[i - t - 1] = (q.dp[i - t - 1] - 1UL) & MP_MASK;
    }
  }

  /* now q is the quotient and x is the remainder 
   * [which we have to normalize] 
   */
  
  /* get sign before writing to c */
  x.sign = x.used == 0 ? MP_ZPOS : a->sign;

  if (c != NULL) {
    mp_clamp (&q);
    mp_exch (&q, c);
    c->sign = neg;
  }

  if (d != NULL) {
    mp_div_2d (&x, norm, &x, NULL);


    mp_exch (&x, d);
  }

  res = MP_OKAY;

LBL_Y:mp_clear (&y);
LBL_X:mp_clear (&x);
LBL_T2:mp_clear (&t2);
LBL_T1:mp_clear (&t1);
LBL_Q:mp_clear (&q);
  return res;
}

#endif

#endif











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  /* step 3. for i from n down to (t + 1) */
  for (i = n; i >= (t + 1); i--) {
    if (i > x.used) {
      continue;
    }

    /* step 3.1 if xi == yt then set q{i-t-1} to b-1,
     * otherwise set q{i-t-1} to (xi*b + x{i-1})/yt */
    if (x.dp[i] == y.dp[t]) {
      q.dp[(i - t) - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1);
    } else {
      mp_word tmp;
      tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT);
      tmp |= ((mp_word) x.dp[i - 1]);
      tmp /= ((mp_word) y.dp[t]);
      if (tmp > (mp_word) MP_MASK) {
        tmp = MP_MASK;
      }
      q.dp[(i - t) - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK));
    }

    /* while (q{i-t-1} * (yt * b + y{t-1})) >
             xi * b**2 + xi-1 * b + xi-2

       do q{i-t-1} -= 1;
    */
    q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] + 1) & MP_MASK;
    do {
      q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] - 1) & MP_MASK;

      /* find left hand */
      mp_zero (&t1);
      t1.dp[0] = ((t - 1) < 0) ? 0 : y.dp[t - 1];
      t1.dp[1] = y.dp[t];
      t1.used = 2;
      if ((res = mp_mul_d (&t1, q.dp[(i - t) - 1], &t1)) != MP_OKAY) {
        goto LBL_Y;
      }

      /* find right hand */
      t2.dp[0] = ((i - 2) < 0) ? 0 : x.dp[i - 2];
      t2.dp[1] = ((i - 1) < 0) ? 0 : x.dp[i - 1];
      t2.dp[2] = x.dp[i];
      t2.used = 3;
    } while (mp_cmp_mag(&t1, &t2) == MP_GT);

    /* step 3.3 x = x - q{i-t-1} * y * b**{i-t-1} */
    if ((res = mp_mul_d (&y, q.dp[(i - t) - 1], &t1)) != MP_OKAY) {
      goto LBL_Y;
    }

    if ((res = mp_lshd (&t1, (i - t) - 1)) != MP_OKAY) {
      goto LBL_Y;
    }

    if ((res = mp_sub (&x, &t1, &x)) != MP_OKAY) {
      goto LBL_Y;
    }

    /* if x < 0 then { x = x + y*b**{i-t-1}; q{i-t-1} -= 1; } */
    if (x.sign == MP_NEG) {
      if ((res = mp_copy (&y, &t1)) != MP_OKAY) {
        goto LBL_Y;
      }
      if ((res = mp_lshd (&t1, (i - t) - 1)) != MP_OKAY) {
        goto LBL_Y;
      }
      if ((res = mp_add (&x, &t1, &x)) != MP_OKAY) {
        goto LBL_Y;
      }

      q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] - 1UL) & MP_MASK;
    }
  }

  /* now q is the quotient and x is the remainder
   * [which we have to normalize]
   */

  /* get sign before writing to c */
  x.sign = (x.used == 0) ? MP_ZPOS : a->sign;

  if (c != NULL) {
    mp_clamp (&q);
    mp_exch (&q, c);
    c->sign = neg;
  }

  if (d != NULL) {
    if ((res = mp_div_2d (&x, norm, &x, NULL)) != MP_OKAY) {
      goto LBL_Y;
    }
    mp_exch (&x, d);
  }

  res = MP_OKAY;

LBL_Y:mp_clear (&y);
LBL_X:mp_clear (&x);
LBL_T2:mp_clear (&t2);
LBL_T1:mp_clear (&t1);
LBL_Q:mp_clear (&q);
  return res;
}

#endif

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_div_2.c.
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#include <tommath.h>
#ifdef BN_MP_DIV_2_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* b = a/2 */
int mp_div_2(mp_int * a, mp_int * b)
{
  int     x, res, oldused;

  /* copy */
  if (b->alloc < a->used) {
    if ((res = mp_grow (b, a->used)) != MP_OKAY) {
      return res;
    }
  }

  oldused = b->used;
  b->used = a->used;
  {
    register mp_digit r, rr, *tmpa, *tmpb;

    /* source alias */
    tmpa = a->dp + b->used - 1;

    /* dest alias */
    tmpb = b->dp + b->used - 1;

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#include <tommath_private.h>
#ifdef BN_MP_DIV_2_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* b = a/2 */
int mp_div_2(mp_int * a, mp_int * b)
{
  int     x, res, oldused;

  /* copy */
  if (b->alloc < a->used) {
    if ((res = mp_grow (b, a->used)) != MP_OKAY) {
      return res;
    }
  }

  oldused = b->used;
  b->used = a->used;
  {
    mp_digit r, rr, *tmpa, *tmpb;

    /* source alias */
    tmpa = a->dp + b->used - 1;

    /* dest alias */
    tmpb = b->dp + b->used - 1;

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    }
  }
  b->sign = a->sign;
  mp_clamp (b);
  return MP_OKAY;
}
#endif











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    }
  }
  b->sign = a->sign;
  mp_clamp (b);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_div_2d.c.
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#include <tommath.h>
#ifdef BN_MP_DIV_2D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* shift right by a certain bit count (store quotient in c, optional remainder in d) */
int mp_div_2d (const mp_int * a, int b, mp_int * c, mp_int * d)
{
  mp_digit D, r, rr;
  int     x, res;
|













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#include <tommath_private.h>
#ifdef BN_MP_DIV_2D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* shift right by a certain bit count (store quotient in c, optional remainder in d) */
int mp_div_2d (const mp_int * a, int b, mp_int * c, mp_int * d)
{
  mp_digit D, r, rr;
  int     x, res;
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  if (b >= (int)DIGIT_BIT) {
    mp_rshd (c, b / DIGIT_BIT);
  }

  /* shift any bit count < DIGIT_BIT */
  D = (mp_digit) (b % DIGIT_BIT);
  if (D != 0) {
    register mp_digit *tmpc, mask, shift;

    /* mask */
    mask = (((mp_digit)1) << D) - 1;

    /* shift for lsb */
    shift = DIGIT_BIT - D;








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  if (b >= (int)DIGIT_BIT) {
    mp_rshd (c, b / DIGIT_BIT);
  }

  /* shift any bit count < DIGIT_BIT */
  D = (mp_digit) (b % DIGIT_BIT);
  if (D != 0) {
    mp_digit *tmpc, mask, shift;

    /* mask */
    mask = (((mp_digit)1) << D) - 1;

    /* shift for lsb */
    shift = DIGIT_BIT - D;

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  if (d != NULL) {
    mp_exch (&t, d);
  }
  mp_clear (&t);
  return MP_OKAY;
}
#endif











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  if (d != NULL) {
    mp_exch (&t, d);
  }
  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_div_3.c.
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#include <tommath.h>
#ifdef BN_MP_DIV_3_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* divide by three (based on routine from MPI and the GMP manual) */
int
mp_div_3 (mp_int * a, mp_int *c, mp_digit * d)
{
  mp_int   q;
|













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#include <tommath_private.h>
#ifdef BN_MP_DIV_3_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* divide by three (based on routine from MPI and the GMP manual) */
int
mp_div_3 (mp_int * a, mp_int *c, mp_digit * d)
{
  mp_int   q;
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  }
  mp_clear(&q);
  
  return res;
}

#endif











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  }
  mp_clear(&q);
  
  return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_div_d.c.
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#include <tommath.h>
#ifdef BN_MP_DIV_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

static int s_is_power_of_two(mp_digit b, int *p)
{
   int x;

   /* quick out - if (b & (b-1)) isn't zero, b isn't a power of two */
   if ((b==0) || (b & (b-1))) {
       return 0;
   }
   for (x = 1; x < DIGIT_BIT; x++) {
      if (b == (((mp_digit)1)<<x)) {
         *p = x;
         return 1;
      }
|













|







|







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#include <tommath_private.h>
#ifdef BN_MP_DIV_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

static int s_is_power_of_two(mp_digit b, int *p)
{
   int x;

   /* quick out - if (b & (b-1)) isn't zero, b isn't a power of two */
   if ((b == 0) || ((b & (b-1)) != 0)) {
       return 0;
   }
   for (x = 1; x < DIGIT_BIT; x++) {
      if (b == (((mp_digit)1)<<x)) {
         *p = x;
         return 1;
      }
42
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51
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56

  /* cannot divide by zero */
  if (b == 0) {
     return MP_VAL;
  }

  /* quick outs */
  if (b == 1 || mp_iszero(a) == 1) {
     if (d != NULL) {
        *d = 0;
     }
     if (c != NULL) {
        return mp_copy(a, c);
     }
     return MP_OKAY;







|







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  /* cannot divide by zero */
  if (b == 0) {
     return MP_VAL;
  }

  /* quick outs */
  if ((b == 1) || (mp_iszero(a) == MP_YES)) {
     if (d != NULL) {
        *d = 0;
     }
     if (c != NULL) {
        return mp_copy(a, c);
     }
     return MP_OKAY;
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  }
  mp_clear(&q);
  
  return res;
}

#endif











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  }
  mp_clear(&q);
  
  return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_dr_is_modulus.c.
1
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#include <tommath.h>
#ifdef BN_MP_DR_IS_MODULUS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* determines if a number is a valid DR modulus */
int mp_dr_is_modulus(mp_int *a)
{
   int ix;

|













|







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#include <tommath_private.h>
#ifdef BN_MP_DR_IS_MODULUS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* determines if a number is a valid DR modulus */
int mp_dr_is_modulus(mp_int *a)
{
   int ix;

33
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35
36
37
38
39




          return 0;
       }
   }
   return 1;
}

#endif











>
>
>
>
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41
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43
          return 0;
       }
   }
   return 1;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_dr_reduce.c.
1
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#include <tommath.h>
#ifdef BN_MP_DR_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* reduce "x" in place modulo "n" using the Diminished Radix algorithm.
 *
 * Based on algorithm from the paper
 *
 * "Generating Efficient Primes for Discrete Log Cryptosystems"
|













|







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21
22
#include <tommath_private.h>
#ifdef BN_MP_DR_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* reduce "x" in place modulo "n" using the Diminished Radix algorithm.
 *
 * Based on algorithm from the paper
 *
 * "Generating Efficient Primes for Discrete Log Cryptosystems"
36
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46
47
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  mp_word  r;
  mp_digit mu, *tmpx1, *tmpx2;

  /* m = digits in modulus */
  m = n->used;

  /* ensure that "x" has at least 2m digits */
  if (x->alloc < m + m) {
    if ((err = mp_grow (x, m + m)) != MP_OKAY) {
      return err;
    }
  }

/* top of loop, this is where the code resumes if
 * another reduction pass is required.







|







36
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  mp_word  r;
  mp_digit mu, *tmpx1, *tmpx2;

  /* m = digits in modulus */
  m = n->used;

  /* ensure that "x" has at least 2m digits */
  if (x->alloc < (m + m)) {
    if ((err = mp_grow (x, m + m)) != MP_OKAY) {
      return err;
    }
  }

/* top of loop, this is where the code resumes if
 * another reduction pass is required.
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86
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  tmpx2 = x->dp + m;

  /* set carry to zero */
  mu = 0;

  /* compute (x mod B**m) + k * [x/B**m] inline and inplace */
  for (i = 0; i < m; i++) {
      r         = ((mp_word)*tmpx2++) * ((mp_word)k) + *tmpx1 + mu;
      *tmpx1++  = (mp_digit)(r & MP_MASK);
      mu        = (mp_digit)(r >> ((mp_word)DIGIT_BIT));
  }

  /* set final carry */
  *tmpx1++ = mu;

  /* zero words above m */
  for (i = m + 1; i < x->used; i++) {
      *tmpx1++ = 0;
  }

  /* clamp, sub and return */
  mp_clamp (x);

  /* if x >= n then subtract and reduce again
   * Each successive "recursion" makes the input smaller and smaller.
   */
  if (mp_cmp_mag (x, n) != MP_LT) {
    s_mp_sub(x, n, x);


    goto top;
  }
  return MP_OKAY;
}
#endif











|



















|
>
>





>
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>
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  tmpx2 = x->dp + m;

  /* set carry to zero */
  mu = 0;

  /* compute (x mod B**m) + k * [x/B**m] inline and inplace */
  for (i = 0; i < m; i++) {
      r         = (((mp_word)*tmpx2++) * (mp_word)k) + *tmpx1 + mu;
      *tmpx1++  = (mp_digit)(r & MP_MASK);
      mu        = (mp_digit)(r >> ((mp_word)DIGIT_BIT));
  }

  /* set final carry */
  *tmpx1++ = mu;

  /* zero words above m */
  for (i = m + 1; i < x->used; i++) {
      *tmpx1++ = 0;
  }

  /* clamp, sub and return */
  mp_clamp (x);

  /* if x >= n then subtract and reduce again
   * Each successive "recursion" makes the input smaller and smaller.
   */
  if (mp_cmp_mag (x, n) != MP_LT) {
    if ((err = s_mp_sub(x, n, x)) != MP_OKAY) {
      return err;
    }
    goto top;
  }
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_dr_setup.c.
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#include <tommath.h>
#ifdef BN_MP_DR_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* determines the setup value */
void mp_dr_setup(mp_int *a, mp_digit *d)
{
   /* the casts are required if DIGIT_BIT is one less than
    * the number of bits in a mp_digit [e.g. DIGIT_BIT==31]
    */
   *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) - 
        ((mp_word)a->dp[0]));
}

#endif




|













|













>
>
>
>
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#include <tommath_private.h>
#ifdef BN_MP_DR_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* determines the setup value */
void mp_dr_setup(mp_int *a, mp_digit *d)
{
   /* the casts are required if DIGIT_BIT is one less than
    * the number of bits in a mp_digit [e.g. DIGIT_BIT==31]
    */
   *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) - 
        ((mp_word)a->dp[0]));
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_exch.c.
1
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#include <tommath.h>
#ifdef BN_MP_EXCH_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* swap the elements of two integers, for cases where you can't simply swap the 
 * mp_int pointers around
 */
void
mp_exch (mp_int * a, mp_int * b)
{
  mp_int  t;

  t  = *a;
  *a = *b;
  *b = t;
}
#endif




|













|















>
>
>
>
1
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#include <tommath_private.h>
#ifdef BN_MP_EXCH_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* swap the elements of two integers, for cases where you can't simply swap the 
 * mp_int pointers around
 */
void
mp_exch (mp_int * a, mp_int * b)
{
  mp_int  t;

  t  = *a;
  *a = *b;
  *b = t;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_expt_d.c.
1
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#include <tommath.h>
#ifdef BN_MP_EXPT_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* calculate c = a**b  using a square-multiply algorithm */
int mp_expt_d (mp_int * a, mp_digit b, mp_int * c)
{
  int     res, x;
  mp_int  g;

  if ((res = mp_init_copy (&g, a)) != MP_OKAY) {
    return res;
  }

  /* set initial result */
  mp_set (c, 1);

  for (x = 0; x < (int) DIGIT_BIT; x++) {
    /* square */
    if ((res = mp_sqr (c, c)) != MP_OKAY) {
      mp_clear (&g);
      return res;
    }

    /* if the bit is set multiply */
    if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) {
      if ((res = mp_mul (c, &g, c)) != MP_OKAY) {
         mp_clear (&g);
         return res;
      }
    }

    /* shift to next bit */
    b <<= 1;
  }

  mp_clear (&g);
  return MP_OKAY;
}
#endif
|













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#include <tommath_private.h>
#ifdef BN_MP_EXPT_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* wrapper function for mp_expt_d_ex() */
int mp_expt_d (mp_int * a, mp_digit b, mp_int * c)
{




  return mp_expt_d_ex(a, b, c, 0);
}



#endif






/* $Source$ */







/* $Revision$ */



/* $Date$ */




Added libtommath/bn_mp_expt_d_ex.c.






































































































































































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#include <tommath_private.h>
#ifdef BN_MP_EXPT_D_EX_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://libtom.org
 */

/* calculate c = a**b  using a square-multiply algorithm */
int mp_expt_d_ex (mp_int * a, mp_digit b, mp_int * c, int fast)
{
  int     res;
  unsigned int x;

  mp_int  g;

  if ((res = mp_init_copy (&g, a)) != MP_OKAY) {
    return res;
  }

  /* set initial result */
  mp_set (c, 1);

  if (fast != 0) {
    while (b > 0) {
      /* if the bit is set multiply */
      if ((b & 1) != 0) {
        if ((res = mp_mul (c, &g, c)) != MP_OKAY) {
          mp_clear (&g);
          return res;
        }
      }

      /* square */
      if (b > 1) {
        if ((res = mp_sqr (&g, &g)) != MP_OKAY) {
          mp_clear (&g);
          return res;
        }
      }

      /* shift to next bit */
      b >>= 1;
    }
  }
  else {
    for (x = 0; x < DIGIT_BIT; x++) {
      /* square */
      if ((res = mp_sqr (c, c)) != MP_OKAY) {
        mp_clear (&g);
        return res;
      }

      /* if the bit is set multiply */
      if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) {
        if ((res = mp_mul (c, &g, c)) != MP_OKAY) {
           mp_clear (&g);
           return res;
        }
      }

      /* shift to next bit */
      b <<= 1;
    }
  } /* if ... else */

  mp_clear (&g);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_exptmod.c.
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#include <tommath.h>
#ifdef BN_MP_EXPTMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */


/* this is a shell function that calls either the normal or Montgomery
 * exptmod functions.  Originally the call to the montgomery code was
 * embedded in the normal function but that wasted alot of stack space
 * for nothing (since 99% of the time the Montgomery code would be called)
|













|







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#include <tommath_private.h>
#ifdef BN_MP_EXPTMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */


/* this is a shell function that calls either the normal or Montgomery
 * exptmod functions.  Originally the call to the montgomery code was
 * embedded in the normal function but that wasted alot of stack space
 * for nothing (since 99% of the time the Montgomery code would be called)
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104
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107
108




  if (dr == 0) {
     dr = mp_reduce_is_2k(P) << 1;
  }
#endif
    
  /* if the modulus is odd or dr != 0 use the montgomery method */
#ifdef BN_MP_EXPTMOD_FAST_C
  if (mp_isodd (P) == 1 || dr !=  0) {
    return mp_exptmod_fast (G, X, P, Y, dr);
  } else {
#endif
#ifdef BN_S_MP_EXPTMOD_C
    /* otherwise use the generic Barrett reduction technique */
    return s_mp_exptmod (G, X, P, Y, 0);
#else
    /* no exptmod for evens */
    return MP_VAL;
#endif
#ifdef BN_MP_EXPTMOD_FAST_C
  }
#endif
}

#endif











|
















>
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  if (dr == 0) {
     dr = mp_reduce_is_2k(P) << 1;
  }
#endif
    
  /* if the modulus is odd or dr != 0 use the montgomery method */
#ifdef BN_MP_EXPTMOD_FAST_C
  if ((mp_isodd (P) == MP_YES) || (dr !=  0)) {
    return mp_exptmod_fast (G, X, P, Y, dr);
  } else {
#endif
#ifdef BN_S_MP_EXPTMOD_C
    /* otherwise use the generic Barrett reduction technique */
    return s_mp_exptmod (G, X, P, Y, 0);
#else
    /* no exptmod for evens */
    return MP_VAL;
#endif
#ifdef BN_MP_EXPTMOD_FAST_C
  }
#endif
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_exptmod_fast.c.
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#include <tommath.h>
#ifdef BN_MP_EXPTMOD_FAST_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* computes Y == G**X mod P, HAC pp.616, Algorithm 14.85
 *
 * Uses a left-to-right k-ary sliding window to compute the modular exponentiation.
 * The value of k changes based on the size of the exponent.
 *
|













|







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#include <tommath_private.h>
#ifdef BN_MP_EXPTMOD_FAST_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* computes Y == G**X mod P, HAC pp.616, Algorithm 14.85
 *
 * Uses a left-to-right k-ary sliding window to compute the modular exponentiation.
 * The value of k changes based on the size of the exponent.
 *
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#else
     err = MP_VAL;
     goto LBL_M;
#endif

     /* automatically pick the comba one if available (saves quite a few calls/ifs) */
#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C
     if (((P->used * 2 + 1) < MP_WARRAY) &&
          P->used < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) {
        redux = fast_mp_montgomery_reduce;
     } else 
#endif
     {
#ifdef BN_MP_MONTGOMERY_REDUCE_C
        /* use slower baseline Montgomery method */
        redux = mp_montgomery_reduce;







|
|







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#else
     err = MP_VAL;
     goto LBL_M;
#endif

     /* automatically pick the comba one if available (saves quite a few calls/ifs) */
#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C
     if ((((P->used * 2) + 1) < MP_WARRAY) &&
          (P->used < (1 << ((CHAR_BIT * sizeof(mp_word)) - (2 * DIGIT_BIT))))) {
        redux = fast_mp_montgomery_reduce;
     } else 
#endif
     {
#ifdef BN_MP_MONTGOMERY_REDUCE_C
        /* use slower baseline Montgomery method */
        redux = mp_montgomery_reduce;
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    buf <<= (mp_digit)1;

    /* if the bit is zero and mode == 0 then we ignore it
     * These represent the leading zero bits before the first 1 bit
     * in the exponent.  Technically this opt is not required but it
     * does lower the # of trivial squaring/reductions used
     */
    if (mode == 0 && y == 0) {
      continue;
    }

    /* if the bit is zero and mode == 1 then we square */
    if (mode == 1 && y == 0) {
      if ((err = mp_sqr (&res, &res)) != MP_OKAY) {
        goto LBL_RES;
      }
      if ((err = redux (&res, P, mp)) != MP_OKAY) {
        goto LBL_RES;
      }
      continue;







|




|







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    buf <<= (mp_digit)1;

    /* if the bit is zero and mode == 0 then we ignore it
     * These represent the leading zero bits before the first 1 bit
     * in the exponent.  Technically this opt is not required but it
     * does lower the # of trivial squaring/reductions used
     */
    if ((mode == 0) && (y == 0)) {
      continue;
    }

    /* if the bit is zero and mode == 1 then we square */
    if ((mode == 1) && (y == 0)) {
      if ((err = mp_sqr (&res, &res)) != MP_OKAY) {
        goto LBL_RES;
      }
      if ((err = redux (&res, P, mp)) != MP_OKAY) {
        goto LBL_RES;
      }
      continue;
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      bitcpy = 0;
      bitbuf = 0;
      mode   = 1;
    }
  }

  /* if bits remain then square/multiply */
  if (mode == 2 && bitcpy > 0) {
    /* square then multiply if the bit is set */
    for (x = 0; x < bitcpy; x++) {
      if ((err = mp_sqr (&res, &res)) != MP_OKAY) {
        goto LBL_RES;
      }
      if ((err = redux (&res, P, mp)) != MP_OKAY) {
        goto LBL_RES;







|







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      bitcpy = 0;
      bitbuf = 0;
      mode   = 1;
    }
  }

  /* if bits remain then square/multiply */
  if ((mode == 2) && (bitcpy > 0)) {
    /* square then multiply if the bit is set */
    for (x = 0; x < bitcpy; x++) {
      if ((err = mp_sqr (&res, &res)) != MP_OKAY) {
        goto LBL_RES;
      }
      if ((err = redux (&res, P, mp)) != MP_OKAY) {
        goto LBL_RES;
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316





  mp_clear(&M[1]);
  for (x = 1<<(winsize-1); x < (1 << winsize); x++) {
    mp_clear (&M[x]);
  }
  return err;
}
#endif












>
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  mp_clear(&M[1]);
  for (x = 1<<(winsize-1); x < (1 << winsize); x++) {
    mp_clear (&M[x]);
  }
  return err;
}
#endif


/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_exteuclid.c.
1
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#include <tommath.h>
#ifdef BN_MP_EXTEUCLID_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* Extended euclidean algorithm of (a, b) produces 
   a*u1 + b*u2 = u3
 */
int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3)
{
   mp_int u1,u2,u3,v1,v2,v3,t1,t2,t3,q,tmp;
   int err;

|













|


|







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#include <tommath_private.h>
#ifdef BN_MP_EXTEUCLID_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* Extended euclidean algorithm of (a, b) produces
   a*u1 + b*u2 = u3
 */
int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3)
{
   mp_int u1,u2,u3,v1,v2,v3,t1,t2,t3,q,tmp;
   int err;

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       if ((err = mp_copy(&t1, &v1)) != MP_OKAY)                                  { goto _ERR; }
       if ((err = mp_copy(&t2, &v2)) != MP_OKAY)                                  { goto _ERR; }
       if ((err = mp_copy(&t3, &v3)) != MP_OKAY)                                  { goto _ERR; }
   }

   /* make sure U3 >= 0 */
   if (u3.sign == MP_NEG) {
      mp_neg(&u1, &u1);
      mp_neg(&u2, &u2);
      mp_neg(&u3, &u3);
   }

   /* copy result out */
   if (U1 != NULL) { mp_exch(U1, &u1); }
   if (U2 != NULL) { mp_exch(U2, &u2); }
   if (U3 != NULL) { mp_exch(U3, &u3); }

   err = MP_OKAY;
_ERR: mp_clear_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL);
   return err;
}
#endif











|
|
|












>
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       if ((err = mp_copy(&t1, &v1)) != MP_OKAY)                                  { goto _ERR; }
       if ((err = mp_copy(&t2, &v2)) != MP_OKAY)                                  { goto _ERR; }
       if ((err = mp_copy(&t3, &v3)) != MP_OKAY)                                  { goto _ERR; }
   }

   /* make sure U3 >= 0 */
   if (u3.sign == MP_NEG) {
       if ((err = mp_neg(&u1, &u1)) != MP_OKAY)                                   { goto _ERR; }
       if ((err = mp_neg(&u2, &u2)) != MP_OKAY)                                   { goto _ERR; }
       if ((err = mp_neg(&u3, &u3)) != MP_OKAY)                                   { goto _ERR; }
   }

   /* copy result out */
   if (U1 != NULL) { mp_exch(U1, &u1); }
   if (U2 != NULL) { mp_exch(U2, &u2); }
   if (U3 != NULL) { mp_exch(U3, &u3); }

   err = MP_OKAY;
_ERR: mp_clear_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL);
   return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_fread.c.
1
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#include <tommath.h>
#ifdef BN_MP_FREAD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* read a bigint from a file stream in ASCII */
int mp_fread(mp_int *a, int radix, FILE *stream)
{
   int err, ch, neg, y;
   
|













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#include <tommath_private.h>
#ifdef BN_MP_FREAD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* read a bigint from a file stream in ASCII */
int mp_fread(mp_int *a, int radix, FILE *stream)
{
   int err, ch, neg, y;
   
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61
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63




      a->sign = neg;
   }
   
   return MP_OKAY;
}

#endif











>
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>
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67
      a->sign = neg;
   }
   
   return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_fwrite.c.
1
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#include <tommath.h>
#ifdef BN_MP_FWRITE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

int mp_fwrite(mp_int *a, int radix, FILE *stream)
{
   char *buf;
   int err, len, x;
   
|













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#include <tommath_private.h>
#ifdef BN_MP_FWRITE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

int mp_fwrite(mp_int *a, int radix, FILE *stream)
{
   char *buf;
   int err, len, x;
   
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   }
   
   XFREE (buf);
   return MP_OKAY;
}

#endif











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   }
   
   XFREE (buf);
   return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_gcd.c.
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#include <tommath.h>
#ifdef BN_MP_GCD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* Greatest Common Divisor using the binary method */
int mp_gcd (mp_int * a, mp_int * b, mp_int * c)
{
  mp_int  u, v;
  int     k, u_lsb, v_lsb, res;
|













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#include <tommath_private.h>
#ifdef BN_MP_GCD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* Greatest Common Divisor using the binary method */
int mp_gcd (mp_int * a, mp_int * b, mp_int * c)
{
  mp_int  u, v;
  int     k, u_lsb, v_lsb, res;
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  if (v_lsb != k) {
     if ((res = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) {
        goto LBL_V;
     }
  }

  while (mp_iszero(&v) == 0) {
     /* make sure v is the largest */
     if (mp_cmp_mag(&u, &v) == MP_GT) {
        /* swap u and v to make sure v is >= u */
        mp_exch(&u, &v);
     }
     
     /* subtract smallest from largest */







|







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  if (v_lsb != k) {
     if ((res = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) {
        goto LBL_V;
     }
  }

  while (mp_iszero(&v) == MP_NO) {
     /* make sure v is the largest */
     if (mp_cmp_mag(&u, &v) == MP_GT) {
        /* swap u and v to make sure v is >= u */
        mp_exch(&u, &v);
     }
     
     /* subtract smallest from largest */
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  c->sign = MP_ZPOS;
  res = MP_OKAY;
LBL_V:mp_clear (&u);
LBL_U:mp_clear (&v);
  return res;
}
#endif











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  c->sign = MP_ZPOS;
  res = MP_OKAY;
LBL_V:mp_clear (&u);
LBL_U:mp_clear (&v);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_get_int.c.
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#include <tommath.h>
#ifdef BN_MP_GET_INT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* get the lower 32-bits of an mp_int */
unsigned long mp_get_int(mp_int * a) 
{
  int i;
  unsigned long res;

  if (a->used == 0) {
     return 0;
  }

  /* get number of digits of the lsb we have to read */
  i = MIN(a->used,(int)((sizeof(unsigned long)*CHAR_BIT+DIGIT_BIT-1)/DIGIT_BIT))-1;

  /* get most significant digit of result */
  res = DIGIT(a,i);
   
  while (--i >= 0) {
    res = (res << DIGIT_BIT) | DIGIT(a,i);
  }

  /* force result to 32-bits always so it is consistent on non 32-bit platforms */
  return res & 0xFFFFFFFFUL;
}
#endif




|













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|






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#include <tommath_private.h>
#ifdef BN_MP_GET_INT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* get the lower 32-bits of an mp_int */
unsigned long mp_get_int(mp_int * a)
{
  int i;
  mp_min_u32 res;

  if (a->used == 0) {
     return 0;
  }

  /* get number of digits of the lsb we have to read */
  i = MIN(a->used,(int)(((sizeof(unsigned long) * CHAR_BIT) + DIGIT_BIT - 1) / DIGIT_BIT)) - 1;

  /* get most significant digit of result */
  res = DIGIT(a,i);

  while (--i >= 0) {
    res = (res << DIGIT_BIT) | DIGIT(a,i);
  }

  /* force result to 32-bits always so it is consistent on non 32-bit platforms */
  return res & 0xFFFFFFFFUL;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_grow.c.
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#include <tommath.h>
#ifdef BN_MP_GROW_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* grow as required */
int mp_grow (mp_int * a, int size)
{
  int     i;
  mp_digit *tmp;
|













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#include <tommath_private.h>
#ifdef BN_MP_GROW_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* grow as required */
int mp_grow (mp_int * a, int size)
{
  int     i;
  mp_digit *tmp;
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    for (; i < a->alloc; i++) {
      a->dp[i] = 0;
    }
  }
  return MP_OKAY;
}
#endif











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    for (; i < a->alloc; i++) {
      a->dp[i] = 0;
    }
  }
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_init.c.
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#include <tommath.h>
#ifdef BN_MP_INIT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* init a new mp_int */
int mp_init (mp_int * a)
{
  int i;

|













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#include <tommath_private.h>
#ifdef BN_MP_INIT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* init a new mp_int */
int mp_init (mp_int * a)
{
  int i;

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  a->used  = 0;
  a->alloc = MP_PREC;
  a->sign  = MP_ZPOS;

  return MP_OKAY;
}
#endif











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  a->used  = 0;
  a->alloc = MP_PREC;
  a->sign  = MP_ZPOS;

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_init_copy.c.
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#include <tommath.h>
#ifdef BN_MP_INIT_COPY_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* creates "a" then copies b into it */
int mp_init_copy (mp_int * a, mp_int * b)
{
  int     res;

  if ((res = mp_init (a)) != MP_OKAY) {
    return res;
  }
  return mp_copy (b, a);
}
#endif




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|





>
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#include <tommath_private.h>
#ifdef BN_MP_INIT_COPY_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* creates "a" then copies b into it */
int mp_init_copy (mp_int * a, mp_int * b)
{
  int     res;

  if ((res = mp_init_size (a, b->used)) != MP_OKAY) {
    return res;
  }
  return mp_copy (b, a);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_init_multi.c.
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#include <tommath.h>
#ifdef BN_MP_INIT_MULTI_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */
#include <stdarg.h>

int mp_init_multi(mp_int *mp, ...) 
{
    mp_err res = MP_OKAY;      /* Assume ok until proven otherwise */
    int n = 0;                 /* Number of ok inits */
|













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#include <tommath_private.h>
#ifdef BN_MP_INIT_MULTI_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */
#include <stdarg.h>

int mp_init_multi(mp_int *mp, ...) 
{
    mp_err res = MP_OKAY;      /* Assume ok until proven otherwise */
    int n = 0;                 /* Number of ok inits */
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            /* end the current list */
            va_end(args);
            
            /* now start cleaning up */            
            cur_arg = mp;
            va_start(clean_args, mp);
            while (n--) {
                mp_clear(cur_arg);
                cur_arg = va_arg(clean_args, mp_int*);
            }
            va_end(clean_args);
            res = MP_MEM;
            break;
        }
        n++;
        cur_arg = va_arg(args, mp_int*);
    }
    va_end(args);
    return res;                /* Assumed ok, if error flagged above. */
}

#endif











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>
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            /* end the current list */
            va_end(args);
            
            /* now start cleaning up */            
            cur_arg = mp;
            va_start(clean_args, mp);
            while (n-- != 0) {
                mp_clear(cur_arg);
                cur_arg = va_arg(clean_args, mp_int*);
            }
            va_end(clean_args);
            res = MP_MEM;
            break;
        }
        n++;
        cur_arg = va_arg(args, mp_int*);
    }
    va_end(args);
    return res;                /* Assumed ok, if error flagged above. */
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_init_set.c.
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#include <tommath.h>
#ifdef BN_MP_INIT_SET_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* initialize and set a digit */
int mp_init_set (mp_int * a, mp_digit b)
{
  int err;
  if ((err = mp_init(a)) != MP_OKAY) {
     return err;
  }
  mp_set(a, b);
  return err;
}
#endif




|













|













>
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#include <tommath_private.h>
#ifdef BN_MP_INIT_SET_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* initialize and set a digit */
int mp_init_set (mp_int * a, mp_digit b)
{
  int err;
  if ((err = mp_init(a)) != MP_OKAY) {
     return err;
  }
  mp_set(a, b);
  return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_init_set_int.c.
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#include <tommath.h>
#ifdef BN_MP_INIT_SET_INT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* initialize and set a digit */
int mp_init_set_int (mp_int * a, unsigned long b)
{
  int err;
  if ((err = mp_init(a)) != MP_OKAY) {
     return err;
  }
  return mp_set_int(a, b);
}
#endif




|













|












>
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>
>
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#include <tommath_private.h>
#ifdef BN_MP_INIT_SET_INT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* initialize and set a digit */
int mp_init_set_int (mp_int * a, unsigned long b)
{
  int err;
  if ((err = mp_init(a)) != MP_OKAY) {
     return err;
  }
  return mp_set_int(a, b);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_init_size.c.
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#include <tommath.h>
#ifdef BN_MP_INIT_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* init an mp_init for a given size */
int mp_init_size (mp_int * a, int size)
{
  int x;

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#include <tommath_private.h>
#ifdef BN_MP_INIT_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* init an mp_init for a given size */
int mp_init_size (mp_int * a, int size)
{
  int x;

38
39
40
41
42
43
44




  for (x = 0; x < size; x++) {
      a->dp[x] = 0;
  }

  return MP_OKAY;
}
#endif











>
>
>
>
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48
  for (x = 0; x < size; x++) {
      a->dp[x] = 0;
  }

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_invmod.c.
1
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5
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8
9
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37

38
39




#include <tommath.h>
#ifdef BN_MP_INVMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* hac 14.61, pp608 */
int mp_invmod (mp_int * a, mp_int * b, mp_int * c)
{
  /* b cannot be negative */
  if (b->sign == MP_NEG || mp_iszero(b) == 1) {
    return MP_VAL;
  }

#ifdef BN_FAST_MP_INVMOD_C
  /* if the modulus is odd we can use a faster routine instead */
  if (mp_isodd (b) == 1) {
    return fast_mp_invmod (a, b, c);
  }
#endif

#ifdef BN_MP_INVMOD_SLOW_C
  return mp_invmod_slow(a, b, c);
#endif

  return MP_VAL;

}
#endif




|













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|
<

>


>
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>
>
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35

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#include <tommath_private.h>
#ifdef BN_MP_INVMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* hac 14.61, pp608 */
int mp_invmod (mp_int * a, mp_int * b, mp_int * c)
{
  /* b cannot be negative */
  if ((b->sign == MP_NEG) || (mp_iszero(b) == MP_YES)) {
    return MP_VAL;
  }

#ifdef BN_FAST_MP_INVMOD_C
  /* if the modulus is odd we can use a faster routine instead */
  if (mp_isodd (b) == MP_YES) {
    return fast_mp_invmod (a, b, c);
  }
#endif

#ifdef BN_MP_INVMOD_SLOW_C
  return mp_invmod_slow(a, b, c);
#else

  return MP_VAL;
#endif
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_invmod_slow.c.
1
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#include <tommath.h>
#ifdef BN_MP_INVMOD_SLOW_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* hac 14.61, pp608 */
int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c)
{
  mp_int  x, y, u, v, A, B, C, D;
  int     res;

  /* b cannot be negative */
  if (b->sign == MP_NEG || mp_iszero(b) == 1) {
    return MP_VAL;
  }

  /* init temps */
  if ((res = mp_init_multi(&x, &y, &u, &v, 
                           &A, &B, &C, &D, NULL)) != MP_OKAY) {
     return res;
  }

  /* x = a, y = b */
  if ((res = mp_mod(a, b, &x)) != MP_OKAY) {
      goto LBL_ERR;
  }
  if ((res = mp_copy (b, &y)) != MP_OKAY) {
    goto LBL_ERR;
  }

  /* 2. [modified] if x,y are both even then return an error! */
  if (mp_iseven (&x) == 1 && mp_iseven (&y) == 1) {
    res = MP_VAL;
    goto LBL_ERR;
  }

  /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */
  if ((res = mp_copy (&x, &u)) != MP_OKAY) {
    goto LBL_ERR;
  }
  if ((res = mp_copy (&y, &v)) != MP_OKAY) {
    goto LBL_ERR;
  }
  mp_set (&A, 1);
  mp_set (&D, 1);

top:
  /* 4.  while u is even do */
  while (mp_iseven (&u) == 1) {
    /* 4.1 u = u/2 */
    if ((res = mp_div_2 (&u, &u)) != MP_OKAY) {
      goto LBL_ERR;
    }
    /* 4.2 if A or B is odd then */
    if (mp_isodd (&A) == 1 || mp_isodd (&B) == 1) {
      /* A = (A+y)/2, B = (B-x)/2 */
      if ((res = mp_add (&A, &y, &A)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) {
         goto LBL_ERR;
      }
    }
    /* A = A/2, B = B/2 */
    if ((res = mp_div_2 (&A, &A)) != MP_OKAY) {
      goto LBL_ERR;
    }
    if ((res = mp_div_2 (&B, &B)) != MP_OKAY) {
      goto LBL_ERR;
    }
  }

  /* 5.  while v is even do */
  while (mp_iseven (&v) == 1) {
    /* 5.1 v = v/2 */
    if ((res = mp_div_2 (&v, &v)) != MP_OKAY) {
      goto LBL_ERR;
    }
    /* 5.2 if C or D is odd then */
    if (mp_isodd (&C) == 1 || mp_isodd (&D) == 1) {
      /* C = (C+y)/2, D = (D-x)/2 */
      if ((res = mp_add (&C, &y, &C)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) {
         goto LBL_ERR;
      }
|













|









|


















|
















|





|


















|





|







1
2
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5
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#include <tommath_private.h>
#ifdef BN_MP_INVMOD_SLOW_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* hac 14.61, pp608 */
int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c)
{
  mp_int  x, y, u, v, A, B, C, D;
  int     res;

  /* b cannot be negative */
  if ((b->sign == MP_NEG) || (mp_iszero(b) == MP_YES)) {
    return MP_VAL;
  }

  /* init temps */
  if ((res = mp_init_multi(&x, &y, &u, &v, 
                           &A, &B, &C, &D, NULL)) != MP_OKAY) {
     return res;
  }

  /* x = a, y = b */
  if ((res = mp_mod(a, b, &x)) != MP_OKAY) {
      goto LBL_ERR;
  }
  if ((res = mp_copy (b, &y)) != MP_OKAY) {
    goto LBL_ERR;
  }

  /* 2. [modified] if x,y are both even then return an error! */
  if ((mp_iseven (&x) == MP_YES) && (mp_iseven (&y) == MP_YES)) {
    res = MP_VAL;
    goto LBL_ERR;
  }

  /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */
  if ((res = mp_copy (&x, &u)) != MP_OKAY) {
    goto LBL_ERR;
  }
  if ((res = mp_copy (&y, &v)) != MP_OKAY) {
    goto LBL_ERR;
  }
  mp_set (&A, 1);
  mp_set (&D, 1);

top:
  /* 4.  while u is even do */
  while (mp_iseven (&u) == MP_YES) {
    /* 4.1 u = u/2 */
    if ((res = mp_div_2 (&u, &u)) != MP_OKAY) {
      goto LBL_ERR;
    }
    /* 4.2 if A or B is odd then */
    if ((mp_isodd (&A) == MP_YES) || (mp_isodd (&B) == MP_YES)) {
      /* A = (A+y)/2, B = (B-x)/2 */
      if ((res = mp_add (&A, &y, &A)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) {
         goto LBL_ERR;
      }
    }
    /* A = A/2, B = B/2 */
    if ((res = mp_div_2 (&A, &A)) != MP_OKAY) {
      goto LBL_ERR;
    }
    if ((res = mp_div_2 (&B, &B)) != MP_OKAY) {
      goto LBL_ERR;
    }
  }

  /* 5.  while v is even do */
  while (mp_iseven (&v) == MP_YES) {
    /* 5.1 v = v/2 */
    if ((res = mp_div_2 (&v, &v)) != MP_OKAY) {
      goto LBL_ERR;
    }
    /* 5.2 if C or D is odd then */
    if ((mp_isodd (&C) == MP_YES) || (mp_isodd (&D) == MP_YES)) {
      /* C = (C+y)/2, D = (D-x)/2 */
      if ((res = mp_add (&C, &y, &C)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) {
         goto LBL_ERR;
      }
133
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135
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137
138
139
140
141
142
143
144
145
146
147

    if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) {
      goto LBL_ERR;
    }
  }

  /* if not zero goto step 4 */
  if (mp_iszero (&u) == 0)
    goto top;

  /* now a = C, b = D, gcd == g*v */

  /* if v != 1 then there is no inverse */
  if (mp_cmp_d (&v, 1) != MP_EQ) {
    res = MP_VAL;







|







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144
145
146
147

    if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) {
      goto LBL_ERR;
    }
  }

  /* if not zero goto step 4 */
  if (mp_iszero (&u) == MP_NO)
    goto top;

  /* now a = C, b = D, gcd == g*v */

  /* if v != 1 then there is no inverse */
  if (mp_cmp_d (&v, 1) != MP_EQ) {
    res = MP_VAL;
165
166
167
168
169
170
171




  /* C is now the inverse */
  mp_exch (&C, c);
  res = MP_OKAY;
LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &A, &B, &C, &D, NULL);
  return res;
}
#endif











>
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>
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175
  /* C is now the inverse */
  mp_exch (&C, c);
  res = MP_OKAY;
LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &A, &B, &C, &D, NULL);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_is_square.c.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
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18
19
20
21
22
#include <tommath.h>
#ifdef BN_MP_IS_SQUARE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* Check if remainders are possible squares - fast exclude non-squares */
static const char rem_128[128] = {
 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
|













|







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11
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14
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20
21
22
#include <tommath_private.h>
#ifdef BN_MP_IS_SQUARE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* Check if remainders are possible squares - fast exclude non-squares */
static const char rem_128[128] = {
 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
78
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     goto ERR;
  }
  r = mp_get_int(&t);
  /* Check for other prime modules, note it's not an ERROR but we must
   * free "t" so the easiest way is to goto ERR.  We know that res
   * is already equal to MP_OKAY from the mp_mod call 
   */ 
  if ( (1L<<(r%11)) & 0x5C4L )             goto ERR;
  if ( (1L<<(r%13)) & 0x9E4L )             goto ERR;
  if ( (1L<<(r%17)) & 0x5CE8L )            goto ERR;
  if ( (1L<<(r%19)) & 0x4F50CL )           goto ERR;
  if ( (1L<<(r%23)) & 0x7ACCA0L )          goto ERR;
  if ( (1L<<(r%29)) & 0xC2EDD0CL )         goto ERR;
  if ( (1L<<(r%31)) & 0x6DE2B848L )        goto ERR;

  /* Final check - is sqr(sqrt(arg)) == arg ? */
  if ((res = mp_sqrt(arg,&t)) != MP_OKAY) {
     goto ERR;
  }
  if ((res = mp_sqr(&t,&t)) != MP_OKAY) {
     goto ERR;
  }

  *ret = (mp_cmp_mag(&t,arg) == MP_EQ) ? MP_YES : MP_NO;
ERR:mp_clear(&t);
  return res;
}
#endif











|
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|














>
>
>
>
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     goto ERR;
  }
  r = mp_get_int(&t);
  /* Check for other prime modules, note it's not an ERROR but we must
   * free "t" so the easiest way is to goto ERR.  We know that res
   * is already equal to MP_OKAY from the mp_mod call 
   */ 
  if (((1L<<(r%11)) & 0x5C4L) != 0L)       goto ERR;
  if (((1L<<(r%13)) & 0x9E4L) != 0L)       goto ERR;
  if (((1L<<(r%17)) & 0x5CE8L) != 0L)      goto ERR;
  if (((1L<<(r%19)) & 0x4F50CL) != 0L)     goto ERR;
  if (((1L<<(r%23)) & 0x7ACCA0L) != 0L)    goto ERR;
  if (((1L<<(r%29)) & 0xC2EDD0CL) != 0L)   goto ERR;
  if (((1L<<(r%31)) & 0x6DE2B848L) != 0L)  goto ERR;

  /* Final check - is sqr(sqrt(arg)) == arg ? */
  if ((res = mp_sqrt(arg,&t)) != MP_OKAY) {
     goto ERR;
  }
  if ((res = mp_sqr(&t,&t)) != MP_OKAY) {
     goto ERR;
  }

  *ret = (mp_cmp_mag(&t,arg) == MP_EQ) ? MP_YES : MP_NO;
ERR:mp_clear(&t);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_jacobi.c.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19


20
21
22
23
24
25
26





27
28
29
30
31
32
33




34

35
36
37
38
39
40
41
42
#include <tommath.h>
#ifdef BN_MP_JACOBI_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* computes the jacobi c = (a | n) (or Legendre if n is prime)
 * HAC pp. 73 Algorithm 2.149


 */
int mp_jacobi (mp_int * a, mp_int * p, int *c)
{
  mp_int  a1, p1;
  int     k, s, r, res;
  mp_digit residue;






  /* if p <= 0 return MP_VAL */
  if (mp_cmp_d(p, 0) != MP_GT) {
     return MP_VAL;
  }

  /* step 1.  if a == 0, return 0 */
  if (mp_iszero (a) == 1) {




    *c = 0;

    return MP_OKAY;
  }

  /* step 2.  if a == 1, return 1 */
  if (mp_cmp_d (a, 1) == MP_EQ) {
    *c = 1;
    return MP_OKAY;
  }
|













|




>
>

|





>
>
>
>
>
|
|



|
|
>
>
>
>
|
>
|







1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
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27
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35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
#include <tommath_private.h>
#ifdef BN_MP_JACOBI_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* computes the jacobi c = (a | n) (or Legendre if n is prime)
 * HAC pp. 73 Algorithm 2.149
 * HAC is wrong here, as the special case of (0 | 1) is not
 * handled correctly.
 */
int mp_jacobi (mp_int * a, mp_int * n, int *c)
{
  mp_int  a1, p1;
  int     k, s, r, res;
  mp_digit residue;

  /* if a < 0 return MP_VAL */
  if (mp_isneg(a) == MP_YES) {
     return MP_VAL;
  }

  /* if n <= 0 return MP_VAL */
  if (mp_cmp_d(n, 0) != MP_GT) {
     return MP_VAL;
  }

  /* step 1. handle case of a == 0 */
  if (mp_iszero (a) == MP_YES) {
     /* special case of a == 0 and n == 1 */
     if (mp_cmp_d (n, 1) == MP_EQ) {
       *c = 1;
     } else {
       *c = 0;
     }
     return MP_OKAY;
  }

  /* step 2.  if a == 1, return 1 */
  if (mp_cmp_d (a, 1) == MP_EQ) {
    *c = 1;
    return MP_OKAY;
  }
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
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96
97
98
99
100
101




  }

  /* step 4.  if e is even set s=1 */
  if ((k & 1) == 0) {
    s = 1;
  } else {
    /* else set s=1 if p = 1/7 (mod 8) or s=-1 if p = 3/5 (mod 8) */
    residue = p->dp[0] & 7;

    if (residue == 1 || residue == 7) {
      s = 1;
    } else if (residue == 3 || residue == 5) {
      s = -1;
    }
  }

  /* step 5.  if p == 3 (mod 4) *and* a1 == 3 (mod 4) then s = -s */
  if ( ((p->dp[0] & 3) == 3) && ((a1.dp[0] & 3) == 3)) {
    s = -s;
  }

  /* if a1 == 1 we're done */
  if (mp_cmp_d (&a1, 1) == MP_EQ) {
    *c = s;
  } else {
    /* n1 = n mod a1 */
    if ((res = mp_mod (p, &a1, &p1)) != MP_OKAY) {
      goto LBL_P1;
    }
    if ((res = mp_jacobi (&p1, &a1, &r)) != MP_OKAY) {
      goto LBL_P1;
    }
    *c = s * r;
  }

  /* done */
  res = MP_OKAY;
LBL_P1:mp_clear (&p1);
LBL_A1:mp_clear (&a1);
  return res;
}
#endif











|

|

|





|








|















>
>
>
>
72
73
74
75
76
77
78
79
80
81
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103
104
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112
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115
116
117
  }

  /* step 4.  if e is even set s=1 */
  if ((k & 1) == 0) {
    s = 1;
  } else {
    /* else set s=1 if p = 1/7 (mod 8) or s=-1 if p = 3/5 (mod 8) */
    residue = n->dp[0] & 7;

    if ((residue == 1) || (residue == 7)) {
      s = 1;
    } else if ((residue == 3) || (residue == 5)) {
      s = -1;
    }
  }

  /* step 5.  if p == 3 (mod 4) *and* a1 == 3 (mod 4) then s = -s */
  if ( ((n->dp[0] & 3) == 3) && ((a1.dp[0] & 3) == 3)) {
    s = -s;
  }

  /* if a1 == 1 we're done */
  if (mp_cmp_d (&a1, 1) == MP_EQ) {
    *c = s;
  } else {
    /* n1 = n mod a1 */
    if ((res = mp_mod (n, &a1, &p1)) != MP_OKAY) {
      goto LBL_P1;
    }
    if ((res = mp_jacobi (&p1, &a1, &r)) != MP_OKAY) {
      goto LBL_P1;
    }
    *c = s * r;
  }

  /* done */
  res = MP_OKAY;
LBL_P1:mp_clear (&p1);
LBL_A1:mp_clear (&a1);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_karatsuba_mul.c.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
#include <tommath.h>
#ifdef BN_MP_KARATSUBA_MUL_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* c = |a| * |b| using Karatsuba Multiplication using 
 * three half size multiplications
 *
 * Let B represent the radix [e.g. 2**DIGIT_BIT] and 
 * let n represent half of the number of digits in 
|













|







1
2
3
4
5
6
7
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9
10
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12
13
14
15
16
17
18
19
20
21
22
#include <tommath_private.h>
#ifdef BN_MP_KARATSUBA_MUL_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* c = |a| * |b| using Karatsuba Multiplication using 
 * three half size multiplications
 *
 * Let B represent the radix [e.g. 2**DIGIT_BIT] and 
 * let n represent half of the number of digits in 
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93

  /* now shift the digits */
  x0.used = y0.used = B;
  x1.used = a->used - B;
  y1.used = b->used - B;

  {
    register int x;
    register mp_digit *tmpa, *tmpb, *tmpx, *tmpy;

    /* we copy the digits directly instead of using higher level functions
     * since we also need to shift the digits
     */
    tmpa = a->dp;
    tmpb = b->dp;








|
|







78
79
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88
89
90
91
92
93

  /* now shift the digits */
  x0.used = y0.used = B;
  x1.used = a->used - B;
  y1.used = b->used - B;

  {
    int x;
    mp_digit *tmpa, *tmpb, *tmpx, *tmpy;

    /* we copy the digits directly instead of using higher level functions
     * since we also need to shift the digits
     */
    tmpa = a->dp;
    tmpb = b->dp;

157
158
159
160
161
162
163




Y0:mp_clear (&y0);
X1:mp_clear (&x1);
X0:mp_clear (&x0);
ERR:
  return err;
}
#endif











>
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>
>
157
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159
160
161
162
163
164
165
166
167
Y0:mp_clear (&y0);
X1:mp_clear (&x1);
X0:mp_clear (&x0);
ERR:
  return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_karatsuba_sqr.c.
1
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7
8
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#include <tommath.h>
#ifdef BN_MP_KARATSUBA_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* Karatsuba squaring, computes b = a*a using three 
 * half size squarings
 *
 * See comments of karatsuba_mul for details.  It 
 * is essentially the same algorithm but merely 
|













|







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20
21
22
#include <tommath_private.h>
#ifdef BN_MP_KARATSUBA_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* Karatsuba squaring, computes b = a*a using three 
 * half size squarings
 *
 * See comments of karatsuba_mul for details.  It 
 * is essentially the same algorithm but merely 
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    goto T1;
  if (mp_init_size (&x0x0, B * 2) != MP_OKAY)
    goto T2;
  if (mp_init_size (&x1x1, (a->used - B) * 2) != MP_OKAY)
    goto X0X0;

  {
    register int x;
    register mp_digit *dst, *src;

    src = a->dp;

    /* now shift the digits */
    dst = x0.dp;
    for (x = 0; x < B; x++) {
      *dst++ = *src++;







|
|







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    goto T1;
  if (mp_init_size (&x0x0, B * 2) != MP_OKAY)
    goto T2;
  if (mp_init_size (&x1x1, (a->used - B) * 2) != MP_OKAY)
    goto X0X0;

  {
    int x;
    mp_digit *dst, *src;

    src = a->dp;

    /* now shift the digits */
    dst = x0.dp;
    for (x = 0; x < B; x++) {
      *dst++ = *src++;
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T1:mp_clear (&t1);
X1:mp_clear (&x1);
X0:mp_clear (&x0);
ERR:
  return err;
}
#endif











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T1:mp_clear (&t1);
X1:mp_clear (&x1);
X0:mp_clear (&x0);
ERR:
  return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_lcm.c.
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#include <tommath.h>
#ifdef BN_MP_LCM_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* computes least common multiple as |a*b|/(a, b) */
int mp_lcm (mp_int * a, mp_int * b, mp_int * c)
{
  int     res;
  mp_int  t1, t2;
|













|







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#include <tommath_private.h>
#ifdef BN_MP_LCM_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* computes least common multiple as |a*b|/(a, b) */
int mp_lcm (mp_int * a, mp_int * b, mp_int * c)
{
  int     res;
  mp_int  t1, t2;
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55
56




  c->sign = MP_ZPOS;

LBL_T:
  mp_clear_multi (&t1, &t2, NULL);
  return res;
}
#endif











>
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  c->sign = MP_ZPOS;

LBL_T:
  mp_clear_multi (&t1, &t2, NULL);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_lshd.c.
1
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5
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8
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#include <tommath.h>
#ifdef BN_MP_LSHD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* shift left a certain amount of digits */
int mp_lshd (mp_int * a, int b)
{
  int     x, res;

  /* if its less than zero return */
  if (b <= 0) {
    return MP_OKAY;
  }

  /* grow to fit the new digits */
  if (a->alloc < a->used + b) {
     if ((res = mp_grow (a, a->used + b)) != MP_OKAY) {
       return res;
     }
  }

  {
    register mp_digit *top, *bottom;

    /* increment the used by the shift amount then copy upwards */
    a->used += b;

    /* top */
    top = a->dp + a->used - 1;

    /* base */
    bottom = a->dp + a->used - 1 - b;

    /* much like mp_rshd this is implemented using a sliding window
     * except the window goes the otherway around.  Copying from
     * the bottom to the top.  see bn_mp_rshd.c for more info.
     */
    for (x = a->used - 1; x >= b; x--) {
      *top-- = *bottom--;
    }

    /* zero the lower digits */
    top = a->dp;
    for (x = 0; x < b; x++) {
      *top++ = 0;
    }
  }
  return MP_OKAY;
}
#endif




|













|













|






|








|


















>
>
>
>
1
2
3
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5
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67
#include <tommath_private.h>
#ifdef BN_MP_LSHD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* shift left a certain amount of digits */
int mp_lshd (mp_int * a, int b)
{
  int     x, res;

  /* if its less than zero return */
  if (b <= 0) {
    return MP_OKAY;
  }

  /* grow to fit the new digits */
  if (a->alloc < (a->used + b)) {
     if ((res = mp_grow (a, a->used + b)) != MP_OKAY) {
       return res;
     }
  }

  {
    mp_digit *top, *bottom;

    /* increment the used by the shift amount then copy upwards */
    a->used += b;

    /* top */
    top = a->dp + a->used - 1;

    /* base */
    bottom = (a->dp + a->used - 1) - b;

    /* much like mp_rshd this is implemented using a sliding window
     * except the window goes the otherway around.  Copying from
     * the bottom to the top.  see bn_mp_rshd.c for more info.
     */
    for (x = a->used - 1; x >= b; x--) {
      *top-- = *bottom--;
    }

    /* zero the lower digits */
    top = a->dp;
    for (x = 0; x < b; x++) {
      *top++ = 0;
    }
  }
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_mod.c.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
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31
32
33
34
35
36
37
38


39
40
41
42
43
44




#include <tommath.h>
#ifdef BN_MP_MOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* c = a mod b, 0 <= c < b */
int
mp_mod (mp_int * a, mp_int * b, mp_int * c)
{
  mp_int  t;
  int     res;

  if ((res = mp_init (&t)) != MP_OKAY) {
    return res;
  }

  if ((res = mp_div (a, b, NULL, &t)) != MP_OKAY) {
    mp_clear (&t);
    return res;
  }

  if (t.sign != b->sign) {
    res = mp_add (b, &t, c);
  } else {
    res = MP_OKAY;
    mp_exch (&t, c);


  }

  mp_clear (&t);
  return res;
}
#endif




|













|


|















|
<
<


>
>






>
>
>
>
1
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34


35
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48
#include <tommath_private.h>
#ifdef BN_MP_MOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* c = a mod b, 0 <= c < b if b > 0, b < c <= 0 if b < 0 */
int
mp_mod (mp_int * a, mp_int * b, mp_int * c)
{
  mp_int  t;
  int     res;

  if ((res = mp_init (&t)) != MP_OKAY) {
    return res;
  }

  if ((res = mp_div (a, b, NULL, &t)) != MP_OKAY) {
    mp_clear (&t);
    return res;
  }

  if ((mp_iszero(&t) != MP_NO) || (t.sign == b->sign)) {


    res = MP_OKAY;
    mp_exch (&t, c);
  } else {
    res = mp_add (b, &t, c);
  }

  mp_clear (&t);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_mod_2d.c.
1
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22
#include <tommath.h>
#ifdef BN_MP_MOD_2D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* calc a value mod 2**b */
int
mp_mod_2d (const mp_int * a, int b, mp_int * c)
{
  int     x, res;
|













|







1
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6
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8
9
10
11
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13
14
15
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22
#include <tommath_private.h>
#ifdef BN_MP_MOD_2D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* calc a value mod 2**b */
int
mp_mod_2d (const mp_int * a, int b, mp_int * c)
{
  int     x, res;
35
36
37
38
39
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41
42
43
44
45
46
47
48
49
50
51





  /* copy */
  if ((res = mp_copy (a, c)) != MP_OKAY) {
    return res;
  }

  /* zero digits above the last digit of the modulus */
  for (x = (b / DIGIT_BIT) + ((b % DIGIT_BIT) == 0 ? 0 : 1); x < c->used; x++) {
    c->dp[x] = 0;
  }
  /* clear the digit that is not completely outside/inside the modulus */
  c->dp[b / DIGIT_BIT] &=
    (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digit) 1));
  mp_clamp (c);
  return MP_OKAY;
}
#endif











|









>
>
>
>
35
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50
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55

  /* copy */
  if ((res = mp_copy (a, c)) != MP_OKAY) {
    return res;
  }

  /* zero digits above the last digit of the modulus */
  for (x = (b / DIGIT_BIT) + (((b % DIGIT_BIT) == 0) ? 0 : 1); x < c->used; x++) {
    c->dp[x] = 0;
  }
  /* clear the digit that is not completely outside/inside the modulus */
  c->dp[b / DIGIT_BIT] &=
    (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digit) 1));
  mp_clamp (c);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_mod_d.c.
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#include <tommath.h>
#ifdef BN_MP_MOD_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

int
mp_mod_d (mp_int * a, mp_digit b, mp_digit * c)
{
  return mp_div_d(a, b, NULL, c);
}
#endif




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#include <tommath_private.h>
#ifdef BN_MP_MOD_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

int
mp_mod_d (mp_int * a, mp_digit b, mp_digit * c)
{
  return mp_div_d(a, b, NULL, c);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_montgomery_calc_normalization.c.
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#include <tommath.h>
#ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/*
 * shifts with subtractions when the result is greater than b.
 *
 * The method is slightly modified to shift B unconditionally upto just under
 * the leading bit of b.  This saves alot of multiple precision shifting.
 */
int mp_montgomery_calc_normalization (mp_int * a, mp_int * b)
{
  int     x, bits, res;

  /* how many bits of last digit does b use */
  bits = mp_count_bits (b) % DIGIT_BIT;

  if (b->used > 1) {
     if ((res = mp_2expt (a, (b->used - 1) * DIGIT_BIT + bits - 1)) != MP_OKAY) {
        return res;
     }
  } else {
     mp_set(a, 1);
     bits = 1;
  }

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#include <tommath_private.h>
#ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/*
 * shifts with subtractions when the result is greater than b.
 *
 * The method is slightly modified to shift B unconditionally upto just under
 * the leading bit of b.  This saves alot of multiple precision shifting.
 */
int mp_montgomery_calc_normalization (mp_int * a, mp_int * b)
{
  int     x, bits, res;

  /* how many bits of last digit does b use */
  bits = mp_count_bits (b) % DIGIT_BIT;

  if (b->used > 1) {
     if ((res = mp_2expt (a, ((b->used - 1) * DIGIT_BIT) + bits - 1)) != MP_OKAY) {
        return res;
     }
  } else {
     mp_set(a, 1);
     bits = 1;
  }

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      }
    }
  }

  return MP_OKAY;
}
#endif











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      }
    }
  }

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_montgomery_reduce.c.
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#include <tommath.h>
#ifdef BN_MP_MONTGOMERY_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* computes xR**-1 == x (mod N) via Montgomery Reduction */
int
mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
{
  int     ix, res, digs;
  mp_digit mu;

  /* can the fast reduction [comba] method be used?
   *
   * Note that unlike in mul you're safely allowed *less*
   * than the available columns [255 per default] since carries
   * are fixed up in the inner loop.
   */
  digs = n->used * 2 + 1;
  if ((digs < MP_WARRAY) &&
      n->used <
      (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) {
    return fast_mp_montgomery_reduce (x, n, rho);
  }

  /* grow the input as required */
  if (x->alloc < digs) {
    if ((res = mp_grow (x, digs)) != MP_OKAY) {
      return res;
    }
  }
  x->used = digs;

  for (ix = 0; ix < n->used; ix++) {
    /* mu = ai * rho mod b
     *
     * The value of rho must be precalculated via
     * montgomery_setup() such that
     * it equals -1/n0 mod b this allows the
     * following inner loop to reduce the
     * input one digit at a time
     */
    mu = (mp_digit) (((mp_word)x->dp[ix]) * ((mp_word)rho) & MP_MASK);

    /* a = a + mu * m * b**i */
    {
      register int iy;
      register mp_digit *tmpn, *tmpx, u;
      register mp_word r;

      /* alias for digits of the modulus */
      tmpn = n->dp;

      /* alias for the digits of x [the input] */
      tmpx = x->dp + ix;

      /* set the carry to zero */
      u = 0;

      /* Multiply and add in place */
      for (iy = 0; iy < n->used; iy++) {
        /* compute product and sum */
        r       = ((mp_word)mu) * ((mp_word)*tmpn++) +
                  ((mp_word) u) + ((mp_word) * tmpx);

        /* get carry */
        u       = (mp_digit)(r >> ((mp_word) DIGIT_BIT));

        /* fix digit */
        *tmpx++ = (mp_digit)(r & ((mp_word) MP_MASK));
      }
      /* At this point the ix'th digit of x should be zero */


      /* propagate carries upwards as required*/
      while (u) {
        *tmpx   += u;
        u        = *tmpx >> DIGIT_BIT;
        *tmpx++ &= MP_MASK;
      }
    }
  }

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#include <tommath_private.h>
#ifdef BN_MP_MONTGOMERY_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* computes xR**-1 == x (mod N) via Montgomery Reduction */
int
mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
{
  int     ix, res, digs;
  mp_digit mu;

  /* can the fast reduction [comba] method be used?
   *
   * Note that unlike in mul you're safely allowed *less*
   * than the available columns [255 per default] since carries
   * are fixed up in the inner loop.
   */
  digs = (n->used * 2) + 1;
  if ((digs < MP_WARRAY) &&
      (n->used <
      (1 << ((CHAR_BIT * sizeof(mp_word)) - (2 * DIGIT_BIT))))) {
    return fast_mp_montgomery_reduce (x, n, rho);
  }

  /* grow the input as required */
  if (x->alloc < digs) {
    if ((res = mp_grow (x, digs)) != MP_OKAY) {
      return res;
    }
  }
  x->used = digs;

  for (ix = 0; ix < n->used; ix++) {
    /* mu = ai * rho mod b
     *
     * The value of rho must be precalculated via
     * montgomery_setup() such that
     * it equals -1/n0 mod b this allows the
     * following inner loop to reduce the
     * input one digit at a time
     */
    mu = (mp_digit) (((mp_word)x->dp[ix] * (mp_word)rho) & MP_MASK);

    /* a = a + mu * m * b**i */
    {
      int iy;
      mp_digit *tmpn, *tmpx, u;
      mp_word r;

      /* alias for digits of the modulus */
      tmpn = n->dp;

      /* alias for the digits of x [the input] */
      tmpx = x->dp + ix;

      /* set the carry to zero */
      u = 0;

      /* Multiply and add in place */
      for (iy = 0; iy < n->used; iy++) {
        /* compute product and sum */
        r       = ((mp_word)mu * (mp_word)*tmpn++) +
                   (mp_word) u + (mp_word) *tmpx;

        /* get carry */
        u       = (mp_digit)(r >> ((mp_word) DIGIT_BIT));

        /* fix digit */
        *tmpx++ = (mp_digit)(r & ((mp_word) MP_MASK));
      }
      /* At this point the ix'th digit of x should be zero */


      /* propagate carries upwards as required*/
      while (u != 0) {
        *tmpx   += u;
        u        = *tmpx >> DIGIT_BIT;
        *tmpx++ &= MP_MASK;
      }
    }
  }

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  if (mp_cmp_mag (x, n) != MP_LT) {
    return s_mp_sub (x, n, x);
  }

  return MP_OKAY;
}
#endif











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  if (mp_cmp_mag (x, n) != MP_LT) {
    return s_mp_sub (x, n, x);
  }

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_montgomery_setup.c.
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#include <tommath.h>
#ifdef BN_MP_MONTGOMERY_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* setups the montgomery reduction stuff */
int
mp_montgomery_setup (mp_int * n, mp_digit * rho)
{
  mp_digit x, b;
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#include <tommath_private.h>
#ifdef BN_MP_MONTGOMERY_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* setups the montgomery reduction stuff */
int
mp_montgomery_setup (mp_int * n, mp_digit * rho)
{
  mp_digit x, b;
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  b = n->dp[0];

  if ((b & 1) == 0) {
    return MP_VAL;
  }

  x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */
  x *= 2 - b * x;               /* here x*a==1 mod 2**8 */
#if !defined(MP_8BIT)
  x *= 2 - b * x;               /* here x*a==1 mod 2**16 */
#endif
#if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT))
  x *= 2 - b * x;               /* here x*a==1 mod 2**32 */
#endif
#ifdef MP_64BIT
  x *= 2 - b * x;               /* here x*a==1 mod 2**64 */
#endif

  /* rho = -1/m mod b */
  *rho = (unsigned long)(((mp_word)1 << ((mp_word) DIGIT_BIT)) - x) & MP_MASK;

  return MP_OKAY;
}
#endif











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  b = n->dp[0];

  if ((b & 1) == 0) {
    return MP_VAL;
  }

  x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */
  x *= 2 - (b * x);             /* here x*a==1 mod 2**8 */
#if !defined(MP_8BIT)
  x *= 2 - (b * x);             /* here x*a==1 mod 2**16 */
#endif
#if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT))
  x *= 2 - (b * x);             /* here x*a==1 mod 2**32 */
#endif
#ifdef MP_64BIT
  x *= 2 - (b * x);             /* here x*a==1 mod 2**64 */
#endif

  /* rho = -1/m mod b */
  *rho = (mp_digit)(((mp_word)1 << ((mp_word) DIGIT_BIT)) - x) & MP_MASK;

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_mul.c.
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#include <tommath.h>
#ifdef BN_MP_MUL_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* high level multiplication (handles sign) */
int mp_mul (mp_int * a, mp_int * b, mp_int * c)
{
  int     res, neg;
  neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;
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#include <tommath_private.h>
#ifdef BN_MP_MUL_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* high level multiplication (handles sign) */
int mp_mul (mp_int * a, mp_int * b, mp_int * c)
{
  int     res, neg;
  neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;
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     * have less than MP_WARRAY digits and the number of 
     * digits won't affect carry propagation
     */
    int     digs = a->used + b->used + 1;

#ifdef BN_FAST_S_MP_MUL_DIGS_C
    if ((digs < MP_WARRAY) &&
        MIN(a->used, b->used) <= 
        (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) {
      res = fast_s_mp_mul_digs (a, b, c, digs);
    } else 
#endif

#ifdef BN_S_MP_MUL_DIGS_C
      res = s_mp_mul (a, b, c); /* uses s_mp_mul_digs */
#else
      res = MP_VAL;
#endif

  }
  c->sign = (c->used > 0) ? neg : MP_ZPOS;
  return res;
}
#endif











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     * have less than MP_WARRAY digits and the number of 
     * digits won't affect carry propagation
     */
    int     digs = a->used + b->used + 1;

#ifdef BN_FAST_S_MP_MUL_DIGS_C
    if ((digs < MP_WARRAY) &&
        (MIN(a->used, b->used) <= 
         (1 << ((CHAR_BIT * sizeof(mp_word)) - (2 * DIGIT_BIT))))) {
      res = fast_s_mp_mul_digs (a, b, c, digs);
    } else 
#endif
    {
#ifdef BN_S_MP_MUL_DIGS_C
      res = s_mp_mul (a, b, c); /* uses s_mp_mul_digs */
#else
      res = MP_VAL;
#endif
    }
  }
  c->sign = (c->used > 0) ? neg : MP_ZPOS;
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_mul_2.c.
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#include <tommath.h>
#ifdef BN_MP_MUL_2_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* b = a*2 */
int mp_mul_2(mp_int * a, mp_int * b)
{
  int     x, res, oldused;

  /* grow to accomodate result */
  if (b->alloc < a->used + 1) {
    if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) {
      return res;
    }
  }

  oldused = b->used;
  b->used = a->used;

  {
    register mp_digit r, rr, *tmpa, *tmpb;

    /* alias for source */
    tmpa = a->dp;
    
    /* alias for dest */
    tmpb = b->dp;

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#include <tommath_private.h>
#ifdef BN_MP_MUL_2_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* b = a*2 */
int mp_mul_2(mp_int * a, mp_int * b)
{
  int     x, res, oldused;

  /* grow to accomodate result */
  if (b->alloc < (a->used + 1)) {
    if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) {
      return res;
    }
  }

  oldused = b->used;
  b->used = a->used;

  {
    mp_digit r, rr, *tmpa, *tmpb;

    /* alias for source */
    tmpa = a->dp;
    
    /* alias for dest */
    tmpb = b->dp;

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      *tmpb++ = 0;
    }
  }
  b->sign = a->sign;
  return MP_OKAY;
}
#endif











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      *tmpb++ = 0;
    }
  }
  b->sign = a->sign;
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_mul_2d.c.
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#include <tommath.h>
#ifdef BN_MP_MUL_2D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* shift left by a certain bit count */
int mp_mul_2d (const mp_int * a, int b, mp_int * c)
{
  mp_digit d;
  int      res;

  /* copy */
  if (a != c) {
     if ((res = mp_copy (a, c)) != MP_OKAY) {
       return res;
     }
  }

  if (c->alloc < (int)(c->used + b/DIGIT_BIT + 1)) {
     if ((res = mp_grow (c, c->used + b / DIGIT_BIT + 1)) != MP_OKAY) {
       return res;
     }
  }

  /* shift by as many digits in the bit count */
  if (b >= (int)DIGIT_BIT) {
    if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) {
      return res;
    }
  }

  /* shift any bit count < DIGIT_BIT */
  d = (mp_digit) (b % DIGIT_BIT);
  if (d != 0) {
    register mp_digit *tmpc, shift, mask, r, rr;
    register int x;

    /* bitmask for carries */
    mask = (((mp_digit)1) << d) - 1;

    /* shift for msbs */
    shift = DIGIT_BIT - d;

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#include <tommath_private.h>
#ifdef BN_MP_MUL_2D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* shift left by a certain bit count */
int mp_mul_2d (const mp_int * a, int b, mp_int * c)
{
  mp_digit d;
  int      res;

  /* copy */
  if (a != c) {
     if ((res = mp_copy (a, c)) != MP_OKAY) {
       return res;
     }
  }

  if (c->alloc < (int)(c->used + (b / DIGIT_BIT) + 1)) {
     if ((res = mp_grow (c, c->used + (b / DIGIT_BIT) + 1)) != MP_OKAY) {
       return res;
     }
  }

  /* shift by as many digits in the bit count */
  if (b >= (int)DIGIT_BIT) {
    if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) {
      return res;
    }
  }

  /* shift any bit count < DIGIT_BIT */
  d = (mp_digit) (b % DIGIT_BIT);
  if (d != 0) {
    mp_digit *tmpc, shift, mask, r, rr;
    int x;

    /* bitmask for carries */
    mask = (((mp_digit)1) << d) - 1;

    /* shift for msbs */
    shift = DIGIT_BIT - d;

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       c->dp[(c->used)++] = r;
    }
  }
  mp_clamp (c);
  return MP_OKAY;
}
#endif











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       c->dp[(c->used)++] = r;
    }
  }
  mp_clamp (c);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_mul_d.c.
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#include <tommath.h>
#ifdef BN_MP_MUL_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* multiply by a digit */
int
mp_mul_d (mp_int * a, mp_digit b, mp_int * c)
{
  mp_digit u, *tmpa, *tmpc;
  mp_word  r;
  int      ix, res, olduse;

  /* make sure c is big enough to hold a*b */
  if (c->alloc < a->used + 1) {
    if ((res = mp_grow (c, a->used + 1)) != MP_OKAY) {
      return res;
    }
  }

  /* get the original destinations used count */
  olduse = c->used;
|













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|







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#include <tommath_private.h>
#ifdef BN_MP_MUL_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* multiply by a digit */
int
mp_mul_d (mp_int * a, mp_digit b, mp_int * c)
{
  mp_digit u, *tmpa, *tmpc;
  mp_word  r;
  int      ix, res, olduse;

  /* make sure c is big enough to hold a*b */
  if (c->alloc < (a->used + 1)) {
    if ((res = mp_grow (c, a->used + 1)) != MP_OKAY) {
      return res;
    }
  }

  /* get the original destinations used count */
  olduse = c->used;
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  /* zero carry */
  u = 0;

  /* compute columns */
  for (ix = 0; ix < a->used; ix++) {
    /* compute product and carry sum for this term */
    r       = ((mp_word) u) + ((mp_word)*tmpa++) * ((mp_word)b);

    /* mask off higher bits to get a single digit */
    *tmpc++ = (mp_digit) (r & ((mp_word) MP_MASK));

    /* send carry into next iteration */
    u       = (mp_digit) (r >> ((mp_word) DIGIT_BIT));
  }







|







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  /* zero carry */
  u = 0;

  /* compute columns */
  for (ix = 0; ix < a->used; ix++) {
    /* compute product and carry sum for this term */
    r       = (mp_word)u + ((mp_word)*tmpa++ * (mp_word)b);

    /* mask off higher bits to get a single digit */
    *tmpc++ = (mp_digit) (r & ((mp_word) MP_MASK));

    /* send carry into next iteration */
    u       = (mp_digit) (r >> ((mp_word) DIGIT_BIT));
  }
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  /* set used count */
  c->used = a->used + 1;
  mp_clamp(c);

  return MP_OKAY;
}
#endif











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  /* set used count */
  c->used = a->used + 1;
  mp_clamp(c);

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_mulmod.c.
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#include <tommath.h>
#ifdef BN_MP_MULMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* d = a * b (mod c) */
int mp_mulmod (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
{
  int     res;
  mp_int  t;
|













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#include <tommath_private.h>
#ifdef BN_MP_MULMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* d = a * b (mod c) */
int mp_mulmod (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
{
  int     res;
  mp_int  t;
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    return res;
  }
  res = mp_mod (&t, c, d);
  mp_clear (&t);
  return res;
}
#endif











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    return res;
  }
  res = mp_mod (&t, c, d);
  mp_clear (&t);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_n_root.c.
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#include <tommath.h>
#ifdef BN_MP_N_ROOT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* find the n'th root of an integer 
 *
 * Result found such that (c)**b <= a and (c+1)**b > a 
 *
 * This algorithm uses Newton's approximation 
 * x[i+1] = x[i] - f(x[i])/f'(x[i]) 
 * which will find the root in log(N) time where 
 * each step involves a fair bit.  This is not meant to 
 * find huge roots [square and cube, etc].
 */
int mp_n_root (mp_int * a, mp_digit b, mp_int * c)
{
  mp_int  t1, t2, t3;
  int     res, neg;

  /* input must be positive if b is even */
  if ((b & 1) == 0 && a->sign == MP_NEG) {
    return MP_VAL;
  }

  if ((res = mp_init (&t1)) != MP_OKAY) {
    return res;
  }

  if ((res = mp_init (&t2)) != MP_OKAY) {
    goto LBL_T1;
  }

  if ((res = mp_init (&t3)) != MP_OKAY) {
    goto LBL_T2;
  }

  /* if a is negative fudge the sign but keep track */
  neg     = a->sign;
  a->sign = MP_ZPOS;

  /* t2 = 2 */
  mp_set (&t2, 2);

  do {
    /* t1 = t2 */
    if ((res = mp_copy (&t2, &t1)) != MP_OKAY) {
      goto LBL_T3;
    }

    /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */
    
    /* t3 = t1**(b-1) */
    if ((res = mp_expt_d (&t1, b - 1, &t3)) != MP_OKAY) {   
      goto LBL_T3;
    }

    /* numerator */
    /* t2 = t1**b */
    if ((res = mp_mul (&t3, &t1, &t2)) != MP_OKAY) {    
      goto LBL_T3;
    }

    /* t2 = t1**b - a */
    if ((res = mp_sub (&t2, a, &t2)) != MP_OKAY) {  
      goto LBL_T3;
    }

    /* denominator */
    /* t3 = t1**(b-1) * b  */
    if ((res = mp_mul_d (&t3, b, &t3)) != MP_OKAY) {    
      goto LBL_T3;
    }

    /* t3 = (t1**b - a)/(b * t1**(b-1)) */
    if ((res = mp_div (&t2, &t3, &t3, NULL)) != MP_OKAY) {  
      goto LBL_T3;
    }

    if ((res = mp_sub (&t1, &t3, &t2)) != MP_OKAY) {
      goto LBL_T3;
    }
  }  while (mp_cmp (&t1, &t2) != MP_EQ);

  /* result can be off by a few so check */
  for (;;) {
    if ((res = mp_expt_d (&t1, b, &t2)) != MP_OKAY) {
      goto LBL_T3;
    }

    if (mp_cmp (&t2, a) == MP_GT) {
      if ((res = mp_sub_d (&t1, 1, &t1)) != MP_OKAY) {
         goto LBL_T3;
      }
    } else {
      break;
    }
  }

  /* reset the sign of a first */
  a->sign = neg;

  /* set the result */
  mp_exch (&t1, c);

  /* set the sign of the result */
  c->sign = neg;

  res = MP_OKAY;

LBL_T3:mp_clear (&t3);
LBL_T2:mp_clear (&t2);
LBL_T1:mp_clear (&t1);
  return res;
}
#endif
|













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#include <tommath_private.h>
#ifdef BN_MP_N_ROOT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* wrapper function for mp_n_root_ex()

 * computes c = (a)**(1/b) such that (c)**b <= a and (c+1)**b > a






 */
int mp_n_root (mp_int * a, mp_digit b, mp_int * c)
{









  return mp_n_root_ex(a, b, c, 0);
}




#endif



/* $Source$ */



/* $Revision$ */


/* $Date$ */








































































Changes to libtommath/bn_mp_neg.c.
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#include <tommath.h>
#ifdef BN_MP_NEG_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* b = -a */
int mp_neg (const mp_int * a, mp_int * b)
{
  int     res;
  if (a != b) {
|













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#include <tommath_private.h>
#ifdef BN_MP_NEG_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* b = -a */
int mp_neg (const mp_int * a, mp_int * b)
{
  int     res;
  if (a != b) {
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36




  } else {
     b->sign = MP_ZPOS;
  }

  return MP_OKAY;
}
#endif











>
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  } else {
     b->sign = MP_ZPOS;
  }

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_or.c.
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#include <tommath.h>
#ifdef BN_MP_OR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* OR two ints together */
int mp_or (mp_int * a, mp_int * b, mp_int * c)
{
  int     res, ix, px;
  mp_int  t, *x;
|













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#include <tommath_private.h>
#ifdef BN_MP_OR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* OR two ints together */
int mp_or (mp_int * a, mp_int * b, mp_int * c)
{
  int     res, ix, px;
  mp_int  t, *x;
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  }
  mp_clamp (&t);
  mp_exch (c, &t);
  mp_clear (&t);
  return MP_OKAY;
}
#endif











>
>
>
>
40
41
42
43
44
45
46
47
48
49
50
  }
  mp_clamp (&t);
  mp_exch (c, &t);
  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_prime_fermat.c.
1
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18
19
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22
#include <tommath.h>
#ifdef BN_MP_PRIME_FERMAT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* performs one Fermat test.
 * 
 * If "a" were prime then b**a == b (mod a) since the order of
 * the multiplicative sub-group would be phi(a) = a-1.  That means
 * it would be the same as b**(a mod (a-1)) == b**1 == b (mod a).
|













|







1
2
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5
6
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8
9
10
11
12
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14
15
16
17
18
19
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21
22
#include <tommath_private.h>
#ifdef BN_MP_PRIME_FERMAT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* performs one Fermat test.
 * 
 * If "a" were prime then b**a == b (mod a) since the order of
 * the multiplicative sub-group would be phi(a) = a-1.  That means
 * it would be the same as b**(a mod (a-1)) == b**1 == b (mod a).
52
53
54
55
56
57
58




  }

  err = MP_OKAY;
LBL_T:mp_clear (&t);
  return err;
}
#endif











>
>
>
>
52
53
54
55
56
57
58
59
60
61
62
  }

  err = MP_OKAY;
LBL_T:mp_clear (&t);
  return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_prime_is_divisible.c.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
#include <tommath.h>
#ifdef BN_MP_PRIME_IS_DIVISIBLE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* determines if an integers is divisible by one 
 * of the first PRIME_SIZE primes or not
 *
 * sets result to 0 if not, 1 if yes
 */
|













|







1
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3
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5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
#include <tommath_private.h>
#ifdef BN_MP_PRIME_IS_DIVISIBLE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* determines if an integers is divisible by one 
 * of the first PRIME_SIZE primes or not
 *
 * sets result to 0 if not, 1 if yes
 */
40
41
42
43
44
45
46




      return MP_OKAY;
    }
  }

  return MP_OKAY;
}
#endif











>
>
>
>
40
41
42
43
44
45
46
47
48
49
50
      return MP_OKAY;
    }
  }

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_prime_is_prime.c.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
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18
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31
32
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35
36
37
38
39
40
41
#include <tommath.h>
#ifdef BN_MP_PRIME_IS_PRIME_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* performs a variable number of rounds of Miller-Rabin
 *
 * Probability of error after t rounds is no more than

 *
 * Sets result to 1 if probably prime, 0 otherwise
 */
int mp_prime_is_prime (mp_int * a, int t, int *result)
{
  mp_int  b;
  int     ix, err, res;

  /* default to no */
  *result = MP_NO;

  /* valid value of t? */
  if (t <= 0 || t > PRIME_SIZE) {
    return MP_VAL;
  }

  /* is the input equal to one of the primes in the table? */
  for (ix = 0; ix < PRIME_SIZE; ix++) {
      if (mp_cmp_d(a, ltm_prime_tab[ix]) == MP_EQ) {
         *result = 1;
|













|


















|







1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
#include <tommath_private.h>
#ifdef BN_MP_PRIME_IS_PRIME_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* performs a variable number of rounds of Miller-Rabin
 *
 * Probability of error after t rounds is no more than

 *
 * Sets result to 1 if probably prime, 0 otherwise
 */
int mp_prime_is_prime (mp_int * a, int t, int *result)
{
  mp_int  b;
  int     ix, err, res;

  /* default to no */
  *result = MP_NO;

  /* valid value of t? */
  if ((t <= 0) || (t > PRIME_SIZE)) {
    return MP_VAL;
  }

  /* is the input equal to one of the primes in the table? */
  for (ix = 0; ix < PRIME_SIZE; ix++) {
      if (mp_cmp_d(a, ltm_prime_tab[ix]) == MP_EQ) {
         *result = 1;
73
74
75
76
77
78
79





  /* passed the test */
  *result = MP_YES;
LBL_B:mp_clear (&b);
  return err;
}
#endif











>
>
>
>
73
74
75
76
77
78
79
80
81
82
83

  /* passed the test */
  *result = MP_YES;
LBL_B:mp_clear (&b);
  return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_prime_miller_rabin.c.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
#include <tommath.h>
#ifdef BN_MP_PRIME_MILLER_RABIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* Miller-Rabin test of "a" to the base of "b" as described in 
 * HAC pp. 139 Algorithm 4.24
 *
 * Sets result to 0 if definitely composite or 1 if probably prime.
 * Randomly the chance of error is no more than 1/4 and often 
|













|







1
2
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4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
#include <tommath_private.h>
#ifdef BN_MP_PRIME_MILLER_RABIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* Miller-Rabin test of "a" to the base of "b" as described in 
 * HAC pp. 139 Algorithm 4.24
 *
 * Sets result to 0 if definitely composite or 1 if probably prime.
 * Randomly the chance of error is no more than 1/4 and often 
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
    goto LBL_R;
  }
  if ((err = mp_exptmod (b, &r, a, &y)) != MP_OKAY) {
    goto LBL_Y;
  }

  /* if y != 1 and y != n1 do */
  if (mp_cmp_d (&y, 1) != MP_EQ && mp_cmp (&y, &n1) != MP_EQ) {
    j = 1;
    /* while j <= s-1 and y != n1 */
    while ((j <= (s - 1)) && mp_cmp (&y, &n1) != MP_EQ) {
      if ((err = mp_sqrmod (&y, a, &y)) != MP_OKAY) {
         goto LBL_Y;
      }

      /* if y == 1 then composite */
      if (mp_cmp_d (&y, 1) == MP_EQ) {
         goto LBL_Y;







|


|







63
64
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74
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77
78
79
80
    goto LBL_R;
  }
  if ((err = mp_exptmod (b, &r, a, &y)) != MP_OKAY) {
    goto LBL_Y;
  }

  /* if y != 1 and y != n1 do */
  if ((mp_cmp_d (&y, 1) != MP_EQ) && (mp_cmp (&y, &n1) != MP_EQ)) {
    j = 1;
    /* while j <= s-1 and y != n1 */
    while ((j <= (s - 1)) && (mp_cmp (&y, &n1) != MP_EQ)) {
      if ((err = mp_sqrmod (&y, a, &y)) != MP_OKAY) {
         goto LBL_Y;
      }

      /* if y == 1 then composite */
      if (mp_cmp_d (&y, 1) == MP_EQ) {
         goto LBL_Y;
93
94
95
96
97
98
99




  *result = MP_YES;
LBL_Y:mp_clear (&y);
LBL_R:mp_clear (&r);
LBL_N1:mp_clear (&n1);
  return err;
}
#endif











>
>
>
>
93
94
95
96
97
98
99
100
101
102
103
  *result = MP_YES;
LBL_Y:mp_clear (&y);
LBL_R:mp_clear (&r);
LBL_N1:mp_clear (&n1);
  return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_prime_next_prime.c.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
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26
27
28
29
30
31
32
33
34
35
36
37
#include <tommath.h>
#ifdef BN_MP_PRIME_NEXT_PRIME_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* finds the next prime after the number "a" using "t" trials
 * of Miller-Rabin.
 *
 * bbs_style = 1 means the prime must be congruent to 3 mod 4
 */
int mp_prime_next_prime(mp_int *a, int t, int bbs_style)
{
   int      err, res, x, y;
   mp_digit res_tab[PRIME_SIZE], step, kstep;
   mp_int   b;

   /* ensure t is valid */
   if (t <= 0 || t > PRIME_SIZE) {
      return MP_VAL;
   }

   /* force positive */
   a->sign = MP_ZPOS;

   /* simple algo if a is less than the largest prime in the table */
|













|









|




|







1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
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21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
#include <tommath_private.h>
#ifdef BN_MP_PRIME_NEXT_PRIME_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* finds the next prime after the number "a" using "t" trials
 * of Miller-Rabin.
 *
 * bbs_style = 1 means the prime must be congruent to 3 mod 4
 */
int mp_prime_next_prime(mp_int *a, int t, int bbs_style)
{
   int      err, res = MP_NO, x, y;
   mp_digit res_tab[PRIME_SIZE], step, kstep;
   mp_int   b;

   /* ensure t is valid */
   if ((t <= 0) || (t > PRIME_SIZE)) {
      return MP_VAL;
   }

   /* force positive */
   a->sign = MP_ZPOS;

   /* simple algo if a is less than the largest prime in the table */
80
81
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83
84
85
86
87
88
89
90
91
92
93
94

   if (bbs_style == 1) {
      /* if a mod 4 != 3 subtract the correct value to make it so */
      if ((a->dp[0] & 3) != 3) {
         if ((err = mp_sub_d(a, (a->dp[0] & 3) + 1, a)) != MP_OKAY) { return err; };
      }
   } else {
      if (mp_iseven(a) == 1) {
         /* force odd */
         if ((err = mp_sub_d(a, 1, a)) != MP_OKAY) {
            return err;
         }
      }
   }








|







80
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89
90
91
92
93
94

   if (bbs_style == 1) {
      /* if a mod 4 != 3 subtract the correct value to make it so */
      if ((a->dp[0] & 3) != 3) {
         if ((err = mp_sub_d(a, (a->dp[0] & 3) + 1, a)) != MP_OKAY) { return err; };
      }
   } else {
      if (mp_iseven(a) == MP_YES) {
         /* force odd */
         if ((err = mp_sub_d(a, 1, a)) != MP_OKAY) {
            return err;
         }
      }
   }

125
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147
             }

             /* set flag if zero */
             if (res_tab[x] == 0) {
                y = 1;
             }
         }
      } while (y == 1 && step < ((((mp_digit)1)<<DIGIT_BIT) - kstep));

      /* add the step */
      if ((err = mp_add_d(a, step, a)) != MP_OKAY) {
         goto LBL_ERR;
      }

      /* if didn't pass sieve and step == MAX then skip test */
      if (y == 1 && step >= ((((mp_digit)1)<<DIGIT_BIT) - kstep)) {
         continue;
      }

      /* is this prime? */
      for (x = 0; x < t; x++) {
          mp_set(&b, ltm_prime_tab[x]);
          if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {







|







|







125
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139
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143
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             }

             /* set flag if zero */
             if (res_tab[x] == 0) {
                y = 1;
             }
         }
      } while ((y == 1) && (step < ((((mp_digit)1) << DIGIT_BIT) - kstep)));

      /* add the step */
      if ((err = mp_add_d(a, step, a)) != MP_OKAY) {
         goto LBL_ERR;
      }

      /* if didn't pass sieve and step == MAX then skip test */
      if ((y == 1) && (step >= ((((mp_digit)1) << DIGIT_BIT) - kstep))) {
         continue;
      }

      /* is this prime? */
      for (x = 0; x < t; x++) {
          mp_set(&b, ltm_prime_tab[x]);
          if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
160
161
162
163
164
165
166




   err = MP_OKAY;
LBL_ERR:
   mp_clear(&b);
   return err;
}

#endif











>
>
>
>
160
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166
167
168
169
170
   err = MP_OKAY;
LBL_ERR:
   mp_clear(&b);
   return err;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_prime_rabin_miller_trials.c.
1
2
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5
6
7
8
9
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14
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18
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21
22
#include <tommath.h>
#ifdef BN_MP_PRIME_RABIN_MILLER_TRIALS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */


static const struct {
   int k, t;
} sizes[] = {
{   128,    28 },
|













|







1
2
3
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5
6
7
8
9
10
11
12
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14
15
16
17
18
19
20
21
22
#include <tommath_private.h>
#ifdef BN_MP_PRIME_RABIN_MILLER_TRIALS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */


static const struct {
   int k, t;
} sizes[] = {
{   128,    28 },
42
43
44
45
46
47
48




       }
   }
   return sizes[x-1].t + 1;
}


#endif











>
>
>
>
42
43
44
45
46
47
48
49
50
51
52
       }
   }
   return sizes[x-1].t + 1;
}


#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_prime_random_ex.c.
1
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3
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5
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7
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77
#include <tommath.h>
#ifdef BN_MP_PRIME_RANDOM_EX_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* makes a truly random prime of a given size (bits),
 *
 * Flags are as follows:
 * 
 *   LTM_PRIME_BBS      - make prime congruent to 3 mod 4
 *   LTM_PRIME_SAFE     - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS)
 *   LTM_PRIME_2MSB_OFF - make the 2nd highest bit zero
 *   LTM_PRIME_2MSB_ON  - make the 2nd highest bit one
 *
 * You have to supply a callback which fills in a buffer with random bytes.  "dat" is a parameter you can
 * have passed to the callback (e.g. a state or something).  This function doesn't use "dat" itself
 * so it can be NULL
 *
 */

/* This is possibly the mother of all prime generation functions, muahahahahaha! */
int mp_prime_random_ex(mp_int *a, int t, int size, int flags, ltm_prime_callback cb, void *dat)
{
   unsigned char *tmp, maskAND, maskOR_msb, maskOR_lsb;
   int res, err, bsize, maskOR_msb_offset;

   /* sanity check the input */
   if (size <= 1 || t <= 0) {
      return MP_VAL;
   }

   /* LTM_PRIME_SAFE implies LTM_PRIME_BBS */
   if (flags & LTM_PRIME_SAFE) {
      flags |= LTM_PRIME_BBS;
   }

   /* calc the byte size */
   bsize = (size>>3) + ((size&7)?1:0);

   /* we need a buffer of bsize bytes */
   tmp = OPT_CAST(unsigned char) XMALLOC(bsize);
   if (tmp == NULL) {
      return MP_MEM;
   }

   /* calc the maskAND value for the MSbyte*/
   maskAND = ((size&7) == 0) ? 0xFF : (0xFF >> (8 - (size & 7)));

   /* calc the maskOR_msb */
   maskOR_msb        = 0;
   maskOR_msb_offset = ((size & 7) == 1) ? 1 : 0;
   if (flags & LTM_PRIME_2MSB_ON) {
      maskOR_msb       |= 0x80 >> ((9 - size) & 7);
   }  

   /* get the maskOR_lsb */
   maskOR_lsb         = 1;
   if (flags & LTM_PRIME_BBS) {
      maskOR_lsb     |= 3;
   }

   do {
      /* read the bytes */
      if (cb(tmp, bsize, dat) != bsize) {
         err = MP_VAL;
|













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<















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#include <tommath_private.h>
#ifdef BN_MP_PRIME_RANDOM_EX_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* makes a truly random prime of a given size (bits),
 *
 * Flags are as follows:
 * 
 *   LTM_PRIME_BBS      - make prime congruent to 3 mod 4
 *   LTM_PRIME_SAFE     - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS)

 *   LTM_PRIME_2MSB_ON  - make the 2nd highest bit one
 *
 * You have to supply a callback which fills in a buffer with random bytes.  "dat" is a parameter you can
 * have passed to the callback (e.g. a state or something).  This function doesn't use "dat" itself
 * so it can be NULL
 *
 */

/* This is possibly the mother of all prime generation functions, muahahahahaha! */
int mp_prime_random_ex(mp_int *a, int t, int size, int flags, ltm_prime_callback cb, void *dat)
{
   unsigned char *tmp, maskAND, maskOR_msb, maskOR_lsb;
   int res, err, bsize, maskOR_msb_offset;

   /* sanity check the input */
   if ((size <= 1) || (t <= 0)) {
      return MP_VAL;
   }

   /* LTM_PRIME_SAFE implies LTM_PRIME_BBS */
   if ((flags & LTM_PRIME_SAFE) != 0) {
      flags |= LTM_PRIME_BBS;
   }

   /* calc the byte size */
   bsize = (size>>3) + ((size&7)?1:0);

   /* we need a buffer of bsize bytes */
   tmp = OPT_CAST(unsigned char) XMALLOC(bsize);
   if (tmp == NULL) {
      return MP_MEM;
   }

   /* calc the maskAND value for the MSbyte*/
   maskAND = ((size&7) == 0) ? 0xFF : (0xFF >> (8 - (size & 7)));

   /* calc the maskOR_msb */
   maskOR_msb        = 0;
   maskOR_msb_offset = ((size & 7) == 1) ? 1 : 0;
   if ((flags & LTM_PRIME_2MSB_ON) != 0) {
      maskOR_msb       |= 0x80 >> ((9 - size) & 7);
   }  

   /* get the maskOR_lsb */
   maskOR_lsb         = 1;
   if ((flags & LTM_PRIME_BBS) != 0) {
      maskOR_lsb     |= 3;
   }

   do {
      /* read the bytes */
      if (cb(tmp, bsize, dat) != bsize) {
         err = MP_VAL;
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      /* is it prime? */
      if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY)           { goto error; }
      if (res == MP_NO) {  
         continue;
      }

      if (flags & LTM_PRIME_SAFE) {
         /* see if (a-1)/2 is prime */
         if ((err = mp_sub_d(a, 1, a)) != MP_OKAY)                    { goto error; }
         if ((err = mp_div_2(a, a)) != MP_OKAY)                       { goto error; }
 
         /* is it prime? */
         if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY)        { goto error; }
      }
   } while (res == MP_NO);

   if (flags & LTM_PRIME_SAFE) {
      /* restore a to the original value */
      if ((err = mp_mul_2(a, a)) != MP_OKAY)                          { goto error; }
      if ((err = mp_add_d(a, 1, a)) != MP_OKAY)                       { goto error; }
   }

   err = MP_OKAY;
error:
   XFREE(tmp);
   return err;
}


#endif











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      /* is it prime? */
      if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY)           { goto error; }
      if (res == MP_NO) {  
         continue;
      }

      if ((flags & LTM_PRIME_SAFE) != 0) {
         /* see if (a-1)/2 is prime */
         if ((err = mp_sub_d(a, 1, a)) != MP_OKAY)                    { goto error; }
         if ((err = mp_div_2(a, a)) != MP_OKAY)                       { goto error; }
 
         /* is it prime? */
         if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY)        { goto error; }
      }
   } while (res == MP_NO);

   if ((flags & LTM_PRIME_SAFE) != 0) {
      /* restore a to the original value */
      if ((err = mp_mul_2(a, a)) != MP_OKAY)                          { goto error; }
      if ((err = mp_add_d(a, 1, a)) != MP_OKAY)                       { goto error; }
   }

   err = MP_OKAY;
error:
   XFREE(tmp);
   return err;
}


#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_radix_size.c.
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#include <tommath.h>
#ifdef BN_MP_RADIX_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* returns size of ASCII reprensentation */
int mp_radix_size (mp_int * a, int radix, int *size)
{
  int     res, digs;
  mp_int  t;
  mp_digit d;

  *size = 0;

  /* special case for binary */
  if (radix == 2) {
    *size = mp_count_bits (a) + (a->sign == MP_NEG ? 1 : 0) + 1;
    return MP_OKAY;
  }

  /* make sure the radix is in range */
  if (radix < 2 || radix > 64) {
    return MP_VAL;
  }

  if (mp_iszero(a) == MP_YES) {
    *size = 2;
    return MP_OKAY;
  }







  /* digs is the digit count */
  digs = 0;

  /* if it's negative add one for the sign */
  if (a->sign == MP_NEG) {
    ++digs;
|













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#include <tommath_private.h>
#ifdef BN_MP_RADIX_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* returns size of ASCII reprensentation */
int mp_radix_size (mp_int * a, int radix, int *size)
{
  int     res, digs;
  mp_int  t;
  mp_digit d;

  *size = 0;







  /* make sure the radix is in range */
  if ((radix < 2) || (radix > 64)) {
    return MP_VAL;
  }

  if (mp_iszero(a) == MP_YES) {
    *size = 2;
    return MP_OKAY;
  }

  /* special case for binary */
  if (radix == 2) {
    *size = mp_count_bits (a) + ((a->sign == MP_NEG) ? 1 : 0) + 1;
    return MP_OKAY;
  }

  /* digs is the digit count */
  digs = 0;

  /* if it's negative add one for the sign */
  if (a->sign == MP_NEG) {
    ++digs;
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  } else {
      *size = 3;
  }
  return MP_OKAY;
}

#endif











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  } else {
      *size = 3;
  }
  return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_radix_smap.c.
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#include <tommath.h>
#ifdef BN_MP_RADIX_SMAP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* chars used in radix conversions */
const char *mp_s_rmap = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/";
#endif




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#include <tommath_private.h>
#ifdef BN_MP_RADIX_SMAP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* chars used in radix conversions */
const char *mp_s_rmap = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/";
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_rand.c.
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#include <tommath.h>
#ifdef BN_MP_RAND_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* makes a pseudo-random int of a given size */
int
mp_rand (mp_int * a, int digits)
{
  int     res;
  mp_digit d;

  mp_zero (a);
  if (digits <= 0) {
    return MP_OKAY;
  }

  /* first place a random non-zero digit */
  do {
    d = ((mp_digit) abs (rand ())) & MP_MASK;
  } while (d == 0);

  if ((res = mp_add_d (a, d, a)) != MP_OKAY) {
    return res;
  }

  while (--digits > 0) {
    if ((res = mp_lshd (a, 1)) != MP_OKAY) {
      return res;
    }

    if ((res = mp_add_d (a, ((mp_digit) abs (rand ())), a)) != MP_OKAY) {
      return res;
    }
  }

  return MP_OKAY;
}
#endif




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#include <tommath_private.h>
#ifdef BN_MP_RAND_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* makes a pseudo-random int of a given size */
int
mp_rand (mp_int * a, int digits)
{
  int     res;
  mp_digit d;

  mp_zero (a);
  if (digits <= 0) {
    return MP_OKAY;
  }

  /* first place a random non-zero digit */
  do {
    d = ((mp_digit) abs (MP_GEN_RANDOM())) & MP_MASK;
  } while (d == 0);

  if ((res = mp_add_d (a, d, a)) != MP_OKAY) {
    return res;
  }

  while (--digits > 0) {
    if ((res = mp_lshd (a, 1)) != MP_OKAY) {
      return res;
    }

    if ((res = mp_add_d (a, ((mp_digit) abs (MP_GEN_RANDOM())), a)) != MP_OKAY) {
      return res;
    }
  }

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_read_radix.c.
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#include <tommath.h>
#ifdef BN_MP_READ_RADIX_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* read a string [ASCII] in a given radix */
int mp_read_radix (mp_int * a, const char *str, int radix)
{
  int     y, res, neg;
  char    ch;

  /* zero the digit bignum */
  mp_zero(a);

  /* make sure the radix is ok */
  if (radix < 2 || radix > 64) {
    return MP_VAL;
  }

  /* if the leading digit is a 
   * minus set the sign to negative. 
   */
  if (*str == '-') {
    ++str;
    neg = MP_NEG;
  } else {
    neg = MP_ZPOS;
  }

  /* set the integer to the default of zero */
  mp_zero (a);
  
  /* process each digit of the string */
  while (*str) {
    /* if the radix < 36 the conversion is case insensitive
     * this allows numbers like 1AB and 1ab to represent the same  value
     * [e.g. in hex]
     */
    ch = (char) ((radix < 36) ? toupper ((unsigned char) *str) : *str);
    for (y = 0; y < 64; y++) {
      if (ch == mp_s_rmap[y]) {
         break;
      }
    }

    /* if the char was found in the map 
|













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#include <tommath_private.h>
#ifdef BN_MP_READ_RADIX_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* read a string [ASCII] in a given radix */
int mp_read_radix (mp_int * a, const char *str, int radix)
{
  int     y, res, neg;
  char    ch;

  /* zero the digit bignum */
  mp_zero(a);

  /* make sure the radix is ok */
  if ((radix < 2) || (radix > 64)) {
    return MP_VAL;
  }

  /* if the leading digit is a 
   * minus set the sign to negative. 
   */
  if (*str == '-') {
    ++str;
    neg = MP_NEG;
  } else {
    neg = MP_ZPOS;
  }

  /* set the integer to the default of zero */
  mp_zero (a);
  
  /* process each digit of the string */
  while (*str != '\0') {
    /* if the radix <= 36 the conversion is case insensitive
     * this allows numbers like 1AB and 1ab to represent the same  value
     * [e.g. in hex]
     */
    ch = (radix <= 36) ? (char)toupper((unsigned char)*str) : *str;
    for (y = 0; y < 64; y++) {
      if (ch == mp_s_rmap[y]) {
         break;
      }
    }

    /* if the char was found in the map 
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  if ( *str != '\0' ) {
      mp_zero( a );
      return MP_VAL;
  }

  /* set the sign only if a != 0 */
  if (mp_iszero(a) != 1) {
     a->sign = neg;
  }
  return MP_OKAY;
}
#endif











|





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  if ( *str != '\0' ) {
      mp_zero( a );
      return MP_VAL;
  }

  /* set the sign only if a != 0 */
  if (mp_iszero(a) != MP_YES) {
     a->sign = neg;
  }
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_read_signed_bin.c.
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#include <tommath.h>
#ifdef BN_MP_READ_SIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* read signed bin, big endian, first byte is 0==positive or 1==negative */
int mp_read_signed_bin (mp_int * a, const unsigned char *b, int c)
{
  int     res;

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#include <tommath_private.h>
#ifdef BN_MP_READ_SIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* read signed bin, big endian, first byte is 0==positive or 1==negative */
int mp_read_signed_bin (mp_int * a, const unsigned char *b, int c)
{
  int     res;

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  } else {
     a->sign = MP_NEG;
  }

  return MP_OKAY;
}
#endif











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41
  } else {
     a->sign = MP_NEG;
  }

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_read_unsigned_bin.c.
1
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4
5
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7
8
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10
11
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13
14
15
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#include <tommath.h>
#ifdef BN_MP_READ_UNSIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* reads a unsigned char array, assumes the msb is stored first [big endian] */
int mp_read_unsigned_bin (mp_int * a, const unsigned char *b, int c)
{
  int     res;

|













|







1
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13
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16
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20
21
22
#include <tommath_private.h>
#ifdef BN_MP_READ_UNSIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* reads a unsigned char array, assumes the msb is stored first [big endian] */
int mp_read_unsigned_bin (mp_int * a, const unsigned char *b, int c)
{
  int     res;

33
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42
43
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45
46
47
48
49
50
51




  /* read the bytes in */
  while (c-- > 0) {
    if ((res = mp_mul_2d (a, 8, a)) != MP_OKAY) {
      return res;
    }

#ifndef MP_8BIT
      a->dp[0] |= *b++;
      a->used += 1;
#else
      a->dp[0] = (*b & MP_MASK);
      a->dp[1] |= ((*b++ >> 7U) & 1);
      a->used += 2;
#endif
  }
  mp_clamp (a);
  return MP_OKAY;
}
#endif











|
|

|
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|






>
>
>
>
33
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54
55
  /* read the bytes in */
  while (c-- > 0) {
    if ((res = mp_mul_2d (a, 8, a)) != MP_OKAY) {
      return res;
    }

#ifndef MP_8BIT
    a->dp[0] |= *b++;
    a->used += 1;
#else
    a->dp[0] = (*b & MP_MASK);
    a->dp[1] |= ((*b++ >> 7U) & 1);
    a->used += 2;
#endif
  }
  mp_clamp (a);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_reduce.c.
1
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5
6
7
8
9
10
11
12
13
14
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58
59
60
61
62
63
64
65
#include <tommath.h>
#ifdef BN_MP_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* reduces x mod m, assumes 0 < x < m**2, mu is 
 * precomputed via mp_reduce_setup.
 * From HAC pp.604 Algorithm 14.42
 */
int mp_reduce (mp_int * x, mp_int * m, mp_int * mu)
{
  mp_int  q;
  int     res, um = m->used;

  /* q = x */
  if ((res = mp_init_copy (&q, x)) != MP_OKAY) {
    return res;
  }

  /* q1 = x / b**(k-1)  */
  mp_rshd (&q, um - 1);         

  /* according to HAC this optimization is ok */
  if (((unsigned long) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) {
    if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) {
      goto CLEANUP;
    }
  } else {
#ifdef BN_S_MP_MUL_HIGH_DIGS_C
    if ((res = s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) {
      goto CLEANUP;
    }
#elif defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C)
    if ((res = fast_s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) {
      goto CLEANUP;
    }
#else 
    { 
      res = MP_VAL;
      goto CLEANUP;
    }
#endif
  }

  /* q3 = q2 / b**(k+1) */
  mp_rshd (&q, um + 1);         

  /* x = x mod b**(k+1), quick (no division) */
  if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) {
    goto CLEANUP;
  }

  /* q = q * m mod b**(k+1), quick (no division) */
|













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|







1
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65
#include <tommath_private.h>
#ifdef BN_MP_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* reduces x mod m, assumes 0 < x < m**2, mu is
 * precomputed via mp_reduce_setup.
 * From HAC pp.604 Algorithm 14.42
 */
int mp_reduce (mp_int * x, mp_int * m, mp_int * mu)
{
  mp_int  q;
  int     res, um = m->used;

  /* q = x */
  if ((res = mp_init_copy (&q, x)) != MP_OKAY) {
    return res;
  }

  /* q1 = x / b**(k-1)  */
  mp_rshd (&q, um - 1);

  /* according to HAC this optimization is ok */
  if (((mp_digit) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) {
    if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) {
      goto CLEANUP;
    }
  } else {
#ifdef BN_S_MP_MUL_HIGH_DIGS_C
    if ((res = s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) {
      goto CLEANUP;
    }
#elif defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C)
    if ((res = fast_s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) {
      goto CLEANUP;
    }
#else
    {
      res = MP_VAL;
      goto CLEANUP;
    }
#endif
  }

  /* q3 = q2 / b**(k+1) */
  mp_rshd (&q, um + 1);

  /* x = x mod b**(k+1), quick (no division) */
  if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) {
    goto CLEANUP;
  }

  /* q = q * m mod b**(k+1), quick (no division) */
83
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96





  /* Back off if it's too big */
  while (mp_cmp (x, m) != MP_LT) {
    if ((res = s_mp_sub (x, m, x)) != MP_OKAY) {
      goto CLEANUP;
    }
  }
  
CLEANUP:
  mp_clear (&q);

  return res;
}
#endif











|






>
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>
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100

  /* Back off if it's too big */
  while (mp_cmp (x, m) != MP_LT) {
    if ((res = s_mp_sub (x, m, x)) != MP_OKAY) {
      goto CLEANUP;
    }
  }

CLEANUP:
  mp_clear (&q);

  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_reduce_2k.c.
1
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3
4
5
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10
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49
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#include <tommath.h>
#ifdef BN_MP_REDUCE_2K_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* reduces a modulo n where n is of the form 2**p - d */
int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d)
{
   mp_int q;
   int    p, res;
   
   if ((res = mp_init(&q)) != MP_OKAY) {
      return res;
   }
   
   p = mp_count_bits(n);    
top:
   /* q = a/2**p, a = a mod 2**p */
   if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) {
      goto ERR;
   }
   
   if (d != 1) {
      /* q = q * d */
      if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) { 
         goto ERR;
      }
   }
   
   /* a = a + q */
   if ((res = s_mp_add(a, &q, a)) != MP_OKAY) {
      goto ERR;
   }
   
   if (mp_cmp_mag(a, n) != MP_LT) {
      s_mp_sub(a, n, a);


      goto top;
   }
   
ERR:
   mp_clear(&q);
   return res;
}

#endif




|













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>
>


|






>
>
>
>
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51
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55
56
57
58
59
60
61
62
63
#include <tommath_private.h>
#ifdef BN_MP_REDUCE_2K_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* reduces a modulo n where n is of the form 2**p - d */
int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d)
{
   mp_int q;
   int    p, res;

   if ((res = mp_init(&q)) != MP_OKAY) {
      return res;
   }

   p = mp_count_bits(n);
top:
   /* q = a/2**p, a = a mod 2**p */
   if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) {
      goto ERR;
   }

   if (d != 1) {
      /* q = q * d */
      if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) {
         goto ERR;
      }
   }

   /* a = a + q */
   if ((res = s_mp_add(a, &q, a)) != MP_OKAY) {
      goto ERR;
   }

   if (mp_cmp_mag(a, n) != MP_LT) {
      if ((res = s_mp_sub(a, n, a)) != MP_OKAY) {
         goto ERR;
      }
      goto top;
   }

ERR:
   mp_clear(&q);
   return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_reduce_2k_l.c.
1
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3
4
5
6
7
8
9
10
11
12
13
14
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48
49


50
51
52
53
54
55
56
57
58




#include <tommath.h>
#ifdef BN_MP_REDUCE_2K_L_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* reduces a modulo n where n is of the form 2**p - d 
   This differs from reduce_2k since "d" can be larger
   than a single digit.
*/
int mp_reduce_2k_l(mp_int *a, mp_int *n, mp_int *d)
{
   mp_int q;
   int    p, res;
   
   if ((res = mp_init(&q)) != MP_OKAY) {
      return res;
   }
   
   p = mp_count_bits(n);    
top:
   /* q = a/2**p, a = a mod 2**p */
   if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) {
      goto ERR;
   }
   
   /* q = q * d */
   if ((res = mp_mul(&q, d, &q)) != MP_OKAY) { 
      goto ERR;
   }
   
   /* a = a + q */
   if ((res = s_mp_add(a, &q, a)) != MP_OKAY) {
      goto ERR;
   }
   
   if (mp_cmp_mag(a, n) != MP_LT) {
      s_mp_sub(a, n, a);


      goto top;
   }
   
ERR:
   mp_clear(&q);
   return res;
}

#endif




|













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>
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>
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61
62
63
64
#include <tommath_private.h>
#ifdef BN_MP_REDUCE_2K_L_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* reduces a modulo n where n is of the form 2**p - d
   This differs from reduce_2k since "d" can be larger
   than a single digit.
*/
int mp_reduce_2k_l(mp_int *a, mp_int *n, mp_int *d)
{
   mp_int q;
   int    p, res;

   if ((res = mp_init(&q)) != MP_OKAY) {
      return res;
   }

   p = mp_count_bits(n);
top:
   /* q = a/2**p, a = a mod 2**p */
   if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) {
      goto ERR;
   }

   /* q = q * d */
   if ((res = mp_mul(&q, d, &q)) != MP_OKAY) {
      goto ERR;
   }

   /* a = a + q */
   if ((res = s_mp_add(a, &q, a)) != MP_OKAY) {
      goto ERR;
   }

   if (mp_cmp_mag(a, n) != MP_LT) {
      if ((res = s_mp_sub(a, n, a)) != MP_OKAY) {
         goto ERR;
      }
      goto top;
   }

ERR:
   mp_clear(&q);
   return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_reduce_2k_setup.c.
1
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#include <tommath.h>
#ifdef BN_MP_REDUCE_2K_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* determines the setup value */
int mp_reduce_2k_setup(mp_int *a, mp_digit *d)
{
   int res, p;
   mp_int tmp;
|













|







1
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13
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20
21
22
#include <tommath_private.h>
#ifdef BN_MP_REDUCE_2K_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* determines the setup value */
int mp_reduce_2k_setup(mp_int *a, mp_digit *d)
{
   int res, p;
   mp_int tmp;
37
38
39
40
41
42
43




   }
   
   *d = tmp.dp[0];
   mp_clear(&tmp);
   return MP_OKAY;
}
#endif











>
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37
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44
45
46
47
   }
   
   *d = tmp.dp[0];
   mp_clear(&tmp);
   return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_reduce_2k_setup_l.c.
1
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#include <tommath.h>
#ifdef BN_MP_REDUCE_2K_SETUP_L_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* determines the setup value */
int mp_reduce_2k_setup_l(mp_int *a, mp_int *d)
{
   int    res;
   mp_int tmp;
|













|







1
2
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5
6
7
8
9
10
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13
14
15
16
17
18
19
20
21
22
#include <tommath_private.h>
#ifdef BN_MP_REDUCE_2K_SETUP_L_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* determines the setup value */
int mp_reduce_2k_setup_l(mp_int *a, mp_int *d)
{
   int    res;
   mp_int tmp;
34
35
36
37
38
39
40




   }
   
ERR:
   mp_clear(&tmp);
   return res;
}
#endif











>
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>
>
34
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36
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43
44
   }
   
ERR:
   mp_clear(&tmp);
   return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_reduce_is_2k.c.
1
2
3
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5
6
7
8
9
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15
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19
20
21
22
#include <tommath.h>
#ifdef BN_MP_REDUCE_IS_2K_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* determines if mp_reduce_2k can be used */
int mp_reduce_is_2k(mp_int *a)
{
   int ix, iy, iw;
   mp_digit iz;
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#include <tommath_private.h>
#ifdef BN_MP_REDUCE_IS_2K_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* determines if mp_reduce_2k can be used */
int mp_reduce_is_2k(mp_int *a)
{
   int ix, iy, iw;
   mp_digit iz;
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          }
      }
   }
   return MP_YES;
}

#endif











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          }
      }
   }
   return MP_YES;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_reduce_is_2k_l.c.
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#include <tommath.h>
#ifdef BN_MP_REDUCE_IS_2K_L_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* determines if reduce_2k_l can be used */
int mp_reduce_is_2k_l(mp_int *a)
{
   int ix, iy;
   
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#include <tommath_private.h>
#ifdef BN_MP_REDUCE_IS_2K_L_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* determines if reduce_2k_l can be used */
int mp_reduce_is_2k_l(mp_int *a)
{
   int ix, iy;
   
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      return (iy >= (a->used/2)) ? MP_YES : MP_NO;
      
   }
   return MP_NO;
}

#endif











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      return (iy >= (a->used/2)) ? MP_YES : MP_NO;
      
   }
   return MP_NO;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_reduce_setup.c.
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#include <tommath.h>
#ifdef BN_MP_REDUCE_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* pre-calculate the value required for Barrett reduction
 * For a given modulus "b" it calulates the value required in "a"
 */
int mp_reduce_setup (mp_int * a, mp_int * b)
{
  int     res;
  
  if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) {
    return res;
  }
  return mp_div (a, b, a, NULL);
}
#endif




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#include <tommath_private.h>
#ifdef BN_MP_REDUCE_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* pre-calculate the value required for Barrett reduction
 * For a given modulus "b" it calulates the value required in "a"
 */
int mp_reduce_setup (mp_int * a, mp_int * b)
{
  int     res;
  
  if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) {
    return res;
  }
  return mp_div (a, b, a, NULL);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_rshd.c.
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#include <tommath.h>
#ifdef BN_MP_RSHD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* shift right a certain amount of digits */
void mp_rshd (mp_int * a, int b)
{
  int     x;

  /* if b <= 0 then ignore it */
  if (b <= 0) {
    return;
  }

  /* if b > used then simply zero it and return */
  if (a->used <= b) {
    mp_zero (a);
    return;
  }

  {
    register mp_digit *bottom, *top;

    /* shift the digits down */

    /* bottom */
    bottom = a->dp;

    /* top [offset into digits] */
|













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#include <tommath_private.h>
#ifdef BN_MP_RSHD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* shift right a certain amount of digits */
void mp_rshd (mp_int * a, int b)
{
  int     x;

  /* if b <= 0 then ignore it */
  if (b <= 0) {
    return;
  }

  /* if b > used then simply zero it and return */
  if (a->used <= b) {
    mp_zero (a);
    return;
  }

  {
    mp_digit *bottom, *top;

    /* shift the digits down */

    /* bottom */
    bottom = a->dp;

    /* top [offset into digits] */
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    }
  }
  
  /* remove excess digits */
  a->used -= b;
}
#endif











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    }
  }
  
  /* remove excess digits */
  a->used -= b;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_set.c.
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#include <tommath.h>
#ifdef BN_MP_SET_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* set to a digit */
void mp_set (mp_int * a, mp_digit b)
{
  mp_zero (a);
  a->dp[0] = b & MP_MASK;
  a->used  = (a->dp[0] != 0) ? 1 : 0;
}
#endif




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#include <tommath_private.h>
#ifdef BN_MP_SET_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* set to a digit */
void mp_set (mp_int * a, mp_digit b)
{
  mp_zero (a);
  a->dp[0] = b & MP_MASK;
  a->used  = (a->dp[0] != 0) ? 1 : 0;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_set_int.c.
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#include <tommath.h>
#ifdef BN_MP_SET_INT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* set a 32-bit const */
int mp_set_int (mp_int * a, unsigned long b)
{
  int     x, res;

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#include <tommath_private.h>
#ifdef BN_MP_SET_INT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* set a 32-bit const */
int mp_set_int (mp_int * a, unsigned long b)
{
  int     x, res;

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    /* ensure that digits are not clamped off */
    a->used += 1;
  }
  mp_clamp (a);
  return MP_OKAY;
}
#endif











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    /* ensure that digits are not clamped off */
    a->used += 1;
  }
  mp_clamp (a);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_shrink.c.
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#include <tommath.h>
#ifdef BN_MP_SHRINK_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* shrink a bignum */
int mp_shrink (mp_int * a)
{
  mp_digit *tmp;
  int used = 1;
  
  if(a->used > 0)
    used = a->used;

  
  if (a->alloc != used) {
    if ((tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * used)) == NULL) {
      return MP_MEM;
    }
    a->dp    = tmp;
    a->alloc = used;
  }
  return MP_OKAY;
}
#endif




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#include <tommath_private.h>
#ifdef BN_MP_SHRINK_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* shrink a bignum */
int mp_shrink (mp_int * a)
{
  mp_digit *tmp;
  int used = 1;
  
  if(a->used > 0) {
    used = a->used;
  }
  
  if (a->alloc != used) {
    if ((tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * used)) == NULL) {
      return MP_MEM;
    }
    a->dp    = tmp;
    a->alloc = used;
  }
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_signed_bin_size.c.
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#include <tommath.h>
#ifdef BN_MP_SIGNED_BIN_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* get the size for an signed equivalent */
int mp_signed_bin_size (mp_int * a)
{
  return 1 + mp_unsigned_bin_size (a);
}
#endif




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#include <tommath_private.h>
#ifdef BN_MP_SIGNED_BIN_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* get the size for an signed equivalent */
int mp_signed_bin_size (mp_int * a)
{
  return 1 + mp_unsigned_bin_size (a);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_sqr.c.
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#include <tommath.h>
#ifdef BN_MP_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* computes b = a*a */
int
mp_sqr (mp_int * a, mp_int * b)
{
  int     res;

#ifdef BN_MP_TOOM_SQR_C
  /* use Toom-Cook? */
  if (a->used >= TOOM_SQR_CUTOFF) {
    res = mp_toom_sqr(a, b);
  /* Karatsuba? */
  } else 
#endif
#ifdef BN_MP_KARATSUBA_SQR_C
if (a->used >= KARATSUBA_SQR_CUTOFF) {
    res = mp_karatsuba_sqr (a, b);
  } else 
#endif
  {
#ifdef BN_FAST_S_MP_SQR_C
    /* can we use the fast comba multiplier? */
    if ((a->used * 2 + 1) < MP_WARRAY && 
         a->used < 
         (1 << (sizeof(mp_word) * CHAR_BIT - 2*DIGIT_BIT - 1))) {
      res = fast_s_mp_sqr (a, b);
    } else
#endif

#ifdef BN_S_MP_SQR_C
      res = s_mp_sqr (a, b);
#else
      res = MP_VAL;
#endif

  }
  b->sign = MP_ZPOS;
  return res;
}
#endif




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#include <tommath_private.h>
#ifdef BN_MP_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* computes b = a*a */
int
mp_sqr (mp_int * a, mp_int * b)
{
  int     res;

#ifdef BN_MP_TOOM_SQR_C
  /* use Toom-Cook? */
  if (a->used >= TOOM_SQR_CUTOFF) {
    res = mp_toom_sqr(a, b);
  /* Karatsuba? */
  } else 
#endif
#ifdef BN_MP_KARATSUBA_SQR_C
  if (a->used >= KARATSUBA_SQR_CUTOFF) {
    res = mp_karatsuba_sqr (a, b);
  } else 
#endif
  {
#ifdef BN_FAST_S_MP_SQR_C
    /* can we use the fast comba multiplier? */
    if ((((a->used * 2) + 1) < MP_WARRAY) &&
         (a->used <
         (1 << (((sizeof(mp_word) * CHAR_BIT) - (2 * DIGIT_BIT)) - 1)))) {
      res = fast_s_mp_sqr (a, b);
    } else
#endif
    {
#ifdef BN_S_MP_SQR_C
      res = s_mp_sqr (a, b);
#else
      res = MP_VAL;
#endif
    }
  }
  b->sign = MP_ZPOS;
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_sqrmod.c.
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#include <tommath.h>
#ifdef BN_MP_SQRMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* c = a * a (mod b) */
int
mp_sqrmod (mp_int * a, mp_int * b, mp_int * c)
{
  int     res;
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#include <tommath_private.h>
#ifdef BN_MP_SQRMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* c = a * a (mod b) */
int
mp_sqrmod (mp_int * a, mp_int * b, mp_int * c)
{
  int     res;
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    return res;
  }
  res = mp_mod (&t, b, c);
  mp_clear (&t);
  return res;
}
#endif











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    return res;
  }
  res = mp_mod (&t, b, c);
  mp_clear (&t);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_sqrt.c.
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#include <tommath.h>

#ifdef BN_MP_SQRT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

#ifndef NO_FLOATING_POINT
#include <math.h>
#endif

/* this function is less generic than mp_n_root, simpler and faster */
|














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#include <tommath_private.h>

#ifdef BN_MP_SQRT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

#ifndef NO_FLOATING_POINT
#include <math.h>
#endif

/* this function is less generic than mp_n_root, simpler and faster */
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E1: mp_clear(&t2);
E2: mp_clear(&t1);
  return res;
}

#endif











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E1: mp_clear(&t2);
E2: mp_clear(&t1);
  return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_sub.c.
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#include <tommath.h>
#ifdef BN_MP_SUB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* high level subtraction (handles signs) */
int
mp_sub (mp_int * a, mp_int * b, mp_int * c)
{
  int     sa, sb, res;
|













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#include <tommath_private.h>
#ifdef BN_MP_SUB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* high level subtraction (handles signs) */
int
mp_sub (mp_int * a, mp_int * b, mp_int * c)
{
  int     sa, sb, res;
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      res = s_mp_sub (b, a, c);
    }
  }
  return res;
}

#endif











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      res = s_mp_sub (b, a, c);
    }
  }
  return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_sub_d.c.
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#include <tommath.h>
#ifdef BN_MP_SUB_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* single digit subtraction */
int
mp_sub_d (mp_int * a, mp_digit b, mp_int * c)
{
  mp_digit *tmpa, *tmpc, mu;
  int       res, ix, oldused;

  /* grow c as required */
  if (c->alloc < a->used + 1) {
     if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) {
        return res;
     }
  }

  /* if a is negative just do an unsigned
   * addition [with fudged signs]
|













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#include <tommath_private.h>
#ifdef BN_MP_SUB_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* single digit subtraction */
int
mp_sub_d (mp_int * a, mp_digit b, mp_int * c)
{
  mp_digit *tmpa, *tmpc, mu;
  int       res, ix, oldused;

  /* grow c as required */
  if (c->alloc < (a->used + 1)) {
     if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) {
        return res;
     }
  }

  /* if a is negative just do an unsigned
   * addition [with fudged signs]
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  /* setup regs */
  oldused = c->used;
  tmpa    = a->dp;
  tmpc    = c->dp;

  /* if a <= b simply fix the single digit */
  if ((a->used == 1 && a->dp[0] <= b) || a->used == 0) {
     if (a->used == 1) {
        *tmpc++ = b - *tmpa;
     } else {
        *tmpc++ = b;
     }
     ix      = 1;

     /* negative/1digit */
     c->sign = MP_NEG;
     c->used = 1;
  } else {
     /* positive/size */
     c->sign = MP_ZPOS;
     c->used = a->used;

     /* subtract first digit */
     *tmpc    = *tmpa++ - b;
     mu       = *tmpc >> (sizeof(mp_digit) * CHAR_BIT - 1);
     *tmpc++ &= MP_MASK;

     /* handle rest of the digits */
     for (ix = 1; ix < a->used; ix++) {
        *tmpc    = *tmpa++ - mu;
        mu       = *tmpc >> (sizeof(mp_digit) * CHAR_BIT - 1);
        *tmpc++ &= MP_MASK;
     }
  }

  /* zero excess digits */
  while (ix++ < oldused) {
     *tmpc++ = 0;
  }
  mp_clamp(c);
  return MP_OKAY;
}

#endif











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  /* setup regs */
  oldused = c->used;
  tmpa    = a->dp;
  tmpc    = c->dp;

  /* if a <= b simply fix the single digit */
  if (((a->used == 1) && (a->dp[0] <= b)) || (a->used == 0)) {
     if (a->used == 1) {
        *tmpc++ = b - *tmpa;
     } else {
        *tmpc++ = b;
     }
     ix      = 1;

     /* negative/1digit */
     c->sign = MP_NEG;
     c->used = 1;
  } else {
     /* positive/size */
     c->sign = MP_ZPOS;
     c->used = a->used;

     /* subtract first digit */
     *tmpc    = *tmpa++ - b;
     mu       = *tmpc >> ((sizeof(mp_digit) * CHAR_BIT) - 1);
     *tmpc++ &= MP_MASK;

     /* handle rest of the digits */
     for (ix = 1; ix < a->used; ix++) {
        *tmpc    = *tmpa++ - mu;
        mu       = *tmpc >> ((sizeof(mp_digit) * CHAR_BIT) - 1);
        *tmpc++ &= MP_MASK;
     }
  }

  /* zero excess digits */
  while (ix++ < oldused) {
     *tmpc++ = 0;
  }
  mp_clamp(c);
  return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_submod.c.
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#include <tommath.h>
#ifdef BN_MP_SUBMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* d = a - b (mod c) */
int
mp_submod (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
{
  int     res;
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#include <tommath_private.h>
#ifdef BN_MP_SUBMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* d = a - b (mod c) */
int
mp_submod (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
{
  int     res;
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    return res;
  }
  res = mp_mod (&t, c, d);
  mp_clear (&t);
  return res;
}
#endif











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    return res;
  }
  res = mp_mod (&t, c, d);
  mp_clear (&t);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_to_signed_bin.c.
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#include <tommath.h>
#ifdef BN_MP_TO_SIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* store in signed [big endian] format */
int mp_to_signed_bin (mp_int * a, unsigned char *b)
{
  int     res;

  if ((res = mp_to_unsigned_bin (a, b + 1)) != MP_OKAY) {
    return res;
  }
  b[0] = (unsigned char) ((a->sign == MP_ZPOS) ? 0 : 1);
  return MP_OKAY;
}
#endif




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#include <tommath_private.h>
#ifdef BN_MP_TO_SIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* store in signed [big endian] format */
int mp_to_signed_bin (mp_int * a, unsigned char *b)
{
  int     res;

  if ((res = mp_to_unsigned_bin (a, b + 1)) != MP_OKAY) {
    return res;
  }
  b[0] = (a->sign == MP_ZPOS) ? (unsigned char)0 : (unsigned char)1;
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_to_signed_bin_n.c.
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#include <tommath.h>
#ifdef BN_MP_TO_SIGNED_BIN_N_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* store in signed [big endian] format */
int mp_to_signed_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen)
{
   if (*outlen < (unsigned long)mp_signed_bin_size(a)) {
      return MP_VAL;
   }
   *outlen = mp_signed_bin_size(a);
   return mp_to_signed_bin(a, b);
}
#endif




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#include <tommath_private.h>
#ifdef BN_MP_TO_SIGNED_BIN_N_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* store in signed [big endian] format */
int mp_to_signed_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen)
{
   if (*outlen < (unsigned long)mp_signed_bin_size(a)) {
      return MP_VAL;
   }
   *outlen = mp_signed_bin_size(a);
   return mp_to_signed_bin(a, b);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_to_unsigned_bin.c.
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#include <tommath.h>
#ifdef BN_MP_TO_UNSIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* store in unsigned [big endian] format */
int mp_to_unsigned_bin (mp_int * a, unsigned char *b)
{
  int     x, res;
  mp_int  t;

  if ((res = mp_init_copy (&t, a)) != MP_OKAY) {
    return res;
  }

  x = 0;
  while (mp_iszero (&t) == 0) {
#ifndef MP_8BIT
      b[x++] = (unsigned char) (t.dp[0] & 255);
#else
      b[x++] = (unsigned char) (t.dp[0] | ((t.dp[1] & 0x01) << 7));
#endif
    if ((res = mp_div_2d (&t, 8, &t, NULL)) != MP_OKAY) {
      mp_clear (&t);
      return res;
    }
  }
  bn_reverse (b, x);
  mp_clear (&t);
  return MP_OKAY;
}
#endif




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#include <tommath_private.h>
#ifdef BN_MP_TO_UNSIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* store in unsigned [big endian] format */
int mp_to_unsigned_bin (mp_int * a, unsigned char *b)
{
  int     x, res;
  mp_int  t;

  if ((res = mp_init_copy (&t, a)) != MP_OKAY) {
    return res;
  }

  x = 0;
  while (mp_iszero (&t) == MP_NO) {
#ifndef MP_8BIT
      b[x++] = (unsigned char) (t.dp[0] & 255);
#else
      b[x++] = (unsigned char) (t.dp[0] | ((t.dp[1] & 0x01) << 7));
#endif
    if ((res = mp_div_2d (&t, 8, &t, NULL)) != MP_OKAY) {
      mp_clear (&t);
      return res;
    }
  }
  bn_reverse (b, x);
  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_to_unsigned_bin_n.c.
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#include <tommath.h>
#ifdef BN_MP_TO_UNSIGNED_BIN_N_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* store in unsigned [big endian] format */
int mp_to_unsigned_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen)
{
   if (*outlen < (unsigned long)mp_unsigned_bin_size(a)) {
      return MP_VAL;
   }
   *outlen = mp_unsigned_bin_size(a);
   return mp_to_unsigned_bin(a, b);
}
#endif




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#include <tommath_private.h>
#ifdef BN_MP_TO_UNSIGNED_BIN_N_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* store in unsigned [big endian] format */
int mp_to_unsigned_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen)
{
   if (*outlen < (unsigned long)mp_unsigned_bin_size(a)) {
      return MP_VAL;
   }
   *outlen = mp_unsigned_bin_size(a);
   return mp_to_unsigned_bin(a, b);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_toom_mul.c.
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#include <tommath.h>
#ifdef BN_MP_TOOM_MUL_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* multiplication using the Toom-Cook 3-way algorithm 
 *
 * Much more complicated than Karatsuba but has a lower 
 * asymptotic running time of O(N**1.464).  This algorithm is 
 * only particularly useful on VERY large inputs 
 * (we're talking 1000s of digits here...).
*/
int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c)
{
    mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2;
    int res, B;
        
    /* init temps */
    if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, 
                             &a0, &a1, &a2, &b0, &b1, 
                             &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) {
       return res;
    }
    
    /* B */
    B = MIN(a->used, b->used) / 3;
    
    /* a = a2 * B**2 + a1 * B + a0 */
    if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_copy(a, &a1)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&a1, B);
    mp_mod_2d(&a1, DIGIT_BIT * B, &a1);



    if ((res = mp_copy(a, &a2)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&a2, B*2);
    
    /* b = b2 * B**2 + b1 * B + b0 */
    if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_copy(b, &b1)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&b1, B);
    mp_mod_2d(&b1, DIGIT_BIT * B, &b1);

    if ((res = mp_copy(b, &b2)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&b2, B*2);
    
    /* w0 = a0*b0 */
    if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) {
       goto ERR;
    }
    
    /* w4 = a2 * b2 */
    if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) {
       goto ERR;
    }
    
    /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */
    if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    
    if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    
    if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) {
       goto ERR;
    }
    
    /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */
    if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    
    if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    
    if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) {
       goto ERR;
    }
    

    /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */
    if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) {
       goto ERR;
    }
    
    /* now solve the matrix 
    
       0  0  0  0  1
       1  2  4  8  16
       1  1  1  1  1
       16 8  4  2  1
       1  0  0  0  0
       
       using 12 subtractions, 4 shifts, 
              2 small divisions and 1 small multiplication 
     */
     
     /* r1 - r4 */
     if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) {
        goto ERR;
     }
     /* r3 - r0 */
     if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) {
        goto ERR;
     }
     /* r1/2 */
     if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) {
        goto ERR;
     }
     /* r3/2 */
     if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) {
        goto ERR;
     }
     /* r2 - r0 - r4 */
     if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) {
        goto ERR;
     }
     /* r1 - r2 */
     if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) {
        goto ERR;
     }
     /* r3 - r2 */
     if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) {
        goto ERR;
     }
     /* r1 - 8r0 */
     if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) {
        goto ERR;
     }
     /* r3 - 8r4 */
     if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) {
        goto ERR;
     }
     /* 3r2 - r1 - r3 */
     if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) {
        goto ERR;
     }
     /* r1 - r2 */
     if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) {
        goto ERR;
     }
     /* r3 - r2 */
     if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) {
        goto ERR;
     }
     /* r1/3 */
     if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) {
        goto ERR;
     }
     /* r3/3 */
     if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) {
        goto ERR;
     }
     
     /* at this point shift W[n] by B*n */
     if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) {
        goto ERR;
     }     
     
     if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) {
        goto ERR;
     }     
     
ERR:
     mp_clear_multi(&w0, &w1, &w2, &w3, &w4, 
                    &a0, &a1, &a2, &b0, &b1, 
                    &b2, &tmp1, &tmp2, NULL);
     return res;
}     
     
#endif




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#include <tommath_private.h>
#ifdef BN_MP_TOOM_MUL_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* multiplication using the Toom-Cook 3-way algorithm
 *
 * Much more complicated than Karatsuba but has a lower
 * asymptotic running time of O(N**1.464).  This algorithm is
 * only particularly useful on VERY large inputs
 * (we're talking 1000s of digits here...).
*/
int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c)
{
    mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2;
    int res, B;

    /* init temps */
    if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4,
                             &a0, &a1, &a2, &b0, &b1,
                             &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) {
       return res;
    }

    /* B */
    B = MIN(a->used, b->used) / 3;

    /* a = a2 * B**2 + a1 * B + a0 */
    if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_copy(a, &a1)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&a1, B);
    if ((res = mp_mod_2d(&a1, DIGIT_BIT * B, &a1)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_copy(a, &a2)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&a2, B*2);

    /* b = b2 * B**2 + b1 * B + b0 */
    if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_copy(b, &b1)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&b1, B);
    (void)mp_mod_2d(&b1, DIGIT_BIT * B, &b1);

    if ((res = mp_copy(b, &b2)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&b2, B*2);

    /* w0 = a0*b0 */
    if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) {
       goto ERR;
    }

    /* w4 = a2 * b2 */
    if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) {
       goto ERR;
    }

    /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */
    if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) {
       goto ERR;
    }

    /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */
    if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) {
       goto ERR;
    }


    /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */
    if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) {
       goto ERR;
    }

    /* now solve the matrix

       0  0  0  0  1
       1  2  4  8  16
       1  1  1  1  1
       16 8  4  2  1
       1  0  0  0  0

       using 12 subtractions, 4 shifts,
              2 small divisions and 1 small multiplication
     */

    /* r1 - r4 */
    if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) {
       goto ERR;
    }
    /* r3 - r0 */
    if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) {
       goto ERR;
    }
    /* r1/2 */
    if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) {
       goto ERR;
    }
    /* r3/2 */
    if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) {
       goto ERR;
    }
    /* r2 - r0 - r4 */
    if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) {
       goto ERR;
    }
    /* r1 - r2 */
    if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) {
       goto ERR;
    }
    /* r3 - r2 */
    if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) {
       goto ERR;
    }
    /* r1 - 8r0 */
    if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) {
       goto ERR;
    }
    /* r3 - 8r4 */
    if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) {
       goto ERR;
    }
    /* 3r2 - r1 - r3 */
    if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) {
       goto ERR;
    }
    /* r1 - r2 */
    if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) {
       goto ERR;
    }
    /* r3 - r2 */
    if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) {
       goto ERR;
    }
    /* r1/3 */
    if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) {
       goto ERR;
    }
    /* r3/3 */
    if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) {
       goto ERR;
    }

    /* at this point shift W[n] by B*n */
    if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) {
       goto ERR;
    }

ERR:
    mp_clear_multi(&w0, &w1, &w2, &w3, &w4,
                   &a0, &a1, &a2, &b0, &b1,
                   &b2, &tmp1, &tmp2, NULL);
    return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_toom_sqr.c.
1
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#include <tommath.h>
#ifdef BN_MP_TOOM_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* squaring using Toom-Cook 3-way algorithm */
int
mp_toom_sqr(mp_int *a, mp_int *b)
{
    mp_int w0, w1, w2, w3, w4, tmp1, a0, a1, a2;
|













|







1
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#include <tommath_private.h>
#ifdef BN_MP_TOOM_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* squaring using Toom-Cook 3-way algorithm */
int
mp_toom_sqr(mp_int *a, mp_int *b)
{
    mp_int w0, w1, w2, w3, w4, tmp1, a0, a1, a2;
35
36
37
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39
40
41
42


43
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45
46
47
48
49
       goto ERR;
    }

    if ((res = mp_copy(a, &a1)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&a1, B);
    mp_mod_2d(&a1, DIGIT_BIT * B, &a1);



    if ((res = mp_copy(a, &a2)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&a2, B*2);

    /* w0 = a0*a0 */







|
>
>







35
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       goto ERR;
    }

    if ((res = mp_copy(a, &a1)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&a1, B);
    if ((res = mp_mod_2d(&a1, DIGIT_BIT * B, &a1)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_copy(a, &a2)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&a2, B*2);

    /* w0 = a0*a0 */
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       1  1  1  1  1
       16 8  4  2  1
       1  0  0  0  0

       using 12 subtractions, 4 shifts, 2 small divisions and 1 small multiplication.
     */

     /* r1 - r4 */
     if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) {
        goto ERR;
     }
     /* r3 - r0 */
     if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) {
        goto ERR;
     }
     /* r1/2 */
     if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) {
        goto ERR;
     }
     /* r3/2 */
     if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) {
        goto ERR;
     }
     /* r2 - r0 - r4 */
     if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) {
        goto ERR;
     }
     /* r1 - r2 */
     if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) {
        goto ERR;
     }
     /* r3 - r2 */
     if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) {
        goto ERR;
     }
     /* r1 - 8r0 */
     if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) {
        goto ERR;
     }
     /* r3 - 8r4 */
     if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) {
        goto ERR;
     }
     /* 3r2 - r1 - r3 */
     if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) {
        goto ERR;
     }
     /* r1 - r2 */
     if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) {
        goto ERR;
     }
     /* r3 - r2 */
     if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) {
        goto ERR;
     }
     /* r1/3 */
     if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) {
        goto ERR;
     }
     /* r3/3 */
     if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) {
        goto ERR;
     }

     /* at this point shift W[n] by B*n */
     if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) {
        goto ERR;
     }

     if ((res = mp_add(&w0, &w1, b)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_add(&tmp1, b, b)) != MP_OKAY) {
        goto ERR;
     }

ERR:
     mp_clear_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &tmp1, NULL);
     return res;
}

#endif











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>
>
>
>
113
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130
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226
227
228
       1  1  1  1  1
       16 8  4  2  1
       1  0  0  0  0

       using 12 subtractions, 4 shifts, 2 small divisions and 1 small multiplication.
     */

    /* r1 - r4 */
    if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) {
       goto ERR;
    }
    /* r3 - r0 */
    if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) {
       goto ERR;
    }
    /* r1/2 */
    if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) {
       goto ERR;
    }
    /* r3/2 */
    if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) {
       goto ERR;
    }
    /* r2 - r0 - r4 */
    if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) {
       goto ERR;
    }
    /* r1 - r2 */
    if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) {
       goto ERR;
    }
    /* r3 - r2 */
    if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) {
       goto ERR;
    }
    /* r1 - 8r0 */
    if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) {
       goto ERR;
    }
    /* r3 - 8r4 */
    if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) {
       goto ERR;
    }
    /* 3r2 - r1 - r3 */
    if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) {
       goto ERR;
    }
    /* r1 - r2 */
    if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) {
       goto ERR;
    }
    /* r3 - r2 */
    if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) {
       goto ERR;
    }
    /* r1/3 */
    if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) {
       goto ERR;
    }
    /* r3/3 */
    if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) {
       goto ERR;
    }

    /* at this point shift W[n] by B*n */
    if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_add(&w0, &w1, b)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, b, b)) != MP_OKAY) {
       goto ERR;
    }

ERR:
    mp_clear_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &tmp1, NULL);
    return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_toradix.c.
1
2
3
4
5
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7
8
9
10
11
12
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14
15
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#include <tommath.h>
#ifdef BN_MP_TORADIX_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* stores a bignum as a ASCII string in a given radix (2..64) */
int mp_toradix (mp_int * a, char *str, int radix)
{
  int     res, digs;
  mp_int  t;
  mp_digit d;
  char   *_s = str;

  /* check range of the radix */
  if (radix < 2 || radix > 64) {
    return MP_VAL;
  }

  /* quick out if its zero */
  if (mp_iszero(a) == 1) {
     *str++ = '0';
     *str = '\0';
     return MP_OKAY;
  }

  if ((res = mp_init_copy (&t, a)) != MP_OKAY) {
    return res;
  }

  /* if it is negative output a - */
  if (t.sign == MP_NEG) {
    ++_s;
    *str++ = '-';
    t.sign = MP_ZPOS;
  }

  digs = 0;
  while (mp_iszero (&t) == 0) {
    if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) {
      mp_clear (&t);
      return res;
    }
    *str++ = mp_s_rmap[d];
    ++digs;
  }
|













|











|




|

















|







1
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#include <tommath_private.h>
#ifdef BN_MP_TORADIX_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* stores a bignum as a ASCII string in a given radix (2..64) */
int mp_toradix (mp_int * a, char *str, int radix)
{
  int     res, digs;
  mp_int  t;
  mp_digit d;
  char   *_s = str;

  /* check range of the radix */
  if ((radix < 2) || (radix > 64)) {
    return MP_VAL;
  }

  /* quick out if its zero */
  if (mp_iszero(a) == MP_YES) {
     *str++ = '0';
     *str = '\0';
     return MP_OKAY;
  }

  if ((res = mp_init_copy (&t, a)) != MP_OKAY) {
    return res;
  }

  /* if it is negative output a - */
  if (t.sign == MP_NEG) {
    ++_s;
    *str++ = '-';
    t.sign = MP_ZPOS;
  }

  digs = 0;
  while (mp_iszero (&t) == MP_NO) {
    if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) {
      mp_clear (&t);
      return res;
    }
    *str++ = mp_s_rmap[d];
    ++digs;
  }
65
66
67
68
69
70
71




  *str = '\0';

  mp_clear (&t);
  return MP_OKAY;
}

#endif











>
>
>
>
65
66
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68
69
70
71
72
73
74
75
  *str = '\0';

  mp_clear (&t);
  return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_toradix_n.c.
1
2
3
4
5
6
7
8
9
10
11
12
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14
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31
32
33
34
35
36
37
#include <tommath.h>
#ifdef BN_MP_TORADIX_N_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* stores a bignum as a ASCII string in a given radix (2..64) 
 *
 * Stores upto maxlen-1 chars and always a NULL byte 
 */
int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen)
{
  int     res, digs;
  mp_int  t;
  mp_digit d;
  char   *_s = str;

  /* check range of the maxlen, radix */
  if (maxlen < 2 || radix < 2 || radix > 64) {
    return MP_VAL;
  }

  /* quick out if its zero */
  if (mp_iszero(a) == MP_YES) {
     *str++ = '0';
     *str = '\0';
|













|














|







1
2
3
4
5
6
7
8
9
10
11
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14
15
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37
#include <tommath_private.h>
#ifdef BN_MP_TORADIX_N_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* stores a bignum as a ASCII string in a given radix (2..64) 
 *
 * Stores upto maxlen-1 chars and always a NULL byte 
 */
int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen)
{
  int     res, digs;
  mp_int  t;
  mp_digit d;
  char   *_s = str;

  /* check range of the maxlen, radix */
  if ((maxlen < 2) || (radix < 2) || (radix > 64)) {
    return MP_VAL;
  }

  /* quick out if its zero */
  if (mp_iszero(a) == MP_YES) {
     *str++ = '0';
     *str = '\0';
52
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58
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66
    t.sign = MP_ZPOS;
 
    /* subtract a char */
    --maxlen;
  }

  digs = 0;
  while (mp_iszero (&t) == 0) {
    if (--maxlen < 1) {
       /* no more room */
       break;
    }
    if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) {
      mp_clear (&t);
      return res;







|







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    t.sign = MP_ZPOS;
 
    /* subtract a char */
    --maxlen;
  }

  digs = 0;
  while (mp_iszero (&t) == MP_NO) {
    if (--maxlen < 1) {
       /* no more room */
       break;
    }
    if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) {
      mp_clear (&t);
      return res;
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  *str = '\0';

  mp_clear (&t);
  return MP_OKAY;
}

#endif











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>
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  *str = '\0';

  mp_clear (&t);
  return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_unsigned_bin_size.c.
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#include <tommath.h>
#ifdef BN_MP_UNSIGNED_BIN_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* get the size for an unsigned equivalent */
int mp_unsigned_bin_size (mp_int * a)
{
  int     size = mp_count_bits (a);
  return (size / 8 + ((size & 7) != 0 ? 1 : 0));
}
#endif




|













|






|


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#include <tommath_private.h>
#ifdef BN_MP_UNSIGNED_BIN_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* get the size for an unsigned equivalent */
int mp_unsigned_bin_size (mp_int * a)
{
  int     size = mp_count_bits (a);
  return (size / 8) + (((size & 7) != 0) ? 1 : 0);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_xor.c.
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#include <tommath.h>
#ifdef BN_MP_XOR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* XOR two ints together */
int
mp_xor (mp_int * a, mp_int * b, mp_int * c)
{
  int     res, ix, px;
|













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#include <tommath_private.h>
#ifdef BN_MP_XOR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* XOR two ints together */
int
mp_xor (mp_int * a, mp_int * b, mp_int * c)
{
  int     res, ix, px;
41
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45
46
47




  }
  mp_clamp (&t);
  mp_exch (c, &t);
  mp_clear (&t);
  return MP_OKAY;
}
#endif











>
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>
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50
51
  }
  mp_clamp (&t);
  mp_exch (c, &t);
  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_mp_zero.c.
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#include <tommath.h>
#ifdef BN_MP_ZERO_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* set to zero */
void mp_zero (mp_int * a)
{
  int       n;
  mp_digit *tmp;

  a->sign = MP_ZPOS;
  a->used = 0;

  tmp = a->dp;
  for (n = 0; n < a->alloc; n++) {
     *tmp++ = 0;
  }
}
#endif




|













|

















>
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#include <tommath_private.h>
#ifdef BN_MP_ZERO_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* set to zero */
void mp_zero (mp_int * a)
{
  int       n;
  mp_digit *tmp;

  a->sign = MP_ZPOS;
  a->used = 0;

  tmp = a->dp;
  for (n = 0; n < a->alloc; n++) {
     *tmp++ = 0;
  }
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_prime_tab.c.
1
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#include <tommath.h>
#ifdef BN_PRIME_TAB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */
const mp_digit ltm_prime_tab[] = {
  0x0002, 0x0003, 0x0005, 0x0007, 0x000B, 0x000D, 0x0011, 0x0013,
  0x0017, 0x001D, 0x001F, 0x0025, 0x0029, 0x002B, 0x002F, 0x0035,
  0x003B, 0x003D, 0x0043, 0x0047, 0x0049, 0x004F, 0x0053, 0x0059,
  0x0061, 0x0065, 0x0067, 0x006B, 0x006D, 0x0071, 0x007F,
#ifndef MP_8BIT
|













|







1
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#include <tommath_private.h>
#ifdef BN_PRIME_TAB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */
const mp_digit ltm_prime_tab[] = {
  0x0002, 0x0003, 0x0005, 0x0007, 0x000B, 0x000D, 0x0011, 0x0013,
  0x0017, 0x001D, 0x001F, 0x0025, 0x0029, 0x002B, 0x002F, 0x0035,
  0x003B, 0x003D, 0x0043, 0x0047, 0x0049, 0x004F, 0x0053, 0x0059,
  0x0061, 0x0065, 0x0067, 0x006B, 0x006D, 0x0071, 0x007F,
#ifndef MP_8BIT
51
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55
56
57




  0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3,
  0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7,
  0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623,
  0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653
#endif
};
#endif











>
>
>
>
51
52
53
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55
56
57
58
59
60
61
  0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3,
  0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7,
  0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623,
  0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653
#endif
};
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_reverse.c.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
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22
23
24
25
26
27
28
29
30
31
32
33
34
35




#include <tommath.h>
#ifdef BN_REVERSE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* reverse an array, used for radix code */
void
bn_reverse (unsigned char *s, int len)
{
  int     ix, iy;
  unsigned char t;

  ix = 0;
  iy = len - 1;
  while (ix < iy) {
    t     = s[ix];
    s[ix] = s[iy];
    s[iy] = t;
    ++ix;
    --iy;
  }
}
#endif




|













|




















>
>
>
>
1
2
3
4
5
6
7
8
9
10
11
12
13
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16
17
18
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23
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27
28
29
30
31
32
33
34
35
36
37
38
39
#include <tommath_private.h>
#ifdef BN_REVERSE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* reverse an array, used for radix code */
void
bn_reverse (unsigned char *s, int len)
{
  int     ix, iy;
  unsigned char t;

  ix = 0;
  iy = len - 1;
  while (ix < iy) {
    t     = s[ix];
    s[ix] = s[iy];
    s[iy] = t;
    ++ix;
    --iy;
  }
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_s_mp_add.c.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
#include <tommath.h>
#ifdef BN_S_MP_ADD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* low level addition, based on HAC pp.594, Algorithm 14.7 */
int
s_mp_add (mp_int * a, mp_int * b, mp_int * c)
{
  mp_int *x;
|













|







1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
#include <tommath_private.h>
#ifdef BN_S_MP_ADD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* low level addition, based on HAC pp.594, Algorithm 14.7 */
int
s_mp_add (mp_int * a, mp_int * b, mp_int * c)
{
  mp_int *x;
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
  } else {
    min = a->used;
    max = b->used;
    x = b;
  }

  /* init result */
  if (c->alloc < max + 1) {
    if ((res = mp_grow (c, max + 1)) != MP_OKAY) {
      return res;
    }
  }

  /* get old used digit count and set new one */
  olduse = c->used;
  c->used = max + 1;

  {
    register mp_digit u, *tmpa, *tmpb, *tmpc;
    register int i;

    /* alias for digit pointers */

    /* first input */
    tmpa = a->dp;

    /* second input */







|










|
|







32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
  } else {
    min = a->used;
    max = b->used;
    x = b;
  }

  /* init result */
  if (c->alloc < (max + 1)) {
    if ((res = mp_grow (c, max + 1)) != MP_OKAY) {
      return res;
    }
  }

  /* get old used digit count and set new one */
  olduse = c->used;
  c->used = max + 1;

  {
    mp_digit u, *tmpa, *tmpb, *tmpc;
    int i;

    /* alias for digit pointers */

    /* first input */
    tmpa = a->dp;

    /* second input */
99
100
101
102
103
104
105




    }
  }

  mp_clamp (c);
  return MP_OKAY;
}
#endif











>
>
>
>
99
100
101
102
103
104
105
106
107
108
109
    }
  }

  mp_clamp (c);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_s_mp_exptmod.c.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
#include <tommath.h>
#ifdef BN_S_MP_EXPTMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */
#ifdef MP_LOW_MEM
   #define TAB_SIZE 32
#else
   #define TAB_SIZE 256
#endif

|













|







1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
#include <tommath_private.h>
#ifdef BN_S_MP_EXPTMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */
#ifdef MP_LOW_MEM
   #define TAB_SIZE 32
#else
   #define TAB_SIZE 256
#endif

160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
    buf <<= (mp_digit)1;

    /* if the bit is zero and mode == 0 then we ignore it
     * These represent the leading zero bits before the first 1 bit
     * in the exponent.  Technically this opt is not required but it
     * does lower the # of trivial squaring/reductions used
     */
    if (mode == 0 && y == 0) {
      continue;
    }

    /* if the bit is zero and mode == 1 then we square */
    if (mode == 1 && y == 0) {
      if ((err = mp_sqr (&res, &res)) != MP_OKAY) {
        goto LBL_RES;
      }
      if ((err = redux (&res, P, &mu)) != MP_OKAY) {
        goto LBL_RES;
      }
      continue;







|




|







160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
    buf <<= (mp_digit)1;

    /* if the bit is zero and mode == 0 then we ignore it
     * These represent the leading zero bits before the first 1 bit
     * in the exponent.  Technically this opt is not required but it
     * does lower the # of trivial squaring/reductions used
     */
    if ((mode == 0) && (y == 0)) {
      continue;
    }

    /* if the bit is zero and mode == 1 then we square */
    if ((mode == 1) && (y == 0)) {
      if ((err = mp_sqr (&res, &res)) != MP_OKAY) {
        goto LBL_RES;
      }
      if ((err = redux (&res, P, &mu)) != MP_OKAY) {
        goto LBL_RES;
      }
      continue;
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
      bitcpy = 0;
      bitbuf = 0;
      mode   = 1;
    }
  }

  /* if bits remain then square/multiply */
  if (mode == 2 && bitcpy > 0) {
    /* square then multiply if the bit is set */
    for (x = 0; x < bitcpy; x++) {
      if ((err = mp_sqr (&res, &res)) != MP_OKAY) {
        goto LBL_RES;
      }
      if ((err = redux (&res, P, &mu)) != MP_OKAY) {
        goto LBL_RES;







|







207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
      bitcpy = 0;
      bitbuf = 0;
      mode   = 1;
    }
  }

  /* if bits remain then square/multiply */
  if ((mode == 2) && (bitcpy > 0)) {
    /* square then multiply if the bit is set */
    for (x = 0; x < bitcpy; x++) {
      if ((err = mp_sqr (&res, &res)) != MP_OKAY) {
        goto LBL_RES;
      }
      if ((err = redux (&res, P, &mu)) != MP_OKAY) {
        goto LBL_RES;
242
243
244
245
246
247
248




  mp_clear(&M[1]);
  for (x = 1<<(winsize-1); x < (1 << winsize); x++) {
    mp_clear (&M[x]);
  }
  return err;
}
#endif











>
>
>
>
242
243
244
245
246
247
248
249
250
251
252
  mp_clear(&M[1]);
  for (x = 1<<(winsize-1); x < (1 << winsize); x++) {
    mp_clear (&M[x]);
  }
  return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_s_mp_mul_digs.c.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
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23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
#include <tommath.h>
#ifdef BN_S_MP_MUL_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* multiplies |a| * |b| and only computes upto digs digits of result
 * HAC pp. 595, Algorithm 14.12  Modified so you can control how 
 * many digits of output are created.
 */
int s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
{
  mp_int  t;
  int     res, pa, pb, ix, iy;
  mp_digit u;
  mp_word r;
  mp_digit tmpx, *tmpt, *tmpy;

  /* can we use the fast multiplier? */
  if (((digs) < MP_WARRAY) &&
      MIN (a->used, b->used) < 
          (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) {
    return fast_s_mp_mul_digs (a, b, c, digs);
  }

  if ((res = mp_init_size (&t, digs)) != MP_OKAY) {
    return res;
  }
  t.used = digs;
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#include <tommath_private.h>
#ifdef BN_S_MP_MUL_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* multiplies |a| * |b| and only computes upto digs digits of result
 * HAC pp. 595, Algorithm 14.12  Modified so you can control how 
 * many digits of output are created.
 */
int s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
{
  mp_int  t;
  int     res, pa, pb, ix, iy;
  mp_digit u;
  mp_word r;
  mp_digit tmpx, *tmpt, *tmpy;

  /* can we use the fast multiplier? */
  if (((digs) < MP_WARRAY) &&
      (MIN (a->used, b->used) < 
          (1 << ((CHAR_BIT * sizeof(mp_word)) - (2 * DIGIT_BIT))))) {
    return fast_s_mp_mul_digs (a, b, c, digs);
  }

  if ((res = mp_init_size (&t, digs)) != MP_OKAY) {
    return res;
  }
  t.used = digs;
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    /* an alias for the digits of b */
    tmpy = b->dp;

    /* compute the columns of the output and propagate the carry */
    for (iy = 0; iy < pb; iy++) {
      /* compute the column as a mp_word */
      r       = ((mp_word)*tmpt) +
                ((mp_word)tmpx) * ((mp_word)*tmpy++) +
                ((mp_word) u);

      /* the new column is the lower part of the result */
      *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));

      /* get the carry word from the result */
      u       = (mp_digit) (r >> ((mp_word) DIGIT_BIT));
    }
    /* set carry if it is placed below digs */
    if (ix + iy < digs) {
      *tmpt = u;
    }
  }

  mp_clamp (&t);
  mp_exch (&t, c);

  mp_clear (&t);
  return MP_OKAY;
}
#endif











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    /* an alias for the digits of b */
    tmpy = b->dp;

    /* compute the columns of the output and propagate the carry */
    for (iy = 0; iy < pb; iy++) {
      /* compute the column as a mp_word */
      r       = (mp_word)*tmpt +
                ((mp_word)tmpx * (mp_word)*tmpy++) +
                (mp_word)u;

      /* the new column is the lower part of the result */
      *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));

      /* get the carry word from the result */
      u       = (mp_digit) (r >> ((mp_word) DIGIT_BIT));
    }
    /* set carry if it is placed below digs */
    if ((ix + iy) < digs) {
      *tmpt = u;
    }
  }

  mp_clamp (&t);
  mp_exch (&t, c);

  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_s_mp_mul_high_digs.c.
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#include <tommath.h>
#ifdef BN_S_MP_MUL_HIGH_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* multiplies |a| * |b| and does not compute the lower digs digits
 * [meant to get the higher part of the product]
 */
int
s_mp_mul_high_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
{
  mp_int  t;
  int     res, pa, pb, ix, iy;
  mp_digit u;
  mp_word r;
  mp_digit tmpx, *tmpt, *tmpy;

  /* can we use the fast multiplier? */
#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C
  if (((a->used + b->used + 1) < MP_WARRAY)
      && MIN (a->used, b->used) < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) {
    return fast_s_mp_mul_high_digs (a, b, c, digs);
  }
#endif

  if ((res = mp_init_size (&t, a->used + b->used + 1)) != MP_OKAY) {
    return res;
  }
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#include <tommath_private.h>
#ifdef BN_S_MP_MUL_HIGH_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* multiplies |a| * |b| and does not compute the lower digs digits
 * [meant to get the higher part of the product]
 */
int
s_mp_mul_high_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
{
  mp_int  t;
  int     res, pa, pb, ix, iy;
  mp_digit u;
  mp_word r;
  mp_digit tmpx, *tmpt, *tmpy;

  /* can we use the fast multiplier? */
#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C
  if (((a->used + b->used + 1) < MP_WARRAY)
      && (MIN (a->used, b->used) < (1 << ((CHAR_BIT * sizeof(mp_word)) - (2 * DIGIT_BIT))))) {
    return fast_s_mp_mul_high_digs (a, b, c, digs);
  }
#endif

  if ((res = mp_init_size (&t, a->used + b->used + 1)) != MP_OKAY) {
    return res;
  }
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    tmpt = &(t.dp[digs]);

    /* alias for where to read the right hand side from */
    tmpy = b->dp + (digs - ix);

    for (iy = digs - ix; iy < pb; iy++) {
      /* calculate the double precision result */
      r       = ((mp_word)*tmpt) +
                ((mp_word)tmpx) * ((mp_word)*tmpy++) +
                ((mp_word) u);

      /* get the lower part */
      *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));

      /* carry the carry */
      u       = (mp_digit) (r >> ((mp_word) DIGIT_BIT));
    }
    *tmpt = u;
  }
  mp_clamp (&t);
  mp_exch (&t, c);
  mp_clear (&t);
  return MP_OKAY;
}
#endif











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    tmpt = &(t.dp[digs]);

    /* alias for where to read the right hand side from */
    tmpy = b->dp + (digs - ix);

    for (iy = digs - ix; iy < pb; iy++) {
      /* calculate the double precision result */
      r       = (mp_word)*tmpt +
                ((mp_word)tmpx * (mp_word)*tmpy++) +
                (mp_word)u;

      /* get the lower part */
      *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));

      /* carry the carry */
      u       = (mp_digit) (r >> ((mp_word) DIGIT_BIT));
    }
    *tmpt = u;
  }
  mp_clamp (&t);
  mp_exch (&t, c);
  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_s_mp_sqr.c.
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#include <tommath.h>
#ifdef BN_S_MP_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */
int s_mp_sqr (mp_int * a, mp_int * b)
{
  mp_int  t;
  int     res, ix, iy, pa;
  mp_word r;
  mp_digit u, tmpx, *tmpt;

  pa = a->used;
  if ((res = mp_init_size (&t, 2*pa + 1)) != MP_OKAY) {
    return res;
  }

  /* default used is maximum possible size */
  t.used = 2*pa + 1;

  for (ix = 0; ix < pa; ix++) {
    /* first calculate the digit at 2*ix */
    /* calculate double precision result */
    r = ((mp_word) t.dp[2*ix]) +
        ((mp_word)a->dp[ix])*((mp_word)a->dp[ix]);

    /* store lower part in result */
    t.dp[ix+ix] = (mp_digit) (r & ((mp_word) MP_MASK));

    /* get the carry */
    u           = (mp_digit)(r >> ((mp_word) DIGIT_BIT));

    /* left hand side of A[ix] * A[iy] */
    tmpx        = a->dp[ix];

    /* alias for where to store the results */
    tmpt        = t.dp + (2*ix + 1);
    
    for (iy = ix + 1; iy < pa; iy++) {
      /* first calculate the product */
      r       = ((mp_word)tmpx) * ((mp_word)a->dp[iy]);

      /* now calculate the double precision result, note we use
       * addition instead of *2 since it's easier to optimize
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#include <tommath_private.h>
#ifdef BN_S_MP_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */
int s_mp_sqr (mp_int * a, mp_int * b)
{
  mp_int  t;
  int     res, ix, iy, pa;
  mp_word r;
  mp_digit u, tmpx, *tmpt;

  pa = a->used;
  if ((res = mp_init_size (&t, (2 * pa) + 1)) != MP_OKAY) {
    return res;
  }

  /* default used is maximum possible size */
  t.used = (2 * pa) + 1;

  for (ix = 0; ix < pa; ix++) {
    /* first calculate the digit at 2*ix */
    /* calculate double precision result */
    r = (mp_word)t.dp[2*ix] +
        ((mp_word)a->dp[ix] * (mp_word)a->dp[ix]);

    /* store lower part in result */
    t.dp[ix+ix] = (mp_digit) (r & ((mp_word) MP_MASK));

    /* get the carry */
    u           = (mp_digit)(r >> ((mp_word) DIGIT_BIT));

    /* left hand side of A[ix] * A[iy] */
    tmpx        = a->dp[ix];

    /* alias for where to store the results */
    tmpt        = t.dp + ((2 * ix) + 1);
    
    for (iy = ix + 1; iy < pa; iy++) {
      /* first calculate the product */
      r       = ((mp_word)tmpx) * ((mp_word)a->dp[iy]);

      /* now calculate the double precision result, note we use
       * addition instead of *2 since it's easier to optimize
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  mp_clamp (&t);
  mp_exch (&t, b);
  mp_clear (&t);
  return MP_OKAY;
}
#endif











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  mp_clamp (&t);
  mp_exch (&t, b);
  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bn_s_mp_sub.c.
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#include <tommath.h>
#ifdef BN_S_MP_SUB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */
int
s_mp_sub (mp_int * a, mp_int * b, mp_int * c)
{
  int     olduse, res, min, max;
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#include <tommath_private.h>
#ifdef BN_S_MP_SUB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */
int
s_mp_sub (mp_int * a, mp_int * b, mp_int * c)
{
  int     olduse, res, min, max;
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      return res;
    }
  }
  olduse = c->used;
  c->used = max;

  {
    register mp_digit u, *tmpa, *tmpb, *tmpc;
    register int i;

    /* alias for digit pointers */
    tmpa = a->dp;
    tmpb = b->dp;
    tmpc = c->dp;

    /* set carry to zero */
    u = 0;
    for (i = 0; i < min; i++) {
      /* T[i] = A[i] - B[i] - U */
      *tmpc = *tmpa++ - *tmpb++ - u;

      /* U = carry bit of T[i]
       * Note this saves performing an AND operation since
       * if a carry does occur it will propagate all the way to the
       * MSB.  As a result a single shift is enough to get the carry
       */
      u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1));

      /* Clear carry from T[i] */
      *tmpc++ &= MP_MASK;
    }

    /* now copy higher words if any, e.g. if A has more digits than B  */
    for (; i < max; i++) {
      /* T[i] = A[i] - U */
      *tmpc = *tmpa++ - u;

      /* U = carry bit of T[i] */
      u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1));

      /* Clear carry from T[i] */
      *tmpc++ &= MP_MASK;
    }

    /* clear digits above used (since we may not have grown result above) */
    for (i = c->used; i < olduse; i++) {
      *tmpc++ = 0;
    }
  }

  mp_clamp (c);
  return MP_OKAY;
}

#endif











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      return res;
    }
  }
  olduse = c->used;
  c->used = max;

  {
    mp_digit u, *tmpa, *tmpb, *tmpc;
    int i;

    /* alias for digit pointers */
    tmpa = a->dp;
    tmpb = b->dp;
    tmpc = c->dp;

    /* set carry to zero */
    u = 0;
    for (i = 0; i < min; i++) {
      /* T[i] = A[i] - B[i] - U */
      *tmpc = (*tmpa++ - *tmpb++) - u;

      /* U = carry bit of T[i]
       * Note this saves performing an AND operation since
       * if a carry does occur it will propagate all the way to the
       * MSB.  As a result a single shift is enough to get the carry
       */
      u = *tmpc >> ((mp_digit)((CHAR_BIT * sizeof(mp_digit)) - 1));

      /* Clear carry from T[i] */
      *tmpc++ &= MP_MASK;
    }

    /* now copy higher words if any, e.g. if A has more digits than B  */
    for (; i < max; i++) {
      /* T[i] = A[i] - U */
      *tmpc = *tmpa++ - u;

      /* U = carry bit of T[i] */
      u = *tmpc >> ((mp_digit)((CHAR_BIT * sizeof(mp_digit)) - 1));

      /* Clear carry from T[i] */
      *tmpc++ &= MP_MASK;
    }

    /* clear digits above used (since we may not have grown result above) */
    for (i = c->used; i < olduse; i++) {
      *tmpc++ = 0;
    }
  }

  mp_clamp (c);
  return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/bncore.c.
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#include <tommath.h>
#ifdef BNCORE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */

/* Known optimal configurations

 CPU                    /Compiler     /MUL CUTOFF/SQR CUTOFF
-------------------------------------------------------------
 Intel P4 Northwood     /GCC v3.4.1   /        88/       128/LTM 0.32 ;-)
 AMD Athlon64           /GCC v3.4.4   /        80/       120/LTM 0.35
 
*/

int     KARATSUBA_MUL_CUTOFF = 80,      /* Min. number of digits before Karatsuba multiplication is used. */
        KARATSUBA_SQR_CUTOFF = 120,     /* Min. number of digits before Karatsuba squaring is used. */
        
        TOOM_MUL_CUTOFF      = 350,      /* no optimal values of these are known yet so set em high */
        TOOM_SQR_CUTOFF      = 400; 
#endif




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#include <tommath_private.h>
#ifdef BNCORE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://libtom.org
 */

/* Known optimal configurations

 CPU                    /Compiler     /MUL CUTOFF/SQR CUTOFF
-------------------------------------------------------------
 Intel P4 Northwood     /GCC v3.4.1   /        88/       128/LTM 0.32 ;-)
 AMD Athlon64           /GCC v3.4.4   /        80/       120/LTM 0.35
 
*/

int     KARATSUBA_MUL_CUTOFF = 80,      /* Min. number of digits before Karatsuba multiplication is used. */
        KARATSUBA_SQR_CUTOFF = 120,     /* Min. number of digits before Karatsuba squaring is used. */
        
        TOOM_MUL_CUTOFF      = 350,      /* no optimal values of these are known yet so set em high */
        TOOM_SQR_CUTOFF      = 400; 
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/booker.pl.
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#Tom St Denis

#get graphics type
if (shift =~ /PDF/) {
   $graph = "";
} else {
   $graph = ".ps";
}   

open(IN,"<tommath.src") or die "Can't open source file";
open(OUT,">tommath.tex") or die "Can't open destination file";

print "Scanning for sections\n";
$chapter = $section = $subsection = 0;
$x = 0;
while (<IN>) {
   print ".";
   if (!(++$x % 80)) { print "\n"; }
   #update the headings 
   if (~($_ =~ /\*/)) {
      if ($_ =~ /\\chapter{.+}/) {
          ++$chapter;
          $section = $subsection = 0;
      } elsif ($_ =~ /\\section{.+}/) {
          ++$section;
          $subsection = 0;
      } elsif ($_ =~ /\\subsection{.+}/) {
          ++$subsection;
      }
   }      

   if ($_ =~ m/MARK/) {
      @m = split(",",$_);
      chomp(@m[1]);
      $index1{@m[1]} = $chapter;
      $index2{@m[1]} = $section;
      $index3{@m[1]} = $subsection;
   }
}
close(IN);

open(IN,"<tommath.src") or die "Can't open source file";
$readline = $wroteline = 0;
$srcline = 0;

while (<IN>) {
   ++$readline;
   ++$srcline;
   
   if ($_ =~ m/MARK/) {
   } elsif ($_ =~ m/EXAM/ || $_ =~ m/LIST/) {
      if ($_ =~ m/EXAM/) {
         $skipheader = 1;
      } else {
         $skipheader = 0;
      }
      
      # EXAM,file
      chomp($_);
      @m = split(",",$_);
      open(SRC,"<$m[1]") or die "Error:$srcline:Can't open source file $m[1]";
      
      print "$srcline:Inserting $m[1]:";
      
      $line = 0;
      $tmp = $m[1];
      $tmp =~ s/_/"\\_"/ge;
      print OUT "\\vspace{+3mm}\\begin{small}\n\\hspace{-5.1mm}{\\bf File}: $tmp\n\\vspace{-3mm}\n\\begin{alltt}\n";
      $wroteline += 5;
      
      if ($skipheader == 1) {
         # scan till next end of comment, e.g. skip license 
         while (<SRC>) {
            $text[$line++] = $_;
            last if ($_ =~ /math\.libtomcrypt\.org/);
         }
         <SRC>;   
      }
      
      $inline = 0;
      while (<SRC>) {
      next if ($_ =~ /\$Source/);
      next if ($_ =~ /\$Revision/);
      next if ($_ =~ /\$Date/);
         $text[$line++] = $_;
         ++$inline;
         chomp($_);
         $_ =~ s/\t/"    "/ge;
         $_ =~ s/{/"^{"/ge;
         $_ =~ s/}/"^}"/ge;
         $_ =~ s/\\/'\symbol{92}'/ge;
         $_ =~ s/\^/"\\"/ge;
           
         printf OUT ("%03d   ", $line);
         for ($x = 0; $x < length($_); $x++) {
             print OUT chr(vec($_, $x, 8));
             if ($x == 75) { 
                 print OUT "\n      ";
                 ++$wroteline;
             }
         }
         print OUT "\n";
         ++$wroteline;
      }
      $totlines = $line;
      print OUT "\\end{alltt}\n\\end{small}\n";
      close(SRC);
      print "$inline lines\n";
      $wroteline += 2;
   } elsif ($_ =~ m/@\d+,.+@/) {
     # line contains [number,text]
     # e.g. @14,for (ix = 0)@
     $txt = $_;
     while ($txt =~ m/@\d+,.+@/) {
        @m = split("@",$txt);      # splits into text, one, two
        @parms = split(",",$m[1]);  # splits one,two into two elements 
                
        # now search from $parms[0] down for $parms[1] 
        $found1 = 0;
        $found2 = 0;
        for ($i = $parms[0]; $i < $totlines && $found1 == 0; $i++) {
           if ($text[$i] =~ m/\Q$parms[1]\E/) {
              $foundline1 = $i + 1;
              $found1 = 1;
           }
        }
        
        # now search backwards
        for ($i = $parms[0] - 1; $i >= 0 && $found2 == 0; $i--) {
           if ($text[$i] =~ m/\Q$parms[1]\E/) {
              $foundline2 = $i + 1;
              $found2 = 1;
           }
        }
        
        # now use the closest match or the first if tied
        if ($found1 == 1 && $found2 == 0) {
           $found = 1;
           $foundline = $foundline1;
        } elsif ($found1 == 0 && $found2 == 1) {
           $found = 1;
           $foundline = $foundline2;
        } elsif ($found1 == 1 && $found2 == 1) {
           $found = 1;
           if (($foundline1 - $parms[0]) <= ($parms[0] - $foundline2)) {
              $foundline = $foundline1;
           } else {
              $foundline = $foundline2;
           }
        } else {
           $found = 0;
        }
                      
        # if found replace 
        if ($found == 1) {
           $delta = $parms[0] - $foundline;
           print "Found replacement tag for \"$parms[1]\" on line $srcline which refers to line $foundline (delta $delta)\n";
           $_ =~ s/@\Q$m[1]\E@/$foundline/;
        } else {
           print "ERROR:  The tag \"$parms[1]\" on line $srcline was not found in the most recently parsed source!\n";
        }
        
        # remake the rest of the line 
        $cnt = @m;
        $txt = "";
        for ($i = 2; $i < $cnt; $i++) {
            $txt = $txt . $m[$i] . "@";
        }
     }
     print OUT $_;
     ++$wroteline;
   } elsif ($_ =~ /~.+~/) {
      # line contains a ~text~ pair used to refer to indexing :-)
      $txt = $_;
      while ($txt =~ /~.+~/) {
         @m = split("~", $txt);
         
         # word is the second position
         $word = @m[1];
         $a = $index1{$word};
         $b = $index2{$word};
         $c = $index3{$word};
         
         # if chapter (a) is zero it wasn't found
         if ($a == 0) {
            print "ERROR: the tag \"$word\" on line $srcline was not found previously marked.\n";
         } else {
            # format the tag as x, x.y or x.y.z depending on the values
            $str = $a;
            $str = $str . ".$b" if ($b != 0);
            $str = $str . ".$c" if ($c != 0);
            
            if ($b == 0 && $c == 0) {
               # its a chapter
               if ($a <= 10) {
                  if ($a == 1) {
                     $str = "chapter one";
                  } elsif ($a == 2) {
                     $str = "chapter two";







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#Tom St Denis

#get graphics type
if (shift =~ /PDF/) {
   $graph = "";
} else {
   $graph = ".ps";
}

open(IN,"<tommath.src") or die "Can't open source file";
open(OUT,">tommath.tex") or die "Can't open destination file";

print "Scanning for sections\n";
$chapter = $section = $subsection = 0;
$x = 0;
while (<IN>) {
   print ".";
   if (!(++$x % 80)) { print "\n"; }
   #update the headings
   if (~($_ =~ /\*/)) {
      if ($_ =~ /\\chapter\{.+}/) {
          ++$chapter;
          $section = $subsection = 0;
      } elsif ($_ =~ /\\section\{.+}/) {
          ++$section;
          $subsection = 0;
      } elsif ($_ =~ /\\subsection\{.+}/) {
          ++$subsection;
      }
   }

   if ($_ =~ m/MARK/) {
      @m = split(",",$_);
      chomp(@m[1]);
      $index1{@m[1]} = $chapter;
      $index2{@m[1]} = $section;
      $index3{@m[1]} = $subsection;
   }
}
close(IN);

open(IN,"<tommath.src") or die "Can't open source file";
$readline = $wroteline = 0;
$srcline = 0;

while (<IN>) {
   ++$readline;
   ++$srcline;

   if ($_ =~ m/MARK/) {
   } elsif ($_ =~ m/EXAM/ || $_ =~ m/LIST/) {
      if ($_ =~ m/EXAM/) {
         $skipheader = 1;
      } else {
         $skipheader = 0;
      }

      # EXAM,file
      chomp($_);
      @m = split(",",$_);
      open(SRC,"<$m[1]") or die "Error:$srcline:Can't open source file $m[1]";

      print "$srcline:Inserting $m[1]:";

      $line = 0;
      $tmp = $m[1];
      $tmp =~ s/_/"\\_"/ge;
      print OUT "\\vspace{+3mm}\\begin{small}\n\\hspace{-5.1mm}{\\bf File}: $tmp\n\\vspace{-3mm}\n\\begin{alltt}\n";
      $wroteline += 5;

      if ($skipheader == 1) {
         # scan till next end of comment, e.g. skip license
         while (<SRC>) {
            $text[$line++] = $_;
            last if ($_ =~ /libtom\.org/);
         }
         <SRC>;
      }

      $inline = 0;
      while (<SRC>) {
      next if ($_ =~ /\$Source/);
      next if ($_ =~ /\$Revision/);
      next if ($_ =~ /\$Date/);
         $text[$line++] = $_;
         ++$inline;
         chomp($_);
         $_ =~ s/\t/"    "/ge;
         $_ =~ s/{/"^{"/ge;
         $_ =~ s/}/"^}"/ge;
         $_ =~ s/\\/'\symbol{92}'/ge;
         $_ =~ s/\^/"\\"/ge;

         printf OUT ("%03d   ", $line);
         for ($x = 0; $x < length($_); $x++) {
             print OUT chr(vec($_, $x, 8));
             if ($x == 75) {
                 print OUT "\n      ";
                 ++$wroteline;
             }
         }
         print OUT "\n";
         ++$wroteline;
      }
      $totlines = $line;
      print OUT "\\end{alltt}\n\\end{small}\n";
      close(SRC);
      print "$inline lines\n";
      $wroteline += 2;
   } elsif ($_ =~ m/@\d+,.+@/) {
     # line contains [number,text]
     # e.g. @14,for (ix = 0)@
     $txt = $_;
     while ($txt =~ m/@\d+,.+@/) {
        @m = split("@",$txt);      # splits into text, one, two
        @parms = split(",",$m[1]);  # splits one,two into two elements

        # now search from $parms[0] down for $parms[1]
        $found1 = 0;
        $found2 = 0;
        for ($i = $parms[0]; $i < $totlines && $found1 == 0; $i++) {
           if ($text[$i] =~ m/\Q$parms[1]\E/) {
              $foundline1 = $i + 1;
              $found1 = 1;
           }
        }

        # now search backwards
        for ($i = $parms[0] - 1; $i >= 0 && $found2 == 0; $i--) {
           if ($text[$i] =~ m/\Q$parms[1]\E/) {
              $foundline2 = $i + 1;
              $found2 = 1;
           }
        }

        # now use the closest match or the first if tied
        if ($found1 == 1 && $found2 == 0) {
           $found = 1;
           $foundline = $foundline1;
        } elsif ($found1 == 0 && $found2 == 1) {
           $found = 1;
           $foundline = $foundline2;
        } elsif ($found1 == 1 && $found2 == 1) {
           $found = 1;
           if (($foundline1 - $parms[0]) <= ($parms[0] - $foundline2)) {
              $foundline = $foundline1;
           } else {
              $foundline = $foundline2;
           }
        } else {
           $found = 0;
        }

        # if found replace
        if ($found == 1) {
           $delta = $parms[0] - $foundline;
           print "Found replacement tag for \"$parms[1]\" on line $srcline which refers to line $foundline (delta $delta)\n";
           $_ =~ s/@\Q$m[1]\E@/$foundline/;
        } else {
           print "ERROR:  The tag \"$parms[1]\" on line $srcline was not found in the most recently parsed source!\n";
        }

        # remake the rest of the line
        $cnt = @m;
        $txt = "";
        for ($i = 2; $i < $cnt; $i++) {
            $txt = $txt . $m[$i] . "@";
        }
     }
     print OUT $_;
     ++$wroteline;
   } elsif ($_ =~ /~.+~/) {
      # line contains a ~text~ pair used to refer to indexing :-)
      $txt = $_;
      while ($txt =~ /~.+~/) {
         @m = split("~", $txt);

         # word is the second position
         $word = @m[1];
         $a = $index1{$word};
         $b = $index2{$word};
         $c = $index3{$word};

         # if chapter (a) is zero it wasn't found
         if ($a == 0) {
            print "ERROR: the tag \"$word\" on line $srcline was not found previously marked.\n";
         } else {
            # format the tag as x, x.y or x.y.z depending on the values
            $str = $a;
            $str = $str . ".$b" if ($b != 0);
            $str = $str . ".$c" if ($c != 0);

            if ($b == 0 && $c == 0) {
               # its a chapter
               if ($a <= 10) {
                  if ($a == 1) {
                     $str = "chapter one";
                  } elsif ($a == 2) {
                     $str = "chapter two";
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                  } elsif ($a == 10) {
                     $str = "chapter ten";
                  }
               } else {
                  $str = "chapter " . $str;
               }
            } else {
               $str = "section " . $str     if ($b != 0 && $c == 0);            
               $str = "sub-section " . $str if ($b != 0 && $c != 0);
            }
            
            #substitute
            $_ =~ s/~\Q$word\E~/$str/;
            
            print "Found replacement tag for marker \"$word\" on line $srcline which refers to $str\n";
         }
         
         # remake rest of the line
         $cnt = @m;
         $txt = "";
         for ($i = 2; $i < $cnt; $i++) {
             $txt = $txt . $m[$i] . "~";
         }
      }







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                  } elsif ($a == 10) {
                     $str = "chapter ten";
                  }
               } else {
                  $str = "chapter " . $str;
               }
            } else {
               $str = "section " . $str     if ($b != 0 && $c == 0);
               $str = "sub-section " . $str if ($b != 0 && $c != 0);
            }

            #substitute
            $_ =~ s/~\Q$word\E~/$str/;

            print "Found replacement tag for marker \"$word\" on line $srcline which refers to $str\n";
         }

         # remake rest of the line
         $cnt = @m;
         $txt = "";
         for ($i = 2; $i < $cnt; $i++) {
             $txt = $txt . $m[$i] . "~";
         }
      }
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      ++$wroteline;
   }
}
print "Read $readline lines, wrote $wroteline lines\n";

close (OUT);
close (IN);









>
>
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      ++$wroteline;
   }
}
print "Read $readline lines, wrote $wroteline lines\n";

close (OUT);
close (IN);

system('perl -pli -e "s/\s*$//" tommath.tex');
Changes to libtommath/callgraph.txt.

more than 10,000 changes

Changes to libtommath/changes.txt.





















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July 23rd, 2010
v0.42.0
       -- Fix for mp_prime_next_prime() bug when checking generated prime
       -- allow mp_shrink to shrink initialized, but empty MPI's
       -- Added project and solution files for Visual Studio 2005 and Visual Studio 2008. 

March 10th, 2007
v0.41  -- Wolfgang Ehrhardt suggested a quick fix to mp_div_d() which makes the detection of powers of two quicker. 
       -- [CRI] Added libtommath.dsp for Visual C++ users.

December 24th, 2006
v0.40  -- Updated makefile to properly support LIBNAME
       -- Fixed bug in fast_s_mp_mul_high_digs() which overflowed (line 83), thanks Valgrind!

April 4th, 2006
v0.39  -- Jim Wigginton pointed out my Montgomery examples in figures 6.4 and 6.6 were off by one, k should be 9 not 8
       -- Bruce Guenter suggested I use --tag=CC for libtool builds where the compiler may think it's C++.
       -- "mm" from sci.crypt pointed out that my mp_gcd was sub-optimal (I also updated and corrected the book)
       -- updated some of the @@ tags in tommath.src to reflect source changes.
       -- updated email and url info in all source files

Jan 26th, 2006
v0.38  -- broken makefile.shared fixed
       -- removed some carry stores that were not required [updated text]
       
November 18th, 2005
v0.37  -- [Don Porter] reported on a TCL list [HEY SEND ME BUGREPORTS ALREADY!!!] that mp_add_d() would compute -0 with some inputs.  Fixed.
       -- [[email protected]] reported the makefile.bcc was messed up.  Fixed.
       -- [Kevin Kenny] reported some issues with mp_toradix_n().  Now it doesn't require a min of 3 chars of output.  
       -- Made the make command renamable.  Wee

August 1st, 2005
v0.36  -- LTM_PRIME_2MSB_ON was fixed and the "OFF" flag was removed.
       -- [Peter LaDow] found a typo in the XREALLOC macro
       -- [Peter LaDow] pointed out that mp_read_(un)signed_bin should have "const" on the input
       -- Ported LTC patch to fix the prime_random_ex() function to get the bitsize correct [and the maskOR flags]
       -- Kevin Kenny pointed out a stray //
       -- David Hulton pointed out a typo in the textbook [mp_montgomery_setup() pseudo-code]
       -- Neal Hamilton (Elliptic Semiconductor) pointed out that my Karatsuba notation was backwards and that I could use 
          unsigned operations in the routine.  
       -- Paul Schmidt pointed out a linking error in mp_exptmod() when BN_S_MP_EXPTMOD_C is undefined (and another for read_radix)
       -- Updated makefiles to be way more flexible

March 12th, 2005
v0.35  -- Stupid XOR function missing line again... oops.
       -- Fixed bug in invmod not handling negative inputs correctly [Wolfgang Ehrhardt]
       -- Made exteuclid always give positive u3 output...[ Wolfgang Ehrhardt ]
       -- [Wolfgang Ehrhardt] Suggested a fix for mp_reduce() which avoided underruns.  ;-)
       -- mp_rand() would emit one too many digits and it was possible to get a 0 out of it ... oops
       -- Added montgomery to the testing to make sure it handles 1..10 digit moduli correctly
       -- Fixed bug in comba that would lead to possible erroneous outputs when "pa < digs" 
       -- Fixed bug in mp_toradix_size for "0" [Kevin Kenny]
       -- Updated chapters 1-5 of the textbook ;-) It now talks about the new comba code!

February 12th, 2005
v0.34  -- Fixed two more small errors in mp_prime_random_ex()
       -- Fixed overflow in mp_mul_d() [Kevin Kenny]
       -- Added mp_to_(un)signed_bin_n() functions which do bounds checking for ya [and report the size]
       -- Added "large" diminished radix support.  Speeds up things like DSA where the moduli is of the form 2^k - P for some P < 2^(k/2) or so
          Actually is faster than Montgomery on my AMD64 (and probably much faster on a P4)
       -- Updated the manual a bit
       -- Ok so I haven't done the textbook work yet... My current freelance gig has landed me in France till the 
          end of Feb/05.  Once I get back I'll have tons of free time and I plan to go to town on the book.
          As of this release the API will freeze.  At least until the book catches up with all the changes.  I welcome
          bug reports but new algorithms will have to wait.

December 23rd, 2004
v0.33  -- Fixed "small" variant for mp_div() which would munge with negative dividends...
       -- Fixed bug in mp_prime_random_ex() which would set the most significant byte to zero when
          no special flags were set
       -- Fixed overflow [minor] bug in fast_s_mp_sqr()
       -- Made the makefiles easier to configure the group/user that ltm will install as
       -- Fixed "final carry" bug in comba multipliers. (Volkan Ceylan)
       -- Matt Johnston pointed out a missing semi-colon in mp_exptmod

October 29th, 2004
v0.32  -- Added "makefile.shared" for shared object support
       -- Added more to the build options/configs in the manual
       -- Started the Depends framework, wrote dep.pl to scan deps and 
          produce "callgraph.txt" ;-)
       -- Wrote SC_RSA_1 which will enable close to the minimum required to perform
          RSA on 32-bit [or 64-bit] platforms with LibTomCrypt
       -- Merged in the small/slower mp_div replacement.  You can now toggle which
          you want to use as your mp_div() at build time.  Saves roughly 8KB or so.
       -- Renamed a few files and changed some comments to make depends system work better.
          (No changes to function names)
       -- Merged in new Combas that perform 2 reads per inner loop instead of the older 
          3reads/2writes per inner loop of the old code.  Really though if you want speed
          learn to use TomsFastMath ;-)

August 9th, 2004
v0.31  -- "profiled" builds now :-) new timings for Intel Northwoods
       -- Added "pretty" build target
       -- Update mp_init() to actually assign 0's instead of relying on calloc()
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Feb 5th, 2016
v1.0.0
       -- Bump to 1.0.0
       -- Dirkjan Bussink provided a faster version of mp_expt_d()
       -- Moritz Lenz contributed a fix to mp_mod()
          and provided mp_get_long() and mp_set_long()
       -- Fixed bugs in mp_read_radix(), mp_radix_size
          Thanks to shameister, Gerhard R,
       -- Christopher Brown provided mp_export() and mp_import()
       -- Improvements in the code of mp_init_copy()
          Thanks to ramkumarkoppu,
       -- lomereiter provided mp_balance_mul()
       -- Alexander Boström from the heimdal project contributed patches to
          mp_prime_next_prime() and mp_invmod() and added a mp_isneg() macro
       -- Fix build issues for Linux x32 ABI
       -- Added mp_get_long_long() and mp_set_long_long()
       -- Carlin provided a patch to use arc4random() instead of rand()
          on platforms where it is supported
       -- Karel Miko provided mp_sqrtmod_prime()


July 23rd, 2010
v0.42.0
       -- Fix for mp_prime_next_prime() bug when checking generated prime
       -- allow mp_shrink to shrink initialized, but empty MPI's
       -- Added project and solution files for Visual Studio 2005 and Visual Studio 2008.

March 10th, 2007
v0.41  -- Wolfgang Ehrhardt suggested a quick fix to mp_div_d() which makes the detection of powers of two quicker.
       -- [CRI] Added libtommath.dsp for Visual C++ users.

December 24th, 2006
v0.40  -- Updated makefile to properly support LIBNAME
       -- Fixed bug in fast_s_mp_mul_high_digs() which overflowed (line 83), thanks Valgrind!

April 4th, 2006
v0.39  -- Jim Wigginton pointed out my Montgomery examples in figures 6.4 and 6.6 were off by one, k should be 9 not 8
       -- Bruce Guenter suggested I use --tag=CC for libtool builds where the compiler may think it's C++.
       -- "mm" from sci.crypt pointed out that my mp_gcd was sub-optimal (I also updated and corrected the book)
       -- updated some of the @@ tags in tommath.src to reflect source changes.
       -- updated email and url info in all source files

Jan 26th, 2006
v0.38  -- broken makefile.shared fixed
       -- removed some carry stores that were not required [updated text]

November 18th, 2005
v0.37  -- [Don Porter] reported on a TCL list [HEY SEND ME BUGREPORTS ALREADY!!!] that mp_add_d() would compute -0 with some inputs.  Fixed.
       -- [[email protected]] reported the makefile.bcc was messed up.  Fixed.
       -- [Kevin Kenny] reported some issues with mp_toradix_n().  Now it doesn't require a min of 3 chars of output.
       -- Made the make command renamable.  Wee

August 1st, 2005
v0.36  -- LTM_PRIME_2MSB_ON was fixed and the "OFF" flag was removed.
       -- [Peter LaDow] found a typo in the XREALLOC macro
       -- [Peter LaDow] pointed out that mp_read_(un)signed_bin should have "const" on the input
       -- Ported LTC patch to fix the prime_random_ex() function to get the bitsize correct [and the maskOR flags]
       -- Kevin Kenny pointed out a stray //
       -- David Hulton pointed out a typo in the textbook [mp_montgomery_setup() pseudo-code]
       -- Neal Hamilton (Elliptic Semiconductor) pointed out that my Karatsuba notation was backwards and that I could use
          unsigned operations in the routine.
       -- Paul Schmidt pointed out a linking error in mp_exptmod() when BN_S_MP_EXPTMOD_C is undefined (and another for read_radix)
       -- Updated makefiles to be way more flexible

March 12th, 2005
v0.35  -- Stupid XOR function missing line again... oops.
       -- Fixed bug in invmod not handling negative inputs correctly [Wolfgang Ehrhardt]
       -- Made exteuclid always give positive u3 output...[ Wolfgang Ehrhardt ]
       -- [Wolfgang Ehrhardt] Suggested a fix for mp_reduce() which avoided underruns.  ;-)
       -- mp_rand() would emit one too many digits and it was possible to get a 0 out of it ... oops
       -- Added montgomery to the testing to make sure it handles 1..10 digit moduli correctly
       -- Fixed bug in comba that would lead to possible erroneous outputs when "pa < digs"
       -- Fixed bug in mp_toradix_size for "0" [Kevin Kenny]
       -- Updated chapters 1-5 of the textbook ;-) It now talks about the new comba code!

February 12th, 2005
v0.34  -- Fixed two more small errors in mp_prime_random_ex()
       -- Fixed overflow in mp_mul_d() [Kevin Kenny]
       -- Added mp_to_(un)signed_bin_n() functions which do bounds checking for ya [and report the size]
       -- Added "large" diminished radix support.  Speeds up things like DSA where the moduli is of the form 2^k - P for some P < 2^(k/2) or so
          Actually is faster than Montgomery on my AMD64 (and probably much faster on a P4)
       -- Updated the manual a bit
       -- Ok so I haven't done the textbook work yet... My current freelance gig has landed me in France till the
          end of Feb/05.  Once I get back I'll have tons of free time and I plan to go to town on the book.
          As of this release the API will freeze.  At least until the book catches up with all the changes.  I welcome
          bug reports but new algorithms will have to wait.

December 23rd, 2004
v0.33  -- Fixed "small" variant for mp_div() which would munge with negative dividends...
       -- Fixed bug in mp_prime_random_ex() which would set the most significant byte to zero when
          no special flags were set
       -- Fixed overflow [minor] bug in fast_s_mp_sqr()
       -- Made the makefiles easier to configure the group/user that ltm will install as
       -- Fixed "final carry" bug in comba multipliers. (Volkan Ceylan)
       -- Matt Johnston pointed out a missing semi-colon in mp_exptmod

October 29th, 2004
v0.32  -- Added "makefile.shared" for shared object support
       -- Added more to the build options/configs in the manual
       -- Started the Depends framework, wrote dep.pl to scan deps and
          produce "callgraph.txt" ;-)
       -- Wrote SC_RSA_1 which will enable close to the minimum required to perform
          RSA on 32-bit [or 64-bit] platforms with LibTomCrypt
       -- Merged in the small/slower mp_div replacement.  You can now toggle which
          you want to use as your mp_div() at build time.  Saves roughly 8KB or so.
       -- Renamed a few files and changed some comments to make depends system work better.
          (No changes to function names)
       -- Merged in new Combas that perform 2 reads per inner loop instead of the older
          3reads/2writes per inner loop of the old code.  Really though if you want speed
          learn to use TomsFastMath ;-)

August 9th, 2004
v0.31  -- "profiled" builds now :-) new timings for Intel Northwoods
       -- Added "pretty" build target
       -- Update mp_init() to actually assign 0's instead of relying on calloc()
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          is only accurate to byte lengths).  See the new LTM_PRIME_* flags ;-)
       -- Alex Polushin contributed an optimized mp_sqrt() as well as mp_get_int() and mp_is_square().
          I've cleaned them all up to be a little more consistent [along with one bug fix] for this release.
       -- Added mp_init_set and mp_init_set_int to initialize and set small constants with one function
          call.
       -- Removed /etclib directory [um LibTomPoly deprecates this].
       -- Fixed mp_mod() so the sign of the result agrees with the sign of the modulus.
       ++ N.B.  My semester is almost up so expect updates to the textbook to be posted to the libtomcrypt.org 
          website.  

Jan 25th, 2004
v0.29  ++ Note: "Henrik" from the v0.28 changelog refers to Henrik Goldman ;-)
       -- Added fix to mp_shrink to prevent a realloc when used == 0 [e.g. realloc zero bytes???]
       -- Made the mp_prime_rabin_miller_trials() function internal table smaller and also
          set the minimum number of tests to two (sounds a bit safer).
       -- Added a mp_exteuclid() which computes the extended euclidean algorithm.







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          is only accurate to byte lengths).  See the new LTM_PRIME_* flags ;-)
       -- Alex Polushin contributed an optimized mp_sqrt() as well as mp_get_int() and mp_is_square().
          I've cleaned them all up to be a little more consistent [along with one bug fix] for this release.
       -- Added mp_init_set and mp_init_set_int to initialize and set small constants with one function
          call.
       -- Removed /etclib directory [um LibTomPoly deprecates this].
       -- Fixed mp_mod() so the sign of the result agrees with the sign of the modulus.
       ++ N.B.  My semester is almost up so expect updates to the textbook to be posted to the libtomcrypt.org
          website.

Jan 25th, 2004
v0.29  ++ Note: "Henrik" from the v0.28 changelog refers to Henrik Goldman ;-)
       -- Added fix to mp_shrink to prevent a realloc when used == 0 [e.g. realloc zero bytes???]
       -- Made the mp_prime_rabin_miller_trials() function internal table smaller and also
          set the minimum number of tests to two (sounds a bit safer).
       -- Added a mp_exteuclid() which computes the extended euclidean algorithm.
Changes to libtommath/demo/demo.c.

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#include <time.h>

#ifdef IOWNANATHLON
#include <unistd.h>
#define SLEEP sleep(4)
#else
#define SLEEP
#endif























#include "tommath.h"

void ndraw(mp_int * a, char *name)
{
   char buf[16000];

   printf("%s: ", name);
   mp_toradix(a, buf, 10);
   printf("%s\n", buf);


}


static void draw(mp_int * a)
{
   ndraw(a, "");
}



unsigned long lfsr = 0xAAAAAAAAUL;

int lbit(void)
{
   if (lfsr & 0x80000000UL) {
      lfsr = ((lfsr << 1) ^ 0x8000001BUL) & 0xFFFFFFFFUL;
      return 1;
   } else {
      lfsr <<= 1;
      return 0;
   }
}




int myrng(unsigned char *dst, int len, void *dat)
{
   int x;











   for (x = 0; x < len; x++)


      dst[x] = rand() & 0xFF;



   return len;
}










































char cmd[4096], buf[4096];
int main(void)
{

   mp_int a, b, c, d, e, f;

   unsigned long expt_n, add_n, sub_n, mul_n, div_n, sqr_n, mul2d_n, div2d_n,
      gcd_n, lcm_n, inv_n, div2_n, mul2_n, add_d_n, sub_d_n, t;


   unsigned rr;
   int i, n, err, cnt, ix, old_kara_m, old_kara_s;

   mp_digit mp;







   mp_init(&a);











   mp_init(&b);
















   mp_init(&c);




















   mp_init(&d);
   mp_init(&e);































   mp_init(&f);









































































































































   srand(time(NULL));










#if 0


























   // test montgomery 
   printf("Testing montgomery...\n");
   for (i = 1; i < 10; i++) {


      printf("Testing digit size: %d\n", i);

      for (n = 0; n < 1000; n++) {
         mp_rand(&a, i);
         a.dp[0] |= 1;

         // let's see if R is right
         mp_montgomery_calc_normalization(&b, &a);
         mp_montgomery_setup(&a, &mp);

         // now test a random reduction 
         for (ix = 0; ix < 100; ix++) {
             mp_rand(&c, 1 + abs(rand()) % (2*i));
             mp_copy(&c, &d);
             mp_copy(&c, &e);

             mp_mod(&d, &a, &d);
             mp_montgomery_reduce(&c, &a, mp);
             mp_mulmod(&c, &b, &a, &c);

             if (mp_cmp(&c, &d) != MP_EQ) { 
printf("d = e mod a, c = e MOD a\n");
mp_todecimal(&a, buf); printf("a = %s\n", buf);
mp_todecimal(&e, buf); printf("e = %s\n", buf);
mp_todecimal(&d, buf); printf("d = %s\n", buf);
mp_todecimal(&c, buf); printf("c = %s\n", buf);
printf("compare no compare!\n"); exit(EXIT_FAILURE); }
         }
      }
   }
   printf("done\n");

   // test mp_get_int
   printf("Testing: mp_get_int\n");
   for (i = 0; i < 1000; ++i) {
      t = ((unsigned long) rand() * rand() + 1) & 0xFFFFFFFF;
      mp_set_int(&a, t);
      if (t != mp_get_int(&a)) {
	 printf("mp_get_int() bad result!\n");
	 return 1;
      }
   }
   mp_set_int(&a, 0);
   if (mp_get_int(&a) != 0) {
      printf("mp_get_int() bad result!\n");
      return 1;
   }
   mp_set_int(&a, 0xffffffff);
   if (mp_get_int(&a) != 0xffffffff) {
      printf("mp_get_int() bad result!\n");
      return 1;
   }
   // test mp_sqrt
   printf("Testing: mp_sqrt\n");
   for (i = 0; i < 1000; ++i) {
      printf("%6d\r", i);
      fflush(stdout);
      n = (rand() & 15) + 1;
      mp_rand(&a, n);
      if (mp_sqrt(&a, &b) != MP_OKAY) {
	 printf("mp_sqrt() error!\n");
	 return 1;
      }
      mp_n_root(&a, 2, &a);
      if (mp_cmp_mag(&b, &a) != MP_EQ) {
	 printf("mp_sqrt() bad result!\n");
	 return 1;
      }
   }

   printf("\nTesting: mp_is_square\n");
   for (i = 0; i < 1000; ++i) {
      printf("%6d\r", i);
      fflush(stdout);

      /* test mp_is_square false negatives */
      n = (rand() & 7) + 1;
      mp_rand(&a, n);
      mp_sqr(&a, &a);
      if (mp_is_square(&a, &n) != MP_OKAY) {
	 printf("fn:mp_is_square() error!\n");
	 return 1;
      }
      if (n == 0) {
	 printf("fn:mp_is_square() bad result!\n");
	 return 1;
      }

      /* test for false positives */
      mp_add_d(&a, 1, &a);
      if (mp_is_square(&a, &n) != MP_OKAY) {
	 printf("fp:mp_is_square() error!\n");
	 return 1;
      }
      if (n == 1) {
	 printf("fp:mp_is_square() bad result!\n");
	 return 1;
      }

   }
   printf("\n\n");

   /* test for size */
   for (ix = 10; ix < 128; ix++) {
      printf("Testing (not safe-prime): %9d bits    \r", ix);
      fflush(stdout);
      err =
	 mp_prime_random_ex(&a, 8, ix,
			    (rand() & 1) ? LTM_PRIME_2MSB_OFF :
			    LTM_PRIME_2MSB_ON, myrng, NULL);
      if (err != MP_OKAY) {
	 printf("failed with err code %d\n", err);
	 return EXIT_FAILURE;
      }
      if (mp_count_bits(&a) != ix) {
	 printf("Prime is %d not %d bits!!!\n", mp_count_bits(&a), ix);
	 return EXIT_FAILURE;
      }
   }

   for (ix = 16; ix < 128; ix++) {
      printf("Testing (   safe-prime): %9d bits    \r", ix);
      fflush(stdout);
      err =
	 mp_prime_random_ex(&a, 8, ix,
			    ((rand() & 1) ? LTM_PRIME_2MSB_OFF :
			     LTM_PRIME_2MSB_ON) | LTM_PRIME_SAFE, myrng,
			    NULL);
      if (err != MP_OKAY) {
	 printf("failed with err code %d\n", err);
	 return EXIT_FAILURE;
      }
      if (mp_count_bits(&a) != ix) {
	 printf("Prime is %d not %d bits!!!\n", mp_count_bits(&a), ix);
	 return EXIT_FAILURE;
      }
      /* let's see if it's really a safe prime */
      mp_sub_d(&a, 1, &a);
      mp_div_2(&a, &a);
      mp_prime_is_prime(&a, 8, &cnt);
      if (cnt != MP_YES) {
	 printf("sub is not prime!\n");
	 return EXIT_FAILURE;
      }
   }

   printf("\n\n");

   mp_read_radix(&a, "123456", 10);
   mp_toradix_n(&a, buf, 10, 3);
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#include <string.h>
#include <time.h>

#ifdef IOWNANATHLON
#include <unistd.h>
#define SLEEP sleep(4)
#else
#define SLEEP
#endif

/*
 * Configuration
 */
#ifndef LTM_DEMO_TEST_VS_MTEST
#define LTM_DEMO_TEST_VS_MTEST 1
#endif

#ifndef LTM_DEMO_TEST_REDUCE_2K_L
/* This test takes a moment so we disable it by default, but it can be:
 * 0 to disable testing
 * 1 to make the test with P = 2^1024 - 0x2A434 B9FDEC95 D8F9D550 FFFFFFFF FFFFFFFF
 * 2 to make the test with P = 2^2048 - 0x1 00000000 00000000 00000000 00000000 4945DDBF 8EA2A91D 5776399B B83E188F
 */
#define LTM_DEMO_TEST_REDUCE_2K_L 0
#endif

#ifdef LTM_DEMO_REAL_RAND
#define LTM_DEMO_RAND_SEED  time(NULL)
#else
#define LTM_DEMO_RAND_SEED  23
#endif

#include "tommath.h"

void ndraw(mp_int * a, char *name)
{
   char buf[16000];

   printf("%s: ", name);
   mp_toradix(a, buf, 10);
   printf("%s\n", buf);
   mp_toradix(a, buf, 16);
   printf("0x%s\n", buf);
}

#if LTM_DEMO_TEST_VS_MTEST
static void draw(mp_int * a)
{
   ndraw(a, "");
}
#endif


unsigned long lfsr = 0xAAAAAAAAUL;

int lbit(void)
{
   if (lfsr & 0x80000000UL) {
      lfsr = ((lfsr << 1) ^ 0x8000001BUL) & 0xFFFFFFFFUL;
      return 1;
   } else {
      lfsr <<= 1;
      return 0;
   }
}

#if defined(LTM_DEMO_REAL_RAND) && !defined(_WIN32)
static FILE* fd_urandom;
#endif
int myrng(unsigned char *dst, int len, void *dat)
{
   int x;
   (void)dat;
#if defined(LTM_DEMO_REAL_RAND)
   if (!fd_urandom) {
#if !defined(_WIN32)
      fprintf(stderr, "\nno /dev/urandom\n");
#endif
   }
   else {
      return fread(dst, 1, len, fd_urandom);
   }
#endif
   for (x = 0; x < len; ) {
      unsigned int r = (unsigned int)rand();
      do {
         dst[x++] = r & 0xFF;
         r >>= 8;
      } while((r != 0) && (x < len));
   }
   return len;
}

#if LTM_DEMO_TEST_VS_MTEST != 0
static void _panic(int l)
{
  fprintf(stderr, "\n%d: fgets failed\n", l);
  exit(EXIT_FAILURE);
}
#endif

mp_int a, b, c, d, e, f;

static void _cleanup(void)
{
  mp_clear_multi(&a, &b, &c, &d, &e, &f, NULL);
  printf("\n");

#ifdef LTM_DEMO_REAL_RAND
  if(fd_urandom)
     fclose(fd_urandom);
#endif
}
struct mp_sqrtmod_prime_st {
   unsigned long p;
   unsigned long n;
   mp_digit r;
};
struct mp_sqrtmod_prime_st sqrtmod_prime[] = {
      { 5, 14, 3 },
      { 7, 9, 4 },
      { 113, 2, 62 }
};
struct mp_jacobi_st {
   unsigned long n;
   int c[16];
};
struct mp_jacobi_st jacobi[] = {
      { 3, {  1, -1,  0,  1, -1,  0,  1, -1,  0,  1, -1,  0,  1, -1,  0,  1 } },
      { 5, {  0,  1, -1, -1,  1,  0,  1, -1, -1,  1,  0,  1, -1, -1,  1,  0 } },
      { 7, {  1, -1,  1, -1, -1,  0,  1,  1, -1,  1, -1, -1,  0,  1,  1, -1 } },
      { 9, { -1,  1,  0,  1,  1,  0,  1,  1,  0,  1,  1,  0,  1,  1,  0,  1 } },
};

char cmd[4096], buf[4096];
int main(void)
{
   unsigned rr;
   int cnt, ix;
#if LTM_DEMO_TEST_VS_MTEST
   unsigned long expt_n, add_n, sub_n, mul_n, div_n, sqr_n, mul2d_n, div2d_n,
      gcd_n, lcm_n, inv_n, div2_n, mul2_n, add_d_n, sub_d_n;
   char* ret;
#else
   unsigned long s, t;

   unsigned long long q, r;
   mp_digit mp;
   int i, n, err, should;
#endif

   if (mp_init_multi(&a, &b, &c, &d, &e, &f, NULL)!= MP_OKAY)
     return EXIT_FAILURE;

   atexit(_cleanup);

#if defined(LTM_DEMO_REAL_RAND)
   if (!fd_urandom) {
      fd_urandom = fopen("/dev/urandom", "r");
      if (!fd_urandom) {
#if !defined(_WIN32)
         fprintf(stderr, "\ncould not open /dev/urandom\n");
#endif
      }
   }
#endif
   srand(LTM_DEMO_RAND_SEED);

#ifdef MP_8BIT
   printf("Digit size 8 Bit \n");
#endif
#ifdef MP_16BIT
   printf("Digit size 16 Bit \n");
#endif
#ifdef MP_32BIT
   printf("Digit size 32 Bit \n");
#endif
#ifdef MP_64BIT
   printf("Digit size 64 Bit \n");
#endif
   printf("Size of mp_digit: %u\n", (unsigned int)sizeof(mp_digit));
   printf("Size of mp_word: %u\n", (unsigned int)sizeof(mp_word));
   printf("DIGIT_BIT: %d\n", DIGIT_BIT);
   printf("MP_PREC: %d\n", MP_PREC);

#if LTM_DEMO_TEST_VS_MTEST == 0
   // trivial stuff
   mp_set_int(&a, 5);
   mp_neg(&a, &b);
   if (mp_cmp(&a, &b) != MP_GT) {
      return EXIT_FAILURE;
   }
   if (mp_cmp(&b, &a) != MP_LT) {
      return EXIT_FAILURE;
   }
   mp_neg(&a, &a);
   if (mp_cmp(&b, &a) != MP_EQ) {
      return EXIT_FAILURE;
   }
   mp_abs(&a, &b);
   if (mp_isneg(&b) != MP_NO) {
      return EXIT_FAILURE;
   }
   mp_add_d(&a, 1, &b);
   mp_add_d(&a, 6, &b);


   mp_set_int(&a, 0);
   mp_set_int(&b, 1);
   if ((err = mp_jacobi(&a, &b, &i)) != MP_OKAY) {
      printf("Failed executing mp_jacobi(0 | 1) %s.\n", mp_error_to_string(err));
      return EXIT_FAILURE;
   }
   if (i != 1) {
      printf("Failed trivial mp_jacobi(0 | 1) %d != 1\n", i);
      return EXIT_FAILURE;
   }
   for (cnt = 0; cnt < (int)(sizeof(jacobi)/sizeof(jacobi[0])); ++cnt) {
      mp_set_int(&b, jacobi[cnt].n);
      /* only test positive values of a */
      for (n = -5; n <= 10; ++n) {
         mp_set_int(&a, abs(n));
         should = MP_OKAY;
         if (n < 0) {
            mp_neg(&a, &a);
            /* Until #44 is fixed the negative a's must fail */
            should = MP_VAL;
         }
         if ((err = mp_jacobi(&a, &b, &i)) != should) {
            printf("Failed executing mp_jacobi(%d | %lu) %s.\n", n, jacobi[cnt].n, mp_error_to_string(err));
            return EXIT_FAILURE;
         }
         if (err == MP_OKAY && i != jacobi[cnt].c[n + 5]) {
            printf("Failed trivial mp_jacobi(%d | %lu) %d != %d\n", n, jacobi[cnt].n, i, jacobi[cnt].c[n + 5]);
            return EXIT_FAILURE;
         }
      }
   }

   // test mp_get_int
   printf("\n\nTesting: mp_get_int");
   for (i = 0; i < 1000; ++i) {
      t = ((unsigned long) rand () * rand () + 1) & 0xFFFFFFFF;
      mp_set_int (&a, t);
      if (t != mp_get_int (&a)) {
         printf ("\nmp_get_int() bad result!");
         return EXIT_FAILURE;
      }
   }
   mp_set_int(&a, 0);
   if (mp_get_int(&a) != 0) {
      printf("\nmp_get_int() bad result!");
      return EXIT_FAILURE;
   }
   mp_set_int(&a, 0xffffffff);
   if (mp_get_int(&a) != 0xffffffff) {
      printf("\nmp_get_int() bad result!");
      return EXIT_FAILURE;
   }

   printf("\n\nTesting: mp_get_long\n");
   for (i = 0; i < (int)(sizeof(unsigned long)*CHAR_BIT) - 1; ++i) {
      t = (1ULL << (i+1)) - 1;
      if (!t)
         t = -1;
      printf(" t = 0x%lx i = %d\r", t, i);
      do {
         if (mp_set_long(&a, t) != MP_OKAY) {
            printf("\nmp_set_long() error!");
            return EXIT_FAILURE;
         }
         s = mp_get_long(&a);
         if (s != t) {
            printf("\nmp_get_long() bad result! 0x%lx != 0x%lx", s, t);
            return EXIT_FAILURE;
         }
         t <<= 1;
      } while(t);
   }

   printf("\n\nTesting: mp_get_long_long\n");
   for (i = 0; i < (int)(sizeof(unsigned long long)*CHAR_BIT) - 1; ++i) {
      r = (1ULL << (i+1)) - 1;
      if (!r)
         r = -1;
      printf(" r = 0x%llx i = %d\r", r, i);
      do {
         if (mp_set_long_long(&a, r) != MP_OKAY) {
            printf("\nmp_set_long_long() error!");
            return EXIT_FAILURE;
         }
         q = mp_get_long_long(&a);
         if (q != r) {
            printf("\nmp_get_long_long() bad result! 0x%llx != 0x%llx", q, r);
            return EXIT_FAILURE;
         }
         r <<= 1;
      } while(r);
   }

   // test mp_sqrt
   printf("\n\nTesting: mp_sqrt\n");
   for (i = 0; i < 1000; ++i) {
      printf ("%6d\r", i);
      fflush (stdout);
      n = (rand () & 15) + 1;
      mp_rand (&a, n);
      if (mp_sqrt (&a, &b) != MP_OKAY) {
         printf ("\nmp_sqrt() error!");
         return EXIT_FAILURE;
      }
      mp_n_root_ex (&a, 2, &c, 0);
      mp_n_root_ex (&a, 2, &d, 1);
      if (mp_cmp_mag (&c, &d) != MP_EQ) {
         printf ("\nmp_n_root_ex() bad result!");
         return EXIT_FAILURE;
      }
      if (mp_cmp_mag (&b, &c) != MP_EQ) {
         printf ("mp_sqrt() bad result!\n");
         return EXIT_FAILURE;
      }
   }

   printf("\n\nTesting: mp_is_square\n");
   for (i = 0; i < 1000; ++i) {
      printf ("%6d\r", i);
      fflush (stdout);

      /* test mp_is_square false negatives */
      n = (rand () & 7) + 1;
      mp_rand (&a, n);
      mp_sqr (&a, &a);
      if (mp_is_square (&a, &n) != MP_OKAY) {
         printf ("\nfn:mp_is_square() error!");
         return EXIT_FAILURE;
      }
      if (n == 0) {
         printf ("\nfn:mp_is_square() bad result!");
         return EXIT_FAILURE;
      }

      /* test for false positives */
      mp_add_d (&a, 1, &a);
      if (mp_is_square (&a, &n) != MP_OKAY) {
         printf ("\nfp:mp_is_square() error!");
         return EXIT_FAILURE;
      }
      if (n == 1) {
         printf ("\nfp:mp_is_square() bad result!");
         return EXIT_FAILURE;
      }

   }
   printf("\n\n");

   // r^2 = n (mod p)
   for (i = 0; i < (int)(sizeof(sqrtmod_prime)/sizeof(sqrtmod_prime[0])); ++i) {
      mp_set_int(&a, sqrtmod_prime[i].p);
      mp_set_int(&b, sqrtmod_prime[i].n);
      if (mp_sqrtmod_prime(&b, &a, &c) != MP_OKAY) {
         printf("Failed executing %d. mp_sqrtmod_prime\n", (i+1));
         return EXIT_FAILURE;
      }
      if (mp_cmp_d(&c, sqrtmod_prime[i].r) != MP_EQ) {
         printf("Failed %d. trivial mp_sqrtmod_prime\n", (i+1));
         ndraw(&c, "r");
         return EXIT_FAILURE;
      }
   }

   /* test for size */
   for (ix = 10; ix < 128; ix++) {
      printf ("Testing (not safe-prime): %9d bits    \r", ix);
      fflush (stdout);
      err = mp_prime_random_ex (&a, 8, ix,
                                (rand () & 1) ? 0 : LTM_PRIME_2MSB_ON, myrng,
                                NULL);
      if (err != MP_OKAY) {
         printf ("failed with err code %d\n", err);
         return EXIT_FAILURE;
      }
      if (mp_count_bits (&a) != ix) {
         printf ("Prime is %d not %d bits!!!\n", mp_count_bits (&a), ix);
         return EXIT_FAILURE;
      }
   }
   printf("\n");

   for (ix = 16; ix < 128; ix++) {
      printf ("Testing (    safe-prime): %9d bits    \r", ix);
      fflush (stdout);
      err = mp_prime_random_ex (
            &a, 8, ix, ((rand () & 1) ? 0 : LTM_PRIME_2MSB_ON) | LTM_PRIME_SAFE,
            myrng, NULL);
      if (err != MP_OKAY) {
         printf ("failed with err code %d\n", err);
         return EXIT_FAILURE;
      }
      if (mp_count_bits (&a) != ix) {
         printf ("Prime is %d not %d bits!!!\n", mp_count_bits (&a), ix);
         return EXIT_FAILURE;
      }
      /* let's see if it's really a safe prime */
      mp_sub_d (&a, 1, &a);
      mp_div_2 (&a, &a);
      mp_prime_is_prime (&a, 8, &cnt);
      if (cnt != MP_YES) {
         printf ("sub is not prime!\n");
         return EXIT_FAILURE;
      }
   }

   printf("\n\n");

   // test montgomery
   printf("Testing: montgomery...\n");
   for (i = 1; i <= 10; i++) {
      if (i == 10)
         i = 1000;
      printf(" digit size: %2d\r", i);
      fflush(stdout);
      for (n = 0; n < 1000; n++) {
         mp_rand(&a, i);
         a.dp[0] |= 1;

         // let's see if R is right
         mp_montgomery_calc_normalization(&b, &a);
         mp_montgomery_setup(&a, &mp);

         // now test a random reduction
         for (ix = 0; ix < 100; ix++) {
             mp_rand(&c, 1 + abs(rand()) % (2*i));
             mp_copy(&c, &d);
             mp_copy(&c, &e);

             mp_mod(&d, &a, &d);
             mp_montgomery_reduce(&c, &a, mp);
             mp_mulmod(&c, &b, &a, &c);

             if (mp_cmp(&c, &d) != MP_EQ) {
printf("d = e mod a, c = e MOD a\n");
mp_todecimal(&a, buf); printf("a = %s\n", buf);
mp_todecimal(&e, buf); printf("e = %s\n", buf);
mp_todecimal(&d, buf); printf("d = %s\n", buf);
mp_todecimal(&c, buf); printf("c = %s\n", buf);
printf("compare no compare!\n"); return EXIT_FAILURE; }




             /* only one big montgomery reduction */





































             if (i > 10)




             {












                n = 1000;















                ix = 100;









             }



         }

























      }
   }

   printf("\n\n");

   mp_read_radix(&a, "123456", 10);
   mp_toradix_n(&a, buf, 10, 3);
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      mp_prime_next_prime(&a, 5, 1);
      mp_toradix(&a, buf, 10);
      printf("%s, %lu\n", buf, a.dp[0] & 3);
   }
#endif

   /* test mp_cnt_lsb */
   printf("testing mp_cnt_lsb...\n");
   mp_set(&a, 1);
   for (ix = 0; ix < 1024; ix++) {
      if (mp_cnt_lsb(&a) != ix) {
	 printf("Failed at %d, %d\n", ix, mp_cnt_lsb(&a));
	 return 0;
      }
      mp_mul_2(&a, &a);
   }

/* test mp_reduce_2k */
   printf("Testing mp_reduce_2k...\n");
   for (cnt = 3; cnt <= 128; ++cnt) {
      mp_digit tmp;

      mp_2expt(&a, cnt);
      mp_sub_d(&a, 2, &a);	/* a = 2**cnt - 2 */


      printf("\nTesting %4d bits", cnt);
      printf("(%d)", mp_reduce_is_2k(&a));
      mp_reduce_2k_setup(&a, &tmp);
      printf("(%d)", tmp);
      for (ix = 0; ix < 1000; ix++) {
	 if (!(ix & 127)) {
	    printf(".");
	    fflush(stdout);
	 }
	 mp_rand(&b, (cnt / DIGIT_BIT + 1) * 2);
	 mp_copy(&c, &b);
	 mp_mod(&c, &a, &c);
	 mp_reduce_2k(&b, &a, 2);
	 if (mp_cmp(&c, &b)) {
	    printf("FAILED\n");
	    exit(0);

	 }
      }
   }

/* test mp_div_3  */
   printf("Testing mp_div_3...\n");
   mp_set(&d, 3);
   for (cnt = 0; cnt < 10000;) {
      mp_digit r1, r2;

      if (!(++cnt & 127))

	 printf("%9d\r", cnt);


      mp_rand(&a, abs(rand()) % 128 + 1);
      mp_div(&a, &d, &b, &e);
      mp_div_3(&a, &c, &r2);

      if (mp_cmp(&b, &c) || mp_cmp_d(&e, r2)) {
	 printf("\n\nmp_div_3 => Failure\n");
      }
   }
   printf("\n\nPassed div_3 testing\n");

/* test the DR reduction */
   printf("testing mp_dr_reduce...\n");
   for (cnt = 2; cnt < 32; cnt++) {
      printf("%d digit modulus\n", cnt);
      mp_grow(&a, cnt);
      mp_zero(&a);
      for (ix = 1; ix < cnt; ix++) {
	 a.dp[ix] = MP_MASK;
      }
      a.used = cnt;
      a.dp[0] = 3;

      mp_rand(&b, cnt - 1);
      mp_copy(&b, &c);

      rr = 0;
      do {
	 if (!(rr & 127)) {
	    printf("%9lu\r", rr);
	    fflush(stdout);
	 }
	 mp_sqr(&b, &b);
	 mp_add_d(&b, 1, &b);
	 mp_copy(&b, &c);

	 mp_mod(&b, &a, &b);

	 mp_dr_reduce(&c, &a, (((mp_digit) 1) << DIGIT_BIT) - a.dp[0]);

	 if (mp_cmp(&b, &c) != MP_EQ) {
	    printf("Failed on trial %lu\n", rr);
	    exit(-1);

	 }
      } while (++rr < 500);
      printf("Passed DR test for %d digits\n", cnt);

   }

#endif

/* test the mp_reduce_2k_l code */
#if 0
#if 0
/* first load P with 2^1024 - 0x2A434 B9FDEC95 D8F9D550 FFFFFFFF FFFFFFFF */
   mp_2expt(&a, 1024);
   mp_read_radix(&b, "2A434B9FDEC95D8F9D550FFFFFFFFFFFFFFFF", 16);
   mp_sub(&a, &b, &a);
#elif 1
/*  p = 2^2048 - 0x1 00000000 00000000 00000000 00000000 4945DDBF 8EA2A91D 5776399B B83E188F  */
   mp_2expt(&a, 2048);
   mp_read_radix(&b,
		 "1000000000000000000000000000000004945DDBF8EA2A91D5776399BB83E188F",
		 16);
   mp_sub(&a, &b, &a);


#endif

   mp_todecimal(&a, buf);
   printf("p==%s\n", buf);
/* now mp_reduce_is_2k_l() should return */
   if (mp_reduce_is_2k_l(&a) != 1) {
      printf("mp_reduce_is_2k_l() return 0, should be 1\n");
      return EXIT_FAILURE;
   }
   mp_reduce_2k_setup_l(&a, &d);
   /* now do a million square+1 to see if it varies */
   mp_rand(&b, 64);
   mp_mod(&b, &a, &b);
   mp_copy(&b, &c);
   printf("testing mp_reduce_2k_l...");
   fflush(stdout);
   for (cnt = 0; cnt < (1UL << 20); cnt++) {
      mp_sqr(&b, &b);
      mp_add_d(&b, 1, &b);
      mp_reduce_2k_l(&b, &a, &d);
      mp_sqr(&c, &c);
      mp_add_d(&c, 1, &c);
      mp_mod(&c, &a, &c);
      if (mp_cmp(&b, &c) != MP_EQ) {
	 printf("mp_reduce_2k_l() failed at step %lu\n", cnt);
	 mp_tohex(&b, buf);
	 printf("b == %s\n", buf);
	 mp_tohex(&c, buf);
	 printf("c == %s\n", buf);
	 return EXIT_FAILURE;
      }
   }
   printf("...Passed\n");
#endif



   div2_n = mul2_n = inv_n = expt_n = lcm_n = gcd_n = add_n =
      sub_n = mul_n = div_n = sqr_n = mul2d_n = div2d_n = cnt = add_d_n =
      sub_d_n = 0;

   /* force KARA and TOOM to enable despite cutoffs */
   KARATSUBA_SQR_CUTOFF = KARATSUBA_MUL_CUTOFF = 8;







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      mp_prime_next_prime(&a, 5, 1);
      mp_toradix(&a, buf, 10);
      printf("%s, %lu\n", buf, a.dp[0] & 3);
   }
#endif

   /* test mp_cnt_lsb */
   printf("\n\nTesting: mp_cnt_lsb");
   mp_set(&a, 1);
   for (ix = 0; ix < 1024; ix++) {
      if (mp_cnt_lsb (&a) != ix) {
         printf ("Failed at %d, %d\n", ix, mp_cnt_lsb (&a));
         return EXIT_FAILURE;
      }
      mp_mul_2 (&a, &a);
   }

/* test mp_reduce_2k */
   printf("\n\nTesting: mp_reduce_2k\n");
   for (cnt = 3; cnt <= 128; ++cnt) {
      mp_digit tmp;

      mp_2expt (&a, cnt);
      mp_sub_d (&a, 2, &a); /* a = 2**cnt - 2 */


      printf ("\r %4d bits", cnt);
      printf ("(%d)", mp_reduce_is_2k (&a));
      mp_reduce_2k_setup (&a, &tmp);
      printf ("(%lu)", (unsigned long) tmp);
      for (ix = 0; ix < 1000; ix++) {
         if (!(ix & 127)) {
            printf (".");
            fflush (stdout);
         }
         mp_rand (&b, (cnt / DIGIT_BIT + 1) * 2);
         mp_copy (&c, &b);
         mp_mod (&c, &a, &c);
         mp_reduce_2k (&b, &a, 2);
         if (mp_cmp (&c, &b)) {
            printf ("FAILED\n");

            return EXIT_FAILURE;
         }
      }
   }

/* test mp_div_3  */
   printf("\n\nTesting: mp_div_3...\n");
   mp_set(&d, 3);
   for (cnt = 0; cnt < 10000;) {
      mp_digit r2;

      if (!(++cnt & 127))
      {
        printf("%9d\r", cnt);
        fflush(stdout);
      }
      mp_rand(&a, abs(rand()) % 128 + 1);
      mp_div(&a, &d, &b, &e);
      mp_div_3(&a, &c, &r2);

      if (mp_cmp(&b, &c) || mp_cmp_d(&e, r2)) {
	 printf("\nmp_div_3 => Failure\n");
      }
   }
   printf("\nPassed div_3 testing");

/* test the DR reduction */
   printf("\n\nTesting: mp_dr_reduce...\n");
   for (cnt = 2; cnt < 32; cnt++) {
      printf ("\r%d digit modulus", cnt);
      mp_grow (&a, cnt);
      mp_zero (&a);
      for (ix = 1; ix < cnt; ix++) {
         a.dp[ix] = MP_MASK;
      }
      a.used = cnt;
      a.dp[0] = 3;

      mp_rand (&b, cnt - 1);
      mp_copy (&b, &c);

      rr = 0;
      do {
         if (!(rr & 127)) {
            printf (".");
            fflush (stdout);
         }
         mp_sqr (&b, &b);
         mp_add_d (&b, 1, &b);
         mp_copy (&b, &c);

         mp_mod (&b, &a, &b);
         mp_dr_setup(&a, &mp),
         mp_dr_reduce (&c, &a, mp);

         if (mp_cmp (&b, &c) != MP_EQ) {
            printf ("Failed on trial %u\n", rr);

            return EXIT_FAILURE;
         }
      } while (++rr < 500);
      printf (" passed");
      fflush (stdout);
   }


#if LTM_DEMO_TEST_REDUCE_2K_L
/* test the mp_reduce_2k_l code */
#if LTM_DEMO_TEST_REDUCE_2K_L == 1

/* first load P with 2^1024 - 0x2A434 B9FDEC95 D8F9D550 FFFFFFFF FFFFFFFF */
   mp_2expt(&a, 1024);
   mp_read_radix(&b, "2A434B9FDEC95D8F9D550FFFFFFFFFFFFFFFF", 16);
   mp_sub(&a, &b, &a);
#elif LTM_DEMO_TEST_REDUCE_2K_L == 2
/*  p = 2^2048 - 0x1 00000000 00000000 00000000 00000000 4945DDBF 8EA2A91D 5776399B B83E188F  */
   mp_2expt(&a, 2048);
   mp_read_radix(&b,
		 "1000000000000000000000000000000004945DDBF8EA2A91D5776399BB83E188F",
		 16);
   mp_sub(&a, &b, &a);
#else
#error oops
#endif

   mp_todecimal(&a, buf);
   printf("\n\np==%s\n", buf);
/* now mp_reduce_is_2k_l() should return */
   if (mp_reduce_is_2k_l(&a) != 1) {
      printf("mp_reduce_is_2k_l() return 0, should be 1\n");
      return EXIT_FAILURE;
   }
   mp_reduce_2k_setup_l(&a, &d);
   /* now do a million square+1 to see if it varies */
   mp_rand(&b, 64);
   mp_mod(&b, &a, &b);
   mp_copy(&b, &c);
   printf("Testing: mp_reduce_2k_l...");
   fflush(stdout);
   for (cnt = 0; cnt < (int)(1UL << 20); cnt++) {
      mp_sqr(&b, &b);
      mp_add_d(&b, 1, &b);
      mp_reduce_2k_l(&b, &a, &d);
      mp_sqr(&c, &c);
      mp_add_d(&c, 1, &c);
      mp_mod(&c, &a, &c);
      if (mp_cmp(&b, &c) != MP_EQ) {
	 printf("mp_reduce_2k_l() failed at step %d\n", cnt);
	 mp_tohex(&b, buf);
	 printf("b == %s\n", buf);
	 mp_tohex(&c, buf);
	 printf("c == %s\n", buf);
	 return EXIT_FAILURE;
      }
   }
   printf("...Passed\n");
#endif /* LTM_DEMO_TEST_REDUCE_2K_L */

#else

   div2_n = mul2_n = inv_n = expt_n = lcm_n = gcd_n = add_n =
      sub_n = mul_n = div_n = sqr_n = mul2d_n = div2d_n = cnt = add_d_n =
      sub_d_n = 0;

   /* force KARA and TOOM to enable despite cutoffs */
   KARATSUBA_SQR_CUTOFF = KARATSUBA_MUL_CUTOFF = 8;
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      }


      printf
	 ("%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu ",
	  add_n, sub_n, mul_n, div_n, sqr_n, mul2d_n, div2d_n, gcd_n, lcm_n,
	  expt_n, inv_n, div2_n, mul2_n, add_d_n, sub_d_n);
      fgets(cmd, 4095, stdin);
      cmd[strlen(cmd) - 1] = 0;
      printf("%s  ]\r", cmd);
      fflush(stdout);
      if (!strcmp(cmd, "mul2d")) {
	 ++mul2d_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 sscanf(buf, "%d", &rr);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);

	 mp_mul_2d(&a, rr, &a);
	 a.sign = b.sign;
	 if (mp_cmp(&a, &b) != MP_EQ) {
	    printf("mul2d failed, rr == %d\n", rr);
	    draw(&a);
	    draw(&b);
	    return 0;
	 }
      } else if (!strcmp(cmd, "div2d")) {
	 ++div2d_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 sscanf(buf, "%d", &rr);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);

	 mp_div_2d(&a, rr, &a, &e);
	 a.sign = b.sign;
	 if (a.used == b.used && a.used == 0) {
	    a.sign = b.sign = MP_ZPOS;
	 }
	 if (mp_cmp(&a, &b) != MP_EQ) {
	    printf("div2d failed, rr == %d\n", rr);
	    draw(&a);
	    draw(&b);
	    return 0;
	 }
      } else if (!strcmp(cmd, "add")) {
	 ++add_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&c, buf, 64);
	 mp_copy(&a, &d);
	 mp_add(&d, &b, &d);
	 if (mp_cmp(&c, &d) != MP_EQ) {
	    printf("add %lu failure!\n", add_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    return 0;
	 }

	 /* test the sign/unsigned storage functions */

	 rr = mp_signed_bin_size(&c);
	 mp_to_signed_bin(&c, (unsigned char *) cmd);
	 memset(cmd + rr, rand() & 255, sizeof(cmd) - rr);
	 mp_read_signed_bin(&d, (unsigned char *) cmd, rr);
	 if (mp_cmp(&c, &d) != MP_EQ) {
	    printf("mp_signed_bin failure!\n");
	    draw(&c);
	    draw(&d);
	    return 0;
	 }


	 rr = mp_unsigned_bin_size(&c);
	 mp_to_unsigned_bin(&c, (unsigned char *) cmd);
	 memset(cmd + rr, rand() & 255, sizeof(cmd) - rr);
	 mp_read_unsigned_bin(&d, (unsigned char *) cmd, rr);
	 if (mp_cmp_mag(&c, &d) != MP_EQ) {
	    printf("mp_unsigned_bin failure!\n");
	    draw(&c);
	    draw(&d);
	    return 0;
	 }

      } else if (!strcmp(cmd, "sub")) {
	 ++sub_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&c, buf, 64);
	 mp_copy(&a, &d);
	 mp_sub(&d, &b, &d);
	 if (mp_cmp(&c, &d) != MP_EQ) {
	    printf("sub %lu failure!\n", sub_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    return 0;
	 }
      } else if (!strcmp(cmd, "mul")) {
	 ++mul_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&c, buf, 64);
	 mp_copy(&a, &d);
	 mp_mul(&d, &b, &d);
	 if (mp_cmp(&c, &d) != MP_EQ) {
	    printf("mul %lu failure!\n", mul_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    return 0;
	 }
      } else if (!strcmp(cmd, "div")) {
	 ++div_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&c, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&d, buf, 64);

	 mp_div(&a, &b, &e, &f);
	 if (mp_cmp(&c, &e) != MP_EQ || mp_cmp(&d, &f) != MP_EQ) {
	    printf("div %lu %d, %d, failure!\n", div_n, mp_cmp(&c, &e),
		   mp_cmp(&d, &f));
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    draw(&e);
	    draw(&f);
	    return 0;
	 }

      } else if (!strcmp(cmd, "sqr")) {
	 ++sqr_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);
	 mp_copy(&a, &c);
	 mp_sqr(&c, &c);
	 if (mp_cmp(&b, &c) != MP_EQ) {
	    printf("sqr %lu failure!\n", sqr_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    return 0;
	 }
      } else if (!strcmp(cmd, "gcd")) {
	 ++gcd_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&c, buf, 64);
	 mp_copy(&a, &d);
	 mp_gcd(&d, &b, &d);
	 d.sign = c.sign;
	 if (mp_cmp(&c, &d) != MP_EQ) {
	    printf("gcd %lu failure!\n", gcd_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    return 0;
	 }
      } else if (!strcmp(cmd, "lcm")) {
	 ++lcm_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&c, buf, 64);
	 mp_copy(&a, &d);
	 mp_lcm(&d, &b, &d);
	 d.sign = c.sign;
	 if (mp_cmp(&c, &d) != MP_EQ) {
	    printf("lcm %lu failure!\n", lcm_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    return 0;
	 }
      } else if (!strcmp(cmd, "expt")) {
	 ++expt_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&c, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&d, buf, 64);
	 mp_copy(&a, &e);
	 mp_exptmod(&e, &b, &c, &e);
	 if (mp_cmp(&d, &e) != MP_EQ) {
	    printf("expt %lu failure!\n", expt_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    draw(&e);
	    return 0;
	 }
      } else if (!strcmp(cmd, "invmod")) {
	 ++inv_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&c, buf, 64);
	 mp_invmod(&a, &b, &d);
	 mp_mulmod(&d, &a, &b, &e);
	 if (mp_cmp_d(&e, 1) != MP_EQ) {
	    printf("inv [wrong value from MPI?!] failure\n");
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);

	    mp_gcd(&a, &b, &e);
	    draw(&e);
	    return 0;
	 }

      } else if (!strcmp(cmd, "div2")) {
	 ++div2_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);
	 mp_div_2(&a, &c);
	 if (mp_cmp(&c, &b) != MP_EQ) {
	    printf("div_2 %lu failure\n", div2_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    return 0;
	 }
      } else if (!strcmp(cmd, "mul2")) {
	 ++mul2_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);
	 mp_mul_2(&a, &c);
	 if (mp_cmp(&c, &b) != MP_EQ) {
	    printf("mul_2 %lu failure\n", mul2_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    return 0;
	 }
      } else if (!strcmp(cmd, "add_d")) {
	 ++add_d_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 sscanf(buf, "%d", &ix);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);
	 mp_add_d(&a, ix, &c);
	 if (mp_cmp(&b, &c) != MP_EQ) {
	    printf("add_d %lu failure\n", add_d_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    printf("d == %d\n", ix);
	    return 0;
	 }
      } else if (!strcmp(cmd, "sub_d")) {
	 ++sub_d_n;
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&a, buf, 64);
	 fgets(buf, 4095, stdin);
	 sscanf(buf, "%d", &ix);
	 fgets(buf, 4095, stdin);
	 mp_read_radix(&b, buf, 64);
	 mp_sub_d(&a, ix, &c);
	 if (mp_cmp(&b, &c) != MP_EQ) {
	    printf("sub_d %lu failure\n", sub_d_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    printf("d == %d\n", ix);
	    return 0;
	 }



      }
   }

   return 0;
}











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      }


      printf
	 ("%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu ",
	  add_n, sub_n, mul_n, div_n, sqr_n, mul2d_n, div2d_n, gcd_n, lcm_n,
	  expt_n, inv_n, div2_n, mul2_n, add_d_n, sub_d_n);
      ret=fgets(cmd, 4095, stdin); if(!ret){_panic(__LINE__);}
      cmd[strlen(cmd) - 1] = 0;
      printf("%-6s ]\r", cmd);
      fflush(stdout);
      if (!strcmp(cmd, "mul2d")) {
	 ++mul2d_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 sscanf(buf, "%d", &rr);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);

	 mp_mul_2d(&a, rr, &a);
	 a.sign = b.sign;
	 if (mp_cmp(&a, &b) != MP_EQ) {
	    printf("mul2d failed, rr == %d\n", rr);
	    draw(&a);
	    draw(&b);
	    return EXIT_FAILURE;
	 }
      } else if (!strcmp(cmd, "div2d")) {
	 ++div2d_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 sscanf(buf, "%d", &rr);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);

	 mp_div_2d(&a, rr, &a, &e);
	 a.sign = b.sign;
	 if (a.used == b.used && a.used == 0) {
	    a.sign = b.sign = MP_ZPOS;
	 }
	 if (mp_cmp(&a, &b) != MP_EQ) {
	    printf("div2d failed, rr == %d\n", rr);
	    draw(&a);
	    draw(&b);
	    return EXIT_FAILURE;
	 }
      } else if (!strcmp(cmd, "add")) {
	 ++add_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&c, buf, 64);
	 mp_copy(&a, &d);
	 mp_add(&d, &b, &d);
	 if (mp_cmp(&c, &d) != MP_EQ) {
	    printf("add %lu failure!\n", add_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    return EXIT_FAILURE;
	 }

	 /* test the sign/unsigned storage functions */

	 rr = mp_signed_bin_size(&c);
	 mp_to_signed_bin(&c, (unsigned char *) cmd);
	 memset(cmd + rr, rand() & 255, sizeof(cmd) - rr);
	 mp_read_signed_bin(&d, (unsigned char *) cmd, rr);
	 if (mp_cmp(&c, &d) != MP_EQ) {
	    printf("mp_signed_bin failure!\n");
	    draw(&c);
	    draw(&d);
	    return EXIT_FAILURE;
	 }


	 rr = mp_unsigned_bin_size(&c);
	 mp_to_unsigned_bin(&c, (unsigned char *) cmd);
	 memset(cmd + rr, rand() & 255, sizeof(cmd) - rr);
	 mp_read_unsigned_bin(&d, (unsigned char *) cmd, rr);
	 if (mp_cmp_mag(&c, &d) != MP_EQ) {
	    printf("mp_unsigned_bin failure!\n");
	    draw(&c);
	    draw(&d);
	    return EXIT_FAILURE;
	 }

      } else if (!strcmp(cmd, "sub")) {
	 ++sub_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&c, buf, 64);
	 mp_copy(&a, &d);
	 mp_sub(&d, &b, &d);
	 if (mp_cmp(&c, &d) != MP_EQ) {
	    printf("sub %lu failure!\n", sub_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    return EXIT_FAILURE;
	 }
      } else if (!strcmp(cmd, "mul")) {
	 ++mul_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&c, buf, 64);
	 mp_copy(&a, &d);
	 mp_mul(&d, &b, &d);
	 if (mp_cmp(&c, &d) != MP_EQ) {
	    printf("mul %lu failure!\n", mul_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    return EXIT_FAILURE;
	 }
      } else if (!strcmp(cmd, "div")) {
	 ++div_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&c, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&d, buf, 64);

	 mp_div(&a, &b, &e, &f);
	 if (mp_cmp(&c, &e) != MP_EQ || mp_cmp(&d, &f) != MP_EQ) {
	    printf("div %lu %d, %d, failure!\n", div_n, mp_cmp(&c, &e),
		   mp_cmp(&d, &f));
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    draw(&e);
	    draw(&f);
	    return EXIT_FAILURE;
	 }

      } else if (!strcmp(cmd, "sqr")) {
	 ++sqr_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);
	 mp_copy(&a, &c);
	 mp_sqr(&c, &c);
	 if (mp_cmp(&b, &c) != MP_EQ) {
	    printf("sqr %lu failure!\n", sqr_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    return EXIT_FAILURE;
	 }
      } else if (!strcmp(cmd, "gcd")) {
	 ++gcd_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&c, buf, 64);
	 mp_copy(&a, &d);
	 mp_gcd(&d, &b, &d);
	 d.sign = c.sign;
	 if (mp_cmp(&c, &d) != MP_EQ) {
	    printf("gcd %lu failure!\n", gcd_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    return EXIT_FAILURE;
	 }
      } else if (!strcmp(cmd, "lcm")) {
	 ++lcm_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&c, buf, 64);
	 mp_copy(&a, &d);
	 mp_lcm(&d, &b, &d);
	 d.sign = c.sign;
	 if (mp_cmp(&c, &d) != MP_EQ) {
	    printf("lcm %lu failure!\n", lcm_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    return EXIT_FAILURE;
	 }
      } else if (!strcmp(cmd, "expt")) {
	 ++expt_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&c, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&d, buf, 64);
	 mp_copy(&a, &e);
	 mp_exptmod(&e, &b, &c, &e);
	 if (mp_cmp(&d, &e) != MP_EQ) {
	    printf("expt %lu failure!\n", expt_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    draw(&e);
	    return EXIT_FAILURE;
	 }
      } else if (!strcmp(cmd, "invmod")) {
	 ++inv_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&c, buf, 64);
	 mp_invmod(&a, &b, &d);
	 mp_mulmod(&d, &a, &b, &e);
	 if (mp_cmp_d(&e, 1) != MP_EQ) {
	    printf("inv [wrong value from MPI?!] failure\n");
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    draw(&d);
	    draw(&e);
	    mp_gcd(&a, &b, &e);
	    draw(&e);
	    return EXIT_FAILURE;
	 }

      } else if (!strcmp(cmd, "div2")) {
	 ++div2_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);
	 mp_div_2(&a, &c);
	 if (mp_cmp(&c, &b) != MP_EQ) {
	    printf("div_2 %lu failure\n", div2_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    return EXIT_FAILURE;
	 }
      } else if (!strcmp(cmd, "mul2")) {
	 ++mul2_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);
	 mp_mul_2(&a, &c);
	 if (mp_cmp(&c, &b) != MP_EQ) {
	    printf("mul_2 %lu failure\n", mul2_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    return EXIT_FAILURE;
	 }
      } else if (!strcmp(cmd, "add_d")) {
	 ++add_d_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 sscanf(buf, "%d", &ix);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);
	 mp_add_d(&a, ix, &c);
	 if (mp_cmp(&b, &c) != MP_EQ) {
	    printf("add_d %lu failure\n", add_d_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    printf("d == %d\n", ix);
	    return EXIT_FAILURE;
	 }
      } else if (!strcmp(cmd, "sub_d")) {
	 ++sub_d_n;
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&a, buf, 64);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 sscanf(buf, "%d", &ix);
	 ret=fgets(buf, 4095, stdin); if(!ret){_panic(__LINE__);}
	 mp_read_radix(&b, buf, 64);
	 mp_sub_d(&a, ix, &c);
	 if (mp_cmp(&b, &c) != MP_EQ) {
	    printf("sub_d %lu failure\n", sub_d_n);
	    draw(&a);
	    draw(&b);
	    draw(&c);
	    printf("d == %d\n", ix);
	    return EXIT_FAILURE;
	 }
      } else if (!strcmp(cmd, "exit")) {
         printf("\nokay, exiting now\n");
         break;
      }
   }
#endif
   return 0;
}

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/demo/timing.c.
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#include <tommath.h>
#include <time.h>


ulong64 _tt;

#ifdef IOWNANATHLON
#include <unistd.h>
#define SLEEP sleep(4)
#else
#define SLEEP
#endif








void ndraw(mp_int * a, char *name)
{
   char buf[4096];

   printf("%s: ", name);


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#include <tommath.h>
#include <time.h>
#include <unistd.h>

ulong64 _tt;

#ifdef IOWNANATHLON
#include <unistd.h>
#define SLEEP sleep(4)
#else
#define SLEEP
#endif

#ifdef LTM_TIMING_REAL_RAND
#define LTM_TIMING_RAND_SEED  time(NULL)
#else
#define LTM_TIMING_RAND_SEED  23
#endif


void ndraw(mp_int * a, char *name)
{
   char buf[4096];

   printf("%s: ", name);
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}

/* RDTSC from Scott Duplichan */
static ulong64 TIMFUNC(void)
{
#if defined __GNUC__
#if defined(__i386__) || defined(__x86_64__)



   unsigned long long a;
   __asm__ __volatile__("rdtsc\nmovl %%eax,%0\nmovl %%edx,4+%0\n"::
			"m"(a):"%eax", "%edx");
   return a;
#else /* gcc-IA64 version */
   unsigned long result;
   __asm__ __volatile__("mov %0=ar.itc":"=r"(result)::"memory");

   while (__builtin_expect((int) result == -1, 0))
      __asm__ __volatile__("mov %0=ar.itc":"=r"(result)::"memory");








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}

/* RDTSC from Scott Duplichan */
static ulong64 TIMFUNC(void)
{
#if defined __GNUC__
#if defined(__i386__) || defined(__x86_64__)
  /* version from http://www.mcs.anl.gov/~kazutomo/rdtsc.html
   * the old code always got a warning issued by gcc, clang did not complain...
   */
  unsigned hi, lo;
  __asm__ __volatile__ ("rdtsc" : "=a"(lo), "=d"(hi));

  return ((ulong64)lo)|( ((ulong64)hi)<<32);
#else /* gcc-IA64 version */
   unsigned long result;
   __asm__ __volatile__("mov %0=ar.itc":"=r"(result)::"memory");

   while (__builtin_expect((int) result == -1, 0))
      __asm__ __volatile__("mov %0=ar.itc":"=r"(result)::"memory");

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#endif
}

#define DO(x) x; x;
//#define DO4(x) DO2(x); DO2(x);
//#define DO8(x) DO4(x); DO4(x);
//#define DO(x)  DO8(x); DO8(x);













int main(void)
{
   ulong64 tt, gg, CLK_PER_SEC;
   FILE *log, *logb, *logc, *logd;
   mp_int a, b, c, d, e, f;
   int n, cnt, ix, old_kara_m, old_kara_s;
   unsigned rr;

   mp_init(&a);
   mp_init(&b);
   mp_init(&c);
   mp_init(&d);
   mp_init(&e);
   mp_init(&f);

   srand(time(NULL));


   /* temp. turn off TOOM */
   TOOM_MUL_CUTOFF = TOOM_SQR_CUTOFF = 100000;

   CLK_PER_SEC = TIMFUNC();
   sleep(1);
   CLK_PER_SEC = TIMFUNC() - CLK_PER_SEC;

   printf("CLK_PER_SEC == %llu\n", CLK_PER_SEC);
   goto exptmod;
   log = fopen("logs/add.log", "w");
   for (cnt = 8; cnt <= 128; cnt += 8) {
      SLEEP;
      mp_rand(&a, cnt);
      mp_rand(&b, cnt);
      rr = 0;
      tt = -1;
      do {
	 gg = TIMFUNC();
	 DO(mp_add(&a, &b, &c));
	 gg = (TIMFUNC() - gg) >> 1;
	 if (tt > gg)
	    tt = gg;
      } while (++rr < 100000);
      printf("Adding\t\t%4d-bit => %9llu/sec, %9llu cycles\n",
	     mp_count_bits(&a), CLK_PER_SEC / tt, tt);
      fprintf(log, "%d %9llu\n", cnt * DIGIT_BIT, tt);
      fflush(log);
   }
   fclose(log);

   log = fopen("logs/sub.log", "w");
   for (cnt = 8; cnt <= 128; cnt += 8) {
      SLEEP;
      mp_rand(&a, cnt);
      mp_rand(&b, cnt);
      rr = 0;
      tt = -1;
      do {
	 gg = TIMFUNC();
	 DO(mp_sub(&a, &b, &c));
	 gg = (TIMFUNC() - gg) >> 1;
	 if (tt > gg)
	    tt = gg;
      } while (++rr < 100000);

      printf("Subtracting\t\t%4d-bit => %9llu/sec, %9llu cycles\n",
	     mp_count_bits(&a), CLK_PER_SEC / tt, tt);
      fprintf(log, "%d %9llu\n", cnt * DIGIT_BIT, tt);
      fflush(log);
   }
   fclose(log);

   /* do mult/square twice, first without karatsuba and second with */
 multtest:
   old_kara_m = KARATSUBA_MUL_CUTOFF;
   old_kara_s = KARATSUBA_SQR_CUTOFF;



   for (ix = 0; ix < 2; ix++) {
      printf("With%s Karatsuba\n", (ix == 0) ? "out" : "");

      KARATSUBA_MUL_CUTOFF = (ix == 0) ? 9999 : old_kara_m;
      KARATSUBA_SQR_CUTOFF = (ix == 0) ? 9999 : old_kara_s;



      log = fopen((ix == 0) ? "logs/mult.log" : "logs/mult_kara.log", "w");
      for (cnt = 4; cnt <= 10240 / DIGIT_BIT; cnt += 2) {
	 SLEEP;
	 mp_rand(&a, cnt);
	 mp_rand(&b, cnt);
	 rr = 0;
	 tt = -1;
	 do {
	    gg = TIMFUNC();
	    DO(mp_mul(&a, &b, &c));
	    gg = (TIMFUNC() - gg) >> 1;
	    if (tt > gg)
	       tt = gg;
	 } while (++rr < 100);
	 printf("Multiplying\t%4d-bit => %9llu/sec, %9llu cycles\n",
		mp_count_bits(&a), CLK_PER_SEC / tt, tt);
	 fprintf(log, "%d %9llu\n", mp_count_bits(&a), tt);
	 fflush(log);
      }
      fclose(log);

      log = fopen((ix == 0) ? "logs/sqr.log" : "logs/sqr_kara.log", "w");
      for (cnt = 4; cnt <= 10240 / DIGIT_BIT; cnt += 2) {
	 SLEEP;
	 mp_rand(&a, cnt);
	 rr = 0;
	 tt = -1;
	 do {
	    gg = TIMFUNC();
	    DO(mp_sqr(&a, &b));
	    gg = (TIMFUNC() - gg) >> 1;
	    if (tt > gg)
	       tt = gg;
	 } while (++rr < 100);
	 printf("Squaring\t%4d-bit => %9llu/sec, %9llu cycles\n",
		mp_count_bits(&a), CLK_PER_SEC / tt, tt);
	 fprintf(log, "%d %9llu\n", mp_count_bits(&a), tt);
	 fflush(log);
      }
      fclose(log);

   }
 exptmod:

   {
      char *primes[] = {
	 /* 2K large moduli */
	 "179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586239334100047359817950870678242457666208137217",
	 "32317006071311007300714876688669951960444102669715484032130345427524655138867890893197201411522913463688717960921898019494119559150490921095088152386448283120630877367300996091750197750389652106796057638384067568276792218642619756161838094338476170470581645852036305042887575891541065808607552399123930385521914333389668342420684974786564569494856176035326322058077805659331026192708460314150258592864177116725943603718461857357598351152301645904403697613233287231227125684710820209725157101726931323469678542580656697935045997268352998638099733077152121140120031150424541696791951097529546801429027668869927491725169",
	 "1044388881413152506691752710716624382579964249047383780384233483283953907971557456848826811934997558340890106714439262837987573438185793607263236087851365277945956976543709998340361590134383718314428070011855946226376318839397712745672334684344586617496807908705803704071284048740118609114467977783598029006686938976881787785946905630190260940599579453432823469303026696443059025015972399867714215541693835559885291486318237914434496734087811872639496475100189041349008417061675093668333850551032972088269550769983616369411933015213796825837188091833656751221318492846368125550225998300412344784862595674492194617023806505913245610825731835380087608622102834270197698202313169017678006675195485079921636419370285375124784014907159135459982790513399611551794271106831134090584272884279791554849782954323534517065223269061394905987693002122963395687782878948440616007412945674919823050571642377154816321380631045902916136926708342856440730447899971901781465763473223850267253059899795996090799469201774624817718449867455659250178329070473119433165550807568221846571746373296884912819520317457002440926616910874148385078411929804522981857338977648103126085902995208257421855249796721729039744118165938433694823325696642096892124547425283",







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#endif
}

#define DO(x) x; x;
//#define DO4(x) DO2(x); DO2(x);
//#define DO8(x) DO4(x); DO4(x);
//#define DO(x)  DO8(x); DO8(x);

#ifdef TIMING_NO_LOGS
#define FOPEN(a, b)     NULL
#define FPRINTF(a,b,c,d)
#define FFLUSH(a)
#define FCLOSE(a)       (void)(a)
#else
#define FOPEN(a,b)       fopen(a,b)
#define FPRINTF(a,b,c,d) fprintf(a,b,c,d)
#define FFLUSH(a)        fflush(a)
#define FCLOSE(a)        fclose(a)
#endif

int main(void)
{
   ulong64 tt, gg, CLK_PER_SEC;
   FILE *log, *logb, *logc, *logd;
   mp_int a, b, c, d, e, f;
   int n, cnt, ix, old_kara_m, old_kara_s, old_toom_m, old_toom_s;
   unsigned rr;

   mp_init(&a);
   mp_init(&b);
   mp_init(&c);
   mp_init(&d);
   mp_init(&e);
   mp_init(&f);


   srand(LTM_TIMING_RAND_SEED);




   CLK_PER_SEC = TIMFUNC();
   sleep(1);
   CLK_PER_SEC = TIMFUNC() - CLK_PER_SEC;

   printf("CLK_PER_SEC == %llu\n", CLK_PER_SEC);

   log = FOPEN("logs/add.log", "w");
   for (cnt = 8; cnt <= 128; cnt += 8) {
      SLEEP;
      mp_rand(&a, cnt);
      mp_rand(&b, cnt);
      rr = 0;
      tt = -1;
      do {
	 gg = TIMFUNC();
	 DO(mp_add(&a, &b, &c));
	 gg = (TIMFUNC() - gg) >> 1;
	 if (tt > gg)
	    tt = gg;
      } while (++rr < 100000);
      printf("Adding\t\t%4d-bit => %9llu/sec, %9llu cycles\n",
	     mp_count_bits(&a), CLK_PER_SEC / tt, tt);
      FPRINTF(log, "%d %9llu\n", cnt * DIGIT_BIT, tt);
      FFLUSH(log);
   }
   FCLOSE(log);

   log = FOPEN("logs/sub.log", "w");
   for (cnt = 8; cnt <= 128; cnt += 8) {
      SLEEP;
      mp_rand(&a, cnt);
      mp_rand(&b, cnt);
      rr = 0;
      tt = -1;
      do {
	 gg = TIMFUNC();
	 DO(mp_sub(&a, &b, &c));
	 gg = (TIMFUNC() - gg) >> 1;
	 if (tt > gg)
	    tt = gg;
      } while (++rr < 100000);

      printf("Subtracting\t\t%4d-bit => %9llu/sec, %9llu cycles\n",
	     mp_count_bits(&a), CLK_PER_SEC / tt, tt);
      FPRINTF(log, "%d %9llu\n", cnt * DIGIT_BIT, tt);
      FFLUSH(log);
   }
   FCLOSE(log);

   /* do mult/square twice, first without karatsuba and second with */

   old_kara_m = KARATSUBA_MUL_CUTOFF;
   old_kara_s = KARATSUBA_SQR_CUTOFF;
   /* currently toom-cook cut-off is too high to kick in, so we just use the karatsuba values */
   old_toom_m = old_kara_m;
   old_toom_s = old_kara_m;
   for (ix = 0; ix < 3; ix++) {
      printf("With%s Karatsuba, With%s Toom\n", (ix == 0) ? "out" : "", (ix == 1) ? "out" : "");

      KARATSUBA_MUL_CUTOFF = (ix == 1) ? old_kara_m : 9999;
      KARATSUBA_SQR_CUTOFF = (ix == 1) ? old_kara_s : 9999;
      TOOM_MUL_CUTOFF = (ix == 2) ? old_toom_m : 9999;
      TOOM_SQR_CUTOFF = (ix == 2) ? old_toom_s : 9999;

      log = FOPEN((ix == 0) ? "logs/mult.log" : (ix == 1) ? "logs/mult_kara.log" : "logs/mult_toom.log", "w");
      for (cnt = 4; cnt <= 10240 / DIGIT_BIT; cnt += 2) {
	 SLEEP;
	 mp_rand(&a, cnt);
	 mp_rand(&b, cnt);
	 rr = 0;
	 tt = -1;
	 do {
	    gg = TIMFUNC();
	    DO(mp_mul(&a, &b, &c));
	    gg = (TIMFUNC() - gg) >> 1;
	    if (tt > gg)
	       tt = gg;
	 } while (++rr < 100);
	 printf("Multiplying\t%4d-bit => %9llu/sec, %9llu cycles\n",
		mp_count_bits(&a), CLK_PER_SEC / tt, tt);
	 FPRINTF(log, "%d %9llu\n", mp_count_bits(&a), tt);
	 FFLUSH(log);
      }
      FCLOSE(log);

      log = FOPEN((ix == 0) ? "logs/sqr.log" : (ix == 1) ? "logs/sqr_kara.log" : "logs/sqr_toom.log", "w");
      for (cnt = 4; cnt <= 10240 / DIGIT_BIT; cnt += 2) {
	 SLEEP;
	 mp_rand(&a, cnt);
	 rr = 0;
	 tt = -1;
	 do {
	    gg = TIMFUNC();
	    DO(mp_sqr(&a, &b));
	    gg = (TIMFUNC() - gg) >> 1;
	    if (tt > gg)
	       tt = gg;
	 } while (++rr < 100);
	 printf("Squaring\t%4d-bit => %9llu/sec, %9llu cycles\n",
		mp_count_bits(&a), CLK_PER_SEC / tt, tt);
	 FPRINTF(log, "%d %9llu\n", mp_count_bits(&a), tt);
	 FFLUSH(log);
      }
      FCLOSE(log);

   }


   {
      char *primes[] = {
	 /* 2K large moduli */
	 "179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586239334100047359817950870678242457666208137217",
	 "32317006071311007300714876688669951960444102669715484032130345427524655138867890893197201411522913463688717960921898019494119559150490921095088152386448283120630877367300996091750197750389652106796057638384067568276792218642619756161838094338476170470581645852036305042887575891541065808607552399123930385521914333389668342420684974786564569494856176035326322058077805659331026192708460314150258592864177116725943603718461857357598351152301645904403697613233287231227125684710820209725157101726931323469678542580656697935045997268352998638099733077152121140120031150424541696791951097529546801429027668869927491725169",
	 "1044388881413152506691752710716624382579964249047383780384233483283953907971557456848826811934997558340890106714439262837987573438185793607263236087851365277945956976543709998340361590134383718314428070011855946226376318839397712745672334684344586617496807908705803704071284048740118609114467977783598029006686938976881787785946905630190260940599579453432823469303026696443059025015972399867714215541693835559885291486318237914434496734087811872639496475100189041349008417061675093668333850551032972088269550769983616369411933015213796825837188091833656751221318492846368125550225998300412344784862595674492194617023806505913245610825731835380087608622102834270197698202313169017678006675195485079921636419370285375124784014907159135459982790513399611551794271106831134090584272884279791554849782954323534517065223269061394905987693002122963395687782878948440616007412945674919823050571642377154816321380631045902916136926708342856440730447899971901781465763473223850267253059899795996090799469201774624817718449867455659250178329070473119433165550807568221846571746373296884912819520317457002440926616910874148385078411929804522981857338977648103126085902995208257421855249796721729039744118165938433694823325696642096892124547425283",
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	 "347743159439876626079252796797422223177535447388206607607181663903045907591201940478223621722118173270898487582987137708656414344685816179420855160986340457973820182883508387588163122354089264395604796675278966117567294812714812796820596564876450716066283126720010859041484786529056457896367683122960411136319",
	 "47266428956356393164697365098120418976400602706072312735924071745438532218237979333351774907308168340693326687317443721193266215155735814510792148768576498491199122744351399489453533553203833318691678263241941706256996197460424029012419012634671862283532342656309677173602509498417976091509154360039893165037637034737020327399910409885798185771003505320583967737293415979917317338985837385734747478364242020380416892056650841470869294527543597349250299539682430605173321029026555546832473048600327036845781970289288898317888427517364945316709081173840186150794397479045034008257793436817683392375274635794835245695887",
	 "436463808505957768574894870394349739623346440601945961161254440072143298152040105676491048248110146278752857839930515766167441407021501229924721335644557342265864606569000117714935185566842453630868849121480179691838399545644365571106757731317371758557990781880691336695584799313313687287468894148823761785582982549586183756806449017542622267874275103877481475534991201849912222670102069951687572917937634467778042874315463238062009202992087620963771759666448266532858079402669920025224220613419441069718482837399612644978839925207109870840278194042158748845445131729137117098529028886770063736487420613144045836803985635654192482395882603511950547826439092832800532152534003936926017612446606135655146445620623395788978726744728503058670046885876251527122350275750995227",
	 "11424167473351836398078306042624362277956429440521137061889702611766348760692206243140413411077394583180726863277012016602279290144126785129569474909173584789822341986742719230331946072730319555984484911716797058875905400999504305877245849119687509023232790273637466821052576859232452982061831009770786031785669030271542286603956118755585683996118896215213488875253101894663403069677745948305893849505434201763745232895780711972432011344857521691017896316861403206449421332243658855453435784006517202894181640562433575390821384210960117518650374602256601091379644034244332285065935413233557998331562749140202965844219336298970011513882564935538704289446968322281451907487362046511461221329799897350993370560697505809686438782036235372137015731304779072430260986460269894522159103008260495503005267165927542949439526272736586626709581721032189532726389643625590680105784844246152702670169304203783072275089194754889511973916207",
	 "1214855636816562637502584060163403830270705000634713483015101384881871978446801224798536155406895823305035467591632531067547890948695117172076954220727075688048751022421198712032848890056357845974246560748347918630050853933697792254955890439720297560693579400297062396904306270145886830719309296352765295712183040773146419022875165382778007040109957609739589875590885701126197906063620133954893216612678838507540777138437797705602453719559017633986486649523611975865005712371194067612263330335590526176087004421363598470302731349138773205901447704682181517904064735636518462452242791676541725292378925568296858010151852326316777511935037531017413910506921922450666933202278489024521263798482237150056835746454842662048692127173834433089016107854491097456725016327709663199738238442164843147132789153725513257167915555162094970853584447993125488607696008169807374736711297007473812256272245489405898470297178738029484459690836250560495461579533254473316340608217876781986188705928270735695752830825527963838355419762516246028680280988020401914551825487349990306976304093109384451438813251211051597392127491464898797406789175453067960072008590614886532333015881171367104445044718144312416815712216611576221546455968770801413440778423979",
	 NULL
      };
      log = fopen("logs/expt.log", "w");
      logb = fopen("logs/expt_dr.log", "w");
      logc = fopen("logs/expt_2k.log", "w");
      logd = fopen("logs/expt_2kl.log", "w");
      for (n = 0; primes[n]; n++) {
	 SLEEP;
	 mp_read_radix(&a, primes[n], 10);
	 mp_zero(&b);
	 for (rr = 0; rr < (unsigned) mp_count_bits(&a); rr++) {
	    mp_mul_2(&b, &b);
	    b.dp[0] |= lbit();







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	 "347743159439876626079252796797422223177535447388206607607181663903045907591201940478223621722118173270898487582987137708656414344685816179420855160986340457973820182883508387588163122354089264395604796675278966117567294812714812796820596564876450716066283126720010859041484786529056457896367683122960411136319",
	 "47266428956356393164697365098120418976400602706072312735924071745438532218237979333351774907308168340693326687317443721193266215155735814510792148768576498491199122744351399489453533553203833318691678263241941706256996197460424029012419012634671862283532342656309677173602509498417976091509154360039893165037637034737020327399910409885798185771003505320583967737293415979917317338985837385734747478364242020380416892056650841470869294527543597349250299539682430605173321029026555546832473048600327036845781970289288898317888427517364945316709081173840186150794397479045034008257793436817683392375274635794835245695887",
	 "436463808505957768574894870394349739623346440601945961161254440072143298152040105676491048248110146278752857839930515766167441407021501229924721335644557342265864606569000117714935185566842453630868849121480179691838399545644365571106757731317371758557990781880691336695584799313313687287468894148823761785582982549586183756806449017542622267874275103877481475534991201849912222670102069951687572917937634467778042874315463238062009202992087620963771759666448266532858079402669920025224220613419441069718482837399612644978839925207109870840278194042158748845445131729137117098529028886770063736487420613144045836803985635654192482395882603511950547826439092832800532152534003936926017612446606135655146445620623395788978726744728503058670046885876251527122350275750995227",
	 "11424167473351836398078306042624362277956429440521137061889702611766348760692206243140413411077394583180726863277012016602279290144126785129569474909173584789822341986742719230331946072730319555984484911716797058875905400999504305877245849119687509023232790273637466821052576859232452982061831009770786031785669030271542286603956118755585683996118896215213488875253101894663403069677745948305893849505434201763745232895780711972432011344857521691017896316861403206449421332243658855453435784006517202894181640562433575390821384210960117518650374602256601091379644034244332285065935413233557998331562749140202965844219336298970011513882564935538704289446968322281451907487362046511461221329799897350993370560697505809686438782036235372137015731304779072430260986460269894522159103008260495503005267165927542949439526272736586626709581721032189532726389643625590680105784844246152702670169304203783072275089194754889511973916207",
	 "1214855636816562637502584060163403830270705000634713483015101384881871978446801224798536155406895823305035467591632531067547890948695117172076954220727075688048751022421198712032848890056357845974246560748347918630050853933697792254955890439720297560693579400297062396904306270145886830719309296352765295712183040773146419022875165382778007040109957609739589875590885701126197906063620133954893216612678838507540777138437797705602453719559017633986486649523611975865005712371194067612263330335590526176087004421363598470302731349138773205901447704682181517904064735636518462452242791676541725292378925568296858010151852326316777511935037531017413910506921922450666933202278489024521263798482237150056835746454842662048692127173834433089016107854491097456725016327709663199738238442164843147132789153725513257167915555162094970853584447993125488607696008169807374736711297007473812256272245489405898470297178738029484459690836250560495461579533254473316340608217876781986188705928270735695752830825527963838355419762516246028680280988020401914551825487349990306976304093109384451438813251211051597392127491464898797406789175453067960072008590614886532333015881171367104445044718144312416815712216611576221546455968770801413440778423979",
	 NULL
      };
      log = FOPEN("logs/expt.log", "w");
      logb = FOPEN("logs/expt_dr.log", "w");
      logc = FOPEN("logs/expt_2k.log", "w");
      logd = FOPEN("logs/expt_2kl.log", "w");
      for (n = 0; primes[n]; n++) {
	 SLEEP;
	 mp_read_radix(&a, primes[n], 10);
	 mp_zero(&b);
	 for (rr = 0; rr < (unsigned) mp_count_bits(&a); rr++) {
	    mp_mul_2(&b, &b);
	    b.dp[0] |= lbit();
267
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	 if (mp_cmp_d(&d, 1)) {
	    printf("Different (%d)!!!\n", mp_count_bits(&a));
	    draw(&d);
	    exit(0);
	 }
	 printf("Exponentiating\t%4d-bit => %9llu/sec, %9llu cycles\n",
		mp_count_bits(&a), CLK_PER_SEC / tt, tt);
	 fprintf(n < 4 ? logd : (n < 9) ? logc : (n < 16) ? logb : log,
		 "%d %9llu\n", mp_count_bits(&a), tt);
      }
   }
   fclose(log);
   fclose(logb);
   fclose(logc);
   fclose(logd);

   log = fopen("logs/invmod.log", "w");
   for (cnt = 4; cnt <= 128; cnt += 4) {
      SLEEP;
      mp_rand(&a, cnt);
      mp_rand(&b, cnt);

      do {
	 mp_add_d(&b, 1, &b);
	 mp_gcd(&a, &b, &c);







|



|
|
|
|

|
|







287
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	 if (mp_cmp_d(&d, 1)) {
	    printf("Different (%d)!!!\n", mp_count_bits(&a));
	    draw(&d);
	    exit(0);
	 }
	 printf("Exponentiating\t%4d-bit => %9llu/sec, %9llu cycles\n",
		mp_count_bits(&a), CLK_PER_SEC / tt, tt);
	 FPRINTF(n < 4 ? logd : (n < 9) ? logc : (n < 16) ? logb : log,
		 "%d %9llu\n", mp_count_bits(&a), tt);
      }
   }
   FCLOSE(log);
   FCLOSE(logb);
   FCLOSE(logc);
   FCLOSE(logd);

   log = FOPEN("logs/invmod.log", "w");
   for (cnt = 4; cnt <= 32; cnt += 4) {
      SLEEP;
      mp_rand(&a, cnt);
      mp_rand(&b, cnt);

      do {
	 mp_add_d(&b, 1, &b);
	 mp_gcd(&a, &b, &c);
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315




      mp_mulmod(&b, &c, &a, &d);
      if (mp_cmp_d(&d, 1) != MP_EQ) {
	 printf("Failed to invert\n");
	 return 0;
      }
      printf("Inverting mod\t%4d-bit => %9llu/sec, %9llu cycles\n",
	     mp_count_bits(&a), CLK_PER_SEC / tt, tt);
      fprintf(log, "%d %9llu\n", cnt * DIGIT_BIT, tt);
   }
   fclose(log);

   return 0;
}











|

|



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339
      mp_mulmod(&b, &c, &a, &d);
      if (mp_cmp_d(&d, 1) != MP_EQ) {
	 printf("Failed to invert\n");
	 return 0;
      }
      printf("Inverting mod\t%4d-bit => %9llu/sec, %9llu cycles\n",
	     mp_count_bits(&a), CLK_PER_SEC / tt, tt);
      FPRINTF(log, "%d %9llu\n", cnt * DIGIT_BIT, tt);
   }
   FCLOSE(log);

   return 0;
}

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/dep.pl.
64
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78
   # scan for mp_* and make classes
   while (<SRC>) {
      my $line = $_;
      while ($line =~ m/(fast_)*(s_)*mp\_[a-z_0-9]*/) {
          $line = $';
          # now $& is the match, we want to skip over LTM keywords like
          # mp_int, mp_word, mp_digit
          if (!($& eq "mp_digit") && !($& eq "mp_word") && !($& eq "mp_int")) {
             my $a = $&;
             $a =~ tr/[a-z]/[A-Z]/;
             $a = "BN_" . $a . "_C";
             if (!($list =~ /$a/)) {
                print CLASS "   #define $a\n";
             }
             $list = $list . "," . $a;







|







64
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78
   # scan for mp_* and make classes
   while (<SRC>) {
      my $line = $_;
      while ($line =~ m/(fast_)*(s_)*mp\_[a-z_0-9]*/) {
          $line = $';
          # now $& is the match, we want to skip over LTM keywords like
          # mp_int, mp_word, mp_digit
          if (!($& eq "mp_digit") && !($& eq "mp_word") && !($& eq "mp_int") && !($& eq "mp_min_u32")) {
             my $a = $&;
             $a =~ tr/[a-z]/[A-Z]/;
             $a = "BN_" . $a . "_C";
             if (!($list =~ /$a/)) {
                print CLASS "   #define $a\n";
             }
             $list = $list . "," . $a;
Changes to libtommath/etc/2kprime.c.
69
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73
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75









       mp_toradix(&q, buf, 10);
       printf("\n\n%d-bits (k = %lu) = %s\n", sizes[x], z, buf);
       fprintf(out, "%d-bits (k = %lu) = %s\n", sizes[x], z, buf); fflush(out);
   }
   
   return 0;
}   
















>
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       mp_toradix(&q, buf, 10);
       printf("\n\n%d-bits (k = %lu) = %s\n", sizes[x], z, buf);
       fprintf(out, "%d-bits (k = %lu) = %s\n", sizes[x], z, buf); fflush(out);
   }
   
   return 0;
}   
       
         
            
            
          

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/etc/drprime.c.
53
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57
58
59





   fclose(out);
   
   mp_clear(&a);
   mp_clear(&b);
   
   return 0;
}












>
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64
   fclose(out);
   
   mp_clear(&a);
   mp_clear(&b);
   
   return 0;
}


/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/etc/mersenne.c.
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139
140




    /* but make sure its prime */
    while (isprime (k) == 0) {
      k += 2;
    }
  }
  return 0;
}











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    /* but make sure its prime */
    while (isprime (k) == 0) {
      k += 2;
    }
  }
  return 0;
}

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/etc/mont.c.
35
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39
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41









           }
       }
       printf("PASSED\n");
    }
    
    return 0;
}
















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           }
       }
       printf("PASSED\n");
    }
    
    return 0;
}






/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/etc/pprime.c.
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394
395
396




  mp_toradix (&p, buf, 10);
  printf ("P == %s\n", buf);
  mp_toradix (&q, buf, 10);
  printf ("Q == %s\n", buf);

  return 0;
}











>
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>
>
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400
  mp_toradix (&p, buf, 10);
  printf ("P == %s\n", buf);
  mp_toradix (&q, buf, 10);
  printf ("Q == %s\n", buf);

  return 0;
}

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/etc/tune.c.
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11


12
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/* Tune the Karatsuba parameters
 *
 * Tom St Denis, [email protected]
 */
#include <tommath.h>
#include <time.h>

/* how many times todo each size mult.  Depends on your computer.  For slow computers
 * this can be low like 5 or 10.  For fast [re: Athlon] should be 25 - 50 or so 
 */
#define TIMES (1UL<<14UL)



/* RDTSC from Scott Duplichan */
static ulong64 TIMFUNC (void)
   {
   #if defined __GNUC__
      #if defined(__i386__) || defined(__x86_64__)



         unsigned long long a;
         __asm__ __volatile__ ("rdtsc\nmovl %%eax,%0\nmovl %%edx,4+%0\n"::"m"(a):"%eax","%edx");
         return a;
      #else /* gcc-IA64 version */
         unsigned long result;
         __asm__ __volatile__("mov %0=ar.itc" : "=r"(result) :: "memory");
         while (__builtin_expect ((int) result == -1, 0))
         __asm__ __volatile__("mov %0=ar.itc" : "=r"(result) :: "memory");
         return result;
      #endif








|


>
>






>
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>
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|







1
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/* Tune the Karatsuba parameters
 *
 * Tom St Denis, [email protected]
 */
#include <tommath.h>
#include <time.h>

/* how many times todo each size mult.  Depends on your computer.  For slow computers
 * this can be low like 5 or 10.  For fast [re: Athlon] should be 25 - 50 or so
 */
#define TIMES (1UL<<14UL)

#ifndef X86_TIMER

/* RDTSC from Scott Duplichan */
static ulong64 TIMFUNC (void)
   {
   #if defined __GNUC__
      #if defined(__i386__) || defined(__x86_64__)
        /* version from http://www.mcs.anl.gov/~kazutomo/rdtsc.html
         * the old code always got a warning issued by gcc, clang did not complain...
         */
        unsigned hi, lo;
        __asm__ __volatile__ ("rdtsc" : "=a"(lo), "=d"(hi));
        return ((ulong64)lo)|( ((ulong64)hi)<<32);
      #else /* gcc-IA64 version */
         unsigned long result;
         __asm__ __volatile__("mov %0=ar.itc" : "=r"(result) :: "memory");
         while (__builtin_expect ((int) result == -1, 0))
         __asm__ __volatile__("mov %0=ar.itc" : "=r"(result) :: "memory");
         return result;
      #endif
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      return __getReg (3116);
   #else
     #error need rdtsc function for this build
   #endif
   }


#ifndef X86_TIMER

/* generic ISO C timer */
ulong64 LBL_T;
void t_start(void) { LBL_T = TIMFUNC(); }
ulong64 t_read(void) { return TIMFUNC() - LBL_T; }

#else
extern void t_start(void);







<
<







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50
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56
      return __getReg (3116);
   #else
     #error need rdtsc function for this build
   #endif
   }




/* generic ISO C timer */
ulong64 LBL_T;
void t_start(void) { LBL_T = TIMFUNC(); }
ulong64 t_read(void) { return TIMFUNC() - LBL_T; }

#else
extern void t_start(void);
63
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  mp_init (&a);
  mp_init (&b);
  mp_init (&c);

  mp_rand (&a, size);
  mp_rand (&b, size);

  if (s == 1) { 
      KARATSUBA_MUL_CUTOFF = size;
  } else {
      KARATSUBA_MUL_CUTOFF = 100000;
  }

  t_start();
  for (x = 0; x < TIMES; x++) {







|







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80
  mp_init (&a);
  mp_init (&b);
  mp_init (&c);

  mp_rand (&a, size);
  mp_rand (&b, size);

  if (s == 1) {
      KARATSUBA_MUL_CUTOFF = size;
  } else {
      KARATSUBA_MUL_CUTOFF = 100000;
  }

  t_start();
  for (x = 0; x < TIMES; x++) {
91
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94
95
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97
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99
100
101
102
103
104
105
  ulong64 t1;

  mp_init (&a);
  mp_init (&b);

  mp_rand (&a, size);

  if (s == 1) { 
      KARATSUBA_SQR_CUTOFF = size;
  } else {
      KARATSUBA_SQR_CUTOFF = 100000;
  }

  t_start();
  for (x = 0; x < TIMES; x++) {







|







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100
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104
105
106
107
108
  ulong64 t1;

  mp_init (&a);
  mp_init (&b);

  mp_rand (&a, size);

  if (s == 1) {
      KARATSUBA_SQR_CUTOFF = size;
  } else {
      KARATSUBA_SQR_CUTOFF = 100000;
  }

  t_start();
  for (x = 0; x < TIMES; x++) {
113
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130
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134
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137
138





int
main (void)
{
  ulong64 t1, t2;
  int x, y;

  for (x = 8; ; x += 2) { 
     t1 = time_mult(x, 0);
     t2 = time_mult(x, 1);
     printf("%d: %9llu %9llu, %9llu\n", x, t1, t2, t2 - t1);
     if (t2 < t1) break;
  }
  y = x;

  for (x = 8; ; x += 2) { 
     t1 = time_sqr(x, 0);
     t2 = time_sqr(x, 1);
     printf("%d: %9llu %9llu, %9llu\n", x, t1, t2, t2 - t1);
     if (t2 < t1) break;
  }
  printf("KARATSUBA_MUL_CUTOFF = %d\n", y);
  printf("KARATSUBA_SQR_CUTOFF = %d\n", x);

  return 0;
}











|







|










>
>
>
>
116
117
118
119
120
121
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123
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125
126
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128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145

int
main (void)
{
  ulong64 t1, t2;
  int x, y;

  for (x = 8; ; x += 2) {
     t1 = time_mult(x, 0);
     t2 = time_mult(x, 1);
     printf("%d: %9llu %9llu, %9llu\n", x, t1, t2, t2 - t1);
     if (t2 < t1) break;
  }
  y = x;

  for (x = 8; ; x += 2) {
     t1 = time_sqr(x, 0);
     t2 = time_sqr(x, 1);
     printf("%d: %9llu %9llu, %9llu\n", x, t1, t2, t2 - t1);
     if (t2 < t1) break;
  }
  printf("KARATSUBA_MUL_CUTOFF = %d\n", y);
  printf("KARATSUBA_SQR_CUTOFF = %d\n", x);

  return 0;
}

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/gen.pl.
11
12
13
14
15
16
17


   print OUT "/* Start: $filename */\n";
   print OUT while <SRC>;
   print OUT "\n/* End: $filename */\n\n";
   close SRC or die "Error closing $filename after reading: $!";
}
print OUT "\n/* EOF */\n";
close OUT or die "Error closing mpi.c after writing: $!";









>
>
11
12
13
14
15
16
17
18
19
   print OUT "/* Start: $filename */\n";
   print OUT while <SRC>;
   print OUT "\n/* End: $filename */\n\n";
   close SRC or die "Error closing $filename after reading: $!";
}
print OUT "\n/* EOF */\n";
close OUT or die "Error closing mpi.c after writing: $!";

system('perl -pli -e "s/\s*$//" mpi.c');
Changes to libtommath/makefile.
1
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6
7
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16
17
18
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59

60

61
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63
64
65




66
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#Makefile for GCC
#
#Tom St Denis

#version of library 
VERSION=0.42.0

CFLAGS  +=  -I./ -Wall -W -Wshadow -Wsign-compare

ifndef MAKE
   MAKE=make
endif

ifndef IGNORE_SPEED

#for speed 
CFLAGS += -O3 -funroll-loops

#for size 
#CFLAGS += -Os

#x86 optimizations [should be valid for any GCC install though]
CFLAGS  += -fomit-frame-pointer

#debug
#CFLAGS += -g3

endif

#install as this user
ifndef INSTALL_GROUP
   GROUP=wheel
else
   GROUP=$(INSTALL_GROUP)
endif

ifndef INSTALL_USER
   USER=root
else
   USER=$(INSTALL_USER)
endif

#default files to install
ifndef LIBNAME
   LIBNAME=libtommath.a
endif

default: ${LIBNAME}

HEADERS=tommath.h tommath_class.h tommath_superclass.h

#LIBPATH-The directory for libtommath to be installed to.
#INCPATH-The directory to install the header files for libtommath.
#DATAPATH-The directory to install the pdf docs.
DESTDIR=
LIBPATH=/usr/lib
INCPATH=/usr/include
DATAPATH=/usr/share/doc/libtommath/pdf


OBJECTS=bncore.o bn_mp_init.o bn_mp_clear.o bn_mp_exch.o bn_mp_grow.o bn_mp_shrink.o \

bn_mp_clamp.o bn_mp_zero.o  bn_mp_set.o bn_mp_set_int.o bn_mp_init_size.o bn_mp_copy.o \
bn_mp_init_copy.o bn_mp_abs.o bn_mp_neg.o bn_mp_cmp_mag.o bn_mp_cmp.o bn_mp_cmp_d.o \
bn_mp_rshd.o bn_mp_lshd.o bn_mp_mod_2d.o bn_mp_div_2d.o bn_mp_mul_2d.o bn_mp_div_2.o \
bn_mp_mul_2.o bn_s_mp_add.o bn_s_mp_sub.o bn_fast_s_mp_mul_digs.o bn_s_mp_mul_digs.o \
bn_fast_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_s_mp_sqr.o \




bn_mp_add.o bn_mp_sub.o bn_mp_karatsuba_mul.o bn_mp_mul.o bn_mp_karatsuba_sqr.o \
bn_mp_sqr.o bn_mp_div.o bn_mp_mod.o bn_mp_add_d.o bn_mp_sub_d.o bn_mp_mul_d.o \
bn_mp_div_d.o bn_mp_mod_d.o bn_mp_expt_d.o bn_mp_addmod.o bn_mp_submod.o \
bn_mp_mulmod.o bn_mp_sqrmod.o bn_mp_gcd.o bn_mp_lcm.o bn_fast_mp_invmod.o bn_mp_invmod.o \
bn_mp_reduce.o bn_mp_montgomery_setup.o bn_fast_mp_montgomery_reduce.o bn_mp_montgomery_reduce.o \
bn_mp_exptmod_fast.o bn_mp_exptmod.o bn_mp_2expt.o bn_mp_n_root.o bn_mp_jacobi.o bn_reverse.o \
bn_mp_count_bits.o bn_mp_read_unsigned_bin.o bn_mp_read_signed_bin.o bn_mp_to_unsigned_bin.o \
bn_mp_to_signed_bin.o bn_mp_unsigned_bin_size.o bn_mp_signed_bin_size.o  \
bn_mp_xor.o bn_mp_and.o bn_mp_or.o bn_mp_rand.o bn_mp_montgomery_calc_normalization.o \
bn_mp_prime_is_divisible.o bn_prime_tab.o bn_mp_prime_fermat.o bn_mp_prime_miller_rabin.o \
bn_mp_prime_is_prime.o bn_mp_prime_next_prime.o bn_mp_dr_reduce.o \
bn_mp_dr_is_modulus.o bn_mp_dr_setup.o bn_mp_reduce_setup.o \
bn_mp_toom_mul.o bn_mp_toom_sqr.o bn_mp_div_3.o bn_s_mp_exptmod.o \
bn_mp_reduce_2k.o bn_mp_reduce_is_2k.o bn_mp_reduce_2k_setup.o \
bn_mp_reduce_2k_l.o bn_mp_reduce_is_2k_l.o bn_mp_reduce_2k_setup_l.o \
bn_mp_radix_smap.o bn_mp_read_radix.o bn_mp_toradix.o bn_mp_radix_size.o \
bn_mp_fread.o bn_mp_fwrite.o bn_mp_cnt_lsb.o bn_error.o \
bn_mp_init_multi.o bn_mp_clear_multi.o bn_mp_exteuclid.o bn_mp_toradix_n.o \
bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o bn_mp_init_set.o \


bn_mp_init_set_int.o bn_mp_invmod_slow.o bn_mp_prime_rabin_miller_trials.o \
bn_mp_to_signed_bin_n.o bn_mp_to_unsigned_bin_n.o


$(LIBNAME):  $(OBJECTS)
	$(AR) $(ARFLAGS) $@ $(OBJECTS)
	ranlib $@

#make a profiled library (takes a while!!!)
#
# This will build the library with profile generation
# then run the test demo and rebuild the library.
# 
# So far I've seen improvements in the MP math
profiled:
	make CFLAGS="$(CFLAGS) -fprofile-arcs -DTESTING" timing
	./ltmtest
	rm -f *.a *.o ltmtest
	make CFLAGS="$(CFLAGS) -fbranch-probabilities"

#make a single object profiled library 
profiled_single:
	perl gen.pl
	$(CC) $(CFLAGS) -fprofile-arcs -DTESTING -c mpi.c -o mpi.o
	$(CC) $(CFLAGS) -DTESTING -DTIMER demo/timing.c mpi.o -o ltmtest
	./ltmtest
	rm -f *.o ltmtest
	$(CC) $(CFLAGS) -fbranch-probabilities -DTESTING -c mpi.c -o mpi.o
	$(AR) $(ARFLAGS) $(LIBNAME) mpi.o
	ranlib $(LIBNAME)	

install: $(LIBNAME)
	install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(LIBPATH)
	install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(INCPATH)
	install -g $(GROUP) -o $(USER) $(LIBNAME) $(DESTDIR)$(LIBPATH)
	install -g $(GROUP) -o $(USER) $(HEADERS) $(DESTDIR)$(INCPATH)

test: $(LIBNAME) demo/demo.o
	$(CC) $(CFLAGS) demo/demo.o $(LIBNAME) -o test
	




mtest: test	
	cd mtest ; $(CC) $(CFLAGS) mtest.c -o mtest
        
timing: $(LIBNAME)
	$(CC) $(CFLAGS) -DTIMER demo/timing.c $(LIBNAME) -o ltmtest




# makes the LTM book DVI file, requires tetex, perl and makeindex [part of tetex I think]
docdvi: tommath.src
	cd pics ; MAKE=${MAKE} ${MAKE} 
	echo "hello" > tommath.ind
	perl booker.pl
	latex tommath > /dev/null
	latex tommath > /dev/null
	makeindex tommath
	latex tommath > /dev/null

# poster, makes the single page PDF poster
poster: poster.tex









	pdflatex poster


	rm -f poster.aux poster.log 

# makes the LTM book PDF file, requires tetex, cleans up the LaTeX temp files
docs:   docdvi
	dvipdf tommath
	rm -f tommath.log tommath.aux tommath.dvi tommath.idx tommath.toc tommath.lof tommath.ind tommath.ilg
	cd pics ; MAKE=${MAKE} ${MAKE} clean
	
#LTM user manual
mandvi: bn.tex









	echo "hello" > bn.ind
	latex bn > /dev/null
	latex bn > /dev/null
	makeindex bn
	latex bn > /dev/null

#LTM user manual [pdf]
manual:	mandvi
	pdflatex bn >/dev/null


	rm -f bn.aux bn.dvi bn.log bn.idx bn.lof bn.out bn.toc

pretty: 
	perl pretty.build

clean:
	rm -f *.bat *.pdf *.o *.a *.obj *.lib *.exe *.dll etclib/*.o demo/demo.o test ltmtest mpitest mtest/mtest mtest/mtest.exe \
        *.idx *.toc *.log *.aux *.dvi *.lof *.ind *.ilg *.ps *.log *.s mpi.c *.da *.dyn *.dpi tommath.tex `find . -type f | grep [~] | xargs` *.lo *.la
	rm -rf .libs
	cd etc ; MAKE=${MAKE} ${MAKE} clean
	cd pics ; MAKE=${MAKE} ${MAKE} clean

#zipup the project (take that!)
no_oops: clean
	cd .. ; cvs commit 
	echo Scanning for scratch/dirty files
	find . -type f | grep -v CVS | xargs -n 1 bash mess.sh

zipup: clean manual poster docs


	perl gen.pl ; mv mpi.c pre_gen/ ; \




	cd .. ; rm -rf ltm* libtommath-$(VERSION) ; mkdir libtommath-$(VERSION) ; \


	cp -R ./libtommath/* ./libtommath-$(VERSION)/ ; \

	tar -c libtommath-$(VERSION)/* | bzip2 -9vvc > ltm-$(VERSION).tar.bz2 ; \

	zip -9 -r ltm-$(VERSION).zip libtommath-$(VERSION)/* ; \

	mv -f ltm* ~ ; rm -rf libtommath-$(VERSION)







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#Makefile for GCC
#
#Tom St Denis



ifeq ($V,1)

silent=
else
silent=@
endif


%.o: %.c


ifneq ($V,1)


	@echo "   * ${CC} $@"






endif
	${silent} ${CC} -c ${CFLAGS} $^ -o $@













#default files to install
ifndef LIBNAME
   LIBNAME=libtommath.a
endif

coverage: LIBNAME:=-Wl,--whole-archive $(LIBNAME)  -Wl,--no-whole-archive

include makefile.include



LCOV_ARGS=--directory .





#START_INS
OBJECTS=bncore.o bn_error.o bn_fast_mp_invmod.o bn_fast_mp_montgomery_reduce.o bn_fast_s_mp_mul_digs.o \
bn_fast_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_mp_2expt.o bn_mp_abs.o bn_mp_add.o bn_mp_add_d.o \
bn_mp_addmod.o bn_mp_and.o bn_mp_clamp.o bn_mp_clear.o bn_mp_clear_multi.o bn_mp_cmp.o bn_mp_cmp_d.o \
bn_mp_cmp_mag.o bn_mp_cnt_lsb.o bn_mp_copy.o bn_mp_count_bits.o bn_mp_div_2.o bn_mp_div_2d.o bn_mp_div_3.o \
bn_mp_div.o bn_mp_div_d.o bn_mp_dr_is_modulus.o bn_mp_dr_reduce.o bn_mp_dr_setup.o bn_mp_exch.o \


bn_mp_export.o bn_mp_expt_d.o bn_mp_expt_d_ex.o bn_mp_exptmod.o bn_mp_exptmod_fast.o bn_mp_exteuclid.o \
bn_mp_fread.o bn_mp_fwrite.o bn_mp_gcd.o bn_mp_get_int.o bn_mp_get_long.o bn_mp_get_long_long.o \
bn_mp_grow.o bn_mp_import.o bn_mp_init.o bn_mp_init_copy.o bn_mp_init_multi.o bn_mp_init_set.o \
bn_mp_init_set_int.o bn_mp_init_size.o bn_mp_invmod.o bn_mp_invmod_slow.o bn_mp_is_square.o \
bn_mp_jacobi.o bn_mp_karatsuba_mul.o bn_mp_karatsuba_sqr.o bn_mp_lcm.o bn_mp_lshd.o bn_mp_mod_2d.o \

bn_mp_mod.o bn_mp_mod_d.o bn_mp_montgomery_calc_normalization.o bn_mp_montgomery_reduce.o \

bn_mp_montgomery_setup.o bn_mp_mul_2.o bn_mp_mul_2d.o bn_mp_mul.o bn_mp_mul_d.o bn_mp_mulmod.o bn_mp_neg.o \




bn_mp_n_root.o bn_mp_n_root_ex.o bn_mp_or.o bn_mp_prime_fermat.o bn_mp_prime_is_divisible.o \
bn_mp_prime_is_prime.o bn_mp_prime_miller_rabin.o bn_mp_prime_next_prime.o \
bn_mp_prime_rabin_miller_trials.o bn_mp_prime_random_ex.o bn_mp_radix_size.o bn_mp_radix_smap.o \
bn_mp_rand.o bn_mp_read_radix.o bn_mp_read_signed_bin.o bn_mp_read_unsigned_bin.o bn_mp_reduce_2k.o \
bn_mp_reduce_2k_l.o bn_mp_reduce_2k_setup.o bn_mp_reduce_2k_setup_l.o bn_mp_reduce.o \
bn_mp_reduce_is_2k.o bn_mp_reduce_is_2k_l.o bn_mp_reduce_setup.o bn_mp_rshd.o bn_mp_set.o bn_mp_set_int.o \
bn_mp_set_long.o bn_mp_set_long_long.o bn_mp_shrink.o bn_mp_signed_bin_size.o bn_mp_sqr.o bn_mp_sqrmod.o \
bn_mp_sqrt.o bn_mp_sqrtmod_prime.o bn_mp_sub.o bn_mp_sub_d.o bn_mp_submod.o bn_mp_toom_mul.o \
bn_mp_toom_sqr.o bn_mp_toradix.o bn_mp_toradix_n.o bn_mp_to_signed_bin.o bn_mp_to_signed_bin_n.o \

bn_mp_to_unsigned_bin.o bn_mp_to_unsigned_bin_n.o bn_mp_unsigned_bin_size.o bn_mp_xor.o bn_mp_zero.o \
bn_prime_tab.o bn_reverse.o bn_s_mp_add.o bn_s_mp_exptmod.o bn_s_mp_mul_digs.o bn_s_mp_mul_high_digs.o \
bn_s_mp_sqr.o bn_s_mp_sub.o

#END_INS

$(LIBNAME):  $(OBJECTS)
	$(AR) $(ARFLAGS) $@ $(OBJECTS)
	$(RANLIB) $@

#make a profiled library (takes a while!!!)
#
# This will build the library with profile generation
# then run the test demo and rebuild the library.
#
# So far I've seen improvements in the MP math
profiled:
	make CFLAGS="$(CFLAGS) -fprofile-arcs -DTESTING" timing
	./ltmtest
	rm -f *.a *.o ltmtest
	make CFLAGS="$(CFLAGS) -fbranch-probabilities"

#make a single object profiled library
profiled_single:
	perl gen.pl
	$(CC) $(CFLAGS) -fprofile-arcs -DTESTING -c mpi.c -o mpi.o
	$(CC) $(CFLAGS) -DTESTING -DTIMER demo/timing.c mpi.o -lgcov -o ltmtest
	./ltmtest
	rm -f *.o ltmtest
	$(CC) $(CFLAGS) -fbranch-probabilities -DTESTING -c mpi.c -o mpi.o
	$(AR) $(ARFLAGS) $(LIBNAME) mpi.o
	ranlib $(LIBNAME)

install: $(LIBNAME)
	install -d $(DESTDIR)$(LIBPATH)
	install -d $(DESTDIR)$(INCPATH)
	install -m 644 $(LIBNAME) $(DESTDIR)$(LIBPATH)
	install -m 644 $(HEADERS_PUB) $(DESTDIR)$(INCPATH)

test: $(LIBNAME) demo/demo.o
	$(CC) $(CFLAGS) demo/demo.o $(LIBNAME) $(LFLAGS) -o test

test_standalone: $(LIBNAME) demo/demo.o
	$(CC) $(CFLAGS) demo/demo.o $(LIBNAME) $(LFLAGS) -o test

.PHONY: mtest
mtest:
	cd mtest ; $(CC) $(CFLAGS) -O0 mtest.c $(LFLAGS) -o mtest

timing: $(LIBNAME)
	$(CC) $(CFLAGS) -DTIMER demo/timing.c $(LIBNAME) $(LFLAGS) -o ltmtest

coveralls: coverage
	cpp-coveralls

# makes the LTM book DVI file, requires tetex, perl and makeindex [part of tetex I think]
docdvi: tommath.src
	cd pics ; MAKE=${MAKE} ${MAKE}
	echo "hello" > tommath.ind
	perl booker.pl
	latex tommath > /dev/null
	latex tommath > /dev/null
	makeindex tommath
	latex tommath > /dev/null

# poster, makes the single page PDF poster
poster: poster.tex
	cp poster.tex poster.bak
	touch --reference=poster.tex poster.bak
	(printf "%s" "\def\fixedpdfdate{"; date +'D:%Y%m%d%H%M%S%:z' -d @$$(stat --format=%Y poster.tex) | sed "s/:\([0-9][0-9]\)$$/'\1'}/g") > poster-deterministic.tex
	printf "%s\n" "\pdfinfo{" >> poster-deterministic.tex
	printf "%s\n" "  /CreationDate (\fixedpdfdate)" >> poster-deterministic.tex
	printf "%s\n}\n" "  /ModDate (\fixedpdfdate)" >> poster-deterministic.tex
	cat poster.tex >> poster-deterministic.tex
	mv poster-deterministic.tex poster.tex
	touch --reference=poster.bak poster.tex
	pdflatex poster
	sed -b -i 's,^/ID \[.*\]$$,/ID [<0> <0>],g' poster.pdf
	mv poster.bak poster.tex
	rm -f poster.aux poster.log poster.out

# makes the LTM book PDF file, requires tetex, cleans up the LaTeX temp files
docs:   docdvi
	dvipdf tommath
	rm -f tommath.log tommath.aux tommath.dvi tommath.idx tommath.toc tommath.lof tommath.ind tommath.ilg
	cd pics ; MAKE=${MAKE} ${MAKE} clean

#LTM user manual
mandvi: bn.tex
	cp bn.tex bn.bak
	touch --reference=bn.tex bn.bak
	(printf "%s" "\def\fixedpdfdate{"; date +'D:%Y%m%d%H%M%S%:z' -d @$$(stat --format=%Y bn.tex) | sed "s/:\([0-9][0-9]\)$$/'\1'}/g") > bn-deterministic.tex
	printf "%s\n" "\pdfinfo{" >> bn-deterministic.tex
	printf "%s\n" "  /CreationDate (\fixedpdfdate)" >> bn-deterministic.tex
	printf "%s\n}\n" "  /ModDate (\fixedpdfdate)" >> bn-deterministic.tex
	cat bn.tex >> bn-deterministic.tex
	mv bn-deterministic.tex bn.tex
	touch --reference=bn.bak bn.tex
	echo "hello" > bn.ind
	latex bn > /dev/null
	latex bn > /dev/null
	makeindex bn
	latex bn > /dev/null

#LTM user manual [pdf]
manual:	mandvi
	pdflatex bn >/dev/null
	sed -b -i 's,^/ID \[.*\]$$,/ID [<0> <0>],g' bn.pdf
	mv bn.bak bn.tex
	rm -f bn.aux bn.dvi bn.log bn.idx bn.lof bn.out bn.toc

pretty:
	perl pretty.build








#\zipup the project (take that!)
no_oops: clean
	cd .. ; cvs commit
	echo Scanning for scratch/dirty files
	find . -type f | grep -v CVS | xargs -n 1 bash mess.sh


.PHONY: pre_gen
pre_gen:
	perl gen.pl
	sed -e 's/[[:blank:]]*$$//' mpi.c > pre_gen/mpi.c
	rm mpi.c

zipup:
	rm -rf ../libtommath-$(VERSION) \
		&& rm -f ../ltm-$(VERSION).zip ../ltm-$(VERSION).zip.asc ../ltm-$(VERSION).tar.xz ../ltm-$(VERSION).tar.xz.asc
	git archive HEAD --prefix=libtommath-$(VERSION)/ > ../libtommath-$(VERSION).tar
	cd .. ; tar xf libtommath-$(VERSION).tar
	MAKE=${MAKE} ${MAKE} -C ../libtommath-$(VERSION) clean manual poster docs
	tar -c ../libtommath-$(VERSION)/* | xz -9 > ../ltm-$(VERSION).tar.xz
	find ../libtommath-$(VERSION)/ -type f -exec unix2dos -q {} \;
	cd .. ; zip -9r ltm-$(VERSION).zip libtommath-$(VERSION)
	gpg -b -a ../ltm-$(VERSION).tar.xz && gpg -b -a ../ltm-$(VERSION).zip

new_file:
	bash updatemakes.sh
	perl dep.pl
Changes to libtommath/makefile.bcc.
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#
# Borland C++Builder Makefile (makefile.bcc)
#


LIB = tlib
CC = bcc32
CFLAGS = -c -O2 -I.


OBJECTS=bncore.obj bn_mp_init.obj bn_mp_clear.obj bn_mp_exch.obj bn_mp_grow.obj bn_mp_shrink.obj \
bn_mp_clamp.obj bn_mp_zero.obj  bn_mp_set.obj bn_mp_set_int.obj bn_mp_init_size.obj bn_mp_copy.obj \
bn_mp_init_copy.obj bn_mp_abs.obj bn_mp_neg.obj bn_mp_cmp_mag.obj bn_mp_cmp.obj bn_mp_cmp_d.obj \
bn_mp_rshd.obj bn_mp_lshd.obj bn_mp_mod_2d.obj bn_mp_div_2d.obj bn_mp_mul_2d.obj bn_mp_div_2.obj \
bn_mp_mul_2.obj bn_s_mp_add.obj bn_s_mp_sub.obj bn_fast_s_mp_mul_digs.obj bn_s_mp_mul_digs.obj \
bn_fast_s_mp_mul_high_digs.obj bn_s_mp_mul_high_digs.obj bn_fast_s_mp_sqr.obj bn_s_mp_sqr.obj \
bn_mp_add.obj bn_mp_sub.obj bn_mp_karatsuba_mul.obj bn_mp_mul.obj bn_mp_karatsuba_sqr.obj \

bn_mp_sqr.obj bn_mp_div.obj bn_mp_mod.obj bn_mp_add_d.obj bn_mp_sub_d.obj bn_mp_mul_d.obj \
bn_mp_div_d.obj bn_mp_mod_d.obj bn_mp_expt_d.obj bn_mp_addmod.obj bn_mp_submod.obj \
bn_mp_mulmod.obj bn_mp_sqrmod.obj bn_mp_gcd.obj bn_mp_lcm.obj bn_fast_mp_invmod.obj bn_mp_invmod.obj \
bn_mp_reduce.obj bn_mp_montgomery_setup.obj bn_fast_mp_montgomery_reduce.obj bn_mp_montgomery_reduce.obj \
bn_mp_exptmod_fast.obj bn_mp_exptmod.obj bn_mp_2expt.obj bn_mp_n_root.obj bn_mp_jacobi.obj bn_reverse.obj \
bn_mp_count_bits.obj bn_mp_read_unsigned_bin.obj bn_mp_read_signed_bin.obj bn_mp_to_unsigned_bin.obj \
bn_mp_to_signed_bin.obj bn_mp_unsigned_bin_size.obj bn_mp_signed_bin_size.obj  \
bn_mp_xor.obj bn_mp_and.obj bn_mp_or.obj bn_mp_rand.obj bn_mp_montgomery_calc_normalization.obj \

bn_mp_prime_is_divisible.obj bn_prime_tab.obj bn_mp_prime_fermat.obj bn_mp_prime_miller_rabin.obj \
bn_mp_prime_is_prime.obj bn_mp_prime_next_prime.obj bn_mp_dr_reduce.obj \
bn_mp_dr_is_modulus.obj bn_mp_dr_setup.obj bn_mp_reduce_setup.obj \
bn_mp_toom_mul.obj bn_mp_toom_sqr.obj bn_mp_div_3.obj bn_s_mp_exptmod.obj \
bn_mp_reduce_2k.obj bn_mp_reduce_is_2k.obj bn_mp_reduce_2k_setup.obj \
bn_mp_reduce_2k_l.obj bn_mp_reduce_is_2k_l.obj bn_mp_reduce_2k_setup_l.obj \
bn_mp_radix_smap.obj bn_mp_read_radix.obj bn_mp_toradix.obj bn_mp_radix_size.obj \
bn_mp_fread.obj bn_mp_fwrite.obj bn_mp_cnt_lsb.obj bn_error.obj \
bn_mp_init_multi.obj bn_mp_clear_multi.obj bn_mp_exteuclid.obj bn_mp_toradix_n.obj \


bn_mp_prime_random_ex.obj bn_mp_get_int.obj bn_mp_sqrt.obj bn_mp_is_square.obj \
bn_mp_init_set.obj bn_mp_init_set_int.obj bn_mp_invmod_slow.obj bn_mp_prime_rabin_miller_trials.obj \

bn_mp_to_signed_bin_n.obj bn_mp_to_unsigned_bin_n.obj


TARGET = libtommath.lib

$(TARGET): $(OBJECTS)

.c.obj:
	$(CC) $(CFLAGS) $<









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#
# Borland C++Builder Makefile (makefile.bcc)
#


LIB = tlib
CC = bcc32
CFLAGS = -c -O2 -I.

#START_INS
OBJECTS=bncore.obj bn_error.obj bn_fast_mp_invmod.obj bn_fast_mp_montgomery_reduce.obj bn_fast_s_mp_mul_digs.obj \




bn_fast_s_mp_mul_high_digs.obj bn_fast_s_mp_sqr.obj bn_mp_2expt.obj bn_mp_abs.obj bn_mp_add.obj bn_mp_add_d.obj \
bn_mp_addmod.obj bn_mp_and.obj bn_mp_clamp.obj bn_mp_clear.obj bn_mp_clear_multi.obj bn_mp_cmp.obj bn_mp_cmp_d.obj \
bn_mp_cmp_mag.obj bn_mp_cnt_lsb.obj bn_mp_copy.obj bn_mp_count_bits.obj bn_mp_div_2.obj bn_mp_div_2d.obj bn_mp_div_3.obj \
bn_mp_div.obj bn_mp_div_d.obj bn_mp_dr_is_modulus.obj bn_mp_dr_reduce.obj bn_mp_dr_setup.obj bn_mp_exch.obj \
bn_mp_export.obj bn_mp_expt_d.obj bn_mp_expt_d_ex.obj bn_mp_exptmod.obj bn_mp_exptmod_fast.obj bn_mp_exteuclid.obj \
bn_mp_fread.obj bn_mp_fwrite.obj bn_mp_gcd.obj bn_mp_get_int.obj bn_mp_get_long.obj bn_mp_get_long_long.obj \

bn_mp_grow.obj bn_mp_import.obj bn_mp_init.obj bn_mp_init_copy.obj bn_mp_init_multi.obj bn_mp_init_set.obj \
bn_mp_init_set_int.obj bn_mp_init_size.obj bn_mp_invmod.obj bn_mp_invmod_slow.obj bn_mp_is_square.obj \
bn_mp_jacobi.obj bn_mp_karatsuba_mul.obj bn_mp_karatsuba_sqr.obj bn_mp_lcm.obj bn_mp_lshd.obj bn_mp_mod_2d.obj \
bn_mp_mod.obj bn_mp_mod_d.obj bn_mp_montgomery_calc_normalization.obj bn_mp_montgomery_reduce.obj \
bn_mp_montgomery_setup.obj bn_mp_mul_2.obj bn_mp_mul_2d.obj bn_mp_mul.obj bn_mp_mul_d.obj bn_mp_mulmod.obj bn_mp_neg.obj \
bn_mp_n_root.obj bn_mp_n_root_ex.obj bn_mp_or.obj bn_mp_prime_fermat.obj bn_mp_prime_is_divisible.obj \
bn_mp_prime_is_prime.obj bn_mp_prime_miller_rabin.obj bn_mp_prime_next_prime.obj \
bn_mp_prime_rabin_miller_trials.obj bn_mp_prime_random_ex.obj bn_mp_radix_size.obj bn_mp_radix_smap.obj \
bn_mp_rand.obj bn_mp_read_radix.obj bn_mp_read_signed_bin.obj bn_mp_read_unsigned_bin.obj bn_mp_reduce_2k.obj \
bn_mp_reduce_2k_l.obj bn_mp_reduce_2k_setup.obj bn_mp_reduce_2k_setup_l.obj bn_mp_reduce.obj \
bn_mp_reduce_is_2k.obj bn_mp_reduce_is_2k_l.obj bn_mp_reduce_setup.obj bn_mp_rshd.obj bn_mp_set.obj bn_mp_set_int.obj \
bn_mp_set_long.obj bn_mp_set_long_long.obj bn_mp_shrink.obj bn_mp_signed_bin_size.obj bn_mp_sqr.obj bn_mp_sqrmod.obj \
bn_mp_sqrt.obj bn_mp_sqrtmod_prime.obj bn_mp_sub.obj bn_mp_sub_d.obj bn_mp_submod.obj bn_mp_toom_mul.obj \
bn_mp_toom_sqr.obj bn_mp_toradix.obj bn_mp_toradix_n.obj bn_mp_to_signed_bin.obj bn_mp_to_signed_bin_n.obj \
bn_mp_to_unsigned_bin.obj bn_mp_to_unsigned_bin_n.obj bn_mp_unsigned_bin_size.obj bn_mp_xor.obj bn_mp_zero.obj \
bn_prime_tab.obj bn_reverse.obj bn_s_mp_add.obj bn_s_mp_exptmod.obj bn_s_mp_mul_digs.obj bn_s_mp_mul_high_digs.obj \
bn_s_mp_sqr.obj bn_s_mp_sub.obj

#END_INS

HEADERS=tommath.h tommath_class.h tommath_superclass.h

TARGET = libtommath.lib

$(TARGET): $(OBJECTS)

.c.obj:
	$(CC) $(CFLAGS) $<
Changes to libtommath/makefile.cygwin_dll.
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#Makefile for Cygwin-GCC
#
#This makefile will build a Windows DLL [doesn't require cygwin to run] in the file
#libtommath.dll.  The import library is in libtommath.dll.a.  Remember to add
#"-Wl,--enable-auto-import" to your client build to avoid the auto-import warnings
#
#Tom St Denis
CFLAGS  +=  -I./ -Wall -W -Wshadow -O3 -funroll-loops -mno-cygwin

#x86 optimizations [should be valid for any GCC install though]
CFLAGS  += -fomit-frame-pointer 

default: windll


OBJECTS=bncore.o bn_mp_init.o bn_mp_clear.o bn_mp_exch.o bn_mp_grow.o bn_mp_shrink.o \
bn_mp_clamp.o bn_mp_zero.o  bn_mp_set.o bn_mp_set_int.o bn_mp_init_size.o bn_mp_copy.o \
bn_mp_init_copy.o bn_mp_abs.o bn_mp_neg.o bn_mp_cmp_mag.o bn_mp_cmp.o bn_mp_cmp_d.o \
bn_mp_rshd.o bn_mp_lshd.o bn_mp_mod_2d.o bn_mp_div_2d.o bn_mp_mul_2d.o bn_mp_div_2.o \
bn_mp_mul_2.o bn_s_mp_add.o bn_s_mp_sub.o bn_fast_s_mp_mul_digs.o bn_s_mp_mul_digs.o \
bn_fast_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_s_mp_sqr.o \
bn_mp_add.o bn_mp_sub.o bn_mp_karatsuba_mul.o bn_mp_mul.o bn_mp_karatsuba_sqr.o \

bn_mp_sqr.o bn_mp_div.o bn_mp_mod.o bn_mp_add_d.o bn_mp_sub_d.o bn_mp_mul_d.o \
bn_mp_div_d.o bn_mp_mod_d.o bn_mp_expt_d.o bn_mp_addmod.o bn_mp_submod.o \
bn_mp_mulmod.o bn_mp_sqrmod.o bn_mp_gcd.o bn_mp_lcm.o bn_fast_mp_invmod.o bn_mp_invmod.o \
bn_mp_reduce.o bn_mp_montgomery_setup.o bn_fast_mp_montgomery_reduce.o bn_mp_montgomery_reduce.o \
bn_mp_exptmod_fast.o bn_mp_exptmod.o bn_mp_2expt.o bn_mp_n_root.o bn_mp_jacobi.o bn_reverse.o \
bn_mp_count_bits.o bn_mp_read_unsigned_bin.o bn_mp_read_signed_bin.o bn_mp_to_unsigned_bin.o \
bn_mp_to_signed_bin.o bn_mp_unsigned_bin_size.o bn_mp_signed_bin_size.o  \
bn_mp_xor.o bn_mp_and.o bn_mp_or.o bn_mp_rand.o bn_mp_montgomery_calc_normalization.o \

bn_mp_prime_is_divisible.o bn_prime_tab.o bn_mp_prime_fermat.o bn_mp_prime_miller_rabin.o \
bn_mp_prime_is_prime.o bn_mp_prime_next_prime.o bn_mp_dr_reduce.o \
bn_mp_dr_is_modulus.o bn_mp_dr_setup.o bn_mp_reduce_setup.o \
bn_mp_toom_mul.o bn_mp_toom_sqr.o bn_mp_div_3.o bn_s_mp_exptmod.o \
bn_mp_reduce_2k.o bn_mp_reduce_is_2k.o bn_mp_reduce_2k_setup.o \
bn_mp_reduce_2k_l.o bn_mp_reduce_is_2k_l.o bn_mp_reduce_2k_setup_l.o \
bn_mp_radix_smap.o bn_mp_read_radix.o bn_mp_toradix.o bn_mp_radix_size.o \
bn_mp_fread.o bn_mp_fwrite.o bn_mp_cnt_lsb.o bn_error.o \
bn_mp_init_multi.o bn_mp_clear_multi.o bn_mp_exteuclid.o bn_mp_toradix_n.o \


bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o bn_mp_init_set.o \
bn_mp_init_set_int.o bn_mp_invmod_slow.o bn_mp_prime_rabin_miller_trials.o \

bn_mp_to_signed_bin_n.o bn_mp_to_unsigned_bin_n.o


# make a Windows DLL via Cygwin
windll:  $(OBJECTS)
	gcc -mno-cygwin -mdll -o libtommath.dll -Wl,--out-implib=libtommath.dll.a -Wl,--export-all-symbols *.o
	ranlib libtommath.dll.a

# build the test program using the windows DLL










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#Makefile for Cygwin-GCC
#
#This makefile will build a Windows DLL [doesn't require cygwin to run] in the file
#libtommath.dll.  The import library is in libtommath.dll.a.  Remember to add
#"-Wl,--enable-auto-import" to your client build to avoid the auto-import warnings
#
#Tom St Denis
CFLAGS  +=  -I./ -Wall -W -Wshadow -O3 -funroll-loops -mno-cygwin

#x86 optimizations [should be valid for any GCC install though]
CFLAGS  += -fomit-frame-pointer

default: windll

#START_INS
OBJECTS=bncore.o bn_error.o bn_fast_mp_invmod.o bn_fast_mp_montgomery_reduce.o bn_fast_s_mp_mul_digs.o \




bn_fast_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_mp_2expt.o bn_mp_abs.o bn_mp_add.o bn_mp_add_d.o \
bn_mp_addmod.o bn_mp_and.o bn_mp_clamp.o bn_mp_clear.o bn_mp_clear_multi.o bn_mp_cmp.o bn_mp_cmp_d.o \
bn_mp_cmp_mag.o bn_mp_cnt_lsb.o bn_mp_copy.o bn_mp_count_bits.o bn_mp_div_2.o bn_mp_div_2d.o bn_mp_div_3.o \
bn_mp_div.o bn_mp_div_d.o bn_mp_dr_is_modulus.o bn_mp_dr_reduce.o bn_mp_dr_setup.o bn_mp_exch.o \
bn_mp_export.o bn_mp_expt_d.o bn_mp_expt_d_ex.o bn_mp_exptmod.o bn_mp_exptmod_fast.o bn_mp_exteuclid.o \
bn_mp_fread.o bn_mp_fwrite.o bn_mp_gcd.o bn_mp_get_int.o bn_mp_get_long.o bn_mp_get_long_long.o \

bn_mp_grow.o bn_mp_import.o bn_mp_init.o bn_mp_init_copy.o bn_mp_init_multi.o bn_mp_init_set.o \
bn_mp_init_set_int.o bn_mp_init_size.o bn_mp_invmod.o bn_mp_invmod_slow.o bn_mp_is_square.o \
bn_mp_jacobi.o bn_mp_karatsuba_mul.o bn_mp_karatsuba_sqr.o bn_mp_lcm.o bn_mp_lshd.o bn_mp_mod_2d.o \
bn_mp_mod.o bn_mp_mod_d.o bn_mp_montgomery_calc_normalization.o bn_mp_montgomery_reduce.o \
bn_mp_montgomery_setup.o bn_mp_mul_2.o bn_mp_mul_2d.o bn_mp_mul.o bn_mp_mul_d.o bn_mp_mulmod.o bn_mp_neg.o \
bn_mp_n_root.o bn_mp_n_root_ex.o bn_mp_or.o bn_mp_prime_fermat.o bn_mp_prime_is_divisible.o \
bn_mp_prime_is_prime.o bn_mp_prime_miller_rabin.o bn_mp_prime_next_prime.o \
bn_mp_prime_rabin_miller_trials.o bn_mp_prime_random_ex.o bn_mp_radix_size.o bn_mp_radix_smap.o \
bn_mp_rand.o bn_mp_read_radix.o bn_mp_read_signed_bin.o bn_mp_read_unsigned_bin.o bn_mp_reduce_2k.o \
bn_mp_reduce_2k_l.o bn_mp_reduce_2k_setup.o bn_mp_reduce_2k_setup_l.o bn_mp_reduce.o \
bn_mp_reduce_is_2k.o bn_mp_reduce_is_2k_l.o bn_mp_reduce_setup.o bn_mp_rshd.o bn_mp_set.o bn_mp_set_int.o \
bn_mp_set_long.o bn_mp_set_long_long.o bn_mp_shrink.o bn_mp_signed_bin_size.o bn_mp_sqr.o bn_mp_sqrmod.o \
bn_mp_sqrt.o bn_mp_sqrtmod_prime.o bn_mp_sub.o bn_mp_sub_d.o bn_mp_submod.o bn_mp_toom_mul.o \
bn_mp_toom_sqr.o bn_mp_toradix.o bn_mp_toradix_n.o bn_mp_to_signed_bin.o bn_mp_to_signed_bin_n.o \
bn_mp_to_unsigned_bin.o bn_mp_to_unsigned_bin_n.o bn_mp_unsigned_bin_size.o bn_mp_xor.o bn_mp_zero.o \
bn_prime_tab.o bn_reverse.o bn_s_mp_add.o bn_s_mp_exptmod.o bn_s_mp_mul_digs.o bn_s_mp_mul_high_digs.o \
bn_s_mp_sqr.o bn_s_mp_sub.o

#END_INS

HEADERS=tommath.h tommath_class.h tommath_superclass.h

# make a Windows DLL via Cygwin
windll:  $(OBJECTS)
	gcc -mno-cygwin -mdll -o libtommath.dll -Wl,--out-implib=libtommath.dll.a -Wl,--export-all-symbols *.o
	ranlib libtommath.dll.a

# build the test program using the windows DLL
Changes to libtommath/makefile.icc.
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#Makefile for ICC
#
#Tom St Denis
CC=icc

CFLAGS  +=  -I./

# optimize for SPEED
#
# -mcpu= can be pentium, pentiumpro (covers PII through PIII) or pentium4
# -ax?   specifies make code specifically for ? but compatible with IA-32
# -x?    specifies compile solely for ? [not specifically IA-32 compatible]
#
# where ? is 
#   K - PIII
#   W - first P4 [Williamette]
#   N - P4 Northwood
#   P - P4 Prescott
#   B - Blend of P4 and PM [mobile]
#
# Default to just generic max opts
CFLAGS += -O3 -xP -ip

#install as this user
USER=root
GROUP=root

default: libtommath.a

#default files to install
LIBNAME=libtommath.a
HEADERS=tommath.h

#LIBPATH-The directory for libtomcrypt to be installed to.
#INCPATH-The directory to install the header files for libtommath.
#DATAPATH-The directory to install the pdf docs.
DESTDIR=
LIBPATH=/usr/lib
INCPATH=/usr/include
DATAPATH=/usr/share/doc/libtommath/pdf


OBJECTS=bncore.o bn_mp_init.o bn_mp_clear.o bn_mp_exch.o bn_mp_grow.o bn_mp_shrink.o \
bn_mp_clamp.o bn_mp_zero.o  bn_mp_set.o bn_mp_set_int.o bn_mp_init_size.o bn_mp_copy.o \
bn_mp_init_copy.o bn_mp_abs.o bn_mp_neg.o bn_mp_cmp_mag.o bn_mp_cmp.o bn_mp_cmp_d.o \
bn_mp_rshd.o bn_mp_lshd.o bn_mp_mod_2d.o bn_mp_div_2d.o bn_mp_mul_2d.o bn_mp_div_2.o \
bn_mp_mul_2.o bn_s_mp_add.o bn_s_mp_sub.o bn_fast_s_mp_mul_digs.o bn_s_mp_mul_digs.o \
bn_fast_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_s_mp_sqr.o \
bn_mp_add.o bn_mp_sub.o bn_mp_karatsuba_mul.o bn_mp_mul.o bn_mp_karatsuba_sqr.o \

bn_mp_sqr.o bn_mp_div.o bn_mp_mod.o bn_mp_add_d.o bn_mp_sub_d.o bn_mp_mul_d.o \
bn_mp_div_d.o bn_mp_mod_d.o bn_mp_expt_d.o bn_mp_addmod.o bn_mp_submod.o \
bn_mp_mulmod.o bn_mp_sqrmod.o bn_mp_gcd.o bn_mp_lcm.o bn_fast_mp_invmod.o bn_mp_invmod.o \
bn_mp_reduce.o bn_mp_montgomery_setup.o bn_fast_mp_montgomery_reduce.o bn_mp_montgomery_reduce.o \
bn_mp_exptmod_fast.o bn_mp_exptmod.o bn_mp_2expt.o bn_mp_n_root.o bn_mp_jacobi.o bn_reverse.o \
bn_mp_count_bits.o bn_mp_read_unsigned_bin.o bn_mp_read_signed_bin.o bn_mp_to_unsigned_bin.o \
bn_mp_to_signed_bin.o bn_mp_unsigned_bin_size.o bn_mp_signed_bin_size.o  \
bn_mp_xor.o bn_mp_and.o bn_mp_or.o bn_mp_rand.o bn_mp_montgomery_calc_normalization.o \

bn_mp_prime_is_divisible.o bn_prime_tab.o bn_mp_prime_fermat.o bn_mp_prime_miller_rabin.o \
bn_mp_prime_is_prime.o bn_mp_prime_next_prime.o bn_mp_dr_reduce.o \
bn_mp_dr_is_modulus.o bn_mp_dr_setup.o bn_mp_reduce_setup.o \
bn_mp_toom_mul.o bn_mp_toom_sqr.o bn_mp_div_3.o bn_s_mp_exptmod.o \
bn_mp_reduce_2k.o bn_mp_reduce_is_2k.o bn_mp_reduce_2k_setup.o \
bn_mp_reduce_2k_l.o bn_mp_reduce_is_2k_l.o bn_mp_reduce_2k_setup_l.o \
bn_mp_radix_smap.o bn_mp_read_radix.o bn_mp_toradix.o bn_mp_radix_size.o \
bn_mp_fread.o bn_mp_fwrite.o bn_mp_cnt_lsb.o bn_error.o \
bn_mp_init_multi.o bn_mp_clear_multi.o bn_mp_exteuclid.o bn_mp_toradix_n.o \


bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o bn_mp_init_set.o \
bn_mp_init_set_int.o bn_mp_invmod_slow.o bn_mp_prime_rabin_miller_trials.o \

bn_mp_to_signed_bin_n.o bn_mp_to_unsigned_bin_n.o


libtommath.a:  $(OBJECTS)
	$(AR) $(ARFLAGS) libtommath.a $(OBJECTS)
	ranlib libtommath.a

#make a profiled library (takes a while!!!)
#
# This will build the library with profile generation
# then run the test demo and rebuild the library.
# 
# So far I've seen improvements in the MP math
profiled:
	make -f makefile.icc CFLAGS="$(CFLAGS) -prof_gen -DTESTING" timing
	./ltmtest
	rm -f *.a *.o ltmtest
	make -f makefile.icc CFLAGS="$(CFLAGS) -prof_use"

#make a single object profiled library 
profiled_single:
	perl gen.pl
	$(CC) $(CFLAGS) -prof_gen -DTESTING -c mpi.c -o mpi.o
	$(CC) $(CFLAGS) -DTESTING -DTIMER demo/demo.c mpi.o -o ltmtest
	./ltmtest
	rm -f *.o ltmtest
	$(CC) $(CFLAGS) -prof_use -ip -DTESTING -c mpi.c -o mpi.o
	$(AR) $(ARFLAGS) libtommath.a mpi.o
	ranlib libtommath.a	

install: libtommath.a
	install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(LIBPATH)
	install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(INCPATH)
	install -g $(GROUP) -o $(USER) $(LIBNAME) $(DESTDIR)$(LIBPATH)
	install -g $(GROUP) -o $(USER) $(HEADERS) $(DESTDIR)$(INCPATH)

test: libtommath.a demo/demo.o
	$(CC) demo/demo.o libtommath.a -o test
	
mtest: test	
	cd mtest ; $(CC) $(CFLAGS) mtest.c -o mtest
        
timing: libtommath.a
	$(CC) $(CFLAGS) -DTIMER demo/timing.c libtommath.a -o ltmtest

clean:
	rm -f *.bat *.pdf *.o *.a *.obj *.lib *.exe *.dll etclib/*.o demo/demo.o test ltmtest mpitest mtest/mtest mtest/mtest.exe \
        *.idx *.toc *.log *.aux *.dvi *.lof *.ind *.ilg *.ps *.log *.s mpi.c *.il etc/*.il *.dyn
	cd etc ; make clean
	cd pics ; make clean













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#Makefile for ICC
#
#Tom St Denis
CC=icc

CFLAGS  +=  -I./

# optimize for SPEED
#
# -mcpu= can be pentium, pentiumpro (covers PII through PIII) or pentium4
# -ax?   specifies make code specifically for ? but compatible with IA-32
# -x?    specifies compile solely for ? [not specifically IA-32 compatible]
#
# where ? is
#   K - PIII
#   W - first P4 [Williamette]
#   N - P4 Northwood
#   P - P4 Prescott
#   B - Blend of P4 and PM [mobile]
#
# Default to just generic max opts
CFLAGS += -O3 -xP -ip

#install as this user
USER=root
GROUP=root

default: libtommath.a

#default files to install
LIBNAME=libtommath.a


#LIBPATH-The directory for libtomcrypt to be installed to.
#INCPATH-The directory to install the header files for libtommath.
#DATAPATH-The directory to install the pdf docs.
DESTDIR=
LIBPATH=/usr/lib
INCPATH=/usr/include
DATAPATH=/usr/share/doc/libtommath/pdf

#START_INS
OBJECTS=bncore.o bn_error.o bn_fast_mp_invmod.o bn_fast_mp_montgomery_reduce.o bn_fast_s_mp_mul_digs.o \




bn_fast_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_mp_2expt.o bn_mp_abs.o bn_mp_add.o bn_mp_add_d.o \
bn_mp_addmod.o bn_mp_and.o bn_mp_clamp.o bn_mp_clear.o bn_mp_clear_multi.o bn_mp_cmp.o bn_mp_cmp_d.o \
bn_mp_cmp_mag.o bn_mp_cnt_lsb.o bn_mp_copy.o bn_mp_count_bits.o bn_mp_div_2.o bn_mp_div_2d.o bn_mp_div_3.o \
bn_mp_div.o bn_mp_div_d.o bn_mp_dr_is_modulus.o bn_mp_dr_reduce.o bn_mp_dr_setup.o bn_mp_exch.o \
bn_mp_export.o bn_mp_expt_d.o bn_mp_expt_d_ex.o bn_mp_exptmod.o bn_mp_exptmod_fast.o bn_mp_exteuclid.o \
bn_mp_fread.o bn_mp_fwrite.o bn_mp_gcd.o bn_mp_get_int.o bn_mp_get_long.o bn_mp_get_long_long.o \

bn_mp_grow.o bn_mp_import.o bn_mp_init.o bn_mp_init_copy.o bn_mp_init_multi.o bn_mp_init_set.o \
bn_mp_init_set_int.o bn_mp_init_size.o bn_mp_invmod.o bn_mp_invmod_slow.o bn_mp_is_square.o \
bn_mp_jacobi.o bn_mp_karatsuba_mul.o bn_mp_karatsuba_sqr.o bn_mp_lcm.o bn_mp_lshd.o bn_mp_mod_2d.o \
bn_mp_mod.o bn_mp_mod_d.o bn_mp_montgomery_calc_normalization.o bn_mp_montgomery_reduce.o \
bn_mp_montgomery_setup.o bn_mp_mul_2.o bn_mp_mul_2d.o bn_mp_mul.o bn_mp_mul_d.o bn_mp_mulmod.o bn_mp_neg.o \
bn_mp_n_root.o bn_mp_n_root_ex.o bn_mp_or.o bn_mp_prime_fermat.o bn_mp_prime_is_divisible.o \
bn_mp_prime_is_prime.o bn_mp_prime_miller_rabin.o bn_mp_prime_next_prime.o \
bn_mp_prime_rabin_miller_trials.o bn_mp_prime_random_ex.o bn_mp_radix_size.o bn_mp_radix_smap.o \
bn_mp_rand.o bn_mp_read_radix.o bn_mp_read_signed_bin.o bn_mp_read_unsigned_bin.o bn_mp_reduce_2k.o \
bn_mp_reduce_2k_l.o bn_mp_reduce_2k_setup.o bn_mp_reduce_2k_setup_l.o bn_mp_reduce.o \
bn_mp_reduce_is_2k.o bn_mp_reduce_is_2k_l.o bn_mp_reduce_setup.o bn_mp_rshd.o bn_mp_set.o bn_mp_set_int.o \
bn_mp_set_long.o bn_mp_set_long_long.o bn_mp_shrink.o bn_mp_signed_bin_size.o bn_mp_sqr.o bn_mp_sqrmod.o \
bn_mp_sqrt.o bn_mp_sqrtmod_prime.o bn_mp_sub.o bn_mp_sub_d.o bn_mp_submod.o bn_mp_toom_mul.o \
bn_mp_toom_sqr.o bn_mp_toradix.o bn_mp_toradix_n.o bn_mp_to_signed_bin.o bn_mp_to_signed_bin_n.o \
bn_mp_to_unsigned_bin.o bn_mp_to_unsigned_bin_n.o bn_mp_unsigned_bin_size.o bn_mp_xor.o bn_mp_zero.o \
bn_prime_tab.o bn_reverse.o bn_s_mp_add.o bn_s_mp_exptmod.o bn_s_mp_mul_digs.o bn_s_mp_mul_high_digs.o \
bn_s_mp_sqr.o bn_s_mp_sub.o

#END_INS

HEADERS=tommath.h tommath_class.h tommath_superclass.h

libtommath.a:  $(OBJECTS)
	$(AR) $(ARFLAGS) libtommath.a $(OBJECTS)
	ranlib libtommath.a

#make a profiled library (takes a while!!!)
#
# This will build the library with profile generation
# then run the test demo and rebuild the library.
#
# So far I've seen improvements in the MP math
profiled:
	make -f makefile.icc CFLAGS="$(CFLAGS) -prof_gen -DTESTING" timing
	./ltmtest
	rm -f *.a *.o ltmtest
	make -f makefile.icc CFLAGS="$(CFLAGS) -prof_use"

#make a single object profiled library
profiled_single:
	perl gen.pl
	$(CC) $(CFLAGS) -prof_gen -DTESTING -c mpi.c -o mpi.o
	$(CC) $(CFLAGS) -DTESTING -DTIMER demo/demo.c mpi.o -o ltmtest
	./ltmtest
	rm -f *.o ltmtest
	$(CC) $(CFLAGS) -prof_use -ip -DTESTING -c mpi.c -o mpi.o
	$(AR) $(ARFLAGS) libtommath.a mpi.o
	ranlib libtommath.a

install: libtommath.a
	install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(LIBPATH)
	install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(INCPATH)
	install -g $(GROUP) -o $(USER) $(LIBNAME) $(DESTDIR)$(LIBPATH)
	install -g $(GROUP) -o $(USER) $(HEADERS) $(DESTDIR)$(INCPATH)

test: libtommath.a demo/demo.o
	$(CC) demo/demo.o libtommath.a -o test

mtest: test
	cd mtest ; $(CC) $(CFLAGS) mtest.c -o mtest

timing: libtommath.a
	$(CC) $(CFLAGS) -DTIMER demo/timing.c libtommath.a -o ltmtest

clean:
	rm -f *.bat *.pdf *.o *.a *.obj *.lib *.exe *.dll etclib/*.o demo/demo.o test ltmtest mpitest mtest/mtest mtest/mtest.exe \
        *.idx *.toc *.log *.aux *.dvi *.lof *.ind *.ilg *.ps *.log *.s mpi.c *.il etc/*.il *.dyn
	cd etc ; make clean
	cd pics ; make clean
Changes to libtommath/makefile.msvc.
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#MSVC Makefile
#
#Tom St Denis

CFLAGS = /I. /Ox /DWIN32 /W3 /Fo$@

default: library


OBJECTS=bncore.obj bn_mp_init.obj bn_mp_clear.obj bn_mp_exch.obj bn_mp_grow.obj bn_mp_shrink.obj \
bn_mp_clamp.obj bn_mp_zero.obj  bn_mp_set.obj bn_mp_set_int.obj bn_mp_init_size.obj bn_mp_copy.obj \
bn_mp_init_copy.obj bn_mp_abs.obj bn_mp_neg.obj bn_mp_cmp_mag.obj bn_mp_cmp.obj bn_mp_cmp_d.obj \
bn_mp_rshd.obj bn_mp_lshd.obj bn_mp_mod_2d.obj bn_mp_div_2d.obj bn_mp_mul_2d.obj bn_mp_div_2.obj \
bn_mp_mul_2.obj bn_s_mp_add.obj bn_s_mp_sub.obj bn_fast_s_mp_mul_digs.obj bn_s_mp_mul_digs.obj \
bn_fast_s_mp_mul_high_digs.obj bn_s_mp_mul_high_digs.obj bn_fast_s_mp_sqr.obj bn_s_mp_sqr.obj \
bn_mp_add.obj bn_mp_sub.obj bn_mp_karatsuba_mul.obj bn_mp_mul.obj bn_mp_karatsuba_sqr.obj \

bn_mp_sqr.obj bn_mp_div.obj bn_mp_mod.obj bn_mp_add_d.obj bn_mp_sub_d.obj bn_mp_mul_d.obj \
bn_mp_div_d.obj bn_mp_mod_d.obj bn_mp_expt_d.obj bn_mp_addmod.obj bn_mp_submod.obj \
bn_mp_mulmod.obj bn_mp_sqrmod.obj bn_mp_gcd.obj bn_mp_lcm.obj bn_fast_mp_invmod.obj bn_mp_invmod.obj \
bn_mp_reduce.obj bn_mp_montgomery_setup.obj bn_fast_mp_montgomery_reduce.obj bn_mp_montgomery_reduce.obj \
bn_mp_exptmod_fast.obj bn_mp_exptmod.obj bn_mp_2expt.obj bn_mp_n_root.obj bn_mp_jacobi.obj bn_reverse.obj \
bn_mp_count_bits.obj bn_mp_read_unsigned_bin.obj bn_mp_read_signed_bin.obj bn_mp_to_unsigned_bin.obj \
bn_mp_to_signed_bin.obj bn_mp_unsigned_bin_size.obj bn_mp_signed_bin_size.obj  \
bn_mp_xor.obj bn_mp_and.obj bn_mp_or.obj bn_mp_rand.obj bn_mp_montgomery_calc_normalization.obj \

bn_mp_prime_is_divisible.obj bn_prime_tab.obj bn_mp_prime_fermat.obj bn_mp_prime_miller_rabin.obj \
bn_mp_prime_is_prime.obj bn_mp_prime_next_prime.obj bn_mp_dr_reduce.obj \
bn_mp_dr_is_modulus.obj bn_mp_dr_setup.obj bn_mp_reduce_setup.obj \
bn_mp_toom_mul.obj bn_mp_toom_sqr.obj bn_mp_div_3.obj bn_s_mp_exptmod.obj \
bn_mp_reduce_2k.obj bn_mp_reduce_is_2k.obj bn_mp_reduce_2k_setup.obj \
bn_mp_reduce_2k_l.obj bn_mp_reduce_is_2k_l.obj bn_mp_reduce_2k_setup_l.obj \
bn_mp_radix_smap.obj bn_mp_read_radix.obj bn_mp_toradix.obj bn_mp_radix_size.obj \
bn_mp_fread.obj bn_mp_fwrite.obj bn_mp_cnt_lsb.obj bn_error.obj \
bn_mp_init_multi.obj bn_mp_clear_multi.obj bn_mp_exteuclid.obj bn_mp_toradix_n.obj \
bn_mp_prime_random_ex.obj bn_mp_get_int.obj bn_mp_sqrt.obj bn_mp_is_square.obj \

bn_mp_init_set.obj bn_mp_init_set_int.obj bn_mp_invmod_slow.obj bn_mp_prime_rabin_miller_trials.obj \
bn_mp_to_signed_bin_n.obj bn_mp_to_unsigned_bin_n.obj


HEADERS=tommath.h tommath_class.h tommath_superclass.h

library: $(OBJECTS)
	lib /out:tommath.lib $(OBJECTS)








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#MSVC Makefile
#
#Tom St Denis

CFLAGS = /I. /Ox /DWIN32 /W3 /Fo$@

default: library

#START_INS
OBJECTS=bncore.obj bn_error.obj bn_fast_mp_invmod.obj bn_fast_mp_montgomery_reduce.obj bn_fast_s_mp_mul_digs.obj \




bn_fast_s_mp_mul_high_digs.obj bn_fast_s_mp_sqr.obj bn_mp_2expt.obj bn_mp_abs.obj bn_mp_add.obj bn_mp_add_d.obj \
bn_mp_addmod.obj bn_mp_and.obj bn_mp_clamp.obj bn_mp_clear.obj bn_mp_clear_multi.obj bn_mp_cmp.obj bn_mp_cmp_d.obj \
bn_mp_cmp_mag.obj bn_mp_cnt_lsb.obj bn_mp_copy.obj bn_mp_count_bits.obj bn_mp_div_2.obj bn_mp_div_2d.obj bn_mp_div_3.obj \
bn_mp_div.obj bn_mp_div_d.obj bn_mp_dr_is_modulus.obj bn_mp_dr_reduce.obj bn_mp_dr_setup.obj bn_mp_exch.obj \
bn_mp_export.obj bn_mp_expt_d.obj bn_mp_expt_d_ex.obj bn_mp_exptmod.obj bn_mp_exptmod_fast.obj bn_mp_exteuclid.obj \
bn_mp_fread.obj bn_mp_fwrite.obj bn_mp_gcd.obj bn_mp_get_int.obj bn_mp_get_long.obj bn_mp_get_long_long.obj \

bn_mp_grow.obj bn_mp_import.obj bn_mp_init.obj bn_mp_init_copy.obj bn_mp_init_multi.obj bn_mp_init_set.obj \
bn_mp_init_set_int.obj bn_mp_init_size.obj bn_mp_invmod.obj bn_mp_invmod_slow.obj bn_mp_is_square.obj \
bn_mp_jacobi.obj bn_mp_karatsuba_mul.obj bn_mp_karatsuba_sqr.obj bn_mp_lcm.obj bn_mp_lshd.obj bn_mp_mod_2d.obj \
bn_mp_mod.obj bn_mp_mod_d.obj bn_mp_montgomery_calc_normalization.obj bn_mp_montgomery_reduce.obj \
bn_mp_montgomery_setup.obj bn_mp_mul_2.obj bn_mp_mul_2d.obj bn_mp_mul.obj bn_mp_mul_d.obj bn_mp_mulmod.obj bn_mp_neg.obj \
bn_mp_n_root.obj bn_mp_n_root_ex.obj bn_mp_or.obj bn_mp_prime_fermat.obj bn_mp_prime_is_divisible.obj \
bn_mp_prime_is_prime.obj bn_mp_prime_miller_rabin.obj bn_mp_prime_next_prime.obj \
bn_mp_prime_rabin_miller_trials.obj bn_mp_prime_random_ex.obj bn_mp_radix_size.obj bn_mp_radix_smap.obj \
bn_mp_rand.obj bn_mp_read_radix.obj bn_mp_read_signed_bin.obj bn_mp_read_unsigned_bin.obj bn_mp_reduce_2k.obj \
bn_mp_reduce_2k_l.obj bn_mp_reduce_2k_setup.obj bn_mp_reduce_2k_setup_l.obj bn_mp_reduce.obj \
bn_mp_reduce_is_2k.obj bn_mp_reduce_is_2k_l.obj bn_mp_reduce_setup.obj bn_mp_rshd.obj bn_mp_set.obj bn_mp_set_int.obj \
bn_mp_set_long.obj bn_mp_set_long_long.obj bn_mp_shrink.obj bn_mp_signed_bin_size.obj bn_mp_sqr.obj bn_mp_sqrmod.obj \
bn_mp_sqrt.obj bn_mp_sqrtmod_prime.obj bn_mp_sub.obj bn_mp_sub_d.obj bn_mp_submod.obj bn_mp_toom_mul.obj \
bn_mp_toom_sqr.obj bn_mp_toradix.obj bn_mp_toradix_n.obj bn_mp_to_signed_bin.obj bn_mp_to_signed_bin_n.obj \
bn_mp_to_unsigned_bin.obj bn_mp_to_unsigned_bin_n.obj bn_mp_unsigned_bin_size.obj bn_mp_xor.obj bn_mp_zero.obj \
bn_prime_tab.obj bn_reverse.obj bn_s_mp_add.obj bn_s_mp_exptmod.obj bn_s_mp_mul_digs.obj bn_s_mp_mul_high_digs.obj \
bn_s_mp_sqr.obj bn_s_mp_sub.obj

#END_INS

HEADERS=tommath.h tommath_class.h tommath_superclass.h

library: $(OBJECTS)
	lib /out:tommath.lib $(OBJECTS)
Changes to libtommath/makefile.shared.
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#Makefile for GCC
#
#Tom St Denis
VERSION=0:41

CC = libtool --mode=compile --tag=CC gcc

CFLAGS  +=  -I./ -Wall -W -Wshadow -Wsign-compare

ifndef IGNORE_SPEED

#for speed 
CFLAGS += -O3 -funroll-loops

#for size 
#CFLAGS += -Os

#x86 optimizations [should be valid for any GCC install though]
CFLAGS  += -fomit-frame-pointer

endif

#install as this user
ifndef INSTALL_GROUP
   GROUP=wheel
else
   GROUP=$(INSTALL_GROUP)
endif

ifndef INSTALL_USER
   USER=root
else
   USER=$(INSTALL_USER)
endif

default: libtommath.la

#default files to install
ifndef LIBNAME
   LIBNAME=libtommath.la
endif
ifndef LIBNAME_S
   LIBNAME_S=libtommath.a
endif
HEADERS=tommath.h tommath_class.h tommath_superclass.h

#LIBPATH-The directory for libtommath to be installed to.

#INCPATH-The directory to install the header files for libtommath.
#DATAPATH-The directory to install the pdf docs.
DESTDIR=
LIBPATH=/usr/lib
INCPATH=/usr/include
DATAPATH=/usr/share/doc/libtommath/pdf


OBJECTS=bncore.o bn_mp_init.o bn_mp_clear.o bn_mp_exch.o bn_mp_grow.o bn_mp_shrink.o \

bn_mp_clamp.o bn_mp_zero.o  bn_mp_set.o bn_mp_set_int.o bn_mp_init_size.o bn_mp_copy.o \
bn_mp_init_copy.o bn_mp_abs.o bn_mp_neg.o bn_mp_cmp_mag.o bn_mp_cmp.o bn_mp_cmp_d.o \
bn_mp_rshd.o bn_mp_lshd.o bn_mp_mod_2d.o bn_mp_div_2d.o bn_mp_mul_2d.o bn_mp_div_2.o \



bn_mp_mul_2.o bn_s_mp_add.o bn_s_mp_sub.o bn_fast_s_mp_mul_digs.o bn_s_mp_mul_digs.o \
bn_fast_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_s_mp_sqr.o \
bn_mp_add.o bn_mp_sub.o bn_mp_karatsuba_mul.o bn_mp_mul.o bn_mp_karatsuba_sqr.o \
bn_mp_sqr.o bn_mp_div.o bn_mp_mod.o bn_mp_add_d.o bn_mp_sub_d.o bn_mp_mul_d.o \
bn_mp_div_d.o bn_mp_mod_d.o bn_mp_expt_d.o bn_mp_addmod.o bn_mp_submod.o \
bn_mp_mulmod.o bn_mp_sqrmod.o bn_mp_gcd.o bn_mp_lcm.o bn_fast_mp_invmod.o bn_mp_invmod.o \
bn_mp_reduce.o bn_mp_montgomery_setup.o bn_fast_mp_montgomery_reduce.o bn_mp_montgomery_reduce.o \
bn_mp_exptmod_fast.o bn_mp_exptmod.o bn_mp_2expt.o bn_mp_n_root.o bn_mp_jacobi.o bn_reverse.o \
bn_mp_count_bits.o bn_mp_read_unsigned_bin.o bn_mp_read_signed_bin.o bn_mp_to_unsigned_bin.o \
bn_mp_to_signed_bin.o bn_mp_unsigned_bin_size.o bn_mp_signed_bin_size.o  \
bn_mp_xor.o bn_mp_and.o bn_mp_or.o bn_mp_rand.o bn_mp_montgomery_calc_normalization.o \
bn_mp_prime_is_divisible.o bn_prime_tab.o bn_mp_prime_fermat.o bn_mp_prime_miller_rabin.o \
bn_mp_prime_is_prime.o bn_mp_prime_next_prime.o bn_mp_dr_reduce.o \
bn_mp_dr_is_modulus.o bn_mp_dr_setup.o bn_mp_reduce_setup.o \
bn_mp_toom_mul.o bn_mp_toom_sqr.o bn_mp_div_3.o bn_s_mp_exptmod.o \
bn_mp_reduce_2k.o bn_mp_reduce_is_2k.o bn_mp_reduce_2k_setup.o \
bn_mp_reduce_2k_l.o bn_mp_reduce_is_2k_l.o bn_mp_reduce_2k_setup_l.o \
bn_mp_radix_smap.o bn_mp_read_radix.o bn_mp_toradix.o bn_mp_radix_size.o \
bn_mp_fread.o bn_mp_fwrite.o bn_mp_cnt_lsb.o bn_error.o \
bn_mp_init_multi.o bn_mp_clear_multi.o bn_mp_exteuclid.o bn_mp_toradix_n.o \
bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o bn_mp_init_set.o \


bn_mp_init_set_int.o bn_mp_invmod_slow.o bn_mp_prime_rabin_miller_trials.o \
bn_mp_to_signed_bin_n.o bn_mp_to_unsigned_bin_n.o


objs: $(OBJECTS)




$(LIBNAME):  $(OBJECTS)
	libtool --mode=link gcc *.lo -o $(LIBNAME) -rpath $(LIBPATH) -version-info $(VERSION)

install: $(LIBNAME)
	install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(LIBPATH)

	libtool --mode=install install -c $(LIBNAME) $(DESTDIR)$(LIBPATH)/$(LIBNAME)
	install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(INCPATH)
	install -g $(GROUP) -o $(USER) $(HEADERS) $(DESTDIR)$(INCPATH)




test: $(LIBNAME) demo/demo.o
	gcc $(CFLAGS) -c demo/demo.c -o demo/demo.o
	libtool --mode=link gcc -o test demo/demo.o $(LIBNAME_S)
	
mtest: test	
	cd mtest ; gcc $(CFLAGS) mtest.c -o mtest
        
timing: $(LIBNAME)
	gcc $(CFLAGS) -DTIMER demo/timing.c $(LIBNAME_S) -o ltmtest



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#Makefile for GCC
#
#Tom St Denis


































#default files to install
ifndef LIBNAME
   LIBNAME=libtommath.la
endif

include makefile.include



LT	?= libtool
LTCOMPILE = $(LT) --mode=compile --tag=CC $(CC)

LCOV_ARGS=--directory .libs --directory .





#START_INS
OBJECTS=bncore.o bn_error.o bn_fast_mp_invmod.o bn_fast_mp_montgomery_reduce.o bn_fast_s_mp_mul_digs.o \
bn_fast_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_mp_2expt.o bn_mp_abs.o bn_mp_add.o bn_mp_add_d.o \
bn_mp_addmod.o bn_mp_and.o bn_mp_clamp.o bn_mp_clear.o bn_mp_clear_multi.o bn_mp_cmp.o bn_mp_cmp_d.o \
bn_mp_cmp_mag.o bn_mp_cnt_lsb.o bn_mp_copy.o bn_mp_count_bits.o bn_mp_div_2.o bn_mp_div_2d.o bn_mp_div_3.o \
bn_mp_div.o bn_mp_div_d.o bn_mp_dr_is_modulus.o bn_mp_dr_reduce.o bn_mp_dr_setup.o bn_mp_exch.o \
bn_mp_export.o bn_mp_expt_d.o bn_mp_expt_d_ex.o bn_mp_exptmod.o bn_mp_exptmod_fast.o bn_mp_exteuclid.o \
bn_mp_fread.o bn_mp_fwrite.o bn_mp_gcd.o bn_mp_get_int.o bn_mp_get_long.o bn_mp_get_long_long.o \
bn_mp_grow.o bn_mp_import.o bn_mp_init.o bn_mp_init_copy.o bn_mp_init_multi.o bn_mp_init_set.o \
bn_mp_init_set_int.o bn_mp_init_size.o bn_mp_invmod.o bn_mp_invmod_slow.o bn_mp_is_square.o \

bn_mp_jacobi.o bn_mp_karatsuba_mul.o bn_mp_karatsuba_sqr.o bn_mp_lcm.o bn_mp_lshd.o bn_mp_mod_2d.o \

bn_mp_mod.o bn_mp_mod_d.o bn_mp_montgomery_calc_normalization.o bn_mp_montgomery_reduce.o \

bn_mp_montgomery_setup.o bn_mp_mul_2.o bn_mp_mul_2d.o bn_mp_mul.o bn_mp_mul_d.o bn_mp_mulmod.o bn_mp_neg.o \




bn_mp_n_root.o bn_mp_n_root_ex.o bn_mp_or.o bn_mp_prime_fermat.o bn_mp_prime_is_divisible.o \
bn_mp_prime_is_prime.o bn_mp_prime_miller_rabin.o bn_mp_prime_next_prime.o \
bn_mp_prime_rabin_miller_trials.o bn_mp_prime_random_ex.o bn_mp_radix_size.o bn_mp_radix_smap.o \
bn_mp_rand.o bn_mp_read_radix.o bn_mp_read_signed_bin.o bn_mp_read_unsigned_bin.o bn_mp_reduce_2k.o \
bn_mp_reduce_2k_l.o bn_mp_reduce_2k_setup.o bn_mp_reduce_2k_setup_l.o bn_mp_reduce.o \
bn_mp_reduce_is_2k.o bn_mp_reduce_is_2k_l.o bn_mp_reduce_setup.o bn_mp_rshd.o bn_mp_set.o bn_mp_set_int.o \
bn_mp_set_long.o bn_mp_set_long_long.o bn_mp_shrink.o bn_mp_signed_bin_size.o bn_mp_sqr.o bn_mp_sqrmod.o \
bn_mp_sqrt.o bn_mp_sqrtmod_prime.o bn_mp_sub.o bn_mp_sub_d.o bn_mp_submod.o bn_mp_toom_mul.o \
bn_mp_toom_sqr.o bn_mp_toradix.o bn_mp_toradix_n.o bn_mp_to_signed_bin.o bn_mp_to_signed_bin_n.o \

bn_mp_to_unsigned_bin.o bn_mp_to_unsigned_bin_n.o bn_mp_unsigned_bin_size.o bn_mp_xor.o bn_mp_zero.o \
bn_prime_tab.o bn_reverse.o bn_s_mp_add.o bn_s_mp_exptmod.o bn_s_mp_mul_digs.o bn_s_mp_mul_high_digs.o \
bn_s_mp_sqr.o bn_s_mp_sub.o

#END_INS

objs: $(OBJECTS)

.c.o:
	$(LTCOMPILE) $(CFLAGS) $(LDFLAGS) -o $@ -c $<

$(LIBNAME):  $(OBJECTS)
	$(LT) --mode=link --tag=CC $(CC) $(LDFLAGS) *.lo -o $(LIBNAME) -rpath $(LIBPATH) -version-info $(VERSION_SO)

install: $(LIBNAME)
	install -d $(DESTDIR)$(LIBPATH)
	install -d $(DESTDIR)$(INCPATH)
	$(LT) --mode=install install -c $(LIBNAME) $(DESTDIR)$(LIBPATH)/$(LIBNAME)
	install -m 644 $(HEADERS_PUB) $(DESTDIR)$(INCPATH)

test: $(LIBNAME) demo/demo.o
	$(CC) $(CFLAGS) -c demo/demo.c -o demo/demo.o
	$(LT) --mode=link $(CC) $(LDFLAGS) -o test demo/demo.o $(LIBNAME)

test_standalone: $(LIBNAME) demo/demo.o
	$(CC) $(CFLAGS) -c demo/demo.c -o demo/demo.o
	$(LT) --mode=link $(CC) $(LDFLAGS) -o test demo/demo.o $(LIBNAME)

mtest:
	cd mtest ; $(CC) $(CFLAGS) $(LDFLAGS) mtest.c -o mtest

timing: $(LIBNAME)
	$(LT) --mode=link $(CC) $(CFLAGS) $(LDFLAGS) -DTIMER demo/timing.c $(LIBNAME) -o ltmtest
Changes to libtommath/mtest/logtab.h.
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   0.183169251, 0.182087900, 0.181042597, 0.180031327, 	/* 44 45 46 47 */
   0.179052232, 0.178103594, 0.177183820, 0.176291434, 	/* 48 49 50 51 */
   0.175425064, 0.174583430, 0.173765343, 0.172969690, 	/* 52 53 54 55 */
   0.172195434, 0.171441601, 0.170707280, 0.169991616, 	/* 56 57 58 59 */
   0.169293808, 0.168613099, 0.167948779, 0.167300179, 	/* 60 61 62 63 */
   0.166666667
};












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   0.183169251, 0.182087900, 0.181042597, 0.180031327, 	/* 44 45 46 47 */
   0.179052232, 0.178103594, 0.177183820, 0.176291434, 	/* 48 49 50 51 */
   0.175425064, 0.174583430, 0.173765343, 0.172969690, 	/* 52 53 54 55 */
   0.172195434, 0.171441601, 0.170707280, 0.169991616, 	/* 56 57 58 59 */
   0.169293808, 0.168613099, 0.167948779, 0.167300179, 	/* 60 61 62 63 */
   0.166666667
};


/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/mtest/mpi-config.h.
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/* Default configuration for MPI library */


#ifndef MPI_CONFIG_H_
#define MPI_CONFIG_H_

/*
  For boolean options, 
  0 = no

>







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/* Default configuration for MPI library */
/* $Id$ */

#ifndef MPI_CONFIG_H_
#define MPI_CONFIG_H_

/*
  For boolean options, 
  0 = no
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#define MP_COMPAT_MACROS 1   /* define compatibility macros?    */
#endif

#endif /* ifndef MPI_CONFIG_H_ */


/* crc==3287762869, version==2, Sat Feb 02 06:43:53 2002 */











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#define MP_COMPAT_MACROS 1   /* define compatibility macros?    */
#endif

#endif /* ifndef MPI_CONFIG_H_ */


/* crc==3287762869, version==2, Sat Feb 02 06:43:53 2002 */

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/mtest/mpi-types.h.
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#define MP_DIGIT_MAX       USHRT_MAX
#define MP_WORD_BIT        (CHAR_BIT*sizeof(mp_word))
#define MP_WORD_MAX        UINT_MAX

#define MP_DIGIT_SIZE      2
#define DIGIT_FMT          "%04X"
#define RADIX              (MP_DIGIT_MAX+1)












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#define MP_DIGIT_MAX       USHRT_MAX
#define MP_WORD_BIT        (CHAR_BIT*sizeof(mp_word))
#define MP_WORD_MAX        UINT_MAX

#define MP_DIGIT_SIZE      2
#define DIGIT_FMT          "%04X"
#define RADIX              (MP_DIGIT_MAX+1)


/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/mtest/mpi.c.
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7


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/*
    mpi.c

    by Michael J. Fromberger <[email protected]>
    Copyright (C) 1998 Michael J. Fromberger, All Rights Reserved

    Arbitrary precision integer arithmetic library


 */

#include "mpi.h"
#include <stdlib.h>
#include <string.h>
#include <ctype.h>

#if MP_DEBUG
#include <stdio.h>

#define DIAG(T,V) {fprintf(stderr,T);mp_print(V,stderr);fputc('\n',stderr);}
#else
#define DIAG(T,V)
#endif

/* 
   If MP_LOGTAB is not defined, use the math library to compute the
   logarithms on the fly.  Otherwise, use the static table below.
   Pick which works best for your system.
 */
#if MP_LOGTAB

/* {{{ s_logv_2[] - log table for 2 in various bases */

/*
  A table of the logs of 2 for various bases (the 0 and 1 entries of
  this table are meaningless and should not be referenced).  

  This table is used to compute output lengths for the mp_toradix()
  function.  Since a number n in radix r takes up about log_r(n)
  digits, we estimate the output size by taking the least integer
  greater than log_r(n), where:

  log_r(n) = log_2(n) * log_r(2)

  This table, therefore, is a table of log_r(2) for 2 <= r <= 36,
  which are the output bases supported.  
 */

#include "logtab.h"

/* }}} */
#define LOG_V_2(R)  s_logv_2[(R)]








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/*
    mpi.c

    by Michael J. Fromberger <[email protected]>
    Copyright (C) 1998 Michael J. Fromberger, All Rights Reserved

    Arbitrary precision integer arithmetic library

    $Id$
 */

#include "mpi.h"
#include <stdlib.h>
#include <string.h>
#include <ctype.h>

#if MP_DEBUG
#include <stdio.h>

#define DIAG(T,V) {fprintf(stderr,T);mp_print(V,stderr);fputc('\n',stderr);}
#else
#define DIAG(T,V)
#endif

/*
   If MP_LOGTAB is not defined, use the math library to compute the
   logarithms on the fly.  Otherwise, use the static table below.
   Pick which works best for your system.
 */
#if MP_LOGTAB

/* {{{ s_logv_2[] - log table for 2 in various bases */

/*
  A table of the logs of 2 for various bases (the 0 and 1 entries of
  this table are meaningless and should not be referenced).

  This table is used to compute output lengths for the mp_toradix()
  function.  Since a number n in radix r takes up about log_r(n)
  digits, we estimate the output size by taking the least integer
  greater than log_r(n), where:

  log_r(n) = log_2(n) * log_r(2)

  This table, therefore, is a table of log_r(2) for 2 <= r <= 36,
  which are the output bases supported.
 */

#include "logtab.h"

/* }}} */
#define LOG_V_2(R)  s_logv_2[(R)]

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  "invalid input parameter", /* MP_BADARG            */
  "result is undefined"      /* MP_UNDEF             */
};

/* Value to digit maps for radix conversion   */

/* s_dmap_1 - standard digits and letters */
static const char *s_dmap_1 = 
  "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/";

#if 0
/* s_dmap_2 - base64 ordering for digits  */
static const char *s_dmap_2 =
  "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/";
#endif

/* }}} */

/* {{{ Static function declarations */

/* 
   If MP_MACRO is false, these will be defined as actual functions;
   otherwise, suitable macro definitions will be used.  This works
   around the fact that ANSI C89 doesn't support an 'inline' keyword
   (although I hear C9x will ... about bloody time).  At present, the
   macro definitions are identical to the function bodies, but they'll
   expand in place, instead of generating a function call.








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  "invalid input parameter", /* MP_BADARG            */
  "result is undefined"      /* MP_UNDEF             */
};

/* Value to digit maps for radix conversion   */

/* s_dmap_1 - standard digits and letters */
static const char *s_dmap_1 =
  "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/";

#if 0
/* s_dmap_2 - base64 ordering for digits  */
static const char *s_dmap_2 =
  "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/";
#endif

/* }}} */

/* {{{ Static function declarations */

/*
   If MP_MACRO is false, these will be defined as actual functions;
   otherwise, suitable macro definitions will be used.  This works
   around the fact that ANSI C89 doesn't support an 'inline' keyword
   (although I hear C9x will ... about bloody time).  At present, the
   macro definitions are identical to the function bodies, but they'll
   expand in place, instead of generating a function call.

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    if((res = mp_init(&mp[pos])) != MP_OKAY)
      goto CLEANUP;
  }

  return MP_OKAY;

 CLEANUP:
  while(--pos >= 0) 
    mp_clear(&mp[pos]);

  return res;

} /* end mp_init_array() */

/* }}} */







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    if((res = mp_init(&mp[pos])) != MP_OKAY)
      goto CLEANUP;
  }

  return MP_OKAY;

 CLEANUP:
  while(--pos >= 0)
    mp_clear(&mp[pos]);

  return res;

} /* end mp_init_array() */

/* }}} */
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      the memory allocater more than necessary; otherwise, we'd have
      to grow anyway, so we just allocate a hunk and make the copy as
      usual
     */
    if(ALLOC(to) >= USED(from)) {
      s_mp_setz(DIGITS(to) + USED(from), ALLOC(to) - USED(from));
      s_mp_copy(DIGITS(from), DIGITS(to), USED(from));
      
    } else {
      if((tmp = s_mp_alloc(USED(from), sizeof(mp_digit))) == NULL)
	return MP_MEM;

      s_mp_copy(DIGITS(from), tmp, USED(from));

      if(DIGITS(to) != NULL) {







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      the memory allocater more than necessary; otherwise, we'd have
      to grow anyway, so we just allocate a hunk and make the copy as
      usual
     */
    if(ALLOC(to) >= USED(from)) {
      s_mp_setz(DIGITS(to) + USED(from), ALLOC(to) - USED(from));
      s_mp_copy(DIGITS(from), DIGITS(to), USED(from));

    } else {
      if((tmp = s_mp_alloc(USED(from), sizeof(mp_digit))) == NULL)
	return MP_MEM;

      s_mp_copy(DIGITS(from), tmp, USED(from));

      if(DIGITS(to) != NULL) {
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/* {{{ mp_clear_array(mp[], count) */

void   mp_clear_array(mp_int mp[], int count)
{
  ARGCHK(mp != NULL && count > 0, MP_BADARG);

  while(--count >= 0) 
    mp_clear(&mp[count]);

} /* end mp_clear_array() */

/* }}} */

/* {{{ mp_zero(mp) */

/*
  mp_zero(mp) 

  Set mp to zero.  Does not change the allocated size of the structure,
  and therefore cannot fail (except on a bad argument, which we ignore)
 */
void   mp_zero(mp_int *mp)
{
  if(mp == NULL)







|









|







441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465

/* {{{ mp_clear_array(mp[], count) */

void   mp_clear_array(mp_int mp[], int count)
{
  ARGCHK(mp != NULL && count > 0, MP_BADARG);

  while(--count >= 0)
    mp_clear(&mp[count]);

} /* end mp_clear_array() */

/* }}} */

/* {{{ mp_zero(mp) */

/*
  mp_zero(mp)

  Set mp to zero.  Does not change the allocated size of the structure,
  and therefore cannot fail (except on a bad argument, which we ignore)
 */
void   mp_zero(mp_int *mp)
{
  if(mp == NULL)
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
    return MP_OKAY;  /* shortcut for zero */

  for(ix = sizeof(long) - 1; ix >= 0; ix--) {

    if((res = s_mp_mul_2d(mp, CHAR_BIT)) != MP_OKAY)
      return res;

    res = s_mp_add_d(mp, 
		     (mp_digit)((v >> (ix * CHAR_BIT)) & UCHAR_MAX));
    if(res != MP_OKAY)
      return res;

  }

  if(z < 0)







|







502
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510
511
512
513
514
515
516
    return MP_OKAY;  /* shortcut for zero */

  for(ix = sizeof(long) - 1; ix >= 0; ix--) {

    if((res = s_mp_mul_2d(mp, CHAR_BIT)) != MP_OKAY)
      return res;

    res = s_mp_add_d(mp,
		     (mp_digit)((v >> (ix * CHAR_BIT)) & UCHAR_MAX));
    if(res != MP_OKAY)
      return res;

  }

  if(z < 0)
835
836
837
838
839
840
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842
843
844
845
846
847
848
849
850
851
  mp_err   res;

  ARGCHK(a != NULL && b != NULL, MP_BADARG);

  if((res = mp_copy(a, b)) != MP_OKAY)
    return res;

  if(s_mp_cmp_d(b, 0) == MP_EQ) 
    SIGN(b) = MP_ZPOS;
  else 
    SIGN(b) = (SIGN(b) == MP_NEG) ? MP_ZPOS : MP_NEG;

  return MP_OKAY;

} /* end mp_neg() */

/* }}} */







|

|







837
838
839
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844
845
846
847
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850
851
852
853
  mp_err   res;

  ARGCHK(a != NULL && b != NULL, MP_BADARG);

  if((res = mp_copy(a, b)) != MP_OKAY)
    return res;

  if(s_mp_cmp_d(b, 0) == MP_EQ)
    SIGN(b) = MP_ZPOS;
  else
    SIGN(b) = (SIGN(b) == MP_NEG) ? MP_ZPOS : MP_NEG;

  return MP_OKAY;

} /* end mp_neg() */

/* }}} */
864
865
866
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869
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871
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895
  int     cmp;

  ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG);

  if(SIGN(a) == SIGN(b)) { /* same sign:  add values, keep sign */

    /* Commutativity of addition lets us do this in either order,
       so we avoid having to use a temporary even if the result 
       is supposed to replace the output
     */
    if(c == b) {
      if((res = s_mp_add(c, a)) != MP_OKAY)
	return res;
    } else {
      if(c != a && (res = mp_copy(a, c)) != MP_OKAY)
	return res;

      if((res = s_mp_add(c, b)) != MP_OKAY) 
	return res;
    }

  } else if((cmp = s_mp_cmp(a, b)) > 0) {  /* different sign: a > b   */

    /* If the output is going to be clobbered, we will use a temporary
       variable; otherwise, we'll do it without touching the memory 
       allocator at all, if possible
     */
    if(c == b) {
      mp_int  tmp;

      if((res = mp_init_copy(&tmp, a)) != MP_OKAY)
	return res;







|









|






|







866
867
868
869
870
871
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873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
  int     cmp;

  ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG);

  if(SIGN(a) == SIGN(b)) { /* same sign:  add values, keep sign */

    /* Commutativity of addition lets us do this in either order,
       so we avoid having to use a temporary even if the result
       is supposed to replace the output
     */
    if(c == b) {
      if((res = s_mp_add(c, a)) != MP_OKAY)
	return res;
    } else {
      if(c != a && (res = mp_copy(a, c)) != MP_OKAY)
	return res;

      if((res = s_mp_add(c, b)) != MP_OKAY)
	return res;
    }

  } else if((cmp = s_mp_cmp(a, b)) > 0) {  /* different sign: a > b   */

    /* If the output is going to be clobbered, we will use a temporary
       variable; otherwise, we'll do it without touching the memory
       allocator at all, if possible
     */
    if(c == b) {
      mp_int  tmp;

      if((res = mp_init_copy(&tmp, a)) != MP_OKAY)
	return res;
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
	mp_clear(&tmp);
	return res;
      }
      s_mp_exch(&tmp, c);
      mp_clear(&tmp);

    } else {
      if(c != b && ((res = mp_copy(b, c)) != MP_OKAY)) 
	return res;

      if((res = s_mp_sub(c, a)) != MP_OKAY)
	return res;
    }

    SIGN(c) = !SIGN(b);







|







1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
	mp_clear(&tmp);
	return res;
      }
      s_mp_exch(&tmp, c);
      mp_clear(&tmp);

    } else {
      if(c != b && ((res = mp_copy(b, c)) != MP_OKAY))
	return res;

      if((res = s_mp_sub(c, a)) != MP_OKAY)
	return res;
    }

    SIGN(c) = !SIGN(b);
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
  } else {
    if((res = mp_copy(a, c)) != MP_OKAY)
      return res;

    if((res = s_mp_mul(c, b)) != MP_OKAY)
      return res;
  }
  
  if(sgn == MP_ZPOS || s_mp_cmp_d(c, 0) == MP_EQ)
    SIGN(c) = MP_ZPOS;
  else
    SIGN(c) = sgn;
  
  return MP_OKAY;

} /* end mp_mul() */

/* }}} */

/* {{{ mp_mul_2d(a, d, c) */







|




|







1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
  } else {
    if((res = mp_copy(a, c)) != MP_OKAY)
      return res;

    if((res = s_mp_mul(c, b)) != MP_OKAY)
      return res;
  }

  if(sgn == MP_ZPOS || s_mp_cmp_d(c, 0) == MP_EQ)
    SIGN(c) = MP_ZPOS;
  else
    SIGN(c) = sgn;

  return MP_OKAY;

} /* end mp_mul() */

/* }}} */

/* {{{ mp_mul_2d(a, d, c) */
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
   */
  if((cmp = s_mp_cmp(a, b)) < 0) {
    if(r) {
      if((res = mp_copy(a, r)) != MP_OKAY)
	return res;
    }

    if(q) 
      mp_zero(q);

    return MP_OKAY;

  } else if(cmp == 0) {

    /* Set quotient to 1, with appropriate sign */







|







1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
   */
  if((cmp = s_mp_cmp(a, b)) < 0) {
    if(r) {
      if((res = mp_copy(a, r)) != MP_OKAY)
	return res;
    }

    if(q)
      mp_zero(q);

    return MP_OKAY;

  } else if(cmp == 0) {

    /* Set quotient to 1, with appropriate sign */
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217

  if(s_mp_cmp_d(&qtmp, 0) == MP_EQ)
    SIGN(&qtmp) = MP_ZPOS;
  if(s_mp_cmp_d(&rtmp, 0) == MP_EQ)
    SIGN(&rtmp) = MP_ZPOS;

  /* Copy output, if it is needed      */
  if(q) 
    s_mp_exch(&qtmp, q);

  if(r) 
    s_mp_exch(&rtmp, r);

CLEANUP:
  mp_clear(&rtmp);
  mp_clear(&qtmp);

  return res;







|


|







1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219

  if(s_mp_cmp_d(&qtmp, 0) == MP_EQ)
    SIGN(&qtmp) = MP_ZPOS;
  if(s_mp_cmp_d(&rtmp, 0) == MP_EQ)
    SIGN(&rtmp) = MP_ZPOS;

  /* Copy output, if it is needed      */
  if(q)
    s_mp_exch(&qtmp, q);

  if(r)
    s_mp_exch(&rtmp, r);

CLEANUP:
  mp_clear(&rtmp);
  mp_clear(&qtmp);

  return res;
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
 */

mp_err mp_expt(mp_int *a, mp_int *b, mp_int *c)
{
  mp_int   s, x;
  mp_err   res;
  mp_digit d;
  int      dig, bit;

  ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG);

  if(mp_cmp_z(b) < 0)
    return MP_RANGE;

  if((res = mp_init(&s)) != MP_OKAY)







|







1260
1261
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1263
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1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
 */

mp_err mp_expt(mp_int *a, mp_int *b, mp_int *c)
{
  mp_int   s, x;
  mp_err   res;
  mp_digit d;
  unsigned int bit, dig;

  ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG);

  if(mp_cmp_z(b) < 0)
    return MP_RANGE;

  if((res = mp_init(&s)) != MP_OKAY)
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
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1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
  /* Loop over low-order digits in ascending order */
  for(dig = 0; dig < (USED(b) - 1); dig++) {
    d = DIGIT(b, dig);

    /* Loop over bits of each non-maximal digit */
    for(bit = 0; bit < DIGIT_BIT; bit++) {
      if(d & 1) {
	if((res = s_mp_mul(&s, &x)) != MP_OKAY) 
	  goto CLEANUP;
      }

      d >>= 1;
      
      if((res = s_mp_sqr(&x)) != MP_OKAY)
	goto CLEANUP;
    }
  }

  /* Consider now the last digit... */
  d = DIGIT(b, dig);

  while(d) {
    if(d & 1) {
      if((res = s_mp_mul(&s, &x)) != MP_OKAY)
	goto CLEANUP;
    }

    d >>= 1;

    if((res = s_mp_sqr(&x)) != MP_OKAY)
      goto CLEANUP;
  }
  
  if(mp_iseven(b))
    SIGN(&s) = SIGN(a);

  res = mp_copy(&s, c);

CLEANUP:
  mp_clear(&x);







|




|



















|







1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
  /* Loop over low-order digits in ascending order */
  for(dig = 0; dig < (USED(b) - 1); dig++) {
    d = DIGIT(b, dig);

    /* Loop over bits of each non-maximal digit */
    for(bit = 0; bit < DIGIT_BIT; bit++) {
      if(d & 1) {
	if((res = s_mp_mul(&s, &x)) != MP_OKAY)
	  goto CLEANUP;
      }

      d >>= 1;

      if((res = s_mp_sqr(&x)) != MP_OKAY)
	goto CLEANUP;
    }
  }

  /* Consider now the last digit... */
  d = DIGIT(b, dig);

  while(d) {
    if(d & 1) {
      if((res = s_mp_mul(&s, &x)) != MP_OKAY)
	goto CLEANUP;
    }

    d >>= 1;

    if((res = s_mp_sqr(&x)) != MP_OKAY)
      goto CLEANUP;
  }

  if(mp_iseven(b))
    SIGN(&s) = SIGN(a);

  res = mp_copy(&s, c);

CLEANUP:
  mp_clear(&x);
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
  ARGCHK(a != NULL && m != NULL && c != NULL, MP_BADARG);

  if(SIGN(m) == MP_NEG)
    return MP_RANGE;

  /*
     If |a| > m, we need to divide to get the remainder and take the
     absolute value.  

     If |a| < m, we don't need to do any division, just copy and adjust
     the sign (if a is negative).

     If |a| == m, we can simply set the result to zero.

     This order is intended to minimize the average path length of the
     comparison chain on common workloads -- the most frequent cases are
     that |a| != m, so we do those first.
   */
  if((mag = s_mp_cmp(a, m)) > 0) {
    if((res = mp_div(a, m, NULL, c)) != MP_OKAY)
      return res;
    
    if(SIGN(c) == MP_NEG) {
      if((res = mp_add(c, m, c)) != MP_OKAY)
	return res;
    }

  } else if(mag < 0) {
    if((res = mp_copy(a, c)) != MP_OKAY)
      return res;

    if(mp_cmp_z(a) < 0) {
      if((res = mp_add(c, m, c)) != MP_OKAY)
	return res;

    }
    
  } else {
    mp_zero(c);

  }

  return MP_OKAY;








|













|














|







1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
  ARGCHK(a != NULL && m != NULL && c != NULL, MP_BADARG);

  if(SIGN(m) == MP_NEG)
    return MP_RANGE;

  /*
     If |a| > m, we need to divide to get the remainder and take the
     absolute value.

     If |a| < m, we don't need to do any division, just copy and adjust
     the sign (if a is negative).

     If |a| == m, we can simply set the result to zero.

     This order is intended to minimize the average path length of the
     comparison chain on common workloads -- the most frequent cases are
     that |a| != m, so we do those first.
   */
  if((mag = s_mp_cmp(a, m)) > 0) {
    if((res = mp_div(a, m, NULL, c)) != MP_OKAY)
      return res;

    if(SIGN(c) == MP_NEG) {
      if((res = mp_add(c, m, c)) != MP_OKAY)
	return res;
    }

  } else if(mag < 0) {
    if((res = mp_copy(a, c)) != MP_OKAY)
      return res;

    if(mp_cmp_z(a) < 0) {
      if((res = mp_add(c, m, c)) != MP_OKAY)
	return res;

    }

  } else {
    mp_zero(c);

  }

  return MP_OKAY;

1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
  ARGCHK(a != NULL && b != NULL, MP_BADARG);

  /* Cannot take square root of a negative value */
  if(SIGN(a) == MP_NEG)
    return MP_RANGE;

  /* Special cases for zero and one, trivial     */
  if(mp_cmp_d(a, 0) == MP_EQ || mp_cmp_d(a, 1) == MP_EQ) 
    return mp_copy(a, b);
    
  /* Initialize the temporaries we'll use below  */
  if((res = mp_init_size(&t, USED(a))) != MP_OKAY)
    return res;

  /* Compute an initial guess for the iteration as a itself */
  if((res = mp_init_copy(&x, a)) != MP_OKAY)
    goto X;







|

|







1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
  ARGCHK(a != NULL && b != NULL, MP_BADARG);

  /* Cannot take square root of a negative value */
  if(SIGN(a) == MP_NEG)
    return MP_RANGE;

  /* Special cases for zero and one, trivial     */
  if(mp_cmp_d(a, 0) == MP_EQ || mp_cmp_d(a, 1) == MP_EQ)
    return mp_copy(a, b);

  /* Initialize the temporaries we'll use below  */
  if((res = mp_init_size(&t, USED(a))) != MP_OKAY)
    return res;

  /* Compute an initial guess for the iteration as a itself */
  if((res = mp_init_copy(&x, a)) != MP_OKAY)
    goto X;
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
  /* Copy result to output parameter */
  mp_sub_d(&x, 1, &x);
  s_mp_exch(&x, b);

 CLEANUP:
  mp_clear(&x);
 X:
  mp_clear(&t); 

  return res;

} /* end mp_sqrt() */

/* }}} */








|







1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
  /* Copy result to output parameter */
  mp_sub_d(&x, 1, &x);
  s_mp_exch(&x, b);

 CLEANUP:
  mp_clear(&x);
 X:
  mp_clear(&t);

  return res;

} /* end mp_sqrt() */

/* }}} */

1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
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1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663

/*
  mp_exptmod(a, b, m, c)

  Compute c = (a ** b) mod m.  Uses a standard square-and-multiply
  method with modular reductions at each step. (This is basically the
  same code as mp_expt(), except for the addition of the reductions)
  
  The modular reductions are done using Barrett's algorithm (see
  s_mp_reduce() below for details)
 */

mp_err mp_exptmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c)
{
  mp_int   s, x, mu;
  mp_err   res;
  mp_digit d, *db = DIGITS(b);
  mp_size  ub = USED(b);
  int      dig, bit;

  ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG);

  if(mp_cmp_z(b) < 0 || mp_cmp_z(m) <= 0)
    return MP_RANGE;

  if((res = mp_init(&s)) != MP_OKAY)
    return res;
  if((res = mp_init_copy(&x, a)) != MP_OKAY)
    goto X;
  if((res = mp_mod(&x, m, &x)) != MP_OKAY ||
     (res = mp_init(&mu)) != MP_OKAY)
    goto MU;

  mp_set(&s, 1);

  /* mu = b^2k / m */
  s_mp_add_d(&mu, 1); 
  s_mp_lshd(&mu, 2 * USED(m));
  if((res = mp_div(&mu, m, &mu, NULL)) != MP_OKAY)
    goto CLEANUP;

  /* Loop over digits of b in ascending order, except highest order */
  for(dig = 0; dig < (ub - 1); dig++) {
    d = *db++;







|










|

















|







1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
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1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665

/*
  mp_exptmod(a, b, m, c)

  Compute c = (a ** b) mod m.  Uses a standard square-and-multiply
  method with modular reductions at each step. (This is basically the
  same code as mp_expt(), except for the addition of the reductions)

  The modular reductions are done using Barrett's algorithm (see
  s_mp_reduce() below for details)
 */

mp_err mp_exptmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c)
{
  mp_int   s, x, mu;
  mp_err   res;
  mp_digit d, *db = DIGITS(b);
  mp_size  ub = USED(b);
  unsigned int bit, dig;

  ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG);

  if(mp_cmp_z(b) < 0 || mp_cmp_z(m) <= 0)
    return MP_RANGE;

  if((res = mp_init(&s)) != MP_OKAY)
    return res;
  if((res = mp_init_copy(&x, a)) != MP_OKAY)
    goto X;
  if((res = mp_mod(&x, m, &x)) != MP_OKAY ||
     (res = mp_init(&mu)) != MP_OKAY)
    goto MU;

  mp_set(&s, 1);

  /* mu = b^2k / m */
  s_mp_add_d(&mu, 1);
  s_mp_lshd(&mu, 2 * USED(m));
  if((res = mp_div(&mu, m, &mu, NULL)) != MP_OKAY)
    goto CLEANUP;

  /* Loop over digits of b in ascending order, except highest order */
  for(dig = 0; dig < (ub - 1); dig++) {
    d = *db++;
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
 */
int    mp_cmp_int(mp_int *a, long z)
{
  mp_int  tmp;
  int     out;

  ARGCHK(a != NULL, MP_EQ);
  
  mp_init(&tmp); mp_set_int(&tmp, z);
  out = mp_cmp(a, &tmp);
  mp_clear(&tmp);

  return out;

} /* end mp_cmp_int() */







|







1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
 */
int    mp_cmp_int(mp_int *a, long z)
{
  mp_int  tmp;
  int     out;

  ARGCHK(a != NULL, MP_EQ);

  mp_init(&tmp); mp_set_int(&tmp, z);
  out = mp_cmp(a, &tmp);
  mp_clear(&tmp);

  return out;

} /* end mp_cmp_int() */
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
    ++k;
  }

  /* Initialize t */
  if(mp_isodd(&u)) {
    if((res = mp_copy(&v, &t)) != MP_OKAY)
      goto CLEANUP;
    
    /* t = -v */
    if(SIGN(&v) == MP_ZPOS)
      SIGN(&t) = MP_NEG;
    else
      SIGN(&t) = MP_ZPOS;
    
  } else {
    if((res = mp_copy(&u, &t)) != MP_OKAY)
      goto CLEANUP;

  }

  for(;;) {







|





|







1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
    ++k;
  }

  /* Initialize t */
  if(mp_isodd(&u)) {
    if((res = mp_copy(&v, &t)) != MP_OKAY)
      goto CLEANUP;

    /* t = -v */
    if(SIGN(&v) == MP_ZPOS)
      SIGN(&t) = MP_NEG;
    else
      SIGN(&t) = MP_ZPOS;

  } else {
    if((res = mp_copy(&u, &t)) != MP_OKAY)
      goto CLEANUP;

  }

  for(;;) {
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
    /* If we're done, copy results to output */
    if(mp_cmp_z(&u) == 0) {
      if(x)
	if((res = mp_copy(&C, x)) != MP_OKAY) goto CLEANUP;

      if(y)
	if((res = mp_copy(&D, y)) != MP_OKAY) goto CLEANUP;
      
      if(g)
	if((res = mp_mul(&gx, &v, g)) != MP_OKAY) goto CLEANUP;

      break;
    }
  }








|







2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
    /* If we're done, copy results to output */
    if(mp_cmp_z(&u) == 0) {
      if(x)
	if((res = mp_copy(&C, x)) != MP_OKAY) goto CLEANUP;

      if(y)
	if((res = mp_copy(&D, y)) != MP_OKAY) goto CLEANUP;

      if(g)
	if((res = mp_mul(&gx, &v, g)) != MP_OKAY) goto CLEANUP;

      break;
    }
  }

2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
/* }}} */

/*------------------------------------------------------------------------*/
/* {{{ More I/O Functions */

/* {{{ mp_read_signed_bin(mp, str, len) */

/* 
   mp_read_signed_bin(mp, str, len)

   Read in a raw value (base 256) into the given mp_int
 */

mp_err  mp_read_signed_bin(mp_int *mp, unsigned char *str, int len)
{







|







2251
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2253
2254
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2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
/* }}} */

/*------------------------------------------------------------------------*/
/* {{{ More I/O Functions */

/* {{{ mp_read_signed_bin(mp, str, len) */

/*
   mp_read_signed_bin(mp, str, len)

   Read in a raw value (base 256) into the given mp_int
 */

mp_err  mp_read_signed_bin(mp_int *mp, unsigned char *str, int len)
{
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
  for(ix = 0; ix < len; ix++) {
    if((res = s_mp_mul_2d(mp, CHAR_BIT)) != MP_OKAY)
      return res;

    if((res = mp_add_d(mp, str[ix], mp)) != MP_OKAY)
      return res;
  }
  
  return MP_OKAY;
  
} /* end mp_read_unsigned_bin() */

/* }}} */

/* {{{ mp_unsigned_bin_size(mp) */

int     mp_unsigned_bin_size(mp_int *mp) 
{
  mp_digit   topdig;
  int        count;

  ARGCHK(mp != NULL, 0);

  /* Special case for the value zero */







|

|






|







2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
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2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
  for(ix = 0; ix < len; ix++) {
    if((res = s_mp_mul_2d(mp, CHAR_BIT)) != MP_OKAY)
      return res;

    if((res = mp_add_d(mp, str[ix], mp)) != MP_OKAY)
      return res;
  }

  return MP_OKAY;

} /* end mp_read_unsigned_bin() */

/* }}} */

/* {{{ mp_unsigned_bin_size(mp) */

int     mp_unsigned_bin_size(mp_int *mp)
{
  mp_digit   topdig;
  int        count;

  ARGCHK(mp != NULL, 0);

  /* Special case for the value zero */
2381
2382
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
  if(dp == end && *dp == 0) {
    *str = '\0';
    return MP_OKAY;
  }

  /* Generate digits in reverse order */
  while(dp < end) {
    int      ix;

    d = *dp;
    for(ix = 0; ix < sizeof(mp_digit); ++ix) {
      *spos = d & UCHAR_MAX;
      d >>= CHAR_BIT;
      ++spos;
    }







|







2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
  if(dp == end && *dp == 0) {
    *str = '\0';
    return MP_OKAY;
  }

  /* Generate digits in reverse order */
  while(dp < end) {
    unsigned int ix;

    d = *dp;
    for(ix = 0; ix < sizeof(mp_digit); ++ix) {
      *spos = d & UCHAR_MAX;
      d >>= CHAR_BIT;
      ++spos;
    }
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448

  while(d != 0) {
    ++len;
    d >>= 1;
  }

  return len;
  
} /* end mp_count_bits() */

/* }}} */

/* {{{ mp_read_radix(mp, str, radix) */

/*







|







2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450

  while(d != 0) {
    ++len;
    d >>= 1;
  }

  return len;

} /* end mp_count_bits() */

/* }}} */

/* {{{ mp_read_radix(mp, str, radix) */

/*
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477

mp_err  mp_read_radix(mp_int *mp, unsigned char *str, int radix)
{
  int     ix = 0, val = 0;
  mp_err  res;
  mp_sign sig = MP_ZPOS;

  ARGCHK(mp != NULL && str != NULL && radix >= 2 && radix <= MAX_RADIX, 
	 MP_BADARG);

  mp_zero(mp);

  /* Skip leading non-digit characters until a digit or '-' or '+' */
  while(str[ix] && 
	(s_mp_tovalue(str[ix], radix) < 0) && 
	str[ix] != '-' &&
	str[ix] != '+') {
    ++ix;
  }

  if(str[ix] == '-') {
    sig = MP_NEG;







|





|
|







2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479

mp_err  mp_read_radix(mp_int *mp, unsigned char *str, int radix)
{
  int     ix = 0, val = 0;
  mp_err  res;
  mp_sign sig = MP_ZPOS;

  ARGCHK(mp != NULL && str != NULL && radix >= 2 && radix <= MAX_RADIX,
	 MP_BADARG);

  mp_zero(mp);

  /* Skip leading non-digit characters until a digit or '-' or '+' */
  while(str[ix] &&
	(s_mp_tovalue(str[ix], radix) < 0) &&
	str[ix] != '-' &&
	str[ix] != '+') {
    ++ix;
  }

  if(str[ix] == '-') {
    sig = MP_NEG;
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
/* }}} */

/* {{{ mp_value_radix_size(num, qty, radix) */

/* num = number of digits
   qty = number of bits per digit
   radix = target base
   
   Return the number of digits in the specified radix that would be
   needed to express 'num' digits of 'qty' bits each.
 */
int    mp_value_radix_size(int num, int qty, int radix)
{
  ARGCHK(num >= 0 && qty > 0 && radix >= 2 && radix <= MAX_RADIX, 0);

  return s_mp_outlen(num * qty, radix);

} /* end mp_value_radix_size() */

/* }}} */

/* {{{ mp_toradix(mp, str, radix) */

mp_err mp_toradix(mp_int *mp, unsigned char *str, int radix)
{
  int  ix, pos = 0;

  ARGCHK(mp != NULL && str != NULL, MP_BADARG);
  ARGCHK(radix > 1 && radix <= MAX_RADIX, MP_RANGE);

  if(mp_cmp_z(mp) == MP_EQ) {







|















|







2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
2550
2551
/* }}} */

/* {{{ mp_value_radix_size(num, qty, radix) */

/* num = number of digits
   qty = number of bits per digit
   radix = target base

   Return the number of digits in the specified radix that would be
   needed to express 'num' digits of 'qty' bits each.
 */
int    mp_value_radix_size(int num, int qty, int radix)
{
  ARGCHK(num >= 0 && qty > 0 && radix >= 2 && radix <= MAX_RADIX, 0);

  return s_mp_outlen(num * qty, radix);

} /* end mp_value_radix_size() */

/* }}} */

/* {{{ mp_toradix(mp, str, radix) */

mp_err mp_toradix(mp_int *mp, char *str, int radix)
{
  int  ix, pos = 0;

  ARGCHK(mp != NULL && str != NULL, MP_BADARG);
  ARGCHK(radix > 1 && radix <= MAX_RADIX, MP_RANGE);

  if(mp_cmp_z(mp) == MP_EQ) {
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602

    /* Add trailing NUL to end the string        */
    str[pos--] = '\0';

    /* Reverse the digits and sign indicator     */
    ix = 0;
    while(ix < pos) {
      char tmp = str[ix];

      str[ix] = str[pos];
      str[pos] = tmp;
      ++ix;
      --pos;
    }
    
    mp_clear(&tmp);
  }

  return MP_OKAY;

} /* end mp_toradix() */








|


|



|







2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604

    /* Add trailing NUL to end the string        */
    str[pos--] = '\0';

    /* Reverse the digits and sign indicator     */
    ix = 0;
    while(ix < pos) {
      char _tmp = str[ix];

      str[ix] = str[pos];
      str[pos] = _tmp;
      ++ix;
      --pos;
    }

    mp_clear(&tmp);
  }

  return MP_OKAY;

} /* end mp_toradix() */

2800
2801
2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
2840
2841
2842
2843
2844
2845
2846
2847
2848
2849
2850
2851
2852

/* }}} */

/* {{{ Arithmetic helpers */

/* {{{ s_mp_lshd(mp, p) */

/* 
   Shift mp leftward by p digits, growing if needed, and zero-filling
   the in-shifted digits at the right end.  This is a convenient
   alternative to multiplication by powers of the radix
 */   

mp_err   s_mp_lshd(mp_int *mp, mp_size p)
{
  mp_err   res;
  mp_size  pos;
  mp_digit *dp;
  int     ix;

  if(p == 0)
    return MP_OKAY;

  if((res = s_mp_pad(mp, USED(mp) + p)) != MP_OKAY)
    return res;

  pos = USED(mp) - 1;
  dp = DIGITS(mp);

  /* Shift all the significant figures over as needed */
  for(ix = pos - p; ix >= 0; ix--) 
    dp[ix + p] = dp[ix];

  /* Fill the bottom digits with zeroes */
  for(ix = 0; ix < p; ix++)
    dp[ix] = 0;

  return MP_OKAY;

} /* end s_mp_lshd() */

/* }}} */

/* {{{ s_mp_rshd(mp, p) */

/* 
   Shift mp rightward by p digits.  Maintains the invariant that
   digits above the precision are all zero.  Digits shifted off the
   end are lost.  Cannot fail.
 */

void     s_mp_rshd(mp_int *mp, mp_size p)
{







|



|






|











|



|










|







2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
2840
2841
2842
2843
2844
2845
2846
2847
2848
2849
2850
2851
2852
2853
2854

/* }}} */

/* {{{ Arithmetic helpers */

/* {{{ s_mp_lshd(mp, p) */

/*
   Shift mp leftward by p digits, growing if needed, and zero-filling
   the in-shifted digits at the right end.  This is a convenient
   alternative to multiplication by powers of the radix
 */

mp_err   s_mp_lshd(mp_int *mp, mp_size p)
{
  mp_err   res;
  mp_size  pos;
  mp_digit *dp;
  int ix;

  if(p == 0)
    return MP_OKAY;

  if((res = s_mp_pad(mp, USED(mp) + p)) != MP_OKAY)
    return res;

  pos = USED(mp) - 1;
  dp = DIGITS(mp);

  /* Shift all the significant figures over as needed */
  for(ix = pos - p; ix >= 0; ix--)
    dp[ix + p] = dp[ix];

  /* Fill the bottom digits with zeroes */
  for(ix = 0; (unsigned)ix < p; ix++)
    dp[ix] = 0;

  return MP_OKAY;

} /* end s_mp_lshd() */

/* }}} */

/* {{{ s_mp_rshd(mp, p) */

/*
   Shift mp rightward by p digits.  Maintains the invariant that
   digits above the precision are all zero.  Digits shifted off the
   end are lost.  Cannot fail.
 */

void     s_mp_rshd(mp_int *mp, mp_size p)
{
2892
2893
2894
2895
2896
2897
2898
2899
2900
2901
2902
2903
2904
2905
2906

/* }}} */

/* {{{ s_mp_mul_2(mp) */

mp_err s_mp_mul_2(mp_int *mp)
{
  int      ix;
  mp_digit kin = 0, kout, *dp = DIGITS(mp);
  mp_err   res;

  /* Shift digits leftward by 1 bit */
  for(ix = 0; ix < USED(mp); ix++) {
    kout = (dp[ix] >> (DIGIT_BIT - 1)) & 1;
    dp[ix] = (dp[ix] << 1) | kin;







|







2894
2895
2896
2897
2898
2899
2900
2901
2902
2903
2904
2905
2906
2907
2908

/* }}} */

/* {{{ s_mp_mul_2(mp) */

mp_err s_mp_mul_2(mp_int *mp)
{
  unsigned int ix;
  mp_digit kin = 0, kout, *dp = DIGITS(mp);
  mp_err   res;

  /* Shift digits leftward by 1 bit */
  for(ix = 0; ix < USED(mp); ix++) {
    kout = (dp[ix] >> (DIGIT_BIT - 1)) & 1;
    dp[ix] = (dp[ix] << 1) | kin;
2964
2965
2966
2967
2968
2969
2970
2971
2972
2973
2974
2975
2976
2977
2978
  full multiplication code.
 */
mp_err    s_mp_mul_2d(mp_int *mp, mp_digit d)
{
  mp_err   res;
  mp_digit save, next, mask, *dp;
  mp_size  used;
  int      ix;

  if((res = s_mp_lshd(mp, d / DIGIT_BIT)) != MP_OKAY)
    return res;

  dp = DIGITS(mp); used = USED(mp);
  d %= DIGIT_BIT;








|







2966
2967
2968
2969
2970
2971
2972
2973
2974
2975
2976
2977
2978
2979
2980
  full multiplication code.
 */
mp_err    s_mp_mul_2d(mp_int *mp, mp_digit d)
{
  mp_err   res;
  mp_digit save, next, mask, *dp;
  mp_size  used;
  unsigned int ix;

  if((res = s_mp_lshd(mp, d / DIGIT_BIT)) != MP_OKAY)
    return res;

  dp = DIGITS(mp); used = USED(mp);
  d %= DIGIT_BIT;

3048
3049
3050
3051
3052
3053
3054
3055
3056
3057
3058
3059
3060
3061
3062
3063
3064
3065
3066
3067
3068
3069
3070
3071
3072
3073
3074
  that we might make good guesses for quotient digits, we want the
  leading digit of b to be at least half the radix, which we
  accomplish by multiplying a and b by a constant.  This constant is
  returned (so that it can be divided back out of the remainder at the
  end of the division process).

  We multiply by the smallest power of 2 that gives us a leading digit
  at least half the radix.  By choosing a power of 2, we simplify the 
  multiplication and division steps to simple shifts.
 */
mp_digit s_mp_norm(mp_int *a, mp_int *b)
{
  mp_digit  t, d = 0;

  t = DIGIT(b, USED(b) - 1);
  while(t < (RADIX / 2)) {
    t <<= 1;
    ++d;
  }
    
  if(d != 0) {
    s_mp_mul_2d(a, d);
    s_mp_mul_2d(b, d);
  }

  return d;








|











|







3050
3051
3052
3053
3054
3055
3056
3057
3058
3059
3060
3061
3062
3063
3064
3065
3066
3067
3068
3069
3070
3071
3072
3073
3074
3075
3076
  that we might make good guesses for quotient digits, we want the
  leading digit of b to be at least half the radix, which we
  accomplish by multiplying a and b by a constant.  This constant is
  returned (so that it can be divided back out of the remainder at the
  end of the division process).

  We multiply by the smallest power of 2 that gives us a leading digit
  at least half the radix.  By choosing a power of 2, we simplify the
  multiplication and division steps to simple shifts.
 */
mp_digit s_mp_norm(mp_int *a, mp_int *b)
{
  mp_digit  t, d = 0;

  t = DIGIT(b, USED(b) - 1);
  while(t < (RADIX / 2)) {
    t <<= 1;
    ++d;
  }

  if(d != 0) {
    s_mp_mul_2d(a, d);
    s_mp_mul_2d(b, d);
  }

  return d;

3182
3183
3184
3185
3186
3187
3188
3189
3190
3191
3192
3193
3194
3195
3196
3197
3198
3199
3200
3201
3202
3203
    k = CARRYOUT(w);
  }

  /* If there is a precision increase, take care of it here; the above
     test guarantees we have enough storage to do this safely.
   */
  if(k) {
    dp[max] = k; 
    USED(a) = max + 1;
  }

  s_mp_clamp(a);

  return MP_OKAY;
  
} /* end s_mp_mul_d() */

/* }}} */

/* {{{ s_mp_div_d(mp, d, r) */

/*







|






|







3184
3185
3186
3187
3188
3189
3190
3191
3192
3193
3194
3195
3196
3197
3198
3199
3200
3201
3202
3203
3204
3205
    k = CARRYOUT(w);
  }

  /* If there is a precision increase, take care of it here; the above
     test guarantees we have enough storage to do this safely.
   */
  if(k) {
    dp[max] = k;
    USED(a) = max + 1;
  }

  s_mp_clamp(a);

  return MP_OKAY;

} /* end s_mp_mul_d() */

/* }}} */

/* {{{ s_mp_div_d(mp, d, r) */

/*
3283
3284
3285
3286
3287
3288
3289
3290
3291
3292
3293
3294
3295
3296
3297
  for(ix = 0; ix < used; ++ix) {
    w += *pa + *pb++;
    *pa++ = ACCUM(w);
    w = CARRYOUT(w);
  }

  /* If we run out of 'b' digits before we're actually done, make
     sure the carries get propagated upward...  
   */
  used = USED(a);
  while(w && ix < used) {
    w += *pa;
    *pa++ = ACCUM(w);
    w = CARRYOUT(w);
    ++ix;







|







3285
3286
3287
3288
3289
3290
3291
3292
3293
3294
3295
3296
3297
3298
3299
  for(ix = 0; ix < used; ++ix) {
    w += *pa + *pb++;
    *pa++ = ACCUM(w);
    w = CARRYOUT(w);
  }

  /* If we run out of 'b' digits before we're actually done, make
     sure the carries get propagated upward...
   */
  used = USED(a);
  while(w && ix < used) {
    w += *pa;
    *pa++ = ACCUM(w);
    w = CARRYOUT(w);
    ++ix;
3345
3346
3347
3348
3349
3350
3351
3352
3353
3354
3355
3356
3357
3358
3359
    w = CARRYOUT(w) ? 0 : 1;
    ++ix;
  }

  /* Clobber any leading zeroes we created    */
  s_mp_clamp(a);

  /* 
     If there was a borrow out, then |b| > |a| in violation
     of our input invariant.  We've already done the work,
     but we'll at least complain about it...
   */
  if(w)
    return MP_RANGE;
  else







|







3347
3348
3349
3350
3351
3352
3353
3354
3355
3356
3357
3358
3359
3360
3361
    w = CARRYOUT(w) ? 0 : 1;
    ++ix;
  }

  /* Clobber any leading zeroes we created    */
  s_mp_clamp(a);

  /*
     If there was a borrow out, then |b| > |a| in violation
     of our input invariant.  We've already done the work,
     but we'll at least complain about it...
   */
  if(w)
    return MP_RANGE;
  else
3381
3382
3383
3384
3385
3386
3387
3388
3389
3390
3391
3392
3393
3394
3395

  /* q = q * m mod b^(k+1), quick (no division), uses the short multiplier */
#ifndef SHRT_MUL
  s_mp_mul(&q, m);
  s_mp_mod_2d(&q, (mp_digit)(DIGIT_BIT * (um + 1)));
#else
  s_mp_mul_dig(&q, m, um + 1);
#endif  

  /* x = x - q */
  if((res = mp_sub(x, &q, x)) != MP_OKAY)
    goto CLEANUP;

  /* If x < 0, add b^(k+1) to it */
  if(mp_cmp_z(x) < 0) {







|







3383
3384
3385
3386
3387
3388
3389
3390
3391
3392
3393
3394
3395
3396
3397

  /* q = q * m mod b^(k+1), quick (no division), uses the short multiplier */
#ifndef SHRT_MUL
  s_mp_mul(&q, m);
  s_mp_mod_2d(&q, (mp_digit)(DIGIT_BIT * (um + 1)));
#else
  s_mp_mul_dig(&q, m, um + 1);
#endif

  /* x = x - q */
  if((res = mp_sub(x, &q, x)) != MP_OKAY)
    goto CLEANUP;

  /* If x < 0, add b^(k+1) to it */
  if(mp_cmp_z(x) < 0) {
3435
3436
3437
3438
3439
3440
3441
3442
3443
3444
3445
3446
3447
3448
3449
  /* We're going to need the base value each iteration */
  pbt = DIGITS(&tmp);

  /* Outer loop:  Digits of b */

  pb = DIGITS(b);
  for(ix = 0; ix < ub; ++ix, ++pb) {
    if(*pb == 0) 
      continue;

    /* Inner product:  Digits of a */
    pa = DIGITS(a);
    for(jx = 0; jx < ua; ++jx, ++pa) {
      pt = pbt + ix + jx;
      w = *pb * *pa + k + *pt;







|







3437
3438
3439
3440
3441
3442
3443
3444
3445
3446
3447
3448
3449
3450
3451
  /* We're going to need the base value each iteration */
  pbt = DIGITS(&tmp);

  /* Outer loop:  Digits of b */

  pb = DIGITS(b);
  for(ix = 0; ix < ub; ++ix, ++pb) {
    if(*pb == 0)
      continue;

    /* Inner product:  Digits of a */
    pa = DIGITS(a);
    for(jx = 0; jx < ua; ++jx, ++pa) {
      pt = pbt + ix + jx;
      w = *pb * *pa + k + *pt;
3474
3475
3476
3477
3478
3479
3480
3481
3482
3483
3484
3485
3486
3487
3488
  mp_word   w, k = 0;
  mp_size   ix, jx;
  mp_digit *pa, *pt;

  for(ix = 0; ix < len; ++ix, ++b) {
    if(*b == 0)
      continue;
    
    pa = a;
    for(jx = 0; jx < len; ++jx, ++pa) {
      pt = out + ix + jx;
      w = *b * *pa + k + *pt;
      *pt = ACCUM(w);
      k = CARRYOUT(w);
    }







|







3476
3477
3478
3479
3480
3481
3482
3483
3484
3485
3486
3487
3488
3489
3490
  mp_word   w, k = 0;
  mp_size   ix, jx;
  mp_digit *pa, *pt;

  for(ix = 0; ix < len; ++ix, ++b) {
    if(*b == 0)
      continue;

    pa = a;
    for(jx = 0; jx < len; ++jx, ++pa) {
      pt = out + ix + jx;
      w = *b * *pa + k + *pt;
      *pt = ACCUM(w);
      k = CARRYOUT(w);
    }
3541
3542
3543
3544
3545
3546
3547
3548
3549
3550
3551
3552
3553
3554
3555
3556
3557
3558
3559
3560
3561
3562
3563
3564
3565
3566
3567
3568
3569
3570
3571
3572
3573
3574
3575
3576
      This can overflow what can be represented in an mp_word, and
      since C arithmetic does not provide any way to check for
      overflow, we have to check explicitly for overflow conditions
      before they happen.
     */
    for(jx = ix + 1, pa2 = DIGITS(a) + jx; jx < used; ++jx, ++pa2) {
      mp_word  u = 0, v;
      
      /* Store this in a temporary to avoid indirections later */
      pt = pbt + ix + jx;

      /* Compute the multiplicative step */
      w = *pa1 * *pa2;

      /* If w is more than half MP_WORD_MAX, the doubling will
	 overflow, and we need to record a carry out into the next
	 word */
      u = (w >> (MP_WORD_BIT - 1)) & 1;

      /* Double what we've got, overflow will be ignored as defined
	 for C arithmetic (we've already noted if it is to occur)
       */
      w *= 2;

      /* Compute the additive step */
      v = *pt + k;

      /* If we do not already have an overflow carry, check to see
	 if the addition will cause one, and set the carry out if so 
       */
      u |= ((MP_WORD_MAX - v) < w);

      /* Add in the rest, again ignoring overflow */
      w += v;

      /* Set the i,j digit of the output */







|




















|







3543
3544
3545
3546
3547
3548
3549
3550
3551
3552
3553
3554
3555
3556
3557
3558
3559
3560
3561
3562
3563
3564
3565
3566
3567
3568
3569
3570
3571
3572
3573
3574
3575
3576
3577
3578
      This can overflow what can be represented in an mp_word, and
      since C arithmetic does not provide any way to check for
      overflow, we have to check explicitly for overflow conditions
      before they happen.
     */
    for(jx = ix + 1, pa2 = DIGITS(a) + jx; jx < used; ++jx, ++pa2) {
      mp_word  u = 0, v;

      /* Store this in a temporary to avoid indirections later */
      pt = pbt + ix + jx;

      /* Compute the multiplicative step */
      w = *pa1 * *pa2;

      /* If w is more than half MP_WORD_MAX, the doubling will
	 overflow, and we need to record a carry out into the next
	 word */
      u = (w >> (MP_WORD_BIT - 1)) & 1;

      /* Double what we've got, overflow will be ignored as defined
	 for C arithmetic (we've already noted if it is to occur)
       */
      w *= 2;

      /* Compute the additive step */
      v = *pt + k;

      /* If we do not already have an overflow carry, check to see
	 if the addition will cause one, and set the carry out if so
       */
      u |= ((MP_WORD_MAX - v) < w);

      /* Add in the rest, again ignoring overflow */
      w += v;

      /* Set the i,j digit of the output */
3586
3587
3588
3589
3590
3591
3592
3593
3594
3595
3596
3597
3598
3599
3600
    k = DIGIT(&tmp, ix + jx) + k;
    pbt[ix + jx] = ACCUM(k);
    k = CARRYOUT(k);

    /* If we are carrying out, propagate the carry to the next digit
       in the output.  This may cascade, so we have to be somewhat
       circumspect -- but we will have enough precision in the output
       that we won't overflow 
     */
    kx = 1;
    while(k) {
      k = pbt[ix + jx + kx] + 1;
      pbt[ix + jx + kx] = ACCUM(k);
      k = CARRYOUT(k);
      ++kx;







|







3588
3589
3590
3591
3592
3593
3594
3595
3596
3597
3598
3599
3600
3601
3602
    k = DIGIT(&tmp, ix + jx) + k;
    pbt[ix + jx] = ACCUM(k);
    k = CARRYOUT(k);

    /* If we are carrying out, propagate the carry to the next digit
       in the output.  This may cascade, so we have to be somewhat
       circumspect -- but we will have enough precision in the output
       that we won't overflow
     */
    kx = 1;
    while(k) {
      k = pbt[ix + jx + kx] + 1;
      pbt[ix + jx + kx] = ACCUM(k);
      k = CARRYOUT(k);
      ++kx;
3658
3659
3660
3661
3662
3663
3664
3665
3666
3667
3668
3669
3670
3671
3672
3673
3674
3675
3676
3677
3678
3679
3680
3681
3682
3683
3684
3685
3686
3687
3688
3689
3690
3691
3692
3693
3694
3695
3696
3697
3698
3699
3700
3701
3702
3703

  /* Perform the division itself...woo!   */
  ix = USED(a) - 1;

  while(ix >= 0) {
    /* Find a partial substring of a which is at least b */
    while(s_mp_cmp(&rem, b) < 0 && ix >= 0) {
      if((res = s_mp_lshd(&rem, 1)) != MP_OKAY) 
	goto CLEANUP;

      if((res = s_mp_lshd(&quot, 1)) != MP_OKAY)
	goto CLEANUP;

      DIGIT(&rem, 0) = DIGIT(a, ix);
      s_mp_clamp(&rem);
      --ix;
    }

    /* If we didn't find one, we're finished dividing    */
    if(s_mp_cmp(&rem, b) < 0) 
      break;    

    /* Compute a guess for the next quotient digit       */
    q = DIGIT(&rem, USED(&rem) - 1);
    if(q <= DIGIT(b, USED(b) - 1) && USED(&rem) > 1)
      q = (q << DIGIT_BIT) | DIGIT(&rem, USED(&rem) - 2);

    q /= DIGIT(b, USED(b) - 1);

    /* The guess can be as much as RADIX + 1 */
    if(q >= RADIX)
      q = RADIX - 1;

    /* See what that multiplies out to                   */
    mp_copy(b, &t);
    if((res = s_mp_mul_d(&t, q)) != MP_OKAY)
      goto CLEANUP;

    /* 
       If it's too big, back it off.  We should not have to do this
       more than once, or, in rare cases, twice.  Knuth describes a
       method by which this could be reduced to a maximum of once, but
       I didn't implement that here.
     */
    while(s_mp_cmp(&t, &rem) > 0) {
      --q;







|











|
|

















|







3660
3661
3662
3663
3664
3665
3666
3667
3668
3669
3670
3671
3672
3673
3674
3675
3676
3677
3678
3679
3680
3681
3682
3683
3684
3685
3686
3687
3688
3689
3690
3691
3692
3693
3694
3695
3696
3697
3698
3699
3700
3701
3702
3703
3704
3705

  /* Perform the division itself...woo!   */
  ix = USED(a) - 1;

  while(ix >= 0) {
    /* Find a partial substring of a which is at least b */
    while(s_mp_cmp(&rem, b) < 0 && ix >= 0) {
      if((res = s_mp_lshd(&rem, 1)) != MP_OKAY)
	goto CLEANUP;

      if((res = s_mp_lshd(&quot, 1)) != MP_OKAY)
	goto CLEANUP;

      DIGIT(&rem, 0) = DIGIT(a, ix);
      s_mp_clamp(&rem);
      --ix;
    }

    /* If we didn't find one, we're finished dividing    */
    if(s_mp_cmp(&rem, b) < 0)
      break;

    /* Compute a guess for the next quotient digit       */
    q = DIGIT(&rem, USED(&rem) - 1);
    if(q <= DIGIT(b, USED(b) - 1) && USED(&rem) > 1)
      q = (q << DIGIT_BIT) | DIGIT(&rem, USED(&rem) - 2);

    q /= DIGIT(b, USED(b) - 1);

    /* The guess can be as much as RADIX + 1 */
    if(q >= RADIX)
      q = RADIX - 1;

    /* See what that multiplies out to                   */
    mp_copy(b, &t);
    if((res = s_mp_mul_d(&t, q)) != MP_OKAY)
      goto CLEANUP;

    /*
       If it's too big, back it off.  We should not have to do this
       more than once, or, in rare cases, twice.  Knuth describes a
       method by which this could be reduced to a maximum of once, but
       I didn't implement that here.
     */
    while(s_mp_cmp(&t, &rem) > 0) {
      --q;
3713
3714
3715
3716
3717
3718
3719
3720
3721
3722
3723
3724
3725
3726
3727
3728
3729
3730
3731
3732
3733
3734
3735
      for any quotient we could ever possibly get, so we should not
      have to check for failures here
     */
    DIGIT(&quot, 0) = q;
  }

  /* Denormalize remainder                */
  if(d != 0) 
    s_mp_div_2d(&rem, d);

  s_mp_clamp(&quot);
  s_mp_clamp(&rem);

  /* Copy quotient back to output         */
  s_mp_exch(&quot, a);
  
  /* Copy remainder back to output        */
  s_mp_exch(&rem, b);

CLEANUP:
  mp_clear(&rem);
REM:
  mp_clear(&t);







|







|







3715
3716
3717
3718
3719
3720
3721
3722
3723
3724
3725
3726
3727
3728
3729
3730
3731
3732
3733
3734
3735
3736
3737
      for any quotient we could ever possibly get, so we should not
      have to check for failures here
     */
    DIGIT(&quot, 0) = q;
  }

  /* Denormalize remainder                */
  if(d != 0)
    s_mp_div_2d(&rem, d);

  s_mp_clamp(&quot);
  s_mp_clamp(&rem);

  /* Copy quotient back to output         */
  s_mp_exch(&quot, a);

  /* Copy remainder back to output        */
  s_mp_exch(&rem, b);

CLEANUP:
  mp_clear(&rem);
REM:
  mp_clear(&t);
3751
3752
3753
3754
3755
3756
3757
3758
3759
3760
3761
3762
3763
3764
3765

  dig = k / DIGIT_BIT;
  bit = k % DIGIT_BIT;

  mp_zero(a);
  if((res = s_mp_pad(a, dig + 1)) != MP_OKAY)
    return res;
  
  DIGIT(a, dig) |= (1 << bit);

  return MP_OKAY;

} /* end s_mp_2expt() */

/* }}} */







|







3753
3754
3755
3756
3757
3758
3759
3760
3761
3762
3763
3764
3765
3766
3767

  dig = k / DIGIT_BIT;
  bit = k % DIGIT_BIT;

  mp_zero(a);
  if((res = s_mp_pad(a, dig + 1)) != MP_OKAY)
    return res;

  DIGIT(a, dig) |= (1 << bit);

  return MP_OKAY;

} /* end s_mp_2expt() */

/* }}} */
3809
3810
3811
3812
3813
3814
3815
3816
3817
3818
3819
3820
3821
3822
3823
{
  mp_size  ua = USED(a);
  mp_digit *ap = DIGITS(a);

  if(ua > 1)
    return MP_GT;

  if(*ap < d) 
    return MP_LT;
  else if(*ap > d)
    return MP_GT;
  else
    return MP_EQ;

} /* end s_mp_cmp_d() */







|







3811
3812
3813
3814
3815
3816
3817
3818
3819
3820
3821
3822
3823
3824
3825
{
  mp_size  ua = USED(a);
  mp_digit *ap = DIGITS(a);

  if(ua > 1)
    return MP_GT;

  if(*ap < d)
    return MP_LT;
  else if(*ap > d)
    return MP_GT;
  else
    return MP_EQ;

} /* end s_mp_cmp_d() */
3851
3852
3853
3854
3855
3856
3857
3858
3859
3860
3861
3862
3863
3864
3865
      if(*dp)
	return -1; /* not a power of two */

      --dp; --ix;
    }

    return ((uv - 1) * DIGIT_BIT) + extra;
  } 

  return -1;

} /* end s_mp_ispow2() */

/* }}} */








|







3853
3854
3855
3856
3857
3858
3859
3860
3861
3862
3863
3864
3865
3866
3867
      if(*dp)
	return -1; /* not a power of two */

      --dp; --ix;
    }

    return ((uv - 1) * DIGIT_BIT) + extra;
  }

  return -1;

} /* end s_mp_ispow2() */

/* }}} */

3895
3896
3897
3898
3899
3900
3901
3902
3903
3904
3905
3906
3907
3908
3909
3910
3911
3912
3913
3914
3915
3916
3917
3918
3919
3920
3921
3922
3923
3924
3925

  The results will be odd if you use a radix < 2 or > 62, you are
  expected to know what you're up to.
 */
int      s_mp_tovalue(char ch, int r)
{
  int    val, xch;
  
  if(r > 36)
    xch = ch;
  else
    xch = toupper(ch);

  if(isdigit(xch))
    val = xch - '0';
  else if(isupper(xch))
    val = xch - 'A' + 10;
  else if(islower(xch))
    val = xch - 'a' + 36;
  else if(xch == '+')
    val = 62;
  else if(xch == '/')
    val = 63;
  else 
    return -1;

  if(val < 0 || val >= r)
    return -1;

  return val;








|















|







3897
3898
3899
3900
3901
3902
3903
3904
3905
3906
3907
3908
3909
3910
3911
3912
3913
3914
3915
3916
3917
3918
3919
3920
3921
3922
3923
3924
3925
3926
3927

  The results will be odd if you use a radix < 2 or > 62, you are
  expected to know what you're up to.
 */
int      s_mp_tovalue(char ch, int r)
{
  int    val, xch;

  if(r > 36)
    xch = ch;
  else
    xch = toupper(ch);

  if(isdigit(xch))
    val = xch - '0';
  else if(isupper(xch))
    val = xch - 'A' + 10;
  else if(islower(xch))
    val = xch - 'a' + 36;
  else if(xch == '+')
    val = 62;
  else if(xch == '/')
    val = 63;
  else
    return -1;

  if(val < 0 || val >= r)
    return -1;

  return val;

3933
3934
3935
3936
3937
3938
3939
3940
3941
3942
3943
3944
3945
3946
3947
3948
3949
3950
3951
3952
3953
3954
3955
3956
3957
3958
3959
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  Convert val to a radix-r digit, if possible.  If val is out of range
  for r, returns zero.  Otherwise, returns an ASCII character denoting
  the value in the given radix.

  The results may be odd if you use a radix < 2 or > 64, you are
  expected to know what you're doing.
 */
  
char     s_mp_todigit(int val, int r, int low)
{
  char   ch;

  if(val < 0 || val >= r)
    return 0;

  ch = s_dmap_1[val];

  if(r <= 36 && low)
    ch = tolower(ch);

  return ch;

} /* end s_mp_todigit() */

/* }}} */

/* {{{ s_mp_outlen(bits, radix) */

/* 
   Return an estimate for how long a string is needed to hold a radix
   r representation of a number with 'bits' significant bits.

   Does not include space for a sign or a NUL terminator.
 */
int      s_mp_outlen(int bits, int r)
{
  return (int)((double)bits * LOG_V_2(r));

} /* end s_mp_outlen() */

/* }}} */

/* }}} */

/*------------------------------------------------------------------------*/
/* HERE THERE BE DRAGONS                                                  */
/* crc==4242132123, version==2, Sat Feb 02 06:43:52 2002 */











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  Convert val to a radix-r digit, if possible.  If val is out of range
  for r, returns zero.  Otherwise, returns an ASCII character denoting
  the value in the given radix.

  The results may be odd if you use a radix < 2 or > 64, you are
  expected to know what you're doing.
 */

char     s_mp_todigit(int val, int r, int low)
{
  char   ch;

  if(val < 0 || val >= r)
    return 0;

  ch = s_dmap_1[val];

  if(r <= 36 && low)
    ch = tolower(ch);

  return ch;

} /* end s_mp_todigit() */

/* }}} */

/* {{{ s_mp_outlen(bits, radix) */

/*
   Return an estimate for how long a string is needed to hold a radix
   r representation of a number with 'bits' significant bits.

   Does not include space for a sign or a NUL terminator.
 */
int      s_mp_outlen(int bits, int r)
{
  return (int)((double)bits * LOG_V_2(r));

} /* end s_mp_outlen() */

/* }}} */

/* }}} */

/*------------------------------------------------------------------------*/
/* HERE THERE BE DRAGONS                                                  */
/* crc==4242132123, version==2, Sat Feb 02 06:43:52 2002 */

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/mtest/mpi.h.
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/*
    mpi.h

    by Michael J. Fromberger <[email protected]>
    Copyright (C) 1998 Michael J. Fromberger, All Rights Reserved

    Arbitrary precision integer arithmetic library


 */

#ifndef _H_MPI_
#define _H_MPI_

#include "mpi-config.h"








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/*
    mpi.h

    by Michael J. Fromberger <[email protected]>
    Copyright (C) 1998 Michael J. Fromberger, All Rights Reserved

    Arbitrary precision integer arithmetic library

    $Id$
 */

#ifndef _H_MPI_
#define _H_MPI_

#include "mpi-config.h"

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#define mp_mag_size(mp)           mp_unsigned_bin_size(mp)
#define mp_tomag(mp, str)         mp_to_unsigned_bin((mp), (str))
#endif

mp_err mp_read_radix(mp_int *mp, unsigned char *str, int radix);
int    mp_radix_size(mp_int *mp, int radix);
int    mp_value_radix_size(int num, int qty, int radix);
mp_err mp_toradix(mp_int *mp, unsigned char *str, int radix);

int    mp_char2value(char ch, int r);

#define mp_tobinary(M, S)  mp_toradix((M), (S), 2)
#define mp_tooctal(M, S)   mp_toradix((M), (S), 8)
#define mp_todecimal(M, S) mp_toradix((M), (S), 10)
#define mp_tohex(M, S)     mp_toradix((M), (S), 16)

/*------------------------------------------------------------------------*/
/* Error strings                                                          */

const  char  *mp_strerror(mp_err ec);

#endif /* end _H_MPI_ */











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#define mp_mag_size(mp)           mp_unsigned_bin_size(mp)
#define mp_tomag(mp, str)         mp_to_unsigned_bin((mp), (str))
#endif

mp_err mp_read_radix(mp_int *mp, unsigned char *str, int radix);
int    mp_radix_size(mp_int *mp, int radix);
int    mp_value_radix_size(int num, int qty, int radix);
mp_err mp_toradix(mp_int *mp, char *str, int radix);

int    mp_char2value(char ch, int r);

#define mp_tobinary(M, S)  mp_toradix((M), (S), 2)
#define mp_tooctal(M, S)   mp_toradix((M), (S), 8)
#define mp_todecimal(M, S) mp_toradix((M), (S), 10)
#define mp_tohex(M, S)     mp_toradix((M), (S), 16)

/*------------------------------------------------------------------------*/
/* Error strings                                                          */

const  char  *mp_strerror(mp_err ec);

#endif /* end _H_MPI_ */

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/mtest/mtest.c.
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#endif

#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#include "mpi.c"



FILE *rng;




void rand_num(mp_int *a)
{
   int n, size;
   unsigned char buf[2048];


   size = 1 + ((fgetc(rng)<<8) + fgetc(rng)) % 101;
   buf[0] = (fgetc(rng)&1)?1:0;

   fread(buf+1, 1, size, rng);










   while (buf[1] == 0) buf[1] = fgetc(rng);
   mp_read_raw(a, buf, 1+size);
}

void rand_num2(mp_int *a)
{
   int n, size;
   unsigned char buf[2048];


   size = 10 + ((fgetc(rng)<<8) + fgetc(rng)) % 101;
   buf[0] = (fgetc(rng)&1)?1:0;

   fread(buf+1, 1, size, rng);










   while (buf[1] == 0) buf[1] = fgetc(rng);
   mp_read_raw(a, buf, 1+size);
}

#define mp_to64(a, b) mp_toradix(a, b, 64)

int main(void)
{
   int n, tmp;

   mp_int a, b, c, d, e;

   clock_t t1;

   char buf[4096];

   mp_init(&a);
   mp_init(&b);
   mp_init(&c);
   mp_init(&d);
   mp_init(&e);


















   /* initial (2^n - 1)^2 testing, makes sure the comba multiplier works [it has the new carry code] */
/*
   mp_set(&a, 1);
   for (n = 1; n < 8192; n++) {
       mp_mul(&a, &a, &c);
       printf("mul\n");
       mp_to64(&a, buf);
       printf("%s\n%s\n", buf, buf);
       mp_to64(&c, buf);
       printf("%s\n", buf);

       mp_add_d(&a, 1, &a);
       mp_mul_2(&a, &a);
       mp_sub_d(&a, 1, &a);
   }
*/


   rng = fopen("/dev/urandom", "rb");
   if (rng == NULL) {
      rng = fopen("/dev/random", "rb");
      if (rng == NULL) {
         fprintf(stderr, "\nWarning:  stdin used as random source\n\n");
         rng = stdin;
      }
   }





   t1 = clock();

   for (;;) {
#if 0
      if (clock() - t1 > CLOCKS_PER_SEC) {
         sleep(2);
         t1 = clock();
      }
#endif
       n = fgetc(rng) % 15;







   if (n == 0) {
       /* add tests */
       rand_num(&a);
       rand_num(&b);
       mp_add(&a, &b, &c);
       printf("add\n");







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#endif

#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#include "mpi.c"

#ifdef LTM_MTEST_REAL_RAND
#define getRandChar() fgetc(rng)
FILE *rng;
#else
#define getRandChar() (rand()&0xFF)
#endif

void rand_num(mp_int *a)
{
   int size;
   unsigned char buf[2048];
   size_t sz;

   size = 1 + ((getRandChar()<<8) + getRandChar()) % 101;
   buf[0] = (getRandChar()&1)?1:0;
#ifdef LTM_MTEST_REAL_RAND
   sz = fread(buf+1, 1, size, rng);
#else
   sz = 1;
   while (sz < (unsigned)size) {
       buf[sz] = getRandChar();
       ++sz;
   }
#endif
   if (sz != (unsigned)size) {
       fprintf(stderr, "\nWarning: fread failed\n\n");
   }
   while (buf[1] == 0) buf[1] = getRandChar();
   mp_read_raw(a, buf, 1+size);
}

void rand_num2(mp_int *a)
{
   int size;
   unsigned char buf[2048];
   size_t sz;

   size = 10 + ((getRandChar()<<8) + getRandChar()) % 101;
   buf[0] = (getRandChar()&1)?1:0;
#ifdef LTM_MTEST_REAL_RAND
   sz = fread(buf+1, 1, size, rng);
#else
   sz = 1;
   while (sz < (unsigned)size) {
       buf[sz] = getRandChar();
       ++sz;
   }
#endif
   if (sz != (unsigned)size) {
       fprintf(stderr, "\nWarning: fread failed\n\n");
   }
   while (buf[1] == 0) buf[1] = getRandChar();
   mp_read_raw(a, buf, 1+size);
}

#define mp_to64(a, b) mp_toradix(a, b, 64)

int main(int argc, char *argv[])
{
   int n, tmp;
   long long max;
   mp_int a, b, c, d, e;
#ifdef MTEST_NO_FULLSPEED
   clock_t t1;
#endif
   char buf[4096];

   mp_init(&a);
   mp_init(&b);
   mp_init(&c);
   mp_init(&d);
   mp_init(&e);

   if (argc > 1) {
       max = strtol(argv[1], NULL, 0);
       if (max < 0) {
           if (max > -64) {
               max = (1 << -(max)) + 1;
           } else {
               max = 1;
           }
       } else if (max == 0) {
           max = 1;
       }
   }
   else {
       max = 0;
   }


   /* initial (2^n - 1)^2 testing, makes sure the comba multiplier works [it has the new carry code] */
/*
   mp_set(&a, 1);
   for (n = 1; n < 8192; n++) {
       mp_mul(&a, &a, &c);
       printf("mul\n");
       mp_to64(&a, buf);
       printf("%s\n%s\n", buf, buf);
       mp_to64(&c, buf);
       printf("%s\n", buf);

       mp_add_d(&a, 1, &a);
       mp_mul_2(&a, &a);
       mp_sub_d(&a, 1, &a);
   }
*/

#ifdef LTM_MTEST_REAL_RAND
   rng = fopen("/dev/urandom", "rb");
   if (rng == NULL) {
      rng = fopen("/dev/random", "rb");
      if (rng == NULL) {
         fprintf(stderr, "\nWarning:  stdin used as random source\n\n");
         rng = stdin;
      }
   }
#else
   srand(23);
#endif

#ifdef MTEST_NO_FULLSPEED
   t1 = clock();
#endif
   for (;;) {
#ifdef MTEST_NO_FULLSPEED
      if (clock() - t1 > CLOCKS_PER_SEC) {
         sleep(2);
         t1 = clock();
      }
#endif
       n = getRandChar() % 15;

       if (max != 0) {
           --max;
           if (max == 0)
             n = 255;
       }

   if (n == 0) {
       /* add tests */
       rand_num(&a);
       rand_num(&b);
       mp_add(&a, &b, &c);
       printf("add\n");
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       printf("%s\n", buf);
       mp_to64(&b, buf);
       printf("%s\n", buf);
   } else if (n == 5) {
      /* mul_2d test */
      rand_num(&a);
      mp_copy(&a, &b);
      n = fgetc(rng) & 63;
      mp_mul_2d(&b, n, &b);
      mp_to64(&a, buf);
      printf("mul2d\n");
      printf("%s\n", buf);
      printf("%d\n", n);
      mp_to64(&b, buf);
      printf("%s\n", buf);
   } else if (n == 6) {
      /* div_2d test */
      rand_num(&a);
      mp_copy(&a, &b);
      n = fgetc(rng) & 63;
      mp_div_2d(&b, n, &b, NULL);
      mp_to64(&a, buf);
      printf("div2d\n");
      printf("%s\n", buf);
      printf("%d\n", n);
      mp_to64(&b, buf);
      printf("%s\n", buf);







|











|







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       printf("%s\n", buf);
       mp_to64(&b, buf);
       printf("%s\n", buf);
   } else if (n == 5) {
      /* mul_2d test */
      rand_num(&a);
      mp_copy(&a, &b);
      n = getRandChar() & 63;
      mp_mul_2d(&b, n, &b);
      mp_to64(&a, buf);
      printf("mul2d\n");
      printf("%s\n", buf);
      printf("%d\n", n);
      mp_to64(&b, buf);
      printf("%s\n", buf);
   } else if (n == 6) {
      /* div_2d test */
      rand_num(&a);
      mp_copy(&a, &b);
      n = getRandChar() & 63;
      mp_div_2d(&b, n, &b, NULL);
      mp_to64(&a, buf);
      printf("div2d\n");
      printf("%s\n", buf);
      printf("%d\n", n);
      mp_to64(&b, buf);
      printf("%s\n", buf);
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297
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299



300
301


302

303
304




      tmp = abs(rand()) & THE_MASK;
      mp_sub_d(&a, tmp, &b);
      printf("sub_d\n");
      mp_to64(&a, buf);
      printf("%s\n%d\n", buf, tmp);
      mp_to64(&b, buf);
      printf("%s\n", buf);



   }
   }


   fclose(rng);

   return 0;
}











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>


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      tmp = abs(rand()) & THE_MASK;
      mp_sub_d(&a, tmp, &b);
      printf("sub_d\n");
      mp_to64(&a, buf);
      printf("%s\n%d\n", buf, tmp);
      mp_to64(&b, buf);
      printf("%s\n", buf);
   } else if (n == 255) {
      printf("exit\n");
      break;
   }

   }
#ifdef LTM_MTEST_REAL_RAND
   fclose(rng);
#endif
   return 0;
}

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/pre_gen/mpi.c.
38
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   }

   /* generic reply for invalid code */
   return "Invalid error code";
}

#endif





/* End: bn_error.c */

/* Start: bn_fast_mp_invmod.c */
#include <tommath.h>
#ifdef BN_FAST_MP_INVMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
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   }

   /* generic reply for invalid code */
   return "Invalid error code";
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_error.c */

/* Start: bn_fast_mp_invmod.c */
#include <tommath.h>
#ifdef BN_FAST_MP_INVMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
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 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* computes the modular inverse via binary extended euclidean algorithm, 
 * that is c = 1/a mod b 
 *
 * Based on slow invmod except this is optimized for the case where b is 
 * odd as per HAC Note 14.64 on pp. 610
 */
int fast_mp_invmod (mp_int * a, mp_int * b, mp_int * c)
{
  mp_int  x, y, u, v, B, D;
  int     res, neg;








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 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* computes the modular inverse via binary extended euclidean algorithm,
 * that is c = 1/a mod b
 *
 * Based on slow invmod except this is optimized for the case where b is
 * odd as per HAC Note 14.64 on pp. 610
 */
int fast_mp_invmod (mp_int * a, mp_int * b, mp_int * c)
{
  mp_int  x, y, u, v, B, D;
  int     res, neg;

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  c->sign = neg;
  res = MP_OKAY;

LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &B, &D, NULL);
  return res;
}
#endif





/* End: bn_fast_mp_invmod.c */

/* Start: bn_fast_mp_montgomery_reduce.c */
#include <tommath.h>
#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







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  c->sign = neg;
  res = MP_OKAY;

LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &B, &D, NULL);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_fast_mp_invmod.c */

/* Start: bn_fast_mp_montgomery_reduce.c */
#include <tommath.h>
#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
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  /* if A >= m then A = A - m */
  if (mp_cmp_mag (x, n) != MP_LT) {
    return s_mp_sub (x, n, x);
  }
  return MP_OKAY;
}
#endif





/* End: bn_fast_mp_montgomery_reduce.c */

/* Start: bn_fast_s_mp_mul_digs.c */
#include <tommath.h>
#ifdef BN_FAST_S_MP_MUL_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







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375
376
377
378
379
380
381
382
383
  /* if A >= m then A = A - m */
  if (mp_cmp_mag (x, n) != MP_LT) {
    return s_mp_sub (x, n, x);
  }
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_fast_mp_montgomery_reduce.c */

/* Start: bn_fast_s_mp_mul_digs.c */
#include <tommath.h>
#ifdef BN_FAST_S_MP_MUL_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* Fast (comba) multiplier
 *
 * This is the fast column-array [comba] multiplier.  It is 
 * designed to compute the columns of the product first 
 * then handle the carries afterwards.  This has the effect 
 * of making the nested loops that compute the columns very
 * simple and schedulable on super-scalar processors.
 *
 * This has been modified to produce a variable number of 
 * digits of output so if say only a half-product is required 
 * you don't have to compute the upper half (a feature 
 * required for fast Barrett reduction).
 *
 * Based on Algorithm 14.12 on pp.595 of HAC.
 *
 */
int fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
{







|
|
|



|
|
|







393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* Fast (comba) multiplier
 *
 * This is the fast column-array [comba] multiplier.  It is
 * designed to compute the columns of the product first
 * then handle the carries afterwards.  This has the effect
 * of making the nested loops that compute the columns very
 * simple and schedulable on super-scalar processors.
 *
 * This has been modified to produce a variable number of
 * digits of output so if say only a half-product is required
 * you don't have to compute the upper half (a feature
 * required for fast Barrett reduction).
 *
 * Based on Algorithm 14.12 on pp.595 of HAC.
 *
 */
int fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
{
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
  }

  /* number of output digits to produce */
  pa = MIN(digs, a->used + b->used);

  /* clear the carry */
  _W = 0;
  for (ix = 0; ix < pa; ix++) { 
      int      tx, ty;
      int      iy;
      mp_digit *tmpx, *tmpy;

      /* get offsets into the two bignums */
      ty = MIN(b->used-1, ix);
      tx = ix - ty;

      /* setup temp aliases */
      tmpx = a->dp + tx;
      tmpy = b->dp + ty;

      /* this is the number of times the loop will iterrate, essentially 
         while (tx++ < a->used && ty-- >= 0) { ... }
       */
      iy = MIN(a->used-tx, ty+1);

      /* execute loop */
      for (iz = 0; iz < iy; ++iz) {
         _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--);







|












|







425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
  }

  /* number of output digits to produce */
  pa = MIN(digs, a->used + b->used);

  /* clear the carry */
  _W = 0;
  for (ix = 0; ix < pa; ix++) {
      int      tx, ty;
      int      iy;
      mp_digit *tmpx, *tmpy;

      /* get offsets into the two bignums */
      ty = MIN(b->used-1, ix);
      tx = ix - ty;

      /* setup temp aliases */
      tmpx = a->dp + tx;
      tmpy = b->dp + ty;

      /* this is the number of times the loop will iterrate, essentially
         while (tx++ < a->used && ty-- >= 0) { ... }
       */
      iy = MIN(a->used-tx, ty+1);

      /* execute loop */
      for (iz = 0; iz < iy; ++iz) {
         _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--);
465
466
467
468
469
470
471




472
473
474
475
476
477
478
      *tmpc++ = 0;
    }
  }
  mp_clamp (c);
  return MP_OKAY;
}
#endif





/* End: bn_fast_s_mp_mul_digs.c */

/* Start: bn_fast_s_mp_mul_high_digs.c */
#include <tommath.h>
#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
      *tmpc++ = 0;
    }
  }
  mp_clamp (c);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_fast_s_mp_mul_digs.c */

/* Start: bn_fast_s_mp_mul_high_digs.c */
#include <tommath.h>
#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
      return res;
    }
  }

  /* number of output digits to produce */
  pa = a->used + b->used;
  _W = 0;
  for (ix = digs; ix < pa; ix++) { 
      int      tx, ty, iy;
      mp_digit *tmpx, *tmpy;

      /* get offsets into the two bignums */
      ty = MIN(b->used-1, ix);
      tx = ix - ty;

      /* setup temp aliases */
      tmpx = a->dp + tx;
      tmpy = b->dp + ty;

      /* this is the number of times the loop will iterrate, essentially its 
         while (tx++ < a->used && ty-- >= 0) { ... }
       */
      iy = MIN(a->used-tx, ty+1);

      /* execute loop */
      for (iz = 0; iz < iy; iz++) {
         _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--);
      }

      /* store term */
      W[ix] = ((mp_digit)_W) & MP_MASK;

      /* make next carry */
      _W = _W >> ((mp_word)DIGIT_BIT);
  }
  
  /* setup dest */
  olduse  = c->used;
  c->used = pa;

  {
    register mp_digit *tmpc;








|











|















|







528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
      return res;
    }
  }

  /* number of output digits to produce */
  pa = a->used + b->used;
  _W = 0;
  for (ix = digs; ix < pa; ix++) {
      int      tx, ty, iy;
      mp_digit *tmpx, *tmpy;

      /* get offsets into the two bignums */
      ty = MIN(b->used-1, ix);
      tx = ix - ty;

      /* setup temp aliases */
      tmpx = a->dp + tx;
      tmpy = b->dp + ty;

      /* this is the number of times the loop will iterrate, essentially its
         while (tx++ < a->used && ty-- >= 0) { ... }
       */
      iy = MIN(a->used-tx, ty+1);

      /* execute loop */
      for (iz = 0; iz < iy; iz++) {
         _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--);
      }

      /* store term */
      W[ix] = ((mp_digit)_W) & MP_MASK;

      /* make next carry */
      _W = _W >> ((mp_word)DIGIT_BIT);
  }

  /* setup dest */
  olduse  = c->used;
  c->used = pa;

  {
    register mp_digit *tmpc;

563
564
565
566
567
568
569




570
571
572
573
574
575
576
      *tmpc++ = 0;
    }
  }
  mp_clamp (c);
  return MP_OKAY;
}
#endif





/* End: bn_fast_s_mp_mul_high_digs.c */

/* Start: bn_fast_s_mp_sqr.c */
#include <tommath.h>
#ifdef BN_FAST_S_MP_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
      *tmpc++ = 0;
    }
  }
  mp_clamp (c);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_fast_s_mp_mul_high_digs.c */

/* Start: bn_fast_s_mp_sqr.c */
#include <tommath.h>
#ifdef BN_FAST_S_MP_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* the jist of squaring...
 * you do like mult except the offset of the tmpx [one that 
 * starts closer to zero] can't equal the offset of tmpy.  
 * So basically you set up iy like before then you min it with
 * (ty-tx) so that it never happens.  You double all those 
 * you add in the inner loop

After that loop you do the squares and add them in.
*/

int fast_s_mp_sqr (mp_int * a, mp_int * b)
{







|
|

|







605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* the jist of squaring...
 * you do like mult except the offset of the tmpx [one that
 * starts closer to zero] can't equal the offset of tmpy.
 * So basically you set up iy like before then you min it with
 * (ty-tx) so that it never happens.  You double all those
 * you add in the inner loop

After that loop you do the squares and add them in.
*/

int fast_s_mp_sqr (mp_int * a, mp_int * b)
{
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
    if ((res = mp_grow (b, pa)) != MP_OKAY) {
      return res;
    }
  }

  /* number of output digits to produce */
  W1 = 0;
  for (ix = 0; ix < pa; ix++) { 
      int      tx, ty, iy;
      mp_word  _W;
      mp_digit *tmpy;

      /* clear counter */
      _W = 0;

      /* get offsets into the two bignums */
      ty = MIN(a->used-1, ix);
      tx = ix - ty;

      /* setup temp aliases */
      tmpx = a->dp + tx;
      tmpy = a->dp + ty;

      /* this is the number of times the loop will iterrate, essentially
         while (tx++ < a->used && ty-- >= 0) { ... }
       */
      iy = MIN(a->used-tx, ty+1);

      /* now for squaring tx can never equal ty 
       * we halve the distance since they approach at a rate of 2x
       * and we have to round because odd cases need to be executed
       */
      iy = MIN(iy, (ty-tx+1)>>1);

      /* execute loop */
      for (iz = 0; iz < iy; iz++) {







|




















|







630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
    if ((res = mp_grow (b, pa)) != MP_OKAY) {
      return res;
    }
  }

  /* number of output digits to produce */
  W1 = 0;
  for (ix = 0; ix < pa; ix++) {
      int      tx, ty, iy;
      mp_word  _W;
      mp_digit *tmpy;

      /* clear counter */
      _W = 0;

      /* get offsets into the two bignums */
      ty = MIN(a->used-1, ix);
      tx = ix - ty;

      /* setup temp aliases */
      tmpx = a->dp + tx;
      tmpy = a->dp + ty;

      /* this is the number of times the loop will iterrate, essentially
         while (tx++ < a->used && ty-- >= 0) { ... }
       */
      iy = MIN(a->used-tx, ty+1);

      /* now for squaring tx can never equal ty
       * we halve the distance since they approach at a rate of 2x
       * and we have to round because odd cases need to be executed
       */
      iy = MIN(iy, (ty-tx+1)>>1);

      /* execute loop */
      for (iz = 0; iz < iy; iz++) {
677
678
679
680
681
682
683




684
685
686
687
688
689
690
      *tmpb++ = 0;
    }
  }
  mp_clamp (b);
  return MP_OKAY;
}
#endif





/* End: bn_fast_s_mp_sqr.c */

/* Start: bn_mp_2expt.c */
#include <tommath.h>
#ifdef BN_MP_2EXPT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
      *tmpb++ = 0;
    }
  }
  mp_clamp (b);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_fast_s_mp_sqr.c */

/* Start: bn_mp_2expt.c */
#include <tommath.h>
#ifdef BN_MP_2EXPT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* computes a = 2**b 
 *
 * Simple algorithm which zeroes the int, grows it then just sets one bit
 * as required.
 */
int
mp_2expt (mp_int * a, int b)
{







|







722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* computes a = 2**b
 *
 * Simple algorithm which zeroes the int, grows it then just sets one bit
 * as required.
 */
int
mp_2expt (mp_int * a, int b)
{
725
726
727
728
729
730
731




732
733
734
735
736
737
738

  /* put the single bit in its place */
  a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT);

  return MP_OKAY;
}
#endif





/* End: bn_mp_2expt.c */

/* Start: bn_mp_abs.c */
#include <tommath.h>
#ifdef BN_MP_ABS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766

  /* put the single bit in its place */
  a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT);

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_2expt.c */

/* Start: bn_mp_abs.c */
#include <tommath.h>
#ifdef BN_MP_ABS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* b = |a| 
 *
 * Simple function copies the input and fixes the sign to positive
 */
int
mp_abs (mp_int * a, mp_int * b)
{
  int     res;







|







774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* b = |a|
 *
 * Simple function copies the input and fixes the sign to positive
 */
int
mp_abs (mp_int * a, mp_int * b)
{
  int     res;
768
769
770
771
772
773
774




775
776
777
778
779
780
781

  /* force the sign of b to positive */
  b->sign = MP_ZPOS;

  return MP_OKAY;
}
#endif





/* End: bn_mp_abs.c */

/* Start: bn_mp_add.c */
#include <tommath.h>
#ifdef BN_MP_ADD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813

  /* force the sign of b to positive */
  b->sign = MP_ZPOS;

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_abs.c */

/* Start: bn_mp_add.c */
#include <tommath.h>
#ifdef BN_MP_ADD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
821
822
823
824
825
826
827




828
829
830
831
832
833
834
      res = s_mp_sub (a, b, c);
    }
  }
  return res;
}

#endif





/* End: bn_mp_add.c */

/* Start: bn_mp_add_d.c */
#include <tommath.h>
#ifdef BN_MP_ADD_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
      res = s_mp_sub (a, b, c);
    }
  }
  return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_add.c */

/* Start: bn_mp_add_d.c */
#include <tommath.h>
#ifdef BN_MP_ADD_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
933
934
935
936
937
938
939




940
941
942
943
944
945
946
  }
  mp_clamp(c);

  return MP_OKAY;
}

#endif





/* End: bn_mp_add_d.c */

/* Start: bn_mp_addmod.c */
#include <tommath.h>
#ifdef BN_MP_ADDMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
  }
  mp_clamp(c);

  return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_add_d.c */

/* Start: bn_mp_addmod.c */
#include <tommath.h>
#ifdef BN_MP_ADDMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
974
975
976
977
978
979
980




981
982
983
984
985
986
987
    return res;
  }
  res = mp_mod (&t, c, d);
  mp_clear (&t);
  return res;
}
#endif





/* End: bn_mp_addmod.c */

/* Start: bn_mp_and.c */
#include <tommath.h>
#ifdef BN_MP_AND_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
    return res;
  }
  res = mp_mod (&t, c, d);
  mp_clear (&t);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_addmod.c */

/* Start: bn_mp_and.c */
#include <tommath.h>
#ifdef BN_MP_AND_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
1031
1032
1033
1034
1035
1036
1037




1038
1039
1040
1041
1042
1043
1044

  mp_clamp (&t);
  mp_exch (c, &t);
  mp_clear (&t);
  return MP_OKAY;
}
#endif





/* End: bn_mp_and.c */

/* Start: bn_mp_clamp.c */
#include <tommath.h>
#ifdef BN_MP_CLAMP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092

  mp_clamp (&t);
  mp_exch (c, &t);
  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_and.c */

/* Start: bn_mp_clamp.c */
#include <tommath.h>
#ifdef BN_MP_CLAMP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* trim unused digits 
 *
 * This is used to ensure that leading zero digits are
 * trimed and the leading "used" digit will be non-zero
 * Typically very fast.  Also fixes the sign if there
 * are no more leading digits
 */
void







|







1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* trim unused digits
 *
 * This is used to ensure that leading zero digits are
 * trimed and the leading "used" digit will be non-zero
 * Typically very fast.  Also fixes the sign if there
 * are no more leading digits
 */
void
1075
1076
1077
1078
1079
1080
1081




1082
1083
1084
1085
1086
1087
1088

  /* reset the sign flag if used == 0 */
  if (a->used == 0) {
    a->sign = MP_ZPOS;
  }
}
#endif





/* End: bn_mp_clamp.c */

/* Start: bn_mp_clear.c */
#include <tommath.h>
#ifdef BN_MP_CLEAR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140

  /* reset the sign flag if used == 0 */
  if (a->used == 0) {
    a->sign = MP_ZPOS;
  }
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_clamp.c */

/* Start: bn_mp_clear.c */
#include <tommath.h>
#ifdef BN_MP_CLEAR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
1119
1120
1121
1122
1123
1124
1125




1126
1127
1128
1129
1130
1131
1132
    /* reset members to make debugging easier */
    a->dp    = NULL;
    a->alloc = a->used = 0;
    a->sign  = MP_ZPOS;
  }
}
#endif





/* End: bn_mp_clear.c */

/* Start: bn_mp_clear_multi.c */
#include <tommath.h>
#ifdef BN_MP_CLEAR_MULTI_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
    /* reset members to make debugging easier */
    a->dp    = NULL;
    a->alloc = a->used = 0;
    a->sign  = MP_ZPOS;
  }
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_clear.c */

/* Start: bn_mp_clear_multi.c */
#include <tommath.h>
#ifdef BN_MP_CLEAR_MULTI_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159




1160
1161
1162
1163
1164
1165
1166
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */
#include <stdarg.h>

void mp_clear_multi(mp_int *mp, ...) 
{
    mp_int* next_mp = mp;
    va_list args;
    va_start(args, mp);
    while (next_mp != NULL) {
        mp_clear(next_mp);
        next_mp = va_arg(args, mp_int*);
    }
    va_end(args);
}
#endif





/* End: bn_mp_clear_multi.c */

/* Start: bn_mp_cmp.c */
#include <tommath.h>
#ifdef BN_MP_CMP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|











>
>
>
>







1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */
#include <stdarg.h>

void mp_clear_multi(mp_int *mp, ...)
{
    mp_int* next_mp = mp;
    va_list args;
    va_start(args, mp);
    while (next_mp != NULL) {
        mp_clear(next_mp);
        next_mp = va_arg(args, mp_int*);
    }
    va_end(args);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_clear_multi.c */

/* Start: bn_mp_cmp.c */
#include <tommath.h>
#ifdef BN_MP_CMP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202




1203
1204
1205
1206
1207
1208
1209
  if (a->sign != b->sign) {
     if (a->sign == MP_NEG) {
        return MP_LT;
     } else {
        return MP_GT;
     }
  }
  
  /* compare digits */
  if (a->sign == MP_NEG) {
     /* if negative compare opposite direction */
     return mp_cmp_mag(b, a);
  } else {
     return mp_cmp_mag(a, b);
  }
}
#endif





/* End: bn_mp_cmp.c */

/* Start: bn_mp_cmp_d.c */
#include <tommath.h>
#ifdef BN_MP_CMP_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|









>
>
>
>







1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
  if (a->sign != b->sign) {
     if (a->sign == MP_NEG) {
        return MP_LT;
     } else {
        return MP_GT;
     }
  }

  /* compare digits */
  if (a->sign == MP_NEG) {
     /* if negative compare opposite direction */
     return mp_cmp_mag(b, a);
  } else {
     return mp_cmp_mag(a, b);
  }
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_cmp.c */

/* Start: bn_mp_cmp_d.c */
#include <tommath.h>
#ifdef BN_MP_CMP_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
1240
1241
1242
1243
1244
1245
1246




1247
1248
1249
1250
1251
1252
1253
  } else if (a->dp[0] < b) {
    return MP_LT;
  } else {
    return MP_EQ;
  }
}
#endif





/* End: bn_mp_cmp_d.c */

/* Start: bn_mp_cmp_mag.c */
#include <tommath.h>
#ifdef BN_MP_CMP_MAG_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
  } else if (a->dp[0] < b) {
    return MP_LT;
  } else {
    return MP_EQ;
  }
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_cmp_d.c */

/* Start: bn_mp_cmp_mag.c */
#include <tommath.h>
#ifdef BN_MP_CMP_MAG_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
  int     n;
  mp_digit *tmpa, *tmpb;

  /* compare based on # of non-zero digits */
  if (a->used > b->used) {
    return MP_GT;
  }
  
  if (a->used < b->used) {
    return MP_LT;
  }

  /* alias for a */
  tmpa = a->dp + (a->used - 1);








|







1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
  int     n;
  mp_digit *tmpa, *tmpb;

  /* compare based on # of non-zero digits */
  if (a->used > b->used) {
    return MP_GT;
  }

  if (a->used < b->used) {
    return MP_LT;
  }

  /* alias for a */
  tmpa = a->dp + (a->used - 1);

1295
1296
1297
1298
1299
1300
1301




1302
1303
1304
1305
1306
1307
1308
    if (*tmpa < *tmpb) {
      return MP_LT;
    }
  }
  return MP_EQ;
}
#endif





/* End: bn_mp_cmp_mag.c */

/* Start: bn_mp_cnt_lsb.c */
#include <tommath.h>
#ifdef BN_MP_CNT_LSB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
    if (*tmpa < *tmpb) {
      return MP_LT;
    }
  }
  return MP_EQ;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_cmp_mag.c */

/* Start: bn_mp_cnt_lsb.c */
#include <tommath.h>
#ifdef BN_MP_CNT_LSB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

static const int lnz[16] = { 
   4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0
};

/* Counts the number of lsbs which are zero before the first zero bit */
int mp_cnt_lsb(mp_int *a)
{
   int x;







|







1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

static const int lnz[16] = {
   4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0
};

/* Counts the number of lsbs which are zero before the first zero bit */
int mp_cnt_lsb(mp_int *a)
{
   int x;
1348
1349
1350
1351
1352
1353
1354




1355
1356
1357
1358
1359
1360
1361
         q >>= 4;
      } while (qq == 0);
   }
   return x;
}

#endif





/* End: bn_mp_cnt_lsb.c */

/* Start: bn_mp_copy.c */
#include <tommath.h>
#ifdef BN_MP_COPY_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
         q >>= 4;
      } while (qq == 0);
   }
   return x;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_cnt_lsb.c */

/* Start: bn_mp_copy.c */
#include <tommath.h>
#ifdef BN_MP_COPY_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
1416
1417
1418
1419
1420
1421
1422




1423
1424
1425
1426
1427
1428
1429

  /* copy used count and sign */
  b->used = a->used;
  b->sign = a->sign;
  return MP_OKAY;
}
#endif





/* End: bn_mp_copy.c */

/* Start: bn_mp_count_bits.c */
#include <tommath.h>
#ifdef BN_MP_COUNT_BITS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509

  /* copy used count and sign */
  b->used = a->used;
  b->sign = a->sign;
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_copy.c */

/* Start: bn_mp_count_bits.c */
#include <tommath.h>
#ifdef BN_MP_COUNT_BITS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467




1468
1469
1470
1471
1472
1473
1474
  /* shortcut */
  if (a->used == 0) {
    return 0;
  }

  /* get number of digits and add that */
  r = (a->used - 1) * DIGIT_BIT;
  
  /* take the last digit and count the bits in it */
  q = a->dp[a->used - 1];
  while (q > ((mp_digit) 0)) {
    ++r;
    q >>= ((mp_digit) 1);
  }
  return r;
}
#endif





/* End: bn_mp_count_bits.c */

/* Start: bn_mp_div.c */
#include <tommath.h>
#ifdef BN_MP_DIV_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|









>
>
>
>







1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
  /* shortcut */
  if (a->used == 0) {
    return 0;
  }

  /* get number of digits and add that */
  r = (a->used - 1) * DIGIT_BIT;

  /* take the last digit and count the bits in it */
  q = a->dp[a->used - 1];
  while (q > ((mp_digit) 0)) {
    ++r;
    q >>= ((mp_digit) 1);
  }
  return r;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_count_bits.c */

/* Start: bn_mp_div.c */
#include <tommath.h>
#ifdef BN_MP_DIV_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
     return res;
  }


  mp_set(&tq, 1);
  n = mp_count_bits(a) - mp_count_bits(b);
  if (((res = mp_abs(a, &ta)) != MP_OKAY) ||
      ((res = mp_abs(b, &tb)) != MP_OKAY) || 
      ((res = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) ||
      ((res = mp_mul_2d(&tq, n, &tq)) != MP_OKAY)) {
      goto LBL_ERR;
  }

  while (n-- >= 0) {
     if (mp_cmp(&tb, &ta) != MP_GT) {







|







1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
     return res;
  }


  mp_set(&tq, 1);
  n = mp_count_bits(a) - mp_count_bits(b);
  if (((res = mp_abs(a, &ta)) != MP_OKAY) ||
      ((res = mp_abs(b, &tb)) != MP_OKAY) ||
      ((res = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) ||
      ((res = mp_mul_2d(&tq, n, &tq)) != MP_OKAY)) {
      goto LBL_ERR;
  }

  while (n-- >= 0) {
     if (mp_cmp(&tb, &ta) != MP_GT) {
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
LBL_ERR:
   mp_clear_multi(&ta, &tb, &tq, &q, NULL);
   return res;
}

#else

/* integer signed division. 
 * c*b + d == a [e.g. a/b, c=quotient, d=remainder]
 * HAC pp.598 Algorithm 14.20
 *
 * Note that the description in HAC is horribly 
 * incomplete.  For example, it doesn't consider 
 * the case where digits are removed from 'x' in 
 * the inner loop.  It also doesn't consider the 
 * case that y has fewer than three digits, etc..
 *
 * The overall algorithm is as described as 
 * 14.20 from HAC but fixed to treat these cases.
*/
int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
{
  mp_int  q, x, y, t1, t2;
  int     res, n, t, i, norm, neg;








|



|
|
|
|


|







1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
LBL_ERR:
   mp_clear_multi(&ta, &tb, &tq, &q, NULL);
   return res;
}

#else

/* integer signed division.
 * c*b + d == a [e.g. a/b, c=quotient, d=remainder]
 * HAC pp.598 Algorithm 14.20
 *
 * Note that the description in HAC is horribly
 * incomplete.  For example, it doesn't consider
 * the case where digits are removed from 'x' in
 * the inner loop.  It also doesn't consider the
 * case that y has fewer than three digits, etc..
 *
 * The overall algorithm is as described as
 * 14.20 from HAC but fixed to treat these cases.
*/
int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
{
  mp_int  q, x, y, t1, t2;
  int     res, n, t, i, norm, neg;

1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685

  /* step 3. for i from n down to (t + 1) */
  for (i = n; i >= (t + 1); i--) {
    if (i > x.used) {
      continue;
    }

    /* step 3.1 if xi == yt then set q{i-t-1} to b-1, 
     * otherwise set q{i-t-1} to (xi*b + x{i-1})/yt */
    if (x.dp[i] == y.dp[t]) {
      q.dp[i - t - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1);
    } else {
      mp_word tmp;
      tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT);
      tmp |= ((mp_word) x.dp[i - 1]);
      tmp /= ((mp_word) y.dp[t]);
      if (tmp > (mp_word) MP_MASK)
        tmp = MP_MASK;
      q.dp[i - t - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK));
    }

    /* while (q{i-t-1} * (yt * b + y{t-1})) > 
             xi * b**2 + xi-1 * b + xi-2 
     
       do q{i-t-1} -= 1; 
    */
    q.dp[i - t - 1] = (q.dp[i - t - 1] + 1) & MP_MASK;
    do {
      q.dp[i - t - 1] = (q.dp[i - t - 1] - 1) & MP_MASK;

      /* find left hand */
      mp_zero (&t1);







|













|
|
|
|







1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769

  /* step 3. for i from n down to (t + 1) */
  for (i = n; i >= (t + 1); i--) {
    if (i > x.used) {
      continue;
    }

    /* step 3.1 if xi == yt then set q{i-t-1} to b-1,
     * otherwise set q{i-t-1} to (xi*b + x{i-1})/yt */
    if (x.dp[i] == y.dp[t]) {
      q.dp[i - t - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1);
    } else {
      mp_word tmp;
      tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT);
      tmp |= ((mp_word) x.dp[i - 1]);
      tmp /= ((mp_word) y.dp[t]);
      if (tmp > (mp_word) MP_MASK)
        tmp = MP_MASK;
      q.dp[i - t - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK));
    }

    /* while (q{i-t-1} * (yt * b + y{t-1})) >
             xi * b**2 + xi-1 * b + xi-2

       do q{i-t-1} -= 1;
    */
    q.dp[i - t - 1] = (q.dp[i - t - 1] + 1) & MP_MASK;
    do {
      q.dp[i - t - 1] = (q.dp[i - t - 1] - 1) & MP_MASK;

      /* find left hand */
      mp_zero (&t1);
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
        goto LBL_Y;
      }

      q.dp[i - t - 1] = (q.dp[i - t - 1] - 1UL) & MP_MASK;
    }
  }

  /* now q is the quotient and x is the remainder 
   * [which we have to normalize] 
   */
  
  /* get sign before writing to c */
  x.sign = x.used == 0 ? MP_ZPOS : a->sign;

  if (c != NULL) {
    mp_clamp (&q);
    mp_exch (&q, c);
    c->sign = neg;







|
|

|







1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
        goto LBL_Y;
      }

      q.dp[i - t - 1] = (q.dp[i - t - 1] - 1UL) & MP_MASK;
    }
  }

  /* now q is the quotient and x is the remainder
   * [which we have to normalize]
   */

  /* get sign before writing to c */
  x.sign = x.used == 0 ? MP_ZPOS : a->sign;

  if (c != NULL) {
    mp_clamp (&q);
    mp_exch (&q, c);
    c->sign = neg;
1753
1754
1755
1756
1757
1758
1759




1760
1761
1762
1763
1764
1765
1766
LBL_Q:mp_clear (&q);
  return res;
}

#endif

#endif





/* End: bn_mp_div.c */

/* Start: bn_mp_div_2.c */
#include <tommath.h>
#ifdef BN_MP_DIV_2_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
LBL_Q:mp_clear (&q);
  return res;
}

#endif

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_div.c */

/* Start: bn_mp_div_2.c */
#include <tommath.h>
#ifdef BN_MP_DIV_2_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
1821
1822
1823
1824
1825
1826
1827




1828
1829
1830
1831
1832
1833
1834
    }
  }
  b->sign = a->sign;
  mp_clamp (b);
  return MP_OKAY;
}
#endif





/* End: bn_mp_div_2.c */

/* Start: bn_mp_div_2d.c */
#include <tommath.h>
#ifdef BN_MP_DIV_2D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
    }
  }
  b->sign = a->sign;
  mp_clamp (b);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_div_2.c */

/* Start: bn_mp_div_2d.c */
#include <tommath.h>
#ifdef BN_MP_DIV_2D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
1918
1919
1920
1921
1922
1923
1924




1925
1926
1927
1928
1929
1930
1931
  if (d != NULL) {
    mp_exch (&t, d);
  }
  mp_clear (&t);
  return MP_OKAY;
}
#endif





/* End: bn_mp_div_2d.c */

/* Start: bn_mp_div_3.c */
#include <tommath.h>
#ifdef BN_MP_DIV_3_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
  if (d != NULL) {
    mp_exch (&t, d);
  }
  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_div_2d.c */

/* Start: bn_mp_div_3.c */
#include <tommath.h>
#ifdef BN_MP_DIV_3_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
int
mp_div_3 (mp_int * a, mp_int *c, mp_digit * d)
{
  mp_int   q;
  mp_word  w, t;
  mp_digit b;
  int      res, ix;
  
  /* b = 2**DIGIT_BIT / 3 */
  b = (((mp_word)1) << ((mp_word)DIGIT_BIT)) / ((mp_word)3);

  if ((res = mp_init_size(&q, a->used)) != MP_OKAY) {
     return res;
  }
  
  q.used = a->used;
  q.sign = a->sign;
  w = 0;
  for (ix = a->used - 1; ix >= 0; ix--) {
     w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]);

     if (w >= 3) {







|






|







2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
int
mp_div_3 (mp_int * a, mp_int *c, mp_digit * d)
{
  mp_int   q;
  mp_word  w, t;
  mp_digit b;
  int      res, ix;

  /* b = 2**DIGIT_BIT / 3 */
  b = (((mp_word)1) << ((mp_word)DIGIT_BIT)) / ((mp_word)3);

  if ((res = mp_init_size(&q, a->used)) != MP_OKAY) {
     return res;
  }

  q.used = a->used;
  q.sign = a->sign;
  w = 0;
  for (ix = a->used - 1; ix >= 0; ix--) {
     w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]);

     if (w >= 3) {
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003




2004
2005
2006
2007
2008
2009
2010

  /* [optional] store the quotient */
  if (c != NULL) {
     mp_clamp(&q);
     mp_exch(&q, c);
  }
  mp_clear(&q);
  
  return res;
}

#endif





/* End: bn_mp_div_3.c */

/* Start: bn_mp_div_d.c */
#include <tommath.h>
#ifdef BN_MP_DIV_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|




>
>
>
>







2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110

  /* [optional] store the quotient */
  if (c != NULL) {
     mp_clamp(&q);
     mp_exch(&q, c);
  }
  mp_clear(&q);

  return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_div_3.c */

/* Start: bn_mp_div_d.c */
#include <tommath.h>
#ifdef BN_MP_DIV_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118




2119
2120
2121
2122
2123
2124
2125
  }
#endif

  /* no easy answer [c'est la vie].  Just division */
  if ((res = mp_init_size(&q, a->used)) != MP_OKAY) {
     return res;
  }
  
  q.used = a->used;
  q.sign = a->sign;
  w = 0;
  for (ix = a->used - 1; ix >= 0; ix--) {
     w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]);
     
     if (w >= b) {
        t = (mp_digit)(w / b);
        w -= ((mp_word)t) * ((mp_word)b);
      } else {
        t = 0;
      }
      q.dp[ix] = (mp_digit)t;
  }
  
  if (d != NULL) {
     *d = (mp_digit)w;
  }
  
  if (c != NULL) {
     mp_clamp(&q);
     mp_exch(&q, c);
  }
  mp_clear(&q);
  
  return res;
}

#endif





/* End: bn_mp_div_d.c */

/* Start: bn_mp_dr_is_modulus.c */
#include <tommath.h>
#ifdef BN_MP_DR_IS_MODULUS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|





|








|



|





|




>
>
>
>







2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
  }
#endif

  /* no easy answer [c'est la vie].  Just division */
  if ((res = mp_init_size(&q, a->used)) != MP_OKAY) {
     return res;
  }

  q.used = a->used;
  q.sign = a->sign;
  w = 0;
  for (ix = a->used - 1; ix >= 0; ix--) {
     w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]);

     if (w >= b) {
        t = (mp_digit)(w / b);
        w -= ((mp_word)t) * ((mp_word)b);
      } else {
        t = 0;
      }
      q.dp[ix] = (mp_digit)t;
  }

  if (d != NULL) {
     *d = (mp_digit)w;
  }

  if (c != NULL) {
     mp_clamp(&q);
     mp_exch(&q, c);
  }
  mp_clear(&q);

  return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_div_d.c */

/* Start: bn_mp_dr_is_modulus.c */
#include <tommath.h>
#ifdef BN_MP_DR_IS_MODULUS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
2155
2156
2157
2158
2159
2160
2161




2162
2163
2164
2165
2166
2167
2168
          return 0;
       }
   }
   return 1;
}

#endif





/* End: bn_mp_dr_is_modulus.c */

/* Start: bn_mp_dr_reduce.c */
#include <tommath.h>
#ifdef BN_MP_DR_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
          return 0;
       }
   }
   return 1;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_dr_is_modulus.c */

/* Start: bn_mp_dr_reduce.c */
#include <tommath.h>
#ifdef BN_MP_DR_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
2249
2250
2251
2252
2253
2254
2255




2256
2257
2258
2259
2260
2261
2262
  if (mp_cmp_mag (x, n) != MP_LT) {
    s_mp_sub(x, n, x);
    goto top;
  }
  return MP_OKAY;
}
#endif





/* End: bn_mp_dr_reduce.c */

/* Start: bn_mp_dr_setup.c */
#include <tommath.h>
#ifdef BN_MP_DR_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
  if (mp_cmp_mag (x, n) != MP_LT) {
    s_mp_sub(x, n, x);
    goto top;
  }
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_dr_reduce.c */

/* Start: bn_mp_dr_setup.c */
#include <tommath.h>
#ifdef BN_MP_DR_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287




2288
2289
2290
2291
2292
2293
2294

/* determines the setup value */
void mp_dr_setup(mp_int *a, mp_digit *d)
{
   /* the casts are required if DIGIT_BIT is one less than
    * the number of bits in a mp_digit [e.g. DIGIT_BIT==31]
    */
   *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) - 
        ((mp_word)a->dp[0]));
}

#endif





/* End: bn_mp_dr_setup.c */

/* Start: bn_mp_exch.c */
#include <tommath.h>
#ifdef BN_MP_EXCH_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|




>
>
>
>







2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410

/* determines the setup value */
void mp_dr_setup(mp_int *a, mp_digit *d)
{
   /* the casts are required if DIGIT_BIT is one less than
    * the number of bits in a mp_digit [e.g. DIGIT_BIT==31]
    */
   *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) -
        ((mp_word)a->dp[0]));
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_dr_setup.c */

/* Start: bn_mp_exch.c */
#include <tommath.h>
#ifdef BN_MP_EXCH_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321




2322
2323
2324
2325
2326
2327
2328
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* swap the elements of two integers, for cases where you can't simply swap the 
 * mp_int pointers around
 */
void
mp_exch (mp_int * a, mp_int * b)
{
  mp_int  t;

  t  = *a;
  *a = *b;
  *b = t;
}
#endif





/* End: bn_mp_exch.c */

/* Start: bn_mp_expt_d.c */
#include <tommath.h>
#ifdef BN_MP_EXPT_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|












>
>
>
>







2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* swap the elements of two integers, for cases where you can't simply swap the
 * mp_int pointers around
 */
void
mp_exch (mp_int * a, mp_int * b)
{
  mp_int  t;

  t  = *a;
  *a = *b;
  *b = t;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_exch.c */

/* Start: bn_mp_expt_d.c */
#include <tommath.h>
#ifdef BN_MP_EXPT_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
2372
2373
2374
2375
2376
2377
2378




2379
2380
2381
2382
2383
2384
2385
    b <<= 1;
  }

  mp_clear (&g);
  return MP_OKAY;
}
#endif





/* End: bn_mp_expt_d.c */

/* Start: bn_mp_exptmod.c */
#include <tommath.h>
#ifdef BN_MP_EXPTMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
    b <<= 1;
  }

  mp_clear (&g);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_expt_d.c */

/* Start: bn_mp_exptmod.c */
#include <tommath.h>
#ifdef BN_MP_EXPTMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
        return err;
     }

     /* and now compute (1/G)**|X| instead of G**X [X < 0] */
     err = mp_exptmod(&tmpG, &tmpX, P, Y);
     mp_clear_multi(&tmpG, &tmpX, NULL);
     return err;
#else 
     /* no invmod */
     return MP_VAL;
#endif
  }

/* modified diminished radix reduction */
#if defined(BN_MP_REDUCE_IS_2K_L_C) && defined(BN_MP_REDUCE_2K_L_C) && defined(BN_S_MP_EXPTMOD_C)







|







2561
2562
2563
2564
2565
2566
2567
2568
2569
2570
2571
2572
2573
2574
2575
        return err;
     }

     /* and now compute (1/G)**|X| instead of G**X [X < 0] */
     err = mp_exptmod(&tmpG, &tmpX, P, Y);
     mp_clear_multi(&tmpG, &tmpX, NULL);
     return err;
#else
     /* no invmod */
     return MP_VAL;
#endif
  }

/* modified diminished radix reduction */
#if defined(BN_MP_REDUCE_IS_2K_L_C) && defined(BN_MP_REDUCE_2K_L_C) && defined(BN_S_MP_EXPTMOD_C)
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490




2491
2492
2493
2494
2495
2496
2497

#ifdef BN_MP_REDUCE_IS_2K_C
  /* if not, is it a unrestricted DR modulus? */
  if (dr == 0) {
     dr = mp_reduce_is_2k(P) << 1;
  }
#endif
    
  /* if the modulus is odd or dr != 0 use the montgomery method */
#ifdef BN_MP_EXPTMOD_FAST_C
  if (mp_isodd (P) == 1 || dr !=  0) {
    return mp_exptmod_fast (G, X, P, Y, dr);
  } else {
#endif
#ifdef BN_S_MP_EXPTMOD_C
    /* otherwise use the generic Barrett reduction technique */
    return s_mp_exptmod (G, X, P, Y, 0);
#else
    /* no exptmod for evens */
    return MP_VAL;
#endif
#ifdef BN_MP_EXPTMOD_FAST_C
  }
#endif
}

#endif





/* End: bn_mp_exptmod.c */

/* Start: bn_mp_exptmod_fast.c */
#include <tommath.h>
#ifdef BN_MP_EXPTMOD_FAST_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|



















>
>
>
>







2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604
2605
2606
2607
2608
2609
2610
2611
2612
2613
2614
2615
2616
2617
2618
2619
2620
2621
2622
2623
2624
2625

#ifdef BN_MP_REDUCE_IS_2K_C
  /* if not, is it a unrestricted DR modulus? */
  if (dr == 0) {
     dr = mp_reduce_is_2k(P) << 1;
  }
#endif

  /* if the modulus is odd or dr != 0 use the montgomery method */
#ifdef BN_MP_EXPTMOD_FAST_C
  if (mp_isodd (P) == 1 || dr !=  0) {
    return mp_exptmod_fast (G, X, P, Y, dr);
  } else {
#endif
#ifdef BN_S_MP_EXPTMOD_C
    /* otherwise use the generic Barrett reduction technique */
    return s_mp_exptmod (G, X, P, Y, 0);
#else
    /* no exptmod for evens */
    return MP_VAL;
#endif
#ifdef BN_MP_EXPTMOD_FAST_C
  }
#endif
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_exptmod.c */

/* Start: bn_mp_exptmod_fast.c */
#include <tommath.h>
#ifdef BN_MP_EXPTMOD_FAST_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
2574
2575
2576
2577
2578
2579
2580
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
      mp_clear(&M[1]);
      return err;
    }
  }

  /* determine and setup reduction code */
  if (redmode == 0) {
#ifdef BN_MP_MONTGOMERY_SETUP_C     
     /* now setup montgomery  */
     if ((err = mp_montgomery_setup (P, &mp)) != MP_OKAY) {
        goto LBL_M;
     }
#else
     err = MP_VAL;
     goto LBL_M;
#endif

     /* automatically pick the comba one if available (saves quite a few calls/ifs) */
#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C
     if (((P->used * 2 + 1) < MP_WARRAY) &&
          P->used < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) {
        redux = fast_mp_montgomery_reduce;
     } else 
#endif
     {
#ifdef BN_MP_MONTGOMERY_REDUCE_C
        /* use slower baseline Montgomery method */
        redux = mp_montgomery_reduce;
#else
        err = MP_VAL;







|














|







2702
2703
2704
2705
2706
2707
2708
2709
2710
2711
2712
2713
2714
2715
2716
2717
2718
2719
2720
2721
2722
2723
2724
2725
2726
2727
2728
2729
2730
2731
      mp_clear(&M[1]);
      return err;
    }
  }

  /* determine and setup reduction code */
  if (redmode == 0) {
#ifdef BN_MP_MONTGOMERY_SETUP_C
     /* now setup montgomery  */
     if ((err = mp_montgomery_setup (P, &mp)) != MP_OKAY) {
        goto LBL_M;
     }
#else
     err = MP_VAL;
     goto LBL_M;
#endif

     /* automatically pick the comba one if available (saves quite a few calls/ifs) */
#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C
     if (((P->used * 2 + 1) < MP_WARRAY) &&
          P->used < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) {
        redux = fast_mp_montgomery_reduce;
     } else
#endif
     {
#ifdef BN_MP_MONTGOMERY_REDUCE_C
        /* use slower baseline Montgomery method */
        redux = mp_montgomery_reduce;
#else
        err = MP_VAL;
2640
2641
2642
2643
2644
2645
2646
2647
2648
2649
2650
2651
2652
2653
2654

  if (redmode == 0) {
#ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C
     /* now we need R mod m */
     if ((err = mp_montgomery_calc_normalization (&res, P)) != MP_OKAY) {
       goto LBL_RES;
     }
#else 
     err = MP_VAL;
     goto LBL_RES;
#endif

     /* now set M[1] to G * R mod m */
     if ((err = mp_mulmod (G, &res, P, &M[1])) != MP_OKAY) {
       goto LBL_RES;







|







2768
2769
2770
2771
2772
2773
2774
2775
2776
2777
2778
2779
2780
2781
2782

  if (redmode == 0) {
#ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C
     /* now we need R mod m */
     if ((err = mp_montgomery_calc_normalization (&res, P)) != MP_OKAY) {
       goto LBL_RES;
     }
#else
     err = MP_VAL;
     goto LBL_RES;
#endif

     /* now set M[1] to G * R mod m */
     if ((err = mp_mulmod (G, &res, P, &M[1])) != MP_OKAY) {
       goto LBL_RES;
2805
2806
2807
2808
2809
2810
2811





2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
  for (x = 1<<(winsize-1); x < (1 << winsize); x++) {
    mp_clear (&M[x]);
  }
  return err;
}
#endif






/* End: bn_mp_exptmod_fast.c */

/* Start: bn_mp_exteuclid.c */
#include <tommath.h>
#ifdef BN_MP_EXTEUCLID_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* Extended euclidean algorithm of (a, b) produces 
   a*u1 + b*u2 = u3
 */
int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3)
{
   mp_int u1,u2,u3,v1,v2,v3,t1,t2,t3,q,tmp;
   int err;








>
>
>
>
>




















|







2933
2934
2935
2936
2937
2938
2939
2940
2941
2942
2943
2944
2945
2946
2947
2948
2949
2950
2951
2952
2953
2954
2955
2956
2957
2958
2959
2960
2961
2962
2963
2964
2965
2966
2967
2968
2969
2970
2971
2972
  for (x = 1<<(winsize-1); x < (1 << winsize); x++) {
    mp_clear (&M[x]);
  }
  return err;
}
#endif


/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_exptmod_fast.c */

/* Start: bn_mp_exteuclid.c */
#include <tommath.h>
#ifdef BN_MP_EXTEUCLID_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* Extended euclidean algorithm of (a, b) produces
   a*u1 + b*u2 = u3
 */
int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3)
{
   mp_int u1,u2,u3,v1,v2,v3,t1,t2,t3,q,tmp;
   int err;

2886
2887
2888
2889
2890
2891
2892




2893
2894
2895
2896
2897
2898
2899
   if (U3 != NULL) { mp_exch(U3, &u3); }

   err = MP_OKAY;
_ERR: mp_clear_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL);
   return err;
}
#endif





/* End: bn_mp_exteuclid.c */

/* Start: bn_mp_fread.c */
#include <tommath.h>
#ifdef BN_MP_FREAD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







3019
3020
3021
3022
3023
3024
3025
3026
3027
3028
3029
3030
3031
3032
3033
3034
3035
3036
   if (U3 != NULL) { mp_exch(U3, &u3); }

   err = MP_OKAY;
_ERR: mp_clear_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL);
   return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_exteuclid.c */

/* Start: bn_mp_fread.c */
#include <tommath.h>
#ifdef BN_MP_FREAD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
2911
2912
2913
2914
2915
2916
2917
2918
2919
2920
2921
2922
2923
2924
2925
2926
2927
2928
2929
2930
2931
2932
2933
2934
2935
2936
2937
2938
2939
2940
2941
2942
2943
2944
2945
2946
2947
2948
2949
2950
2951
2952
2953
2954
2955
2956
2957
2958
2959




2960
2961
2962
2963
2964
2965
2966
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* read a bigint from a file stream in ASCII */
int mp_fread(mp_int *a, int radix, FILE *stream)
{
   int err, ch, neg, y;
   
   /* clear a */
   mp_zero(a);
   
   /* if first digit is - then set negative */
   ch = fgetc(stream);
   if (ch == '-') {
      neg = MP_NEG;
      ch = fgetc(stream);
   } else {
      neg = MP_ZPOS;
   }
   
   for (;;) {
      /* find y in the radix map */
      for (y = 0; y < radix; y++) {
          if (mp_s_rmap[y] == ch) {
             break;
          }
      }
      if (y == radix) {
         break;
      }
      
      /* shift up and add */
      if ((err = mp_mul_d(a, radix, a)) != MP_OKAY) {
         return err;
      }
      if ((err = mp_add_d(a, y, a)) != MP_OKAY) {
         return err;
      }
      
      ch = fgetc(stream);
   }
   if (mp_cmp_d(a, 0) != MP_EQ) {
      a->sign = neg;
   }
   
   return MP_OKAY;
}

#endif





/* End: bn_mp_fread.c */

/* Start: bn_mp_fwrite.c */
#include <tommath.h>
#ifdef BN_MP_FWRITE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|


|








|










|







|





|




>
>
>
>







3048
3049
3050
3051
3052
3053
3054
3055
3056
3057
3058
3059
3060
3061
3062
3063
3064
3065
3066
3067
3068
3069
3070
3071
3072
3073
3074
3075
3076
3077
3078
3079
3080
3081
3082
3083
3084
3085
3086
3087
3088
3089
3090
3091
3092
3093
3094
3095
3096
3097
3098
3099
3100
3101
3102
3103
3104
3105
3106
3107
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* read a bigint from a file stream in ASCII */
int mp_fread(mp_int *a, int radix, FILE *stream)
{
   int err, ch, neg, y;

   /* clear a */
   mp_zero(a);

   /* if first digit is - then set negative */
   ch = fgetc(stream);
   if (ch == '-') {
      neg = MP_NEG;
      ch = fgetc(stream);
   } else {
      neg = MP_ZPOS;
   }

   for (;;) {
      /* find y in the radix map */
      for (y = 0; y < radix; y++) {
          if (mp_s_rmap[y] == ch) {
             break;
          }
      }
      if (y == radix) {
         break;
      }

      /* shift up and add */
      if ((err = mp_mul_d(a, radix, a)) != MP_OKAY) {
         return err;
      }
      if ((err = mp_add_d(a, y, a)) != MP_OKAY) {
         return err;
      }

      ch = fgetc(stream);
   }
   if (mp_cmp_d(a, 0) != MP_EQ) {
      a->sign = neg;
   }

   return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_fread.c */

/* Start: bn_mp_fwrite.c */
#include <tommath.h>
#ifdef BN_MP_FWRITE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
2978
2979
2980
2981
2982
2983
2984
2985
2986
2987
2988
2989
2990
2991
2992
2993
2994
2995
2996
2997
2998
2999
3000
3001
3002
3003
3004
3005
3006
3007
3008
3009
3010
3011




3012
3013
3014
3015
3016
3017
3018
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

int mp_fwrite(mp_int *a, int radix, FILE *stream)
{
   char *buf;
   int err, len, x;
   
   if ((err = mp_radix_size(a, radix, &len)) != MP_OKAY) {
      return err;
   }

   buf = OPT_CAST(char) XMALLOC (len);
   if (buf == NULL) {
      return MP_MEM;
   }
   
   if ((err = mp_toradix(a, buf, radix)) != MP_OKAY) {
      XFREE (buf);
      return err;
   }
   
   for (x = 0; x < len; x++) {
       if (fputc(buf[x], stream) == EOF) {
          XFREE (buf);
          return MP_VAL;
       }
   }
   
   XFREE (buf);
   return MP_OKAY;
}

#endif





/* End: bn_mp_fwrite.c */

/* Start: bn_mp_gcd.c */
#include <tommath.h>
#ifdef BN_MP_GCD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|








|




|






|





>
>
>
>







3119
3120
3121
3122
3123
3124
3125
3126
3127
3128
3129
3130
3131
3132
3133
3134
3135
3136
3137
3138
3139
3140
3141
3142
3143
3144
3145
3146
3147
3148
3149
3150
3151
3152
3153
3154
3155
3156
3157
3158
3159
3160
3161
3162
3163
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

int mp_fwrite(mp_int *a, int radix, FILE *stream)
{
   char *buf;
   int err, len, x;

   if ((err = mp_radix_size(a, radix, &len)) != MP_OKAY) {
      return err;
   }

   buf = OPT_CAST(char) XMALLOC (len);
   if (buf == NULL) {
      return MP_MEM;
   }

   if ((err = mp_toradix(a, buf, radix)) != MP_OKAY) {
      XFREE (buf);
      return err;
   }

   for (x = 0; x < len; x++) {
       if (fputc(buf[x], stream) == EOF) {
          XFREE (buf);
          return MP_VAL;
       }
   }

   XFREE (buf);
   return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_fwrite.c */

/* Start: bn_mp_gcd.c */
#include <tommath.h>
#ifdef BN_MP_GCD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
3087
3088
3089
3090
3091
3092
3093
3094
3095
3096
3097
3098
3099
3100
3101
3102
3103
3104
3105
3106
3107
3108
3109
3110
3111
3112
3113
3114
3115
3116




3117
3118
3119
3120
3121
3122
3123

  while (mp_iszero(&v) == 0) {
     /* make sure v is the largest */
     if (mp_cmp_mag(&u, &v) == MP_GT) {
        /* swap u and v to make sure v is >= u */
        mp_exch(&u, &v);
     }
     
     /* subtract smallest from largest */
     if ((res = s_mp_sub(&v, &u, &v)) != MP_OKAY) {
        goto LBL_V;
     }
     
     /* Divide out all factors of two */
     if ((res = mp_div_2d(&v, mp_cnt_lsb(&v), &v, NULL)) != MP_OKAY) {
        goto LBL_V;
     } 
  } 

  /* multiply by 2**k which we divided out at the beginning */
  if ((res = mp_mul_2d (&u, k, c)) != MP_OKAY) {
     goto LBL_V;
  }
  c->sign = MP_ZPOS;
  res = MP_OKAY;
LBL_V:mp_clear (&u);
LBL_U:mp_clear (&v);
  return res;
}
#endif





/* End: bn_mp_gcd.c */

/* Start: bn_mp_get_int.c */
#include <tommath.h>
#ifdef BN_MP_GET_INT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|




|



|
|












>
>
>
>







3232
3233
3234
3235
3236
3237
3238
3239
3240
3241
3242
3243
3244
3245
3246
3247
3248
3249
3250
3251
3252
3253
3254
3255
3256
3257
3258
3259
3260
3261
3262
3263
3264
3265
3266
3267
3268
3269
3270
3271
3272

  while (mp_iszero(&v) == 0) {
     /* make sure v is the largest */
     if (mp_cmp_mag(&u, &v) == MP_GT) {
        /* swap u and v to make sure v is >= u */
        mp_exch(&u, &v);
     }

     /* subtract smallest from largest */
     if ((res = s_mp_sub(&v, &u, &v)) != MP_OKAY) {
        goto LBL_V;
     }

     /* Divide out all factors of two */
     if ((res = mp_div_2d(&v, mp_cnt_lsb(&v), &v, NULL)) != MP_OKAY) {
        goto LBL_V;
     }
  }

  /* multiply by 2**k which we divided out at the beginning */
  if ((res = mp_mul_2d (&u, k, c)) != MP_OKAY) {
     goto LBL_V;
  }
  c->sign = MP_ZPOS;
  res = MP_OKAY;
LBL_V:mp_clear (&u);
LBL_U:mp_clear (&v);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_gcd.c */

/* Start: bn_mp_get_int.c */
#include <tommath.h>
#ifdef BN_MP_GET_INT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
3132
3133
3134
3135
3136
3137
3138
3139
3140
3141
3142
3143
3144
3145
3146
3147
3148
3149
3150
3151
3152
3153
3154
3155
3156
3157
3158
3159
3160
3161




3162
3163
3164
3165
3166
3167
3168
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* get the lower 32-bits of an mp_int */
unsigned long mp_get_int(mp_int * a) 
{
  int i;
  unsigned long res;

  if (a->used == 0) {
     return 0;
  }

  /* get number of digits of the lsb we have to read */
  i = MIN(a->used,(int)((sizeof(unsigned long)*CHAR_BIT+DIGIT_BIT-1)/DIGIT_BIT))-1;

  /* get most significant digit of result */
  res = DIGIT(a,i);
   
  while (--i >= 0) {
    res = (res << DIGIT_BIT) | DIGIT(a,i);
  }

  /* force result to 32-bits always so it is consistent on non 32-bit platforms */
  return res & 0xFFFFFFFFUL;
}
#endif





/* End: bn_mp_get_int.c */

/* Start: bn_mp_grow.c */
#include <tommath.h>
#ifdef BN_MP_GROW_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|













|








>
>
>
>







3281
3282
3283
3284
3285
3286
3287
3288
3289
3290
3291
3292
3293
3294
3295
3296
3297
3298
3299
3300
3301
3302
3303
3304
3305
3306
3307
3308
3309
3310
3311
3312
3313
3314
3315
3316
3317
3318
3319
3320
3321
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* get the lower 32-bits of an mp_int */
unsigned long mp_get_int(mp_int * a)
{
  int i;
  unsigned long res;

  if (a->used == 0) {
     return 0;
  }

  /* get number of digits of the lsb we have to read */
  i = MIN(a->used,(int)((sizeof(unsigned long)*CHAR_BIT+DIGIT_BIT-1)/DIGIT_BIT))-1;

  /* get most significant digit of result */
  res = DIGIT(a,i);

  while (--i >= 0) {
    res = (res << DIGIT_BIT) | DIGIT(a,i);
  }

  /* force result to 32-bits always so it is consistent on non 32-bit platforms */
  return res & 0xFFFFFFFFUL;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_get_int.c */

/* Start: bn_mp_grow.c */
#include <tommath.h>
#ifdef BN_MP_GROW_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
3212
3213
3214
3215
3216
3217
3218




3219
3220
3221
3222
3223
3224
3225
    for (; i < a->alloc; i++) {
      a->dp[i] = 0;
    }
  }
  return MP_OKAY;
}
#endif





/* End: bn_mp_grow.c */

/* Start: bn_mp_init.c */
#include <tommath.h>
#ifdef BN_MP_INIT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







3365
3366
3367
3368
3369
3370
3371
3372
3373
3374
3375
3376
3377
3378
3379
3380
3381
3382
    for (; i < a->alloc; i++) {
      a->dp[i] = 0;
    }
  }
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_grow.c */

/* Start: bn_mp_init.c */
#include <tommath.h>
#ifdef BN_MP_INIT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
3258
3259
3260
3261
3262
3263
3264




3265
3266
3267
3268
3269
3270
3271
  a->used  = 0;
  a->alloc = MP_PREC;
  a->sign  = MP_ZPOS;

  return MP_OKAY;
}
#endif





/* End: bn_mp_init.c */

/* Start: bn_mp_init_copy.c */
#include <tommath.h>
#ifdef BN_MP_INIT_COPY_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







3415
3416
3417
3418
3419
3420
3421
3422
3423
3424
3425
3426
3427
3428
3429
3430
3431
3432
  a->used  = 0;
  a->alloc = MP_PREC;
  a->sign  = MP_ZPOS;

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_init.c */

/* Start: bn_mp_init_copy.c */
#include <tommath.h>
#ifdef BN_MP_INIT_COPY_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
3290
3291
3292
3293
3294
3295
3296




3297
3298
3299
3300
3301
3302
3303

  if ((res = mp_init (a)) != MP_OKAY) {
    return res;
  }
  return mp_copy (b, a);
}
#endif





/* End: bn_mp_init_copy.c */

/* Start: bn_mp_init_multi.c */
#include <tommath.h>
#ifdef BN_MP_INIT_MULTI_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







3451
3452
3453
3454
3455
3456
3457
3458
3459
3460
3461
3462
3463
3464
3465
3466
3467
3468

  if ((res = mp_init (a)) != MP_OKAY) {
    return res;
  }
  return mp_copy (b, a);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_init_copy.c */

/* Start: bn_mp_init_multi.c */
#include <tommath.h>
#ifdef BN_MP_INIT_MULTI_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
3312
3313
3314
3315
3316
3317
3318
3319
3320
3321
3322
3323
3324
3325
3326
3327
3328
3329
3330
3331
3332
3333
3334
3335
3336
3337
3338
3339
3340
3341
3342
3343
3344
3345
3346
3347
3348
3349
3350
3351
3352
3353
3354
3355




3356
3357
3358
3359
3360
3361
3362
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */
#include <stdarg.h>

int mp_init_multi(mp_int *mp, ...) 
{
    mp_err res = MP_OKAY;      /* Assume ok until proven otherwise */
    int n = 0;                 /* Number of ok inits */
    mp_int* cur_arg = mp;
    va_list args;

    va_start(args, mp);        /* init args to next argument from caller */
    while (cur_arg != NULL) {
        if (mp_init(cur_arg) != MP_OKAY) {
            /* Oops - error! Back-track and mp_clear what we already
               succeeded in init-ing, then return error.
            */
            va_list clean_args;
            
            /* end the current list */
            va_end(args);
            
            /* now start cleaning up */            
            cur_arg = mp;
            va_start(clean_args, mp);
            while (n--) {
                mp_clear(cur_arg);
                cur_arg = va_arg(clean_args, mp_int*);
            }
            va_end(clean_args);
            res = MP_MEM;
            break;
        }
        n++;
        cur_arg = va_arg(args, mp_int*);
    }
    va_end(args);
    return res;                /* Assumed ok, if error flagged above. */
}

#endif





/* End: bn_mp_init_multi.c */

/* Start: bn_mp_init_set.c */
#include <tommath.h>
#ifdef BN_MP_INIT_SET_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|













|


|
|


















>
>
>
>







3477
3478
3479
3480
3481
3482
3483
3484
3485
3486
3487
3488
3489
3490
3491
3492
3493
3494
3495
3496
3497
3498
3499
3500
3501
3502
3503
3504
3505
3506
3507
3508
3509
3510
3511
3512
3513
3514
3515
3516
3517
3518
3519
3520
3521
3522
3523
3524
3525
3526
3527
3528
3529
3530
3531
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */
#include <stdarg.h>

int mp_init_multi(mp_int *mp, ...)
{
    mp_err res = MP_OKAY;      /* Assume ok until proven otherwise */
    int n = 0;                 /* Number of ok inits */
    mp_int* cur_arg = mp;
    va_list args;

    va_start(args, mp);        /* init args to next argument from caller */
    while (cur_arg != NULL) {
        if (mp_init(cur_arg) != MP_OKAY) {
            /* Oops - error! Back-track and mp_clear what we already
               succeeded in init-ing, then return error.
            */
            va_list clean_args;

            /* end the current list */
            va_end(args);

            /* now start cleaning up */
            cur_arg = mp;
            va_start(clean_args, mp);
            while (n--) {
                mp_clear(cur_arg);
                cur_arg = va_arg(clean_args, mp_int*);
            }
            va_end(clean_args);
            res = MP_MEM;
            break;
        }
        n++;
        cur_arg = va_arg(args, mp_int*);
    }
    va_end(args);
    return res;                /* Assumed ok, if error flagged above. */
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_init_multi.c */

/* Start: bn_mp_init_set.c */
#include <tommath.h>
#ifdef BN_MP_INIT_SET_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
3381
3382
3383
3384
3385
3386
3387




3388
3389
3390
3391
3392
3393
3394
  if ((err = mp_init(a)) != MP_OKAY) {
     return err;
  }
  mp_set(a, b);
  return err;
}
#endif





/* End: bn_mp_init_set.c */

/* Start: bn_mp_init_set_int.c */
#include <tommath.h>
#ifdef BN_MP_INIT_SET_INT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







3550
3551
3552
3553
3554
3555
3556
3557
3558
3559
3560
3561
3562
3563
3564
3565
3566
3567
  if ((err = mp_init(a)) != MP_OKAY) {
     return err;
  }
  mp_set(a, b);
  return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_init_set.c */

/* Start: bn_mp_init_set_int.c */
#include <tommath.h>
#ifdef BN_MP_INIT_SET_INT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
3412
3413
3414
3415
3416
3417
3418




3419
3420
3421
3422
3423
3424
3425
  int err;
  if ((err = mp_init(a)) != MP_OKAY) {
     return err;
  }
  return mp_set_int(a, b);
}
#endif





/* End: bn_mp_init_set_int.c */

/* Start: bn_mp_init_size.c */
#include <tommath.h>
#ifdef BN_MP_INIT_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







3585
3586
3587
3588
3589
3590
3591
3592
3593
3594
3595
3596
3597
3598
3599
3600
3601
3602
  int err;
  if ((err = mp_init(a)) != MP_OKAY) {
     return err;
  }
  return mp_set_int(a, b);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_init_set_int.c */

/* Start: bn_mp_init_size.c */
#include <tommath.h>
#ifdef BN_MP_INIT_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
3440
3441
3442
3443
3444
3445
3446
3447
3448
3449
3450
3451
3452
3453
3454
3455
3456
3457
3458
3459
3460
3461
3462
3463
3464
3465
3466




3467
3468
3469
3470
3471
3472
3473
/* init an mp_init for a given size */
int mp_init_size (mp_int * a, int size)
{
  int x;

  /* pad size so there are always extra digits */
  size += (MP_PREC * 2) - (size % MP_PREC);	
  
  /* alloc mem */
  a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * size);
  if (a->dp == NULL) {
    return MP_MEM;
  }

  /* set the members */
  a->used  = 0;
  a->alloc = size;
  a->sign  = MP_ZPOS;

  /* zero the digits */
  for (x = 0; x < size; x++) {
      a->dp[x] = 0;
  }

  return MP_OKAY;
}
#endif





/* End: bn_mp_init_size.c */

/* Start: bn_mp_invmod.c */
#include <tommath.h>
#ifdef BN_MP_INVMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|



















>
>
>
>







3617
3618
3619
3620
3621
3622
3623
3624
3625
3626
3627
3628
3629
3630
3631
3632
3633
3634
3635
3636
3637
3638
3639
3640
3641
3642
3643
3644
3645
3646
3647
3648
3649
3650
3651
3652
3653
3654
/* init an mp_init for a given size */
int mp_init_size (mp_int * a, int size)
{
  int x;

  /* pad size so there are always extra digits */
  size += (MP_PREC * 2) - (size % MP_PREC);	

  /* alloc mem */
  a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * size);
  if (a->dp == NULL) {
    return MP_MEM;
  }

  /* set the members */
  a->used  = 0;
  a->alloc = size;
  a->sign  = MP_ZPOS;

  /* zero the digits */
  for (x = 0; x < size; x++) {
      a->dp[x] = 0;
  }

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_init_size.c */

/* Start: bn_mp_invmod.c */
#include <tommath.h>
#ifdef BN_MP_INVMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
3503
3504
3505
3506
3507
3508
3509




3510
3511
3512
3513
3514
3515
3516
#ifdef BN_MP_INVMOD_SLOW_C
  return mp_invmod_slow(a, b, c);
#endif

  return MP_VAL;
}
#endif





/* End: bn_mp_invmod.c */

/* Start: bn_mp_invmod_slow.c */
#include <tommath.h>
#ifdef BN_MP_INVMOD_SLOW_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







3684
3685
3686
3687
3688
3689
3690
3691
3692
3693
3694
3695
3696
3697
3698
3699
3700
3701
#ifdef BN_MP_INVMOD_SLOW_C
  return mp_invmod_slow(a, b, c);
#endif

  return MP_VAL;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_invmod.c */

/* Start: bn_mp_invmod_slow.c */
#include <tommath.h>
#ifdef BN_MP_INVMOD_SLOW_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
3536
3537
3538
3539
3540
3541
3542
3543
3544
3545
3546
3547
3548
3549
3550

  /* b cannot be negative */
  if (b->sign == MP_NEG || mp_iszero(b) == 1) {
    return MP_VAL;
  }

  /* init temps */
  if ((res = mp_init_multi(&x, &y, &u, &v, 
                           &A, &B, &C, &D, NULL)) != MP_OKAY) {
     return res;
  }

  /* x = a, y = b */
  if ((res = mp_mod(a, b, &x)) != MP_OKAY) {
      goto LBL_ERR;







|







3721
3722
3723
3724
3725
3726
3727
3728
3729
3730
3731
3732
3733
3734
3735

  /* b cannot be negative */
  if (b->sign == MP_NEG || mp_iszero(b) == 1) {
    return MP_VAL;
  }

  /* init temps */
  if ((res = mp_init_multi(&x, &y, &u, &v,
                           &A, &B, &C, &D, NULL)) != MP_OKAY) {
     return res;
  }

  /* x = a, y = b */
  if ((res = mp_mod(a, b, &x)) != MP_OKAY) {
      goto LBL_ERR;
3663
3664
3665
3666
3667
3668
3669
3670
3671
3672
3673
3674
3675
3676
3677
3678
3679
3680
3681
3682
3683
3684




3685
3686
3687
3688
3689
3690
3691

  /* if its too low */
  while (mp_cmp_d(&C, 0) == MP_LT) {
      if ((res = mp_add(&C, b, &C)) != MP_OKAY) {
         goto LBL_ERR;
      }
  }
  
  /* too big */
  while (mp_cmp_mag(&C, b) != MP_LT) {
      if ((res = mp_sub(&C, b, &C)) != MP_OKAY) {
         goto LBL_ERR;
      }
  }
  
  /* C is now the inverse */
  mp_exch (&C, c);
  res = MP_OKAY;
LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &A, &B, &C, &D, NULL);
  return res;
}
#endif





/* End: bn_mp_invmod_slow.c */

/* Start: bn_mp_is_square.c */
#include <tommath.h>
#ifdef BN_MP_IS_SQUARE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|






|







>
>
>
>







3848
3849
3850
3851
3852
3853
3854
3855
3856
3857
3858
3859
3860
3861
3862
3863
3864
3865
3866
3867
3868
3869
3870
3871
3872
3873
3874
3875
3876
3877
3878
3879
3880

  /* if its too low */
  while (mp_cmp_d(&C, 0) == MP_LT) {
      if ((res = mp_add(&C, b, &C)) != MP_OKAY) {
         goto LBL_ERR;
      }
  }

  /* too big */
  while (mp_cmp_mag(&C, b) != MP_LT) {
      if ((res = mp_sub(&C, b, &C)) != MP_OKAY) {
         goto LBL_ERR;
      }
  }

  /* C is now the inverse */
  mp_exch (&C, c);
  res = MP_OKAY;
LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &A, &B, &C, &D, NULL);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_invmod_slow.c */

/* Start: bn_mp_is_square.c */
#include <tommath.h>
#ifdef BN_MP_IS_SQUARE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
3722
3723
3724
3725
3726
3727
3728
3729
3730
3731
3732
3733
3734
3735
3736
3737
3738
3739
3740
3741
3742
3743
3744
 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1,
 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1,
 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1,
 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1
};

/* Store non-zero to ret if arg is square, and zero if not */
int mp_is_square(mp_int *arg,int *ret) 
{
  int           res;
  mp_digit      c;
  mp_int        t;
  unsigned long r;

  /* Default to Non-square :) */
  *ret = MP_NO; 

  if (arg->sign == MP_NEG) {
    return MP_VAL;
  }

  /* digits used?  (TSD) */
  if (arg->used == 0) {







|







|







3911
3912
3913
3914
3915
3916
3917
3918
3919
3920
3921
3922
3923
3924
3925
3926
3927
3928
3929
3930
3931
3932
3933
 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1,
 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1,
 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1,
 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1
};

/* Store non-zero to ret if arg is square, and zero if not */
int mp_is_square(mp_int *arg,int *ret)
{
  int           res;
  mp_digit      c;
  mp_int        t;
  unsigned long r;

  /* Default to Non-square :) */
  *ret = MP_NO;

  if (arg->sign == MP_NEG) {
    return MP_VAL;
  }

  /* digits used?  (TSD) */
  if (arg->used == 0) {
3764
3765
3766
3767
3768
3769
3770
3771
3772
3773
3774
3775
3776
3777
3778
3779
  }
  if ((res = mp_mod(arg,&t,&t)) != MP_OKAY) {
     goto ERR;
  }
  r = mp_get_int(&t);
  /* Check for other prime modules, note it's not an ERROR but we must
   * free "t" so the easiest way is to goto ERR.  We know that res
   * is already equal to MP_OKAY from the mp_mod call 
   */ 
  if ( (1L<<(r%11)) & 0x5C4L )             goto ERR;
  if ( (1L<<(r%13)) & 0x9E4L )             goto ERR;
  if ( (1L<<(r%17)) & 0x5CE8L )            goto ERR;
  if ( (1L<<(r%19)) & 0x4F50CL )           goto ERR;
  if ( (1L<<(r%23)) & 0x7ACCA0L )          goto ERR;
  if ( (1L<<(r%29)) & 0xC2EDD0CL )         goto ERR;
  if ( (1L<<(r%31)) & 0x6DE2B848L )        goto ERR;







|
|







3953
3954
3955
3956
3957
3958
3959
3960
3961
3962
3963
3964
3965
3966
3967
3968
  }
  if ((res = mp_mod(arg,&t,&t)) != MP_OKAY) {
     goto ERR;
  }
  r = mp_get_int(&t);
  /* Check for other prime modules, note it's not an ERROR but we must
   * free "t" so the easiest way is to goto ERR.  We know that res
   * is already equal to MP_OKAY from the mp_mod call
   */
  if ( (1L<<(r%11)) & 0x5C4L )             goto ERR;
  if ( (1L<<(r%13)) & 0x9E4L )             goto ERR;
  if ( (1L<<(r%17)) & 0x5CE8L )            goto ERR;
  if ( (1L<<(r%19)) & 0x4F50CL )           goto ERR;
  if ( (1L<<(r%23)) & 0x7ACCA0L )          goto ERR;
  if ( (1L<<(r%29)) & 0xC2EDD0CL )         goto ERR;
  if ( (1L<<(r%31)) & 0x6DE2B848L )        goto ERR;
3787
3788
3789
3790
3791
3792
3793




3794
3795
3796
3797
3798
3799
3800
  }

  *ret = (mp_cmp_mag(&t,arg) == MP_EQ) ? MP_YES : MP_NO;
ERR:mp_clear(&t);
  return res;
}
#endif





/* End: bn_mp_is_square.c */

/* Start: bn_mp_jacobi.c */
#include <tommath.h>
#ifdef BN_MP_JACOBI_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







3976
3977
3978
3979
3980
3981
3982
3983
3984
3985
3986
3987
3988
3989
3990
3991
3992
3993
  }

  *ret = (mp_cmp_mag(&t,arg) == MP_EQ) ? MP_YES : MP_NO;
ERR:mp_clear(&t);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_is_square.c */

/* Start: bn_mp_jacobi.c */
#include <tommath.h>
#ifdef BN_MP_JACOBI_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
3892
3893
3894
3895
3896
3897
3898




3899
3900
3901
3902
3903
3904
3905
  /* done */
  res = MP_OKAY;
LBL_P1:mp_clear (&p1);
LBL_A1:mp_clear (&a1);
  return res;
}
#endif





/* End: bn_mp_jacobi.c */

/* Start: bn_mp_karatsuba_mul.c */
#include <tommath.h>
#ifdef BN_MP_KARATSUBA_MUL_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







4085
4086
4087
4088
4089
4090
4091
4092
4093
4094
4095
4096
4097
4098
4099
4100
4101
4102
  /* done */
  res = MP_OKAY;
LBL_P1:mp_clear (&p1);
LBL_A1:mp_clear (&a1);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_jacobi.c */

/* Start: bn_mp_karatsuba_mul.c */
#include <tommath.h>
#ifdef BN_MP_KARATSUBA_MUL_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
3913
3914
3915
3916
3917
3918
3919
3920
3921
3922
3923
3924
3925
3926
3927
3928
3929
3930
3931
3932
3933
3934
3935
3936
3937
3938
3939
3940
3941
3942
3943
3944
3945
3946
3947
3948
3949
3950
3951
3952
3953
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* c = |a| * |b| using Karatsuba Multiplication using 
 * three half size multiplications
 *
 * Let B represent the radix [e.g. 2**DIGIT_BIT] and 
 * let n represent half of the number of digits in 
 * the min(a,b)
 *
 * a = a1 * B**n + a0
 * b = b1 * B**n + b0
 *
 * Then, a * b => 
   a1b1 * B**2n + ((a1 + a0)(b1 + b0) - (a0b0 + a1b1)) * B + a0b0
 *
 * Note that a1b1 and a0b0 are used twice and only need to be 
 * computed once.  So in total three half size (half # of 
 * digit) multiplications are performed, a0b0, a1b1 and 
 * (a1+b1)(a0+b0)
 *
 * Note that a multiplication of half the digits requires
 * 1/4th the number of single precision multiplications so in 
 * total after one call 25% of the single precision multiplications 
 * are saved.  Note also that the call to mp_mul can end up back 
 * in this function if the a0, a1, b0, or b1 are above the threshold.  
 * This is known as divide-and-conquer and leads to the famous 
 * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than 
 * the standard O(N**2) that the baseline/comba methods use.  
 * Generally though the overhead of this method doesn't pay off 
 * until a certain size (N ~ 80) is reached.
 */
int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c)
{
  mp_int  x0, x1, y0, y1, t1, x0y0, x1y1;
  int     B, err;








|


|
|





|


|
|
|



|
|
|
|
|
|
|
|







4110
4111
4112
4113
4114
4115
4116
4117
4118
4119
4120
4121
4122
4123
4124
4125
4126
4127
4128
4129
4130
4131
4132
4133
4134
4135
4136
4137
4138
4139
4140
4141
4142
4143
4144
4145
4146
4147
4148
4149
4150
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* c = |a| * |b| using Karatsuba Multiplication using
 * three half size multiplications
 *
 * Let B represent the radix [e.g. 2**DIGIT_BIT] and
 * let n represent half of the number of digits in
 * the min(a,b)
 *
 * a = a1 * B**n + a0
 * b = b1 * B**n + b0
 *
 * Then, a * b =>
   a1b1 * B**2n + ((a1 + a0)(b1 + b0) - (a0b0 + a1b1)) * B + a0b0
 *
 * Note that a1b1 and a0b0 are used twice and only need to be
 * computed once.  So in total three half size (half # of
 * digit) multiplications are performed, a0b0, a1b1 and
 * (a1+b1)(a0+b0)
 *
 * Note that a multiplication of half the digits requires
 * 1/4th the number of single precision multiplications so in
 * total after one call 25% of the single precision multiplications
 * are saved.  Note also that the call to mp_mul can end up back
 * in this function if the a0, a1, b0, or b1 are above the threshold.
 * This is known as divide-and-conquer and leads to the famous
 * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than
 * the standard O(N**2) that the baseline/comba methods use.
 * Generally though the overhead of this method doesn't pay off
 * until a certain size (N ~ 80) is reached.
 */
int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c)
{
  mp_int  x0, x1, y0, y1, t1, x0y0, x1y1;
  int     B, err;

4007
4008
4009
4010
4011
4012
4013
4014
4015
4016
4017
4018
4019
4020
4021
4022
4023
4024
4025
4026
4027
4028
4029

    tmpy = y1.dp;
    for (x = B; x < b->used; x++) {
      *tmpy++ = *tmpb++;
    }
  }

  /* only need to clamp the lower words since by definition the 
   * upper words x1/y1 must have a known number of digits
   */
  mp_clamp (&x0);
  mp_clamp (&y0);

  /* now calc the products x0y0 and x1y1 */
  /* after this x0 is no longer required, free temp [x0==t2]! */
  if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY)  
    goto X1Y1;          /* x0y0 = x0*y0 */
  if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY)
    goto X1Y1;          /* x1y1 = x1*y1 */

  /* now calc x1+x0 and y1+y0 */
  if (s_mp_add (&x1, &x0, &t1) != MP_OKAY)
    goto X1Y1;          /* t1 = x1 - x0 */







|







|







4204
4205
4206
4207
4208
4209
4210
4211
4212
4213
4214
4215
4216
4217
4218
4219
4220
4221
4222
4223
4224
4225
4226

    tmpy = y1.dp;
    for (x = B; x < b->used; x++) {
      *tmpy++ = *tmpb++;
    }
  }

  /* only need to clamp the lower words since by definition the
   * upper words x1/y1 must have a known number of digits
   */
  mp_clamp (&x0);
  mp_clamp (&y0);

  /* now calc the products x0y0 and x1y1 */
  /* after this x0 is no longer required, free temp [x0==t2]! */
  if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY)
    goto X1Y1;          /* x0y0 = x0*y0 */
  if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY)
    goto X1Y1;          /* x1y1 = x1*y1 */

  /* now calc x1+x0 and y1+y0 */
  if (s_mp_add (&x1, &x0, &t1) != MP_OKAY)
    goto X1Y1;          /* t1 = x1 - x0 */
4059
4060
4061
4062
4063
4064
4065




4066
4067
4068
4069
4070
4071
4072
Y0:mp_clear (&y0);
X1:mp_clear (&x1);
X0:mp_clear (&x0);
ERR:
  return err;
}
#endif





/* End: bn_mp_karatsuba_mul.c */

/* Start: bn_mp_karatsuba_sqr.c */
#include <tommath.h>
#ifdef BN_MP_KARATSUBA_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







4256
4257
4258
4259
4260
4261
4262
4263
4264
4265
4266
4267
4268
4269
4270
4271
4272
4273
Y0:mp_clear (&y0);
X1:mp_clear (&x1);
X0:mp_clear (&x0);
ERR:
  return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_karatsuba_mul.c */

/* Start: bn_mp_karatsuba_sqr.c */
#include <tommath.h>
#ifdef BN_MP_KARATSUBA_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
4080
4081
4082
4083
4084
4085
4086
4087
4088
4089
4090
4091
4092
4093
4094
4095
4096
4097
4098
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* Karatsuba squaring, computes b = a*a using three 
 * half size squarings
 *
 * See comments of karatsuba_mul for details.  It 
 * is essentially the same algorithm but merely 
 * tuned to perform recursive squarings.
 */
int mp_karatsuba_sqr (mp_int * a, mp_int * b)
{
  mp_int  x0, x1, t1, t2, x0x0, x1x1;
  int     B, err;








|


|
|







4281
4282
4283
4284
4285
4286
4287
4288
4289
4290
4291
4292
4293
4294
4295
4296
4297
4298
4299
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* Karatsuba squaring, computes b = a*a using three
 * half size squarings
 *
 * See comments of karatsuba_mul for details.  It
 * is essentially the same algorithm but merely
 * tuned to perform recursive squarings.
 */
int mp_karatsuba_sqr (mp_int * a, mp_int * b)
{
  mp_int  x0, x1, t1, t2, x0x0, x1x1;
  int     B, err;

4180
4181
4182
4183
4184
4185
4186




4187
4188
4189
4190
4191
4192
4193
T1:mp_clear (&t1);
X1:mp_clear (&x1);
X0:mp_clear (&x0);
ERR:
  return err;
}
#endif





/* End: bn_mp_karatsuba_sqr.c */

/* Start: bn_mp_lcm.c */
#include <tommath.h>
#ifdef BN_MP_LCM_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







4381
4382
4383
4384
4385
4386
4387
4388
4389
4390
4391
4392
4393
4394
4395
4396
4397
4398
T1:mp_clear (&t1);
X1:mp_clear (&x1);
X0:mp_clear (&x0);
ERR:
  return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_karatsuba_sqr.c */

/* Start: bn_mp_lcm.c */
#include <tommath.h>
#ifdef BN_MP_LCM_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
4240
4241
4242
4243
4244
4245
4246




4247
4248
4249
4250
4251
4252
4253
  c->sign = MP_ZPOS;

LBL_T:
  mp_clear_multi (&t1, &t2, NULL);
  return res;
}
#endif





/* End: bn_mp_lcm.c */

/* Start: bn_mp_lshd.c */
#include <tommath.h>
#ifdef BN_MP_LSHD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







4445
4446
4447
4448
4449
4450
4451
4452
4453
4454
4455
4456
4457
4458
4459
4460
4461
4462
  c->sign = MP_ZPOS;

LBL_T:
  mp_clear_multi (&t1, &t2, NULL);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_lcm.c */

/* Start: bn_mp_lshd.c */
#include <tommath.h>
#ifdef BN_MP_LSHD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
4307
4308
4309
4310
4311
4312
4313




4314
4315
4316
4317
4318
4319
4320
    for (x = 0; x < b; x++) {
      *top++ = 0;
    }
  }
  return MP_OKAY;
}
#endif





/* End: bn_mp_lshd.c */

/* Start: bn_mp_mod.c */
#include <tommath.h>
#ifdef BN_MP_MOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







4516
4517
4518
4519
4520
4521
4522
4523
4524
4525
4526
4527
4528
4529
4530
4531
4532
4533
    for (x = 0; x < b; x++) {
      *top++ = 0;
    }
  }
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_lshd.c */

/* Start: bn_mp_mod.c */
#include <tommath.h>
#ifdef BN_MP_MOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
4355
4356
4357
4358
4359
4360
4361




4362
4363
4364
4365
4366
4367
4368
    mp_exch (&t, c);
  }

  mp_clear (&t);
  return res;
}
#endif





/* End: bn_mp_mod.c */

/* Start: bn_mp_mod_2d.c */
#include <tommath.h>
#ifdef BN_MP_MOD_2D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







4568
4569
4570
4571
4572
4573
4574
4575
4576
4577
4578
4579
4580
4581
4582
4583
4584
4585
    mp_exch (&t, c);
  }

  mp_clear (&t);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_mod.c */

/* Start: bn_mp_mod_2d.c */
#include <tommath.h>
#ifdef BN_MP_MOD_2D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
4410
4411
4412
4413
4414
4415
4416




4417
4418
4419
4420
4421
4422
4423
  /* clear the digit that is not completely outside/inside the modulus */
  c->dp[b / DIGIT_BIT] &=
    (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digit) 1));
  mp_clamp (c);
  return MP_OKAY;
}
#endif





/* End: bn_mp_mod_2d.c */

/* Start: bn_mp_mod_d.c */
#include <tommath.h>
#ifdef BN_MP_MOD_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







4627
4628
4629
4630
4631
4632
4633
4634
4635
4636
4637
4638
4639
4640
4641
4642
4643
4644
  /* clear the digit that is not completely outside/inside the modulus */
  c->dp[b / DIGIT_BIT] &=
    (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digit) 1));
  mp_clamp (c);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_mod_2d.c */

/* Start: bn_mp_mod_d.c */
#include <tommath.h>
#ifdef BN_MP_MOD_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
4437
4438
4439
4440
4441
4442
4443




4444
4445
4446
4447
4448
4449
4450

int
mp_mod_d (mp_int * a, mp_digit b, mp_digit * c)
{
  return mp_div_d(a, b, NULL, c);
}
#endif





/* End: bn_mp_mod_d.c */

/* Start: bn_mp_montgomery_calc_normalization.c */
#include <tommath.h>
#ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







4658
4659
4660
4661
4662
4663
4664
4665
4666
4667
4668
4669
4670
4671
4672
4673
4674
4675

int
mp_mod_d (mp_int * a, mp_digit b, mp_digit * c)
{
  return mp_div_d(a, b, NULL, c);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_mod_d.c */

/* Start: bn_mp_montgomery_calc_normalization.c */
#include <tommath.h>
#ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
4496
4497
4498
4499
4500
4501
4502




4503
4504
4505
4506
4507
4508
4509
      }
    }
  }

  return MP_OKAY;
}
#endif





/* End: bn_mp_montgomery_calc_normalization.c */

/* Start: bn_mp_montgomery_reduce.c */
#include <tommath.h>
#ifdef BN_MP_MONTGOMERY_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







4721
4722
4723
4724
4725
4726
4727
4728
4729
4730
4731
4732
4733
4734
4735
4736
4737
4738
      }
    }
  }

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_montgomery_calc_normalization.c */

/* Start: bn_mp_montgomery_reduce.c */
#include <tommath.h>
#ifdef BN_MP_MONTGOMERY_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
4614
4615
4616
4617
4618
4619
4620




4621
4622
4623
4624
4625
4626
4627
  if (mp_cmp_mag (x, n) != MP_LT) {
    return s_mp_sub (x, n, x);
  }

  return MP_OKAY;
}
#endif





/* End: bn_mp_montgomery_reduce.c */

/* Start: bn_mp_montgomery_setup.c */
#include <tommath.h>
#ifdef BN_MP_MONTGOMERY_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







4843
4844
4845
4846
4847
4848
4849
4850
4851
4852
4853
4854
4855
4856
4857
4858
4859
4860
  if (mp_cmp_mag (x, n) != MP_LT) {
    return s_mp_sub (x, n, x);
  }

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_montgomery_reduce.c */

/* Start: bn_mp_montgomery_setup.c */
#include <tommath.h>
#ifdef BN_MP_MONTGOMERY_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
4673
4674
4675
4676
4677
4678
4679




4680
4681
4682
4683
4684
4685
4686

  /* rho = -1/m mod b */
  *rho = (unsigned long)(((mp_word)1 << ((mp_word) DIGIT_BIT)) - x) & MP_MASK;

  return MP_OKAY;
}
#endif





/* End: bn_mp_montgomery_setup.c */

/* Start: bn_mp_mul.c */
#include <tommath.h>
#ifdef BN_MP_MUL_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







4906
4907
4908
4909
4910
4911
4912
4913
4914
4915
4916
4917
4918
4919
4920
4921
4922
4923

  /* rho = -1/m mod b */
  *rho = (unsigned long)(((mp_word)1 << ((mp_word) DIGIT_BIT)) - x) & MP_MASK;

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_montgomery_setup.c */

/* Start: bn_mp_mul.c */
#include <tommath.h>
#ifdef BN_MP_MUL_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
4704
4705
4706
4707
4708
4709
4710
4711
4712
4713
4714
4715
4716
4717
4718
4719
4720
4721
4722
4723
4724
4725
4726
4727
4728
4729
4730
4731
4732
4733
4734
4735
4736
4737
4738
4739
4740
4741
4742
4743
4744
4745




4746
4747
4748
4749
4750
4751
4752
  int     res, neg;
  neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;

  /* use Toom-Cook? */
#ifdef BN_MP_TOOM_MUL_C
  if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) {
    res = mp_toom_mul(a, b, c);
  } else 
#endif
#ifdef BN_MP_KARATSUBA_MUL_C
  /* use Karatsuba? */
  if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) {
    res = mp_karatsuba_mul (a, b, c);
  } else 
#endif
  {
    /* can we use the fast multiplier?
     *
     * The fast multiplier can be used if the output will 
     * have less than MP_WARRAY digits and the number of 
     * digits won't affect carry propagation
     */
    int     digs = a->used + b->used + 1;

#ifdef BN_FAST_S_MP_MUL_DIGS_C
    if ((digs < MP_WARRAY) &&
        MIN(a->used, b->used) <= 
        (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) {
      res = fast_s_mp_mul_digs (a, b, c, digs);
    } else 
#endif
#ifdef BN_S_MP_MUL_DIGS_C
      res = s_mp_mul (a, b, c); /* uses s_mp_mul_digs */
#else
      res = MP_VAL;
#endif

  }
  c->sign = (c->used > 0) ? neg : MP_ZPOS;
  return res;
}
#endif





/* End: bn_mp_mul.c */

/* Start: bn_mp_mul_2.c */
#include <tommath.h>
#ifdef BN_MP_MUL_2_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|





|




|
|






|


|












>
>
>
>







4941
4942
4943
4944
4945
4946
4947
4948
4949
4950
4951
4952
4953
4954
4955
4956
4957
4958
4959
4960
4961
4962
4963
4964
4965
4966
4967
4968
4969
4970
4971
4972
4973
4974
4975
4976
4977
4978
4979
4980
4981
4982
4983
4984
4985
4986
4987
4988
4989
4990
4991
4992
4993
  int     res, neg;
  neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;

  /* use Toom-Cook? */
#ifdef BN_MP_TOOM_MUL_C
  if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) {
    res = mp_toom_mul(a, b, c);
  } else
#endif
#ifdef BN_MP_KARATSUBA_MUL_C
  /* use Karatsuba? */
  if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) {
    res = mp_karatsuba_mul (a, b, c);
  } else
#endif
  {
    /* can we use the fast multiplier?
     *
     * The fast multiplier can be used if the output will
     * have less than MP_WARRAY digits and the number of
     * digits won't affect carry propagation
     */
    int     digs = a->used + b->used + 1;

#ifdef BN_FAST_S_MP_MUL_DIGS_C
    if ((digs < MP_WARRAY) &&
        MIN(a->used, b->used) <=
        (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) {
      res = fast_s_mp_mul_digs (a, b, c, digs);
    } else
#endif
#ifdef BN_S_MP_MUL_DIGS_C
      res = s_mp_mul (a, b, c); /* uses s_mp_mul_digs */
#else
      res = MP_VAL;
#endif

  }
  c->sign = (c->used > 0) ? neg : MP_ZPOS;
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_mul.c */

/* Start: bn_mp_mul_2.c */
#include <tommath.h>
#ifdef BN_MP_MUL_2_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
4780
4781
4782
4783
4784
4785
4786
4787
4788
4789
4790
4791
4792
4793
4794
4795
4796
4797
4798
4799
4800
4801
4802
4803
4804
4805
4806
4807
4808
4809
4810
4811
4812
4813
4814
4815
4816
4817
4818
4819
4820
4821
4822
4823
4824
4825
4826
4827




4828
4829
4830
4831
4832
4833
4834
  b->used = a->used;

  {
    register mp_digit r, rr, *tmpa, *tmpb;

    /* alias for source */
    tmpa = a->dp;
    
    /* alias for dest */
    tmpb = b->dp;

    /* carry */
    r = 0;
    for (x = 0; x < a->used; x++) {
    
      /* get what will be the *next* carry bit from the 
       * MSB of the current digit 
       */
      rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1));
      
      /* now shift up this digit, add in the carry [from the previous] */
      *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK;
      
      /* copy the carry that would be from the source 
       * digit into the next iteration 
       */
      r = rr;
    }

    /* new leading digit? */
    if (r != 0) {
      /* add a MSB which is always 1 at this point */
      *tmpb = 1;
      ++(b->used);
    }

    /* now zero any excess digits on the destination 
     * that we didn't write to 
     */
    tmpb = b->dp + b->used;
    for (x = b->used; x < oldused; x++) {
      *tmpb++ = 0;
    }
  }
  b->sign = a->sign;
  return MP_OKAY;
}
#endif





/* End: bn_mp_mul_2.c */

/* Start: bn_mp_mul_2d.c */
#include <tommath.h>
#ifdef BN_MP_MUL_2D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|






|
|
|


|


|
|
|











|
|










>
>
>
>







5021
5022
5023
5024
5025
5026
5027
5028
5029
5030
5031
5032
5033
5034
5035
5036
5037
5038
5039
5040
5041
5042
5043
5044
5045
5046
5047
5048
5049
5050
5051
5052
5053
5054
5055
5056
5057
5058
5059
5060
5061
5062
5063
5064
5065
5066
5067
5068
5069
5070
5071
5072
5073
5074
5075
5076
5077
5078
5079
  b->used = a->used;

  {
    register mp_digit r, rr, *tmpa, *tmpb;

    /* alias for source */
    tmpa = a->dp;

    /* alias for dest */
    tmpb = b->dp;

    /* carry */
    r = 0;
    for (x = 0; x < a->used; x++) {

      /* get what will be the *next* carry bit from the
       * MSB of the current digit
       */
      rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1));

      /* now shift up this digit, add in the carry [from the previous] */
      *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK;

      /* copy the carry that would be from the source
       * digit into the next iteration
       */
      r = rr;
    }

    /* new leading digit? */
    if (r != 0) {
      /* add a MSB which is always 1 at this point */
      *tmpb = 1;
      ++(b->used);
    }

    /* now zero any excess digits on the destination
     * that we didn't write to
     */
    tmpb = b->dp + b->used;
    for (x = b->used; x < oldused; x++) {
      *tmpb++ = 0;
    }
  }
  b->sign = a->sign;
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_mul_2.c */

/* Start: bn_mp_mul_2d.c */
#include <tommath.h>
#ifdef BN_MP_MUL_2D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
4896
4897
4898
4899
4900
4901
4902
4903
4904
4905
4906
4907
4908
4909
4910
4911
4912




4913
4914
4915
4916
4917
4918
4919
      /* shift the current word and OR in the carry */
      *tmpc = ((*tmpc << d) | r) & MP_MASK;
      ++tmpc;

      /* set the carry to the carry bits of the current word */
      r = rr;
    }
    
    /* set final carry */
    if (r != 0) {
       c->dp[(c->used)++] = r;
    }
  }
  mp_clamp (c);
  return MP_OKAY;
}
#endif





/* End: bn_mp_mul_2d.c */

/* Start: bn_mp_mul_d.c */
#include <tommath.h>
#ifdef BN_MP_MUL_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|









>
>
>
>







5141
5142
5143
5144
5145
5146
5147
5148
5149
5150
5151
5152
5153
5154
5155
5156
5157
5158
5159
5160
5161
5162
5163
5164
5165
5166
5167
5168
      /* shift the current word and OR in the carry */
      *tmpc = ((*tmpc << d) | r) & MP_MASK;
      ++tmpc;

      /* set the carry to the carry bits of the current word */
      r = rr;
    }

    /* set final carry */
    if (r != 0) {
       c->dp[(c->used)++] = r;
    }
  }
  mp_clamp (c);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_mul_2d.c */

/* Start: bn_mp_mul_d.c */
#include <tommath.h>
#ifdef BN_MP_MUL_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
4985
4986
4987
4988
4989
4990
4991




4992
4993
4994
4995
4996
4997
4998
  /* set used count */
  c->used = a->used + 1;
  mp_clamp(c);

  return MP_OKAY;
}
#endif





/* End: bn_mp_mul_d.c */

/* Start: bn_mp_mulmod.c */
#include <tommath.h>
#ifdef BN_MP_MULMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







5234
5235
5236
5237
5238
5239
5240
5241
5242
5243
5244
5245
5246
5247
5248
5249
5250
5251
  /* set used count */
  c->used = a->used + 1;
  mp_clamp(c);

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_mul_d.c */

/* Start: bn_mp_mulmod.c */
#include <tommath.h>
#ifdef BN_MP_MULMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
5025
5026
5027
5028
5029
5030
5031




5032
5033
5034
5035
5036
5037
5038
    return res;
  }
  res = mp_mod (&t, c, d);
  mp_clear (&t);
  return res;
}
#endif





/* End: bn_mp_mulmod.c */

/* Start: bn_mp_n_root.c */
#include <tommath.h>
#ifdef BN_MP_N_ROOT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







5278
5279
5280
5281
5282
5283
5284
5285
5286
5287
5288
5289
5290
5291
5292
5293
5294
5295
    return res;
  }
  res = mp_mod (&t, c, d);
  mp_clear (&t);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_mulmod.c */

/* Start: bn_mp_n_root.c */
#include <tommath.h>
#ifdef BN_MP_N_ROOT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
5046
5047
5048
5049
5050
5051
5052
5053
5054
5055
5056
5057
5058
5059
5060
5061
5062
5063
5064
5065
5066
5067
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* find the n'th root of an integer 
 *
 * Result found such that (c)**b <= a and (c+1)**b > a 
 *
 * This algorithm uses Newton's approximation 
 * x[i+1] = x[i] - f(x[i])/f'(x[i]) 
 * which will find the root in log(N) time where 
 * each step involves a fair bit.  This is not meant to 
 * find huge roots [square and cube, etc].
 */
int mp_n_root (mp_int * a, mp_digit b, mp_int * c)
{
  mp_int  t1, t2, t3;
  int     res, neg;








|

|

|
|
|
|







5303
5304
5305
5306
5307
5308
5309
5310
5311
5312
5313
5314
5315
5316
5317
5318
5319
5320
5321
5322
5323
5324
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* find the n'th root of an integer
 *
 * Result found such that (c)**b <= a and (c+1)**b > a
 *
 * This algorithm uses Newton's approximation
 * x[i+1] = x[i] - f(x[i])/f'(x[i])
 * which will find the root in log(N) time where
 * each step involves a fair bit.  This is not meant to
 * find huge roots [square and cube, etc].
 */
int mp_n_root (mp_int * a, mp_digit b, mp_int * c)
{
  mp_int  t1, t2, t3;
  int     res, neg;

5092
5093
5094
5095
5096
5097
5098
5099
5100
5101
5102
5103
5104
5105
5106
5107
5108
5109
5110
5111
5112
5113
5114
5115
5116
5117
5118
5119
5120
5121
5122
5123
5124
5125
5126
5127
5128
5129
5130
  do {
    /* t1 = t2 */
    if ((res = mp_copy (&t2, &t1)) != MP_OKAY) {
      goto LBL_T3;
    }

    /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */
    
    /* t3 = t1**(b-1) */
    if ((res = mp_expt_d (&t1, b - 1, &t3)) != MP_OKAY) {   
      goto LBL_T3;
    }

    /* numerator */
    /* t2 = t1**b */
    if ((res = mp_mul (&t3, &t1, &t2)) != MP_OKAY) {    
      goto LBL_T3;
    }

    /* t2 = t1**b - a */
    if ((res = mp_sub (&t2, a, &t2)) != MP_OKAY) {  
      goto LBL_T3;
    }

    /* denominator */
    /* t3 = t1**(b-1) * b  */
    if ((res = mp_mul_d (&t3, b, &t3)) != MP_OKAY) {    
      goto LBL_T3;
    }

    /* t3 = (t1**b - a)/(b * t1**(b-1)) */
    if ((res = mp_div (&t2, &t3, &t3, NULL)) != MP_OKAY) {  
      goto LBL_T3;
    }

    if ((res = mp_sub (&t1, &t3, &t2)) != MP_OKAY) {
      goto LBL_T3;
    }
  }  while (mp_cmp (&t1, &t2) != MP_EQ);







|

|





|




|





|




|







5349
5350
5351
5352
5353
5354
5355
5356
5357
5358
5359
5360
5361
5362
5363
5364
5365
5366
5367
5368
5369
5370
5371
5372
5373
5374
5375
5376
5377
5378
5379
5380
5381
5382
5383
5384
5385
5386
5387
  do {
    /* t1 = t2 */
    if ((res = mp_copy (&t2, &t1)) != MP_OKAY) {
      goto LBL_T3;
    }

    /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */

    /* t3 = t1**(b-1) */
    if ((res = mp_expt_d (&t1, b - 1, &t3)) != MP_OKAY) {
      goto LBL_T3;
    }

    /* numerator */
    /* t2 = t1**b */
    if ((res = mp_mul (&t3, &t1, &t2)) != MP_OKAY) {
      goto LBL_T3;
    }

    /* t2 = t1**b - a */
    if ((res = mp_sub (&t2, a, &t2)) != MP_OKAY) {
      goto LBL_T3;
    }

    /* denominator */
    /* t3 = t1**(b-1) * b  */
    if ((res = mp_mul_d (&t3, b, &t3)) != MP_OKAY) {
      goto LBL_T3;
    }

    /* t3 = (t1**b - a)/(b * t1**(b-1)) */
    if ((res = mp_div (&t2, &t3, &t3, NULL)) != MP_OKAY) {
      goto LBL_T3;
    }

    if ((res = mp_sub (&t1, &t3, &t2)) != MP_OKAY) {
      goto LBL_T3;
    }
  }  while (mp_cmp (&t1, &t2) != MP_EQ);
5157
5158
5159
5160
5161
5162
5163




5164
5165
5166
5167
5168
5169
5170

LBL_T3:mp_clear (&t3);
LBL_T2:mp_clear (&t2);
LBL_T1:mp_clear (&t1);
  return res;
}
#endif





/* End: bn_mp_n_root.c */

/* Start: bn_mp_neg.c */
#include <tommath.h>
#ifdef BN_MP_NEG_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







5414
5415
5416
5417
5418
5419
5420
5421
5422
5423
5424
5425
5426
5427
5428
5429
5430
5431

LBL_T3:mp_clear (&t3);
LBL_T2:mp_clear (&t2);
LBL_T1:mp_clear (&t1);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_n_root.c */

/* Start: bn_mp_neg.c */
#include <tommath.h>
#ifdef BN_MP_NEG_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
5197
5198
5199
5200
5201
5202
5203




5204
5205
5206
5207
5208
5209
5210
  } else {
     b->sign = MP_ZPOS;
  }

  return MP_OKAY;
}
#endif





/* End: bn_mp_neg.c */

/* Start: bn_mp_or.c */
#include <tommath.h>
#ifdef BN_MP_OR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







5458
5459
5460
5461
5462
5463
5464
5465
5466
5467
5468
5469
5470
5471
5472
5473
5474
5475
  } else {
     b->sign = MP_ZPOS;
  }

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_neg.c */

/* Start: bn_mp_or.c */
#include <tommath.h>
#ifdef BN_MP_OR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
5247
5248
5249
5250
5251
5252
5253




5254
5255
5256
5257
5258
5259
5260
  }
  mp_clamp (&t);
  mp_exch (c, &t);
  mp_clear (&t);
  return MP_OKAY;
}
#endif





/* End: bn_mp_or.c */

/* Start: bn_mp_prime_fermat.c */
#include <tommath.h>
#ifdef BN_MP_PRIME_FERMAT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







5512
5513
5514
5515
5516
5517
5518
5519
5520
5521
5522
5523
5524
5525
5526
5527
5528
5529
  }
  mp_clamp (&t);
  mp_exch (c, &t);
  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_or.c */

/* Start: bn_mp_prime_fermat.c */
#include <tommath.h>
#ifdef BN_MP_PRIME_FERMAT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
5269
5270
5271
5272
5273
5274
5275
5276
5277
5278
5279
5280
5281
5282
5283
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* performs one Fermat test.
 * 
 * If "a" were prime then b**a == b (mod a) since the order of
 * the multiplicative sub-group would be phi(a) = a-1.  That means
 * it would be the same as b**(a mod (a-1)) == b**1 == b (mod a).
 *
 * Sets result to 1 if the congruence holds, or zero otherwise.
 */
int mp_prime_fermat (mp_int * a, mp_int * b, int *result)







|







5538
5539
5540
5541
5542
5543
5544
5545
5546
5547
5548
5549
5550
5551
5552
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* performs one Fermat test.
 *
 * If "a" were prime then b**a == b (mod a) since the order of
 * the multiplicative sub-group would be phi(a) = a-1.  That means
 * it would be the same as b**(a mod (a-1)) == b**1 == b (mod a).
 *
 * Sets result to 1 if the congruence holds, or zero otherwise.
 */
int mp_prime_fermat (mp_int * a, mp_int * b, int *result)
5309
5310
5311
5312
5313
5314
5315




5316
5317
5318
5319
5320
5321
5322
  }

  err = MP_OKAY;
LBL_T:mp_clear (&t);
  return err;
}
#endif





/* End: bn_mp_prime_fermat.c */

/* Start: bn_mp_prime_is_divisible.c */
#include <tommath.h>
#ifdef BN_MP_PRIME_IS_DIVISIBLE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







5578
5579
5580
5581
5582
5583
5584
5585
5586
5587
5588
5589
5590
5591
5592
5593
5594
5595
  }

  err = MP_OKAY;
LBL_T:mp_clear (&t);
  return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_prime_fermat.c */

/* Start: bn_mp_prime_is_divisible.c */
#include <tommath.h>
#ifdef BN_MP_PRIME_IS_DIVISIBLE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
5330
5331
5332
5333
5334
5335
5336
5337
5338
5339
5340
5341
5342
5343
5344
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* determines if an integers is divisible by one 
 * of the first PRIME_SIZE primes or not
 *
 * sets result to 0 if not, 1 if yes
 */
int mp_prime_is_divisible (mp_int * a, int *result)
{
  int     err, ix;







|







5603
5604
5605
5606
5607
5608
5609
5610
5611
5612
5613
5614
5615
5616
5617
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* determines if an integers is divisible by one
 * of the first PRIME_SIZE primes or not
 *
 * sets result to 0 if not, 1 if yes
 */
int mp_prime_is_divisible (mp_int * a, int *result)
{
  int     err, ix;
5359
5360
5361
5362
5363
5364
5365




5366
5367
5368
5369
5370
5371
5372
      return MP_OKAY;
    }
  }

  return MP_OKAY;
}
#endif





/* End: bn_mp_prime_is_divisible.c */

/* Start: bn_mp_prime_is_prime.c */
#include <tommath.h>
#ifdef BN_MP_PRIME_IS_PRIME_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







5632
5633
5634
5635
5636
5637
5638
5639
5640
5641
5642
5643
5644
5645
5646
5647
5648
5649
      return MP_OKAY;
    }
  }

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_prime_is_divisible.c */

/* Start: bn_mp_prime_is_prime.c */
#include <tommath.h>
#ifdef BN_MP_PRIME_IS_PRIME_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
5442
5443
5444
5445
5446
5447
5448




5449
5450
5451
5452
5453
5454
5455

  /* passed the test */
  *result = MP_YES;
LBL_B:mp_clear (&b);
  return err;
}
#endif





/* End: bn_mp_prime_is_prime.c */

/* Start: bn_mp_prime_miller_rabin.c */
#include <tommath.h>
#ifdef BN_MP_PRIME_MILLER_RABIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







5719
5720
5721
5722
5723
5724
5725
5726
5727
5728
5729
5730
5731
5732
5733
5734
5735
5736

  /* passed the test */
  *result = MP_YES;
LBL_B:mp_clear (&b);
  return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_prime_is_prime.c */

/* Start: bn_mp_prime_miller_rabin.c */
#include <tommath.h>
#ifdef BN_MP_PRIME_MILLER_RABIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
5463
5464
5465
5466
5467
5468
5469
5470
5471
5472
5473
5474
5475
5476
5477
5478
5479
5480
5481
5482
5483
5484
5485
5486
5487
5488
5489
5490
5491
5492
5493
5494
5495
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* Miller-Rabin test of "a" to the base of "b" as described in 
 * HAC pp. 139 Algorithm 4.24
 *
 * Sets result to 0 if definitely composite or 1 if probably prime.
 * Randomly the chance of error is no more than 1/4 and often 
 * very much lower.
 */
int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result)
{
  mp_int  n1, y, r;
  int     s, j, err;

  /* default */
  *result = MP_NO;

  /* ensure b > 1 */
  if (mp_cmp_d(b, 1) != MP_GT) {
     return MP_VAL;
  }     

  /* get n1 = a - 1 */
  if ((err = mp_init_copy (&n1, a)) != MP_OKAY) {
    return err;
  }
  if ((err = mp_sub_d (&n1, 1, &n1)) != MP_OKAY) {
    goto LBL_N1;







|



|













|







5744
5745
5746
5747
5748
5749
5750
5751
5752
5753
5754
5755
5756
5757
5758
5759
5760
5761
5762
5763
5764
5765
5766
5767
5768
5769
5770
5771
5772
5773
5774
5775
5776
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* Miller-Rabin test of "a" to the base of "b" as described in
 * HAC pp. 139 Algorithm 4.24
 *
 * Sets result to 0 if definitely composite or 1 if probably prime.
 * Randomly the chance of error is no more than 1/4 and often
 * very much lower.
 */
int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result)
{
  mp_int  n1, y, r;
  int     s, j, err;

  /* default */
  *result = MP_NO;

  /* ensure b > 1 */
  if (mp_cmp_d(b, 1) != MP_GT) {
     return MP_VAL;
  }

  /* get n1 = a - 1 */
  if ((err = mp_init_copy (&n1, a)) != MP_OKAY) {
    return err;
  }
  if ((err = mp_sub_d (&n1, 1, &n1)) != MP_OKAY) {
    goto LBL_N1;
5545
5546
5547
5548
5549
5550
5551




5552
5553
5554
5555
5556
5557
5558
  *result = MP_YES;
LBL_Y:mp_clear (&y);
LBL_R:mp_clear (&r);
LBL_N1:mp_clear (&n1);
  return err;
}
#endif





/* End: bn_mp_prime_miller_rabin.c */

/* Start: bn_mp_prime_next_prime.c */
#include <tommath.h>
#ifdef BN_MP_PRIME_NEXT_PRIME_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







5826
5827
5828
5829
5830
5831
5832
5833
5834
5835
5836
5837
5838
5839
5840
5841
5842
5843
  *result = MP_YES;
LBL_Y:mp_clear (&y);
LBL_R:mp_clear (&r);
LBL_N1:mp_clear (&n1);
  return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_prime_miller_rabin.c */

/* Start: bn_mp_prime_next_prime.c */
#include <tommath.h>
#ifdef BN_MP_PRIME_NEXT_PRIME_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
5715
5716
5717
5718
5719
5720
5721




5722
5723
5724
5725
5726
5727
5728
   err = MP_OKAY;
LBL_ERR:
   mp_clear(&b);
   return err;
}

#endif





/* End: bn_mp_prime_next_prime.c */

/* Start: bn_mp_prime_rabin_miller_trials.c */
#include <tommath.h>
#ifdef BN_MP_PRIME_RABIN_MILLER_TRIALS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







6000
6001
6002
6003
6004
6005
6006
6007
6008
6009
6010
6011
6012
6013
6014
6015
6016
6017
   err = MP_OKAY;
LBL_ERR:
   mp_clear(&b);
   return err;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_prime_next_prime.c */

/* Start: bn_mp_prime_rabin_miller_trials.c */
#include <tommath.h>
#ifdef BN_MP_PRIME_RABIN_MILLER_TRIALS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
5767
5768
5769
5770
5771
5772
5773




5774
5775
5776
5777
5778
5779
5780
       }
   }
   return sizes[x-1].t + 1;
}


#endif





/* End: bn_mp_prime_rabin_miller_trials.c */

/* Start: bn_mp_prime_random_ex.c */
#include <tommath.h>
#ifdef BN_MP_PRIME_RANDOM_EX_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







6056
6057
6058
6059
6060
6061
6062
6063
6064
6065
6066
6067
6068
6069
6070
6071
6072
6073
       }
   }
   return sizes[x-1].t + 1;
}


#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_prime_rabin_miller_trials.c */

/* Start: bn_mp_prime_random_ex.c */
#include <tommath.h>
#ifdef BN_MP_PRIME_RANDOM_EX_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
5791
5792
5793
5794
5795
5796
5797
5798
5799
5800
5801
5802
5803
5804
5805
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* makes a truly random prime of a given size (bits),
 *
 * Flags are as follows:
 * 
 *   LTM_PRIME_BBS      - make prime congruent to 3 mod 4
 *   LTM_PRIME_SAFE     - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS)
 *   LTM_PRIME_2MSB_OFF - make the 2nd highest bit zero
 *   LTM_PRIME_2MSB_ON  - make the 2nd highest bit one
 *
 * You have to supply a callback which fills in a buffer with random bytes.  "dat" is a parameter you can
 * have passed to the callback (e.g. a state or something).  This function doesn't use "dat" itself







|







6084
6085
6086
6087
6088
6089
6090
6091
6092
6093
6094
6095
6096
6097
6098
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* makes a truly random prime of a given size (bits),
 *
 * Flags are as follows:
 *
 *   LTM_PRIME_BBS      - make prime congruent to 3 mod 4
 *   LTM_PRIME_SAFE     - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS)
 *   LTM_PRIME_2MSB_OFF - make the 2nd highest bit zero
 *   LTM_PRIME_2MSB_ON  - make the 2nd highest bit one
 *
 * You have to supply a callback which fills in a buffer with random bytes.  "dat" is a parameter you can
 * have passed to the callback (e.g. a state or something).  This function doesn't use "dat" itself
5836
5837
5838
5839
5840
5841
5842
5843
5844
5845
5846
5847
5848
5849
5850
5851
5852
5853
5854
5855
5856
5857
5858
5859
5860
5861
5862
5863
5864
5865
5866
5867
5868
5869
5870
5871
5872
5873
5874
5875
5876
5877
5878
5879
5880
5881
5882
5883
5884
5885
5886
5887
5888
5889
5890
5891
5892
5893
5894
5895
5896
5897
5898




5899
5900
5901
5902
5903
5904
5905
   maskAND = ((size&7) == 0) ? 0xFF : (0xFF >> (8 - (size & 7)));

   /* calc the maskOR_msb */
   maskOR_msb        = 0;
   maskOR_msb_offset = ((size & 7) == 1) ? 1 : 0;
   if (flags & LTM_PRIME_2MSB_ON) {
      maskOR_msb       |= 0x80 >> ((9 - size) & 7);
   }  

   /* get the maskOR_lsb */
   maskOR_lsb         = 1;
   if (flags & LTM_PRIME_BBS) {
      maskOR_lsb     |= 3;
   }

   do {
      /* read the bytes */
      if (cb(tmp, bsize, dat) != bsize) {
         err = MP_VAL;
         goto error;
      }
 
      /* work over the MSbyte */
      tmp[0]    &= maskAND;
      tmp[0]    |= 1 << ((size - 1) & 7);

      /* mix in the maskORs */
      tmp[maskOR_msb_offset]   |= maskOR_msb;
      tmp[bsize-1]             |= maskOR_lsb;

      /* read it in */
      if ((err = mp_read_unsigned_bin(a, tmp, bsize)) != MP_OKAY)     { goto error; }

      /* is it prime? */
      if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY)           { goto error; }
      if (res == MP_NO) {  
         continue;
      }

      if (flags & LTM_PRIME_SAFE) {
         /* see if (a-1)/2 is prime */
         if ((err = mp_sub_d(a, 1, a)) != MP_OKAY)                    { goto error; }
         if ((err = mp_div_2(a, a)) != MP_OKAY)                       { goto error; }
 
         /* is it prime? */
         if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY)        { goto error; }
      }
   } while (res == MP_NO);

   if (flags & LTM_PRIME_SAFE) {
      /* restore a to the original value */
      if ((err = mp_mul_2(a, a)) != MP_OKAY)                          { goto error; }
      if ((err = mp_add_d(a, 1, a)) != MP_OKAY)                       { goto error; }
   }

   err = MP_OKAY;
error:
   XFREE(tmp);
   return err;
}


#endif





/* End: bn_mp_prime_random_ex.c */

/* Start: bn_mp_radix_size.c */
#include <tommath.h>
#ifdef BN_MP_RADIX_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|













|













|







|



















>
>
>
>







6129
6130
6131
6132
6133
6134
6135
6136
6137
6138
6139
6140
6141
6142
6143
6144
6145
6146
6147
6148
6149
6150
6151
6152
6153
6154
6155
6156
6157
6158
6159
6160
6161
6162
6163
6164
6165
6166
6167
6168
6169
6170
6171
6172
6173
6174
6175
6176
6177
6178
6179
6180
6181
6182
6183
6184
6185
6186
6187
6188
6189
6190
6191
6192
6193
6194
6195
6196
6197
6198
6199
6200
6201
6202
   maskAND = ((size&7) == 0) ? 0xFF : (0xFF >> (8 - (size & 7)));

   /* calc the maskOR_msb */
   maskOR_msb        = 0;
   maskOR_msb_offset = ((size & 7) == 1) ? 1 : 0;
   if (flags & LTM_PRIME_2MSB_ON) {
      maskOR_msb       |= 0x80 >> ((9 - size) & 7);
   }

   /* get the maskOR_lsb */
   maskOR_lsb         = 1;
   if (flags & LTM_PRIME_BBS) {
      maskOR_lsb     |= 3;
   }

   do {
      /* read the bytes */
      if (cb(tmp, bsize, dat) != bsize) {
         err = MP_VAL;
         goto error;
      }

      /* work over the MSbyte */
      tmp[0]    &= maskAND;
      tmp[0]    |= 1 << ((size - 1) & 7);

      /* mix in the maskORs */
      tmp[maskOR_msb_offset]   |= maskOR_msb;
      tmp[bsize-1]             |= maskOR_lsb;

      /* read it in */
      if ((err = mp_read_unsigned_bin(a, tmp, bsize)) != MP_OKAY)     { goto error; }

      /* is it prime? */
      if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY)           { goto error; }
      if (res == MP_NO) {
         continue;
      }

      if (flags & LTM_PRIME_SAFE) {
         /* see if (a-1)/2 is prime */
         if ((err = mp_sub_d(a, 1, a)) != MP_OKAY)                    { goto error; }
         if ((err = mp_div_2(a, a)) != MP_OKAY)                       { goto error; }

         /* is it prime? */
         if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY)        { goto error; }
      }
   } while (res == MP_NO);

   if (flags & LTM_PRIME_SAFE) {
      /* restore a to the original value */
      if ((err = mp_mul_2(a, a)) != MP_OKAY)                          { goto error; }
      if ((err = mp_add_d(a, 1, a)) != MP_OKAY)                       { goto error; }
   }

   err = MP_OKAY;
error:
   XFREE(tmp);
   return err;
}


#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_prime_random_ex.c */

/* Start: bn_mp_radix_size.c */
#include <tommath.h>
#ifdef BN_MP_RADIX_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
5952
5953
5954
5955
5956
5957
5958
5959
5960
5961
5962
5963
5964
5965
5966
5967
5968
5969
5970
5971
5972
5973
5974
5975
5976




5977
5978
5979
5980
5981
5982
5983

  /* init a copy of the input */
  if ((res = mp_init_copy (&t, a)) != MP_OKAY) {
    return res;
  }

  /* force temp to positive */
  t.sign = MP_ZPOS; 

  /* fetch out all of the digits */
  while (mp_iszero (&t) == MP_NO) {
    if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) {
      mp_clear (&t);
      return res;
    }
    ++digs;
  }
  mp_clear (&t);

  /* return digs + 1, the 1 is for the NULL byte that would be required. */
  *size = digs + 1;
  return MP_OKAY;
}

#endif





/* End: bn_mp_radix_size.c */

/* Start: bn_mp_radix_smap.c */
#include <tommath.h>
#ifdef BN_MP_RADIX_SMAP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|

















>
>
>
>







6249
6250
6251
6252
6253
6254
6255
6256
6257
6258
6259
6260
6261
6262
6263
6264
6265
6266
6267
6268
6269
6270
6271
6272
6273
6274
6275
6276
6277
6278
6279
6280
6281
6282
6283
6284

  /* init a copy of the input */
  if ((res = mp_init_copy (&t, a)) != MP_OKAY) {
    return res;
  }

  /* force temp to positive */
  t.sign = MP_ZPOS;

  /* fetch out all of the digits */
  while (mp_iszero (&t) == MP_NO) {
    if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) {
      mp_clear (&t);
      return res;
    }
    ++digs;
  }
  mp_clear (&t);

  /* return digs + 1, the 1 is for the NULL byte that would be required. */
  *size = digs + 1;
  return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_radix_size.c */

/* Start: bn_mp_radix_smap.c */
#include <tommath.h>
#ifdef BN_MP_RADIX_SMAP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
5994
5995
5996
5997
5998
5999
6000




6001
6002
6003
6004
6005
6006
6007
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* chars used in radix conversions */
const char *mp_s_rmap = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/";
#endif





/* End: bn_mp_radix_smap.c */

/* Start: bn_mp_rand.c */
#include <tommath.h>
#ifdef BN_MP_RAND_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







6295
6296
6297
6298
6299
6300
6301
6302
6303
6304
6305
6306
6307
6308
6309
6310
6311
6312
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* chars used in radix conversions */
const char *mp_s_rmap = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/";
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_radix_smap.c */

/* Start: bn_mp_rand.c */
#include <tommath.h>
#ifdef BN_MP_RAND_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6049
6050
6051
6052
6053
6054
6055




6056
6057
6058
6059
6060
6061
6062
      return res;
    }
  }

  return MP_OKAY;
}
#endif





/* End: bn_mp_rand.c */

/* Start: bn_mp_read_radix.c */
#include <tommath.h>
#ifdef BN_MP_READ_RADIX_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







6354
6355
6356
6357
6358
6359
6360
6361
6362
6363
6364
6365
6366
6367
6368
6369
6370
6371
      return res;
    }
  }

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_rand.c */

/* Start: bn_mp_read_radix.c */
#include <tommath.h>
#ifdef BN_MP_READ_RADIX_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6084
6085
6086
6087
6088
6089
6090
6091
6092
6093
6094
6095
6096
6097
6098
6099
6100
6101
6102
6103
6104
6105
6106
6107
6108
6109
6110
6111
6112
6113
6114
6115
6116
6117
6118
6119
6120
6121
6122
6123
6124
6125
6126
6127
6128
6129
6130
6131
6132
6133
6134
6135
6136
6137
6138
6139
6140




6141
6142
6143
6144
6145
6146
6147
  mp_zero(a);

  /* make sure the radix is ok */
  if (radix < 2 || radix > 64) {
    return MP_VAL;
  }

  /* if the leading digit is a 
   * minus set the sign to negative. 
   */
  if (*str == '-') {
    ++str;
    neg = MP_NEG;
  } else {
    neg = MP_ZPOS;
  }

  /* set the integer to the default of zero */
  mp_zero (a);
  
  /* process each digit of the string */
  while (*str) {
    /* if the radix < 36 the conversion is case insensitive
     * this allows numbers like 1AB and 1ab to represent the same  value
     * [e.g. in hex]
     */
    ch = (char) ((radix < 36) ? toupper (*str) : *str);
    for (y = 0; y < 64; y++) {
      if (ch == mp_s_rmap[y]) {
         break;
      }
    }

    /* if the char was found in the map 
     * and is less than the given radix add it
     * to the number, otherwise exit the loop. 
     */
    if (y < radix) {
      if ((res = mp_mul_d (a, (mp_digit) radix, a)) != MP_OKAY) {
         return res;
      }
      if ((res = mp_add_d (a, (mp_digit) y, a)) != MP_OKAY) {
         return res;
      }
    } else {
      break;
    }
    ++str;
  }
  
  /* set the sign only if a != 0 */
  if (mp_iszero(a) != 1) {
     a->sign = neg;
  }
  return MP_OKAY;
}
#endif





/* End: bn_mp_read_radix.c */

/* Start: bn_mp_read_signed_bin.c */
#include <tommath.h>
#ifdef BN_MP_READ_SIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|
|










|






|






|

|













|







>
>
>
>







6393
6394
6395
6396
6397
6398
6399
6400
6401
6402
6403
6404
6405
6406
6407
6408
6409
6410
6411
6412
6413
6414
6415
6416
6417
6418
6419
6420
6421
6422
6423
6424
6425
6426
6427
6428
6429
6430
6431
6432
6433
6434
6435
6436
6437
6438
6439
6440
6441
6442
6443
6444
6445
6446
6447
6448
6449
6450
6451
6452
6453
6454
6455
6456
6457
6458
6459
6460
  mp_zero(a);

  /* make sure the radix is ok */
  if (radix < 2 || radix > 64) {
    return MP_VAL;
  }

  /* if the leading digit is a
   * minus set the sign to negative.
   */
  if (*str == '-') {
    ++str;
    neg = MP_NEG;
  } else {
    neg = MP_ZPOS;
  }

  /* set the integer to the default of zero */
  mp_zero (a);

  /* process each digit of the string */
  while (*str) {
    /* if the radix < 36 the conversion is case insensitive
     * this allows numbers like 1AB and 1ab to represent the same  value
     * [e.g. in hex]
     */
    ch = (char) ((radix < 36) ? toupper ((int)*str) : *str);
    for (y = 0; y < 64; y++) {
      if (ch == mp_s_rmap[y]) {
         break;
      }
    }

    /* if the char was found in the map
     * and is less than the given radix add it
     * to the number, otherwise exit the loop.
     */
    if (y < radix) {
      if ((res = mp_mul_d (a, (mp_digit) radix, a)) != MP_OKAY) {
         return res;
      }
      if ((res = mp_add_d (a, (mp_digit) y, a)) != MP_OKAY) {
         return res;
      }
    } else {
      break;
    }
    ++str;
  }

  /* set the sign only if a != 0 */
  if (mp_iszero(a) != 1) {
     a->sign = neg;
  }
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_read_radix.c */

/* Start: bn_mp_read_signed_bin.c */
#include <tommath.h>
#ifdef BN_MP_READ_SIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6175
6176
6177
6178
6179
6180
6181




6182
6183
6184
6185
6186
6187
6188
  } else {
     a->sign = MP_NEG;
  }

  return MP_OKAY;
}
#endif





/* End: bn_mp_read_signed_bin.c */

/* Start: bn_mp_read_unsigned_bin.c */
#include <tommath.h>
#ifdef BN_MP_READ_UNSIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







6488
6489
6490
6491
6492
6493
6494
6495
6496
6497
6498
6499
6500
6501
6502
6503
6504
6505
  } else {
     a->sign = MP_NEG;
  }

  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_read_signed_bin.c */

/* Start: bn_mp_read_unsigned_bin.c */
#include <tommath.h>
#ifdef BN_MP_READ_UNSIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6230
6231
6232
6233
6234
6235
6236




6237
6238
6239
6240
6241
6242
6243
      a->used += 2;
#endif
  }
  mp_clamp (a);
  return MP_OKAY;
}
#endif





/* End: bn_mp_read_unsigned_bin.c */

/* Start: bn_mp_reduce.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







6547
6548
6549
6550
6551
6552
6553
6554
6555
6556
6557
6558
6559
6560
6561
6562
6563
6564
      a->used += 2;
#endif
  }
  mp_clamp (a);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_read_unsigned_bin.c */

/* Start: bn_mp_reduce.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6251
6252
6253
6254
6255
6256
6257
6258
6259
6260
6261
6262
6263
6264
6265
6266
6267
6268
6269
6270
6271
6272
6273
6274
6275
6276
6277
6278
6279
6280
6281
6282
6283
6284
6285
6286
6287
6288
6289
6290
6291
6292
6293
6294
6295
6296
6297
6298
6299
6300
6301
6302
6303
6304
6305
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* reduces x mod m, assumes 0 < x < m**2, mu is 
 * precomputed via mp_reduce_setup.
 * From HAC pp.604 Algorithm 14.42
 */
int mp_reduce (mp_int * x, mp_int * m, mp_int * mu)
{
  mp_int  q;
  int     res, um = m->used;

  /* q = x */
  if ((res = mp_init_copy (&q, x)) != MP_OKAY) {
    return res;
  }

  /* q1 = x / b**(k-1)  */
  mp_rshd (&q, um - 1);         

  /* according to HAC this optimization is ok */
  if (((unsigned long) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) {
    if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) {
      goto CLEANUP;
    }
  } else {
#ifdef BN_S_MP_MUL_HIGH_DIGS_C
    if ((res = s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) {
      goto CLEANUP;
    }
#elif defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C)
    if ((res = fast_s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) {
      goto CLEANUP;
    }
#else 
    { 
      res = MP_VAL;
      goto CLEANUP;
    }
#endif
  }

  /* q3 = q2 / b**(k+1) */
  mp_rshd (&q, um + 1);         

  /* x = x mod b**(k+1), quick (no division) */
  if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) {
    goto CLEANUP;
  }

  /* q = q * m mod b**(k+1), quick (no division) */







|














|















|
|







|







6572
6573
6574
6575
6576
6577
6578
6579
6580
6581
6582
6583
6584
6585
6586
6587
6588
6589
6590
6591
6592
6593
6594
6595
6596
6597
6598
6599
6600
6601
6602
6603
6604
6605
6606
6607
6608
6609
6610
6611
6612
6613
6614
6615
6616
6617
6618
6619
6620
6621
6622
6623
6624
6625
6626
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* reduces x mod m, assumes 0 < x < m**2, mu is
 * precomputed via mp_reduce_setup.
 * From HAC pp.604 Algorithm 14.42
 */
int mp_reduce (mp_int * x, mp_int * m, mp_int * mu)
{
  mp_int  q;
  int     res, um = m->used;

  /* q = x */
  if ((res = mp_init_copy (&q, x)) != MP_OKAY) {
    return res;
  }

  /* q1 = x / b**(k-1)  */
  mp_rshd (&q, um - 1);

  /* according to HAC this optimization is ok */
  if (((unsigned long) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) {
    if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) {
      goto CLEANUP;
    }
  } else {
#ifdef BN_S_MP_MUL_HIGH_DIGS_C
    if ((res = s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) {
      goto CLEANUP;
    }
#elif defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C)
    if ((res = fast_s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) {
      goto CLEANUP;
    }
#else
    {
      res = MP_VAL;
      goto CLEANUP;
    }
#endif
  }

  /* q3 = q2 / b**(k+1) */
  mp_rshd (&q, um + 1);

  /* x = x mod b**(k+1), quick (no division) */
  if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) {
    goto CLEANUP;
  }

  /* q = q * m mod b**(k+1), quick (no division) */
6323
6324
6325
6326
6327
6328
6329
6330
6331
6332
6333
6334
6335
6336




6337
6338
6339
6340
6341
6342
6343

  /* Back off if it's too big */
  while (mp_cmp (x, m) != MP_LT) {
    if ((res = s_mp_sub (x, m, x)) != MP_OKAY) {
      goto CLEANUP;
    }
  }
  
CLEANUP:
  mp_clear (&q);

  return res;
}
#endif





/* End: bn_mp_reduce.c */

/* Start: bn_mp_reduce_2k.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_2K_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|






>
>
>
>







6644
6645
6646
6647
6648
6649
6650
6651
6652
6653
6654
6655
6656
6657
6658
6659
6660
6661
6662
6663
6664
6665
6666
6667
6668

  /* Back off if it's too big */
  while (mp_cmp (x, m) != MP_LT) {
    if ((res = s_mp_sub (x, m, x)) != MP_OKAY) {
      goto CLEANUP;
    }
  }

CLEANUP:
  mp_clear (&q);

  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_reduce.c */

/* Start: bn_mp_reduce_2k.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_2K_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6356
6357
6358
6359
6360
6361
6362
6363
6364
6365
6366
6367
6368
6369
6370
6371
6372
6373
6374
6375
6376
6377
6378
6379
6380
6381
6382
6383
6384
6385
6386
6387
6388
6389
6390
6391
6392
6393
6394
6395
6396
6397




6398
6399
6400
6401
6402
6403
6404
 */

/* reduces a modulo n where n is of the form 2**p - d */
int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d)
{
   mp_int q;
   int    p, res;
   
   if ((res = mp_init(&q)) != MP_OKAY) {
      return res;
   }
   
   p = mp_count_bits(n);    
top:
   /* q = a/2**p, a = a mod 2**p */
   if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) {
      goto ERR;
   }
   
   if (d != 1) {
      /* q = q * d */
      if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) { 
         goto ERR;
      }
   }
   
   /* a = a + q */
   if ((res = s_mp_add(a, &q, a)) != MP_OKAY) {
      goto ERR;
   }
   
   if (mp_cmp_mag(a, n) != MP_LT) {
      s_mp_sub(a, n, a);
      goto top;
   }
   
ERR:
   mp_clear(&q);
   return res;
}

#endif





/* End: bn_mp_reduce_2k.c */

/* Start: bn_mp_reduce_2k_l.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_2K_L_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|



|
|





|


|



|




|




|






>
>
>
>







6681
6682
6683
6684
6685
6686
6687
6688
6689
6690
6691
6692
6693
6694
6695
6696
6697
6698
6699
6700
6701
6702
6703
6704
6705
6706
6707
6708
6709
6710
6711
6712
6713
6714
6715
6716
6717
6718
6719
6720
6721
6722
6723
6724
6725
6726
6727
6728
6729
6730
6731
6732
6733
 */

/* reduces a modulo n where n is of the form 2**p - d */
int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d)
{
   mp_int q;
   int    p, res;

   if ((res = mp_init(&q)) != MP_OKAY) {
      return res;
   }

   p = mp_count_bits(n);
top:
   /* q = a/2**p, a = a mod 2**p */
   if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) {
      goto ERR;
   }

   if (d != 1) {
      /* q = q * d */
      if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) {
         goto ERR;
      }
   }

   /* a = a + q */
   if ((res = s_mp_add(a, &q, a)) != MP_OKAY) {
      goto ERR;
   }

   if (mp_cmp_mag(a, n) != MP_LT) {
      s_mp_sub(a, n, a);
      goto top;
   }

ERR:
   mp_clear(&q);
   return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_reduce_2k.c */

/* Start: bn_mp_reduce_2k_l.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_2K_L_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6412
6413
6414
6415
6416
6417
6418
6419
6420
6421
6422
6423
6424
6425
6426
6427
6428
6429
6430
6431
6432
6433
6434
6435
6436
6437
6438
6439
6440
6441
6442
6443
6444
6445
6446
6447
6448
6449
6450
6451
6452
6453
6454
6455
6456
6457
6458
6459




6460
6461
6462
6463
6464
6465
6466
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* reduces a modulo n where n is of the form 2**p - d 
   This differs from reduce_2k since "d" can be larger
   than a single digit.
*/
int mp_reduce_2k_l(mp_int *a, mp_int *n, mp_int *d)
{
   mp_int q;
   int    p, res;
   
   if ((res = mp_init(&q)) != MP_OKAY) {
      return res;
   }
   
   p = mp_count_bits(n);    
top:
   /* q = a/2**p, a = a mod 2**p */
   if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) {
      goto ERR;
   }
   
   /* q = q * d */
   if ((res = mp_mul(&q, d, &q)) != MP_OKAY) { 
      goto ERR;
   }
   
   /* a = a + q */
   if ((res = s_mp_add(a, &q, a)) != MP_OKAY) {
      goto ERR;
   }
   
   if (mp_cmp_mag(a, n) != MP_LT) {
      s_mp_sub(a, n, a);
      goto top;
   }
   
ERR:
   mp_clear(&q);
   return res;
}

#endif





/* End: bn_mp_reduce_2k_l.c */

/* Start: bn_mp_reduce_2k_setup.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_2K_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|







|



|
|





|

|


|




|




|






>
>
>
>







6741
6742
6743
6744
6745
6746
6747
6748
6749
6750
6751
6752
6753
6754
6755
6756
6757
6758
6759
6760
6761
6762
6763
6764
6765
6766
6767
6768
6769
6770
6771
6772
6773
6774
6775
6776
6777
6778
6779
6780
6781
6782
6783
6784
6785
6786
6787
6788
6789
6790
6791
6792
6793
6794
6795
6796
6797
6798
6799
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* reduces a modulo n where n is of the form 2**p - d
   This differs from reduce_2k since "d" can be larger
   than a single digit.
*/
int mp_reduce_2k_l(mp_int *a, mp_int *n, mp_int *d)
{
   mp_int q;
   int    p, res;

   if ((res = mp_init(&q)) != MP_OKAY) {
      return res;
   }

   p = mp_count_bits(n);
top:
   /* q = a/2**p, a = a mod 2**p */
   if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) {
      goto ERR;
   }

   /* q = q * d */
   if ((res = mp_mul(&q, d, &q)) != MP_OKAY) {
      goto ERR;
   }

   /* a = a + q */
   if ((res = s_mp_add(a, &q, a)) != MP_OKAY) {
      goto ERR;
   }

   if (mp_cmp_mag(a, n) != MP_LT) {
      s_mp_sub(a, n, a);
      goto top;
   }

ERR:
   mp_clear(&q);
   return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_reduce_2k_l.c */

/* Start: bn_mp_reduce_2k_setup.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_2K_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6479
6480
6481
6482
6483
6484
6485
6486
6487
6488
6489
6490
6491
6492
6493
6494
6495
6496
6497
6498
6499
6500
6501
6502
6503
6504
6505
6506




6507
6508
6509
6510
6511
6512
6513
 */

/* determines the setup value */
int mp_reduce_2k_setup(mp_int *a, mp_digit *d)
{
   int res, p;
   mp_int tmp;
   
   if ((res = mp_init(&tmp)) != MP_OKAY) {
      return res;
   }
   
   p = mp_count_bits(a);
   if ((res = mp_2expt(&tmp, p)) != MP_OKAY) {
      mp_clear(&tmp);
      return res;
   }
   
   if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) {
      mp_clear(&tmp);
      return res;
   }
   
   *d = tmp.dp[0];
   mp_clear(&tmp);
   return MP_OKAY;
}
#endif





/* End: bn_mp_reduce_2k_setup.c */

/* Start: bn_mp_reduce_2k_setup_l.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_2K_SETUP_L_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|



|





|




|





>
>
>
>







6812
6813
6814
6815
6816
6817
6818
6819
6820
6821
6822
6823
6824
6825
6826
6827
6828
6829
6830
6831
6832
6833
6834
6835
6836
6837
6838
6839
6840
6841
6842
6843
6844
6845
6846
6847
6848
6849
6850
 */

/* determines the setup value */
int mp_reduce_2k_setup(mp_int *a, mp_digit *d)
{
   int res, p;
   mp_int tmp;

   if ((res = mp_init(&tmp)) != MP_OKAY) {
      return res;
   }

   p = mp_count_bits(a);
   if ((res = mp_2expt(&tmp, p)) != MP_OKAY) {
      mp_clear(&tmp);
      return res;
   }

   if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) {
      mp_clear(&tmp);
      return res;
   }

   *d = tmp.dp[0];
   mp_clear(&tmp);
   return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_reduce_2k_setup.c */

/* Start: bn_mp_reduce_2k_setup_l.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_2K_SETUP_L_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6526
6527
6528
6529
6530
6531
6532
6533
6534
6535
6536
6537
6538
6539
6540
6541
6542
6543
6544
6545
6546
6547
6548
6549
6550




6551
6552
6553
6554
6555
6556
6557
 */

/* determines the setup value */
int mp_reduce_2k_setup_l(mp_int *a, mp_int *d)
{
   int    res;
   mp_int tmp;
   
   if ((res = mp_init(&tmp)) != MP_OKAY) {
      return res;
   }
   
   if ((res = mp_2expt(&tmp, mp_count_bits(a))) != MP_OKAY) {
      goto ERR;
   }
   
   if ((res = s_mp_sub(&tmp, a, d)) != MP_OKAY) {
      goto ERR;
   }
   
ERR:
   mp_clear(&tmp);
   return res;
}
#endif





/* End: bn_mp_reduce_2k_setup_l.c */

/* Start: bn_mp_reduce_is_2k.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_IS_2K_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|



|



|



|





>
>
>
>







6863
6864
6865
6866
6867
6868
6869
6870
6871
6872
6873
6874
6875
6876
6877
6878
6879
6880
6881
6882
6883
6884
6885
6886
6887
6888
6889
6890
6891
6892
6893
6894
6895
6896
6897
6898
 */

/* determines the setup value */
int mp_reduce_2k_setup_l(mp_int *a, mp_int *d)
{
   int    res;
   mp_int tmp;

   if ((res = mp_init(&tmp)) != MP_OKAY) {
      return res;
   }

   if ((res = mp_2expt(&tmp, mp_count_bits(a))) != MP_OKAY) {
      goto ERR;
   }

   if ((res = s_mp_sub(&tmp, a, d)) != MP_OKAY) {
      goto ERR;
   }

ERR:
   mp_clear(&tmp);
   return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_reduce_2k_setup_l.c */

/* Start: bn_mp_reduce_is_2k.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_IS_2K_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6570
6571
6572
6573
6574
6575
6576
6577
6578
6579
6580
6581
6582
6583
6584
6585
6586
6587
6588
6589
6590
6591
6592
6593
6594
6595
6596
6597
6598
6599
6600
6601
6602




6603
6604
6605
6606
6607
6608
6609
 */

/* determines if mp_reduce_2k can be used */
int mp_reduce_is_2k(mp_int *a)
{
   int ix, iy, iw;
   mp_digit iz;
   
   if (a->used == 0) {
      return MP_NO;
   } else if (a->used == 1) {
      return MP_YES;
   } else if (a->used > 1) {
      iy = mp_count_bits(a);
      iz = 1;
      iw = 1;
    
      /* Test every bit from the second digit up, must be 1 */
      for (ix = DIGIT_BIT; ix < iy; ix++) {
          if ((a->dp[iw] & iz) == 0) {
             return MP_NO;
          }
          iz <<= 1;
          if (iz > (mp_digit)MP_MASK) {
             ++iw;
             iz = 1;
          }
      }
   }
   return MP_YES;
}

#endif





/* End: bn_mp_reduce_is_2k.c */

/* Start: bn_mp_reduce_is_2k_l.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_IS_2K_L_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|








|
















>
>
>
>







6911
6912
6913
6914
6915
6916
6917
6918
6919
6920
6921
6922
6923
6924
6925
6926
6927
6928
6929
6930
6931
6932
6933
6934
6935
6936
6937
6938
6939
6940
6941
6942
6943
6944
6945
6946
6947
6948
6949
6950
6951
6952
6953
6954
 */

/* determines if mp_reduce_2k can be used */
int mp_reduce_is_2k(mp_int *a)
{
   int ix, iy, iw;
   mp_digit iz;

   if (a->used == 0) {
      return MP_NO;
   } else if (a->used == 1) {
      return MP_YES;
   } else if (a->used > 1) {
      iy = mp_count_bits(a);
      iz = 1;
      iw = 1;

      /* Test every bit from the second digit up, must be 1 */
      for (ix = DIGIT_BIT; ix < iy; ix++) {
          if ((a->dp[iw] & iz) == 0) {
             return MP_NO;
          }
          iz <<= 1;
          if (iz > (mp_digit)MP_MASK) {
             ++iw;
             iz = 1;
          }
      }
   }
   return MP_YES;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_reduce_is_2k.c */

/* Start: bn_mp_reduce_is_2k_l.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_IS_2K_L_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6621
6622
6623
6624
6625
6626
6627
6628
6629
6630
6631
6632
6633
6634
6635
6636
6637
6638
6639
6640
6641
6642
6643
6644
6645
6646




6647
6648
6649
6650
6651
6652
6653
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* determines if reduce_2k_l can be used */
int mp_reduce_is_2k_l(mp_int *a)
{
   int ix, iy;
   
   if (a->used == 0) {
      return MP_NO;
   } else if (a->used == 1) {
      return MP_YES;
   } else if (a->used > 1) {
      /* if more than half of the digits are -1 we're sold */
      for (iy = ix = 0; ix < a->used; ix++) {
          if (a->dp[ix] == MP_MASK) {
              ++iy;
          }
      }
      return (iy >= (a->used/2)) ? MP_YES : MP_NO;
      
   }
   return MP_NO;
}

#endif





/* End: bn_mp_reduce_is_2k_l.c */

/* Start: bn_mp_reduce_setup.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|












|





>
>
>
>







6966
6967
6968
6969
6970
6971
6972
6973
6974
6975
6976
6977
6978
6979
6980
6981
6982
6983
6984
6985
6986
6987
6988
6989
6990
6991
6992
6993
6994
6995
6996
6997
6998
6999
7000
7001
7002
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* determines if reduce_2k_l can be used */
int mp_reduce_is_2k_l(mp_int *a)
{
   int ix, iy;

   if (a->used == 0) {
      return MP_NO;
   } else if (a->used == 1) {
      return MP_YES;
   } else if (a->used > 1) {
      /* if more than half of the digits are -1 we're sold */
      for (iy = ix = 0; ix < a->used; ix++) {
          if (a->dp[ix] == MP_MASK) {
              ++iy;
          }
      }
      return (iy >= (a->used/2)) ? MP_YES : MP_NO;

   }
   return MP_NO;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_reduce_is_2k_l.c */

/* Start: bn_mp_reduce_setup.c */
#include <tommath.h>
#ifdef BN_MP_REDUCE_SETUP_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6667
6668
6669
6670
6671
6672
6673
6674
6675
6676
6677
6678
6679
6680




6681
6682
6683
6684
6685
6686
6687

/* pre-calculate the value required for Barrett reduction
 * For a given modulus "b" it calulates the value required in "a"
 */
int mp_reduce_setup (mp_int * a, mp_int * b)
{
  int     res;
  
  if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) {
    return res;
  }
  return mp_div (a, b, a, NULL);
}
#endif





/* End: bn_mp_reduce_setup.c */

/* Start: bn_mp_rshd.c */
#include <tommath.h>
#ifdef BN_MP_RSHD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|






>
>
>
>







7016
7017
7018
7019
7020
7021
7022
7023
7024
7025
7026
7027
7028
7029
7030
7031
7032
7033
7034
7035
7036
7037
7038
7039
7040

/* pre-calculate the value required for Barrett reduction
 * For a given modulus "b" it calulates the value required in "a"
 */
int mp_reduce_setup (mp_int * a, mp_int * b)
{
  int     res;

  if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) {
    return res;
  }
  return mp_div (a, b, a, NULL);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_reduce_setup.c */

/* Start: bn_mp_rshd.c */
#include <tommath.h>
#ifdef BN_MP_RSHD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6722
6723
6724
6725
6726
6727
6728
6729
6730
6731
6732
6733
6734
6735
6736
6737
6738
6739
6740
6741
6742
6743
6744
6745
6746
6747
6748
6749
6750
6751
6752




6753
6754
6755
6756
6757
6758
6759

    /* bottom */
    bottom = a->dp;

    /* top [offset into digits] */
    top = a->dp + b;

    /* this is implemented as a sliding window where 
     * the window is b-digits long and digits from 
     * the top of the window are copied to the bottom
     *
     * e.g.

     b-2 | b-1 | b0 | b1 | b2 | ... | bb |   ---->
                 /\                   |      ---->
                  \-------------------/      ---->
     */
    for (x = 0; x < (a->used - b); x++) {
      *bottom++ = *top++;
    }

    /* zero the top digits */
    for (; x < a->used; x++) {
      *bottom++ = 0;
    }
  }
  
  /* remove excess digits */
  a->used -= b;
}
#endif





/* End: bn_mp_rshd.c */

/* Start: bn_mp_set.c */
#include <tommath.h>
#ifdef BN_MP_SET_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|
|

















|




>
>
>
>







7075
7076
7077
7078
7079
7080
7081
7082
7083
7084
7085
7086
7087
7088
7089
7090
7091
7092
7093
7094
7095
7096
7097
7098
7099
7100
7101
7102
7103
7104
7105
7106
7107
7108
7109
7110
7111
7112
7113
7114
7115
7116

    /* bottom */
    bottom = a->dp;

    /* top [offset into digits] */
    top = a->dp + b;

    /* this is implemented as a sliding window where
     * the window is b-digits long and digits from
     * the top of the window are copied to the bottom
     *
     * e.g.

     b-2 | b-1 | b0 | b1 | b2 | ... | bb |   ---->
                 /\                   |      ---->
                  \-------------------/      ---->
     */
    for (x = 0; x < (a->used - b); x++) {
      *bottom++ = *top++;
    }

    /* zero the top digits */
    for (; x < a->used; x++) {
      *bottom++ = 0;
    }
  }

  /* remove excess digits */
  a->used -= b;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_rshd.c */

/* Start: bn_mp_set.c */
#include <tommath.h>
#ifdef BN_MP_SET_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6775
6776
6777
6778
6779
6780
6781




6782
6783
6784
6785
6786
6787
6788
void mp_set (mp_int * a, mp_digit b)
{
  mp_zero (a);
  a->dp[0] = b & MP_MASK;
  a->used  = (a->dp[0] != 0) ? 1 : 0;
}
#endif





/* End: bn_mp_set.c */

/* Start: bn_mp_set_int.c */
#include <tommath.h>
#ifdef BN_MP_SET_INT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







7132
7133
7134
7135
7136
7137
7138
7139
7140
7141
7142
7143
7144
7145
7146
7147
7148
7149
void mp_set (mp_int * a, mp_digit b)
{
  mp_zero (a);
  a->dp[0] = b & MP_MASK;
  a->used  = (a->dp[0] != 0) ? 1 : 0;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_set.c */

/* Start: bn_mp_set_int.c */
#include <tommath.h>
#ifdef BN_MP_SET_INT_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6802
6803
6804
6805
6806
6807
6808
6809
6810
6811
6812
6813
6814
6815
6816
6817
6818
6819
6820
6821
6822
6823
6824
6825
6826
6827
6828
6829




6830
6831
6832
6833
6834
6835
6836

/* set a 32-bit const */
int mp_set_int (mp_int * a, unsigned long b)
{
  int     x, res;

  mp_zero (a);
  
  /* set four bits at a time */
  for (x = 0; x < 8; x++) {
    /* shift the number up four bits */
    if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) {
      return res;
    }

    /* OR in the top four bits of the source */
    a->dp[0] |= (b >> 28) & 15;

    /* shift the source up to the next four bits */
    b <<= 4;

    /* ensure that digits are not clamped off */
    a->used += 1;
  }
  mp_clamp (a);
  return MP_OKAY;
}
#endif





/* End: bn_mp_set_int.c */

/* Start: bn_mp_shrink.c */
#include <tommath.h>
#ifdef BN_MP_SHRINK_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|




















>
>
>
>







7163
7164
7165
7166
7167
7168
7169
7170
7171
7172
7173
7174
7175
7176
7177
7178
7179
7180
7181
7182
7183
7184
7185
7186
7187
7188
7189
7190
7191
7192
7193
7194
7195
7196
7197
7198
7199
7200
7201

/* set a 32-bit const */
int mp_set_int (mp_int * a, unsigned long b)
{
  int     x, res;

  mp_zero (a);

  /* set four bits at a time */
  for (x = 0; x < 8; x++) {
    /* shift the number up four bits */
    if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) {
      return res;
    }

    /* OR in the top four bits of the source */
    a->dp[0] |= (b >> 28) & 15;

    /* shift the source up to the next four bits */
    b <<= 4;

    /* ensure that digits are not clamped off */
    a->used += 1;
  }
  mp_clamp (a);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_set_int.c */

/* Start: bn_mp_shrink.c */
#include <tommath.h>
#ifdef BN_MP_SHRINK_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6849
6850
6851
6852
6853
6854
6855
6856
6857
6858
6859
6860
6861
6862
6863
6864
6865
6866
6867
6868
6869




6870
6871
6872
6873
6874
6875
6876
 */

/* shrink a bignum */
int mp_shrink (mp_int * a)
{
  mp_digit *tmp;
  int used = 1;
  
  if(a->used > 0)
    used = a->used;
  
  if (a->alloc != used) {
    if ((tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * used)) == NULL) {
      return MP_MEM;
    }
    a->dp    = tmp;
    a->alloc = used;
  }
  return MP_OKAY;
}
#endif





/* End: bn_mp_shrink.c */

/* Start: bn_mp_signed_bin_size.c */
#include <tommath.h>
#ifdef BN_MP_SIGNED_BIN_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|


|










>
>
>
>







7214
7215
7216
7217
7218
7219
7220
7221
7222
7223
7224
7225
7226
7227
7228
7229
7230
7231
7232
7233
7234
7235
7236
7237
7238
7239
7240
7241
7242
7243
7244
7245
 */

/* shrink a bignum */
int mp_shrink (mp_int * a)
{
  mp_digit *tmp;
  int used = 1;

  if(a->used > 0)
    used = a->used;

  if (a->alloc != used) {
    if ((tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * used)) == NULL) {
      return MP_MEM;
    }
    a->dp    = tmp;
    a->alloc = used;
  }
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_shrink.c */

/* Start: bn_mp_signed_bin_size.c */
#include <tommath.h>
#ifdef BN_MP_SIGNED_BIN_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6890
6891
6892
6893
6894
6895
6896




6897
6898
6899
6900
6901
6902
6903

/* get the size for an signed equivalent */
int mp_signed_bin_size (mp_int * a)
{
  return 1 + mp_unsigned_bin_size (a);
}
#endif





/* End: bn_mp_signed_bin_size.c */

/* Start: bn_mp_sqr.c */
#include <tommath.h>
#ifdef BN_MP_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







7259
7260
7261
7262
7263
7264
7265
7266
7267
7268
7269
7270
7271
7272
7273
7274
7275
7276

/* get the size for an signed equivalent */
int mp_signed_bin_size (mp_int * a)
{
  return 1 + mp_unsigned_bin_size (a);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_signed_bin_size.c */

/* Start: bn_mp_sqr.c */
#include <tommath.h>
#ifdef BN_MP_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6922
6923
6924
6925
6926
6927
6928
6929
6930
6931
6932
6933
6934
6935
6936
6937
6938
6939
6940
6941
6942
6943
6944
6945
6946
6947
6948
6949
6950
6951
6952
6953
6954




6955
6956
6957
6958
6959
6960
6961
  int     res;

#ifdef BN_MP_TOOM_SQR_C
  /* use Toom-Cook? */
  if (a->used >= TOOM_SQR_CUTOFF) {
    res = mp_toom_sqr(a, b);
  /* Karatsuba? */
  } else 
#endif
#ifdef BN_MP_KARATSUBA_SQR_C
if (a->used >= KARATSUBA_SQR_CUTOFF) {
    res = mp_karatsuba_sqr (a, b);
  } else 
#endif
  {
#ifdef BN_FAST_S_MP_SQR_C
    /* can we use the fast comba multiplier? */
    if ((a->used * 2 + 1) < MP_WARRAY && 
         a->used < 
         (1 << (sizeof(mp_word) * CHAR_BIT - 2*DIGIT_BIT - 1))) {
      res = fast_s_mp_sqr (a, b);
    } else
#endif
#ifdef BN_S_MP_SQR_C
      res = s_mp_sqr (a, b);
#else
      res = MP_VAL;
#endif
  }
  b->sign = MP_ZPOS;
  return res;
}
#endif





/* End: bn_mp_sqr.c */

/* Start: bn_mp_sqrmod.c */
#include <tommath.h>
#ifdef BN_MP_SQRMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|




|




|
|














>
>
>
>







7295
7296
7297
7298
7299
7300
7301
7302
7303
7304
7305
7306
7307
7308
7309
7310
7311
7312
7313
7314
7315
7316
7317
7318
7319
7320
7321
7322
7323
7324
7325
7326
7327
7328
7329
7330
7331
7332
7333
7334
7335
7336
7337
7338
  int     res;

#ifdef BN_MP_TOOM_SQR_C
  /* use Toom-Cook? */
  if (a->used >= TOOM_SQR_CUTOFF) {
    res = mp_toom_sqr(a, b);
  /* Karatsuba? */
  } else
#endif
#ifdef BN_MP_KARATSUBA_SQR_C
if (a->used >= KARATSUBA_SQR_CUTOFF) {
    res = mp_karatsuba_sqr (a, b);
  } else
#endif
  {
#ifdef BN_FAST_S_MP_SQR_C
    /* can we use the fast comba multiplier? */
    if ((a->used * 2 + 1) < MP_WARRAY &&
         a->used <
         (1 << (sizeof(mp_word) * CHAR_BIT - 2*DIGIT_BIT - 1))) {
      res = fast_s_mp_sqr (a, b);
    } else
#endif
#ifdef BN_S_MP_SQR_C
      res = s_mp_sqr (a, b);
#else
      res = MP_VAL;
#endif
  }
  b->sign = MP_ZPOS;
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_sqr.c */

/* Start: bn_mp_sqrmod.c */
#include <tommath.h>
#ifdef BN_MP_SQRMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
6989
6990
6991
6992
6993
6994
6995




6996
6997
6998
6999
7000
7001
7002
    return res;
  }
  res = mp_mod (&t, b, c);
  mp_clear (&t);
  return res;
}
#endif





/* End: bn_mp_sqrmod.c */

/* Start: bn_mp_sqrt.c */
#include <tommath.h>

#ifdef BN_MP_SQRT_C







>
>
>
>







7366
7367
7368
7369
7370
7371
7372
7373
7374
7375
7376
7377
7378
7379
7380
7381
7382
7383
    return res;
  }
  res = mp_mod (&t, b, c);
  mp_clear (&t);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_sqrmod.c */

/* Start: bn_mp_sqrt.c */
#include <tommath.h>

#ifdef BN_MP_SQRT_C
7012
7013
7014
7015
7016
7017
7018
7019
7020
7021
7022
7023
7024
7025
7026
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* this function is less generic than mp_n_root, simpler and faster */
int mp_sqrt(mp_int *arg, mp_int *ret) 
{
  int res;
  mp_int t1,t2;

  /* must be positive */
  if (arg->sign == MP_NEG) {
    return MP_VAL;







|







7393
7394
7395
7396
7397
7398
7399
7400
7401
7402
7403
7404
7405
7406
7407
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* this function is less generic than mp_n_root, simpler and faster */
int mp_sqrt(mp_int *arg, mp_int *ret)
{
  int res;
  mp_int t1,t2;

  /* must be positive */
  if (arg->sign == MP_NEG) {
    return MP_VAL;
7039
7040
7041
7042
7043
7044
7045
7046
7047
7048
7049
7050
7051
7052
7053
7054
7055
7056
7057
7058
7059
7060
7061
7062
7063
7064
7065
7066
7067
7068
7069
7070
7071
7072
7073
7074
7075
7076
7077




7078
7079
7080
7081
7082
7083
7084
  if ((res = mp_init(&t2)) != MP_OKAY) {
    goto E2;
  }

  /* First approx. (not very bad for large arg) */
  mp_rshd (&t1,t1.used/2);

  /* t1 > 0  */ 
  if ((res = mp_div(arg,&t1,&t2,NULL)) != MP_OKAY) {
    goto E1;
  }
  if ((res = mp_add(&t1,&t2,&t1)) != MP_OKAY) {
    goto E1;
  }
  if ((res = mp_div_2(&t1,&t1)) != MP_OKAY) {
    goto E1;
  }
  /* And now t1 > sqrt(arg) */
  do { 
    if ((res = mp_div(arg,&t1,&t2,NULL)) != MP_OKAY) {
      goto E1;
    }
    if ((res = mp_add(&t1,&t2,&t1)) != MP_OKAY) {
      goto E1;
    }
    if ((res = mp_div_2(&t1,&t1)) != MP_OKAY) {
      goto E1;
    }
    /* t1 >= sqrt(arg) >= t2 at this point */
  } while (mp_cmp_mag(&t1,&t2) == MP_GT);

  mp_exch(&t1,ret);

E1: mp_clear(&t2);
E2: mp_clear(&t1);
  return res;
}

#endif





/* End: bn_mp_sqrt.c */

/* Start: bn_mp_sub.c */
#include <tommath.h>
#ifdef BN_MP_SUB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|










|




















>
>
>
>







7420
7421
7422
7423
7424
7425
7426
7427
7428
7429
7430
7431
7432
7433
7434
7435
7436
7437
7438
7439
7440
7441
7442
7443
7444
7445
7446
7447
7448
7449
7450
7451
7452
7453
7454
7455
7456
7457
7458
7459
7460
7461
7462
7463
7464
7465
7466
7467
7468
7469
  if ((res = mp_init(&t2)) != MP_OKAY) {
    goto E2;
  }

  /* First approx. (not very bad for large arg) */
  mp_rshd (&t1,t1.used/2);

  /* t1 > 0  */
  if ((res = mp_div(arg,&t1,&t2,NULL)) != MP_OKAY) {
    goto E1;
  }
  if ((res = mp_add(&t1,&t2,&t1)) != MP_OKAY) {
    goto E1;
  }
  if ((res = mp_div_2(&t1,&t1)) != MP_OKAY) {
    goto E1;
  }
  /* And now t1 > sqrt(arg) */
  do {
    if ((res = mp_div(arg,&t1,&t2,NULL)) != MP_OKAY) {
      goto E1;
    }
    if ((res = mp_add(&t1,&t2,&t1)) != MP_OKAY) {
      goto E1;
    }
    if ((res = mp_div_2(&t1,&t1)) != MP_OKAY) {
      goto E1;
    }
    /* t1 >= sqrt(arg) >= t2 at this point */
  } while (mp_cmp_mag(&t1,&t2) == MP_GT);

  mp_exch(&t1,ret);

E1: mp_clear(&t2);
E2: mp_clear(&t1);
  return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_sqrt.c */

/* Start: bn_mp_sub.c */
#include <tommath.h>
#ifdef BN_MP_SUB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
7130
7131
7132
7133
7134
7135
7136




7137
7138
7139
7140
7141
7142
7143
      res = s_mp_sub (b, a, c);
    }
  }
  return res;
}

#endif





/* End: bn_mp_sub.c */

/* Start: bn_mp_sub_d.c */
#include <tommath.h>
#ifdef BN_MP_SUB_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







7515
7516
7517
7518
7519
7520
7521
7522
7523
7524
7525
7526
7527
7528
7529
7530
7531
7532
      res = s_mp_sub (b, a, c);
    }
  }
  return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_sub.c */

/* Start: bn_mp_sub_d.c */
#include <tommath.h>
#ifdef BN_MP_SUB_D_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
7223
7224
7225
7226
7227
7228
7229




7230
7231
7232
7233
7234
7235
7236
     *tmpc++ = 0;
  }
  mp_clamp(c);
  return MP_OKAY;
}

#endif





/* End: bn_mp_sub_d.c */

/* Start: bn_mp_submod.c */
#include <tommath.h>
#ifdef BN_MP_SUBMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







7612
7613
7614
7615
7616
7617
7618
7619
7620
7621
7622
7623
7624
7625
7626
7627
7628
7629
     *tmpc++ = 0;
  }
  mp_clamp(c);
  return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_sub_d.c */

/* Start: bn_mp_submod.c */
#include <tommath.h>
#ifdef BN_MP_SUBMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
7265
7266
7267
7268
7269
7270
7271




7272
7273
7274
7275
7276
7277
7278
    return res;
  }
  res = mp_mod (&t, c, d);
  mp_clear (&t);
  return res;
}
#endif





/* End: bn_mp_submod.c */

/* Start: bn_mp_to_signed_bin.c */
#include <tommath.h>
#ifdef BN_MP_TO_SIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







7658
7659
7660
7661
7662
7663
7664
7665
7666
7667
7668
7669
7670
7671
7672
7673
7674
7675
    return res;
  }
  res = mp_mod (&t, c, d);
  mp_clear (&t);
  return res;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_submod.c */

/* Start: bn_mp_to_signed_bin.c */
#include <tommath.h>
#ifdef BN_MP_TO_SIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
7298
7299
7300
7301
7302
7303
7304




7305
7306
7307
7308
7309
7310
7311
  if ((res = mp_to_unsigned_bin (a, b + 1)) != MP_OKAY) {
    return res;
  }
  b[0] = (unsigned char) ((a->sign == MP_ZPOS) ? 0 : 1);
  return MP_OKAY;
}
#endif





/* End: bn_mp_to_signed_bin.c */

/* Start: bn_mp_to_signed_bin_n.c */
#include <tommath.h>
#ifdef BN_MP_TO_SIGNED_BIN_N_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







7695
7696
7697
7698
7699
7700
7701
7702
7703
7704
7705
7706
7707
7708
7709
7710
7711
7712
  if ((res = mp_to_unsigned_bin (a, b + 1)) != MP_OKAY) {
    return res;
  }
  b[0] = (unsigned char) ((a->sign == MP_ZPOS) ? 0 : 1);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_to_signed_bin.c */

/* Start: bn_mp_to_signed_bin_n.c */
#include <tommath.h>
#ifdef BN_MP_TO_SIGNED_BIN_N_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
7329
7330
7331
7332
7333
7334
7335




7336
7337
7338
7339
7340
7341
7342
   if (*outlen < (unsigned long)mp_signed_bin_size(a)) {
      return MP_VAL;
   }
   *outlen = mp_signed_bin_size(a);
   return mp_to_signed_bin(a, b);
}
#endif





/* End: bn_mp_to_signed_bin_n.c */

/* Start: bn_mp_to_unsigned_bin.c */
#include <tommath.h>
#ifdef BN_MP_TO_UNSIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







7730
7731
7732
7733
7734
7735
7736
7737
7738
7739
7740
7741
7742
7743
7744
7745
7746
7747
   if (*outlen < (unsigned long)mp_signed_bin_size(a)) {
      return MP_VAL;
   }
   *outlen = mp_signed_bin_size(a);
   return mp_to_signed_bin(a, b);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_to_signed_bin_n.c */

/* Start: bn_mp_to_unsigned_bin.c */
#include <tommath.h>
#ifdef BN_MP_TO_UNSIGNED_BIN_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
7377
7378
7379
7380
7381
7382
7383




7384
7385
7386
7387
7388
7389
7390
    }
  }
  bn_reverse (b, x);
  mp_clear (&t);
  return MP_OKAY;
}
#endif





/* End: bn_mp_to_unsigned_bin.c */

/* Start: bn_mp_to_unsigned_bin_n.c */
#include <tommath.h>
#ifdef BN_MP_TO_UNSIGNED_BIN_N_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







7782
7783
7784
7785
7786
7787
7788
7789
7790
7791
7792
7793
7794
7795
7796
7797
7798
7799
    }
  }
  bn_reverse (b, x);
  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_to_unsigned_bin.c */

/* Start: bn_mp_to_unsigned_bin_n.c */
#include <tommath.h>
#ifdef BN_MP_TO_UNSIGNED_BIN_N_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
7408
7409
7410
7411
7412
7413
7414




7415
7416
7417
7418
7419
7420
7421
   if (*outlen < (unsigned long)mp_unsigned_bin_size(a)) {
      return MP_VAL;
   }
   *outlen = mp_unsigned_bin_size(a);
   return mp_to_unsigned_bin(a, b);
}
#endif





/* End: bn_mp_to_unsigned_bin_n.c */

/* Start: bn_mp_toom_mul.c */
#include <tommath.h>
#ifdef BN_MP_TOOM_MUL_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







7817
7818
7819
7820
7821
7822
7823
7824
7825
7826
7827
7828
7829
7830
7831
7832
7833
7834
   if (*outlen < (unsigned long)mp_unsigned_bin_size(a)) {
      return MP_VAL;
   }
   *outlen = mp_unsigned_bin_size(a);
   return mp_to_unsigned_bin(a, b);
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_to_unsigned_bin_n.c */

/* Start: bn_mp_toom_mul.c */
#include <tommath.h>
#ifdef BN_MP_TOOM_MUL_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
7429
7430
7431
7432
7433
7434
7435
7436
7437
7438
7439
7440
7441
7442
7443
7444
7445
7446
7447
7448
7449
7450
7451
7452
7453
7454
7455
7456
7457
7458
7459
7460
7461
7462
7463
7464
7465
7466
7467
7468
7469
7470
7471
7472
7473
7474
7475
7476
7477
7478
7479
7480
7481
7482
7483
7484
7485
7486
7487
7488
7489
7490
7491
7492
7493
7494
7495
7496
7497
7498
7499
7500
7501
7502
7503
7504
7505
7506
7507
7508
7509
7510
7511
7512
7513
7514
7515
7516
7517
7518
7519
7520
7521
7522
7523
7524
7525
7526
7527
7528
7529
7530
7531
7532
7533
7534
7535
7536
7537
7538
7539
7540
7541
7542
7543
7544
7545
7546
7547
7548
7549
7550
7551
7552
7553
7554
7555
7556
7557
7558
7559
7560
7561
7562
7563
7564
7565
7566
7567
7568
7569
7570
7571
7572
7573
7574
7575
7576
7577
7578
7579
7580
7581
7582
7583
7584
7585
7586
7587
7588
7589
7590
7591
7592
7593
7594
7595
7596
7597
7598
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* multiplication using the Toom-Cook 3-way algorithm 
 *
 * Much more complicated than Karatsuba but has a lower 
 * asymptotic running time of O(N**1.464).  This algorithm is 
 * only particularly useful on VERY large inputs 
 * (we're talking 1000s of digits here...).
*/
int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c)
{
    mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2;
    int res, B;
        
    /* init temps */
    if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, 
                             &a0, &a1, &a2, &b0, &b1, 
                             &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) {
       return res;
    }
    
    /* B */
    B = MIN(a->used, b->used) / 3;
    
    /* a = a2 * B**2 + a1 * B + a0 */
    if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_copy(a, &a1)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&a1, B);
    mp_mod_2d(&a1, DIGIT_BIT * B, &a1);

    if ((res = mp_copy(a, &a2)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&a2, B*2);
    
    /* b = b2 * B**2 + b1 * B + b0 */
    if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_copy(b, &b1)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&b1, B);
    mp_mod_2d(&b1, DIGIT_BIT * B, &b1);

    if ((res = mp_copy(b, &b2)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&b2, B*2);
    
    /* w0 = a0*b0 */
    if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) {
       goto ERR;
    }
    
    /* w4 = a2 * b2 */
    if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) {
       goto ERR;
    }
    
    /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */
    if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    
    if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    
    if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) {
       goto ERR;
    }
    
    /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */
    if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    
    if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    
    if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) {
       goto ERR;
    }
    

    /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */
    if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) {
       goto ERR;
    }
    
    /* now solve the matrix 
    
       0  0  0  0  1
       1  2  4  8  16
       1  1  1  1  1
       16 8  4  2  1
       1  0  0  0  0
       
       using 12 subtractions, 4 shifts, 
              2 small divisions and 1 small multiplication 
     */
     
     /* r1 - r4 */
     if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) {
        goto ERR;
     }
     /* r3 - r0 */
     if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) {
        goto ERR;







|

|
|
|






|

|
|



|


|















|















|




|




|













|












|



|













|












|



|

















|
|
|





|
|
|

|







7842
7843
7844
7845
7846
7847
7848
7849
7850
7851
7852
7853
7854
7855
7856
7857
7858
7859
7860
7861
7862
7863
7864
7865
7866
7867
7868
7869
7870
7871
7872
7873
7874
7875
7876
7877
7878
7879
7880
7881
7882
7883
7884
7885
7886
7887
7888
7889
7890
7891
7892
7893
7894
7895
7896
7897
7898
7899
7900
7901
7902
7903
7904
7905
7906
7907
7908
7909
7910
7911
7912
7913
7914
7915
7916
7917
7918
7919
7920
7921
7922
7923
7924
7925
7926
7927
7928
7929
7930
7931
7932
7933
7934
7935
7936
7937
7938
7939
7940
7941
7942
7943
7944
7945
7946
7947
7948
7949
7950
7951
7952
7953
7954
7955
7956
7957
7958
7959
7960
7961
7962
7963
7964
7965
7966
7967
7968
7969
7970
7971
7972
7973
7974
7975
7976
7977
7978
7979
7980
7981
7982
7983
7984
7985
7986
7987
7988
7989
7990
7991
7992
7993
7994
7995
7996
7997
7998
7999
8000
8001
8002
8003
8004
8005
8006
8007
8008
8009
8010
8011
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* multiplication using the Toom-Cook 3-way algorithm
 *
 * Much more complicated than Karatsuba but has a lower
 * asymptotic running time of O(N**1.464).  This algorithm is
 * only particularly useful on VERY large inputs
 * (we're talking 1000s of digits here...).
*/
int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c)
{
    mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2;
    int res, B;

    /* init temps */
    if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4,
                             &a0, &a1, &a2, &b0, &b1,
                             &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) {
       return res;
    }

    /* B */
    B = MIN(a->used, b->used) / 3;

    /* a = a2 * B**2 + a1 * B + a0 */
    if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_copy(a, &a1)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&a1, B);
    mp_mod_2d(&a1, DIGIT_BIT * B, &a1);

    if ((res = mp_copy(a, &a2)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&a2, B*2);

    /* b = b2 * B**2 + b1 * B + b0 */
    if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_copy(b, &b1)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&b1, B);
    mp_mod_2d(&b1, DIGIT_BIT * B, &b1);

    if ((res = mp_copy(b, &b2)) != MP_OKAY) {
       goto ERR;
    }
    mp_rshd(&b2, B*2);

    /* w0 = a0*b0 */
    if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) {
       goto ERR;
    }

    /* w4 = a2 * b2 */
    if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) {
       goto ERR;
    }

    /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */
    if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) {
       goto ERR;
    }

    /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */
    if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) {
       goto ERR;
    }

    if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) {
       goto ERR;
    }


    /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */
    if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) {
       goto ERR;
    }
    if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) {
       goto ERR;
    }

    /* now solve the matrix

       0  0  0  0  1
       1  2  4  8  16
       1  1  1  1  1
       16 8  4  2  1
       1  0  0  0  0

       using 12 subtractions, 4 shifts,
              2 small divisions and 1 small multiplication
     */

     /* r1 - r4 */
     if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) {
        goto ERR;
     }
     /* r3 - r0 */
     if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) {
        goto ERR;
7656
7657
7658
7659
7660
7661
7662
7663
7664
7665
7666
7667
7668
7669
7670
7671
7672
7673
7674
7675
7676
7677
7678
7679
7680
7681
7682
7683
7684
7685
7686
7687
7688
7689
7690
7691
7692
7693
7694
7695
7696
7697
7698




7699
7700
7701
7702
7703
7704
7705
     if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) {
        goto ERR;
     }
     /* r3/3 */
     if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) {
        goto ERR;
     }
     
     /* at this point shift W[n] by B*n */
     if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) {
        goto ERR;
     }     
     
     if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) {
        goto ERR;
     }     
     
ERR:
     mp_clear_multi(&w0, &w1, &w2, &w3, &w4, 
                    &a0, &a1, &a2, &b0, &b1, 
                    &b2, &tmp1, &tmp2, NULL);
     return res;
}     
     
#endif





/* End: bn_mp_toom_mul.c */

/* Start: bn_mp_toom_sqr.c */
#include <tommath.h>
#ifdef BN_MP_TOOM_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







|












|
|











|
|

|
|


|
|

>
>
>
>







8069
8070
8071
8072
8073
8074
8075
8076
8077
8078
8079
8080
8081
8082
8083
8084
8085
8086
8087
8088
8089
8090
8091
8092
8093
8094
8095
8096
8097
8098
8099
8100
8101
8102
8103
8104
8105
8106
8107
8108
8109
8110
8111
8112
8113
8114
8115
8116
8117
8118
8119
8120
8121
8122
     if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) {
        goto ERR;
     }
     /* r3/3 */
     if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) {
        goto ERR;
     }

     /* at this point shift W[n] by B*n */
     if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) {
        goto ERR;
     }

     if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) {
        goto ERR;
     }
     if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) {
        goto ERR;
     }

ERR:
     mp_clear_multi(&w0, &w1, &w2, &w3, &w4,
                    &a0, &a1, &a2, &b0, &b1,
                    &b2, &tmp1, &tmp2, NULL);
     return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_toom_mul.c */

/* Start: bn_mp_toom_sqr.c */
#include <tommath.h>
#ifdef BN_MP_TOOM_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
7918
7919
7920
7921
7922
7923
7924




7925
7926
7927
7928
7929
7930
7931

ERR:
     mp_clear_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &tmp1, NULL);
     return res;
}

#endif





/* End: bn_mp_toom_sqr.c */

/* Start: bn_mp_toradix.c */
#include <tommath.h>
#ifdef BN_MP_TORADIX_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







8335
8336
8337
8338
8339
8340
8341
8342
8343
8344
8345
8346
8347
8348
8349
8350
8351
8352

ERR:
     mp_clear_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &tmp1, NULL);
     return res;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_toom_sqr.c */

/* Start: bn_mp_toradix.c */
#include <tommath.h>
#ifdef BN_MP_TORADIX_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
7993
7994
7995
7996
7997
7998
7999




8000
8001
8002
8003
8004
8005
8006
  *str = '\0';

  mp_clear (&t);
  return MP_OKAY;
}

#endif





/* End: bn_mp_toradix.c */

/* Start: bn_mp_toradix_n.c */
#include <tommath.h>
#ifdef BN_MP_TORADIX_N_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







8414
8415
8416
8417
8418
8419
8420
8421
8422
8423
8424
8425
8426
8427
8428
8429
8430
8431
  *str = '\0';

  mp_clear (&t);
  return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_toradix.c */

/* Start: bn_mp_toradix_n.c */
#include <tommath.h>
#ifdef BN_MP_TORADIX_N_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
8014
8015
8016
8017
8018
8019
8020
8021
8022
8023
8024
8025
8026
8027
8028
8029
8030
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* stores a bignum as a ASCII string in a given radix (2..64) 
 *
 * Stores upto maxlen-1 chars and always a NULL byte 
 */
int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen)
{
  int     res, digs;
  mp_int  t;
  mp_digit d;
  char   *_s = str;







|

|







8439
8440
8441
8442
8443
8444
8445
8446
8447
8448
8449
8450
8451
8452
8453
8454
8455
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* stores a bignum as a ASCII string in a given radix (2..64)
 *
 * Stores upto maxlen-1 chars and always a NULL byte
 */
int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen)
{
  int     res, digs;
  mp_int  t;
  mp_digit d;
  char   *_s = str;
8049
8050
8051
8052
8053
8054
8055
8056
8057
8058
8059
8060
8061
8062
8063
  if (t.sign == MP_NEG) {
    /* we have to reverse our digits later... but not the - sign!! */
    ++_s;

    /* store the flag and mark the number as positive */
    *str++ = '-';
    t.sign = MP_ZPOS;
 
    /* subtract a char */
    --maxlen;
  }

  digs = 0;
  while (mp_iszero (&t) == 0) {
    if (--maxlen < 1) {







|







8474
8475
8476
8477
8478
8479
8480
8481
8482
8483
8484
8485
8486
8487
8488
  if (t.sign == MP_NEG) {
    /* we have to reverse our digits later... but not the - sign!! */
    ++_s;

    /* store the flag and mark the number as positive */
    *str++ = '-';
    t.sign = MP_ZPOS;

    /* subtract a char */
    --maxlen;
  }

  digs = 0;
  while (mp_iszero (&t) == 0) {
    if (--maxlen < 1) {
8081
8082
8083
8084
8085
8086
8087




8088
8089
8090
8091
8092
8093
8094
  *str = '\0';

  mp_clear (&t);
  return MP_OKAY;
}

#endif





/* End: bn_mp_toradix_n.c */

/* Start: bn_mp_unsigned_bin_size.c */
#include <tommath.h>
#ifdef BN_MP_UNSIGNED_BIN_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







8506
8507
8508
8509
8510
8511
8512
8513
8514
8515
8516
8517
8518
8519
8520
8521
8522
8523
  *str = '\0';

  mp_clear (&t);
  return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_toradix_n.c */

/* Start: bn_mp_unsigned_bin_size.c */
#include <tommath.h>
#ifdef BN_MP_UNSIGNED_BIN_SIZE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
8109
8110
8111
8112
8113
8114
8115




8116
8117
8118
8119
8120
8121
8122
/* get the size for an unsigned equivalent */
int mp_unsigned_bin_size (mp_int * a)
{
  int     size = mp_count_bits (a);
  return (size / 8 + ((size & 7) != 0 ? 1 : 0));
}
#endif





/* End: bn_mp_unsigned_bin_size.c */

/* Start: bn_mp_xor.c */
#include <tommath.h>
#ifdef BN_MP_XOR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







8538
8539
8540
8541
8542
8543
8544
8545
8546
8547
8548
8549
8550
8551
8552
8553
8554
8555
/* get the size for an unsigned equivalent */
int mp_unsigned_bin_size (mp_int * a)
{
  int     size = mp_count_bits (a);
  return (size / 8 + ((size & 7) != 0 ? 1 : 0));
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_unsigned_bin_size.c */

/* Start: bn_mp_xor.c */
#include <tommath.h>
#ifdef BN_MP_XOR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
8160
8161
8162
8163
8164
8165
8166




8167
8168
8169
8170
8171
8172
8173
  }
  mp_clamp (&t);
  mp_exch (c, &t);
  mp_clear (&t);
  return MP_OKAY;
}
#endif





/* End: bn_mp_xor.c */

/* Start: bn_mp_zero.c */
#include <tommath.h>
#ifdef BN_MP_ZERO_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







8593
8594
8595
8596
8597
8598
8599
8600
8601
8602
8603
8604
8605
8606
8607
8608
8609
8610
  }
  mp_clamp (&t);
  mp_exch (c, &t);
  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_xor.c */

/* Start: bn_mp_zero.c */
#include <tommath.h>
#ifdef BN_MP_ZERO_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
8196
8197
8198
8199
8200
8201
8202




8203
8204
8205
8206
8207
8208
8209

  tmp = a->dp;
  for (n = 0; n < a->alloc; n++) {
     *tmp++ = 0;
  }
}
#endif





/* End: bn_mp_zero.c */

/* Start: bn_prime_tab.c */
#include <tommath.h>
#ifdef BN_PRIME_TAB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







8633
8634
8635
8636
8637
8638
8639
8640
8641
8642
8643
8644
8645
8646
8647
8648
8649
8650

  tmp = a->dp;
  for (n = 0; n < a->alloc; n++) {
     *tmp++ = 0;
  }
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_mp_zero.c */

/* Start: bn_prime_tab.c */
#include <tommath.h>
#ifdef BN_PRIME_TAB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
8257
8258
8259
8260
8261
8262
8263




8264
8265
8266
8267
8268
8269
8270
  0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3,
  0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7,
  0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623,
  0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653
#endif
};
#endif





/* End: bn_prime_tab.c */

/* Start: bn_reverse.c */
#include <tommath.h>
#ifdef BN_REVERSE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







8698
8699
8700
8701
8702
8703
8704
8705
8706
8707
8708
8709
8710
8711
8712
8713
8714
8715
  0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3,
  0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7,
  0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623,
  0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653
#endif
};
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_prime_tab.c */

/* Start: bn_reverse.c */
#include <tommath.h>
#ifdef BN_REVERSE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
8296
8297
8298
8299
8300
8301
8302




8303
8304
8305
8306
8307
8308
8309
    s[ix] = s[iy];
    s[iy] = t;
    ++ix;
    --iy;
  }
}
#endif





/* End: bn_reverse.c */

/* Start: bn_s_mp_add.c */
#include <tommath.h>
#ifdef BN_S_MP_ADD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







8741
8742
8743
8744
8745
8746
8747
8748
8749
8750
8751
8752
8753
8754
8755
8756
8757
8758
    s[ix] = s[iy];
    s[iy] = t;
    ++ix;
    --iy;
  }
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_reverse.c */

/* Start: bn_s_mp_add.c */
#include <tommath.h>
#ifdef BN_S_MP_ADD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
8376
8377
8378
8379
8380
8381
8382
8383
8384
8385
8386
8387
8388
8389
8390
8391
      /* U = carry bit of T[i] */
      u = *tmpc >> ((mp_digit)DIGIT_BIT);

      /* take away carry bit from T[i] */
      *tmpc++ &= MP_MASK;
    }

    /* now copy higher words if any, that is in A+B 
     * if A or B has more digits add those in 
     */
    if (min != max) {
      for (; i < max; i++) {
        /* T[i] = X[i] + U */
        *tmpc = x->dp[i] + u;

        /* U = carry bit of T[i] */







|
|







8825
8826
8827
8828
8829
8830
8831
8832
8833
8834
8835
8836
8837
8838
8839
8840
      /* U = carry bit of T[i] */
      u = *tmpc >> ((mp_digit)DIGIT_BIT);

      /* take away carry bit from T[i] */
      *tmpc++ &= MP_MASK;
    }

    /* now copy higher words if any, that is in A+B
     * if A or B has more digits add those in
     */
    if (min != max) {
      for (; i < max; i++) {
        /* T[i] = X[i] + U */
        *tmpc = x->dp[i] + u;

        /* U = carry bit of T[i] */
8405
8406
8407
8408
8409
8410
8411




8412
8413
8414
8415
8416
8417
8418
    }
  }

  mp_clamp (c);
  return MP_OKAY;
}
#endif





/* End: bn_s_mp_add.c */

/* Start: bn_s_mp_exptmod.c */
#include <tommath.h>
#ifdef BN_S_MP_EXPTMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







8854
8855
8856
8857
8858
8859
8860
8861
8862
8863
8864
8865
8866
8867
8868
8869
8870
8871
    }
  }

  mp_clamp (c);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_s_mp_add.c */

/* Start: bn_s_mp_exptmod.c */
#include <tommath.h>
#ifdef BN_S_MP_EXPTMOD_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
8465
8466
8467
8468
8469
8470
8471
8472
8473
8474
8475
8476
8477
8478
8479
8480
8481
8482
8483
8484
8485
8486
8487
8488
8489
8490
8491
8492
8493
8494
8495
8496
8497
8498
8499
8500
8501
8502
8503
8504
8505
8506
8507
8508
8509
8510
8511
8512
8513
8514
8515
8516
8517
8518
8519
8520
8521
8522
8523
8524
8525
8526
8527
8528
8529
8530
8531
       winsize = 5;
    }
#endif

  /* init M array */
  /* init first cell */
  if ((err = mp_init(&M[1])) != MP_OKAY) {
     return err; 
  }

  /* now init the second half of the array */
  for (x = 1<<(winsize-1); x < (1 << winsize); x++) {
    if ((err = mp_init(&M[x])) != MP_OKAY) {
      for (y = 1<<(winsize-1); y < x; y++) {
        mp_clear (&M[y]);
      }
      mp_clear(&M[1]);
      return err;
    }
  }

  /* create mu, used for Barrett reduction */
  if ((err = mp_init (&mu)) != MP_OKAY) {
    goto LBL_M;
  }
  
  if (redmode == 0) {
     if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) {
        goto LBL_MU;
     }
     redux = mp_reduce;
  } else {
     if ((err = mp_reduce_2k_setup_l (P, &mu)) != MP_OKAY) {
        goto LBL_MU;
     }
     redux = mp_reduce_2k_l;
  }    

  /* create M table
   *
   * The M table contains powers of the base, 
   * e.g. M[x] = G**x mod P
   *
   * The first half of the table is not 
   * computed though accept for M[0] and M[1]
   */
  if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) {
    goto LBL_MU;
  }

  /* compute the value at M[1<<(winsize-1)] by squaring 
   * M[1] (winsize-1) times 
   */
  if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) {
    goto LBL_MU;
  }

  for (x = 0; x < (winsize - 1); x++) {
    /* square it */
    if ((err = mp_sqr (&M[1 << (winsize - 1)], 
                       &M[1 << (winsize - 1)])) != MP_OKAY) {
      goto LBL_MU;
    }

    /* reduce modulo P */
    if ((err = redux (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) {
      goto LBL_MU;







|

















|










|



|


|






|
|







|







8918
8919
8920
8921
8922
8923
8924
8925
8926
8927
8928
8929
8930
8931
8932
8933
8934
8935
8936
8937
8938
8939
8940
8941
8942
8943
8944
8945
8946
8947
8948
8949
8950
8951
8952
8953
8954
8955
8956
8957
8958
8959
8960
8961
8962
8963
8964
8965
8966
8967
8968
8969
8970
8971
8972
8973
8974
8975
8976
8977
8978
8979
8980
8981
8982
8983
8984
       winsize = 5;
    }
#endif

  /* init M array */
  /* init first cell */
  if ((err = mp_init(&M[1])) != MP_OKAY) {
     return err;
  }

  /* now init the second half of the array */
  for (x = 1<<(winsize-1); x < (1 << winsize); x++) {
    if ((err = mp_init(&M[x])) != MP_OKAY) {
      for (y = 1<<(winsize-1); y < x; y++) {
        mp_clear (&M[y]);
      }
      mp_clear(&M[1]);
      return err;
    }
  }

  /* create mu, used for Barrett reduction */
  if ((err = mp_init (&mu)) != MP_OKAY) {
    goto LBL_M;
  }

  if (redmode == 0) {
     if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) {
        goto LBL_MU;
     }
     redux = mp_reduce;
  } else {
     if ((err = mp_reduce_2k_setup_l (P, &mu)) != MP_OKAY) {
        goto LBL_MU;
     }
     redux = mp_reduce_2k_l;
  }

  /* create M table
   *
   * The M table contains powers of the base,
   * e.g. M[x] = G**x mod P
   *
   * The first half of the table is not
   * computed though accept for M[0] and M[1]
   */
  if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) {
    goto LBL_MU;
  }

  /* compute the value at M[1<<(winsize-1)] by squaring
   * M[1] (winsize-1) times
   */
  if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) {
    goto LBL_MU;
  }

  for (x = 0; x < (winsize - 1); x++) {
    /* square it */
    if ((err = mp_sqr (&M[1 << (winsize - 1)],
                       &M[1 << (winsize - 1)])) != MP_OKAY) {
      goto LBL_MU;
    }

    /* reduce modulo P */
    if ((err = redux (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) {
      goto LBL_MU;
8657
8658
8659
8660
8661
8662
8663




8664
8665
8666
8667
8668
8669
8670
  mp_clear(&M[1]);
  for (x = 1<<(winsize-1); x < (1 << winsize); x++) {
    mp_clear (&M[x]);
  }
  return err;
}
#endif





/* End: bn_s_mp_exptmod.c */

/* Start: bn_s_mp_mul_digs.c */
#include <tommath.h>
#ifdef BN_S_MP_MUL_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







9110
9111
9112
9113
9114
9115
9116
9117
9118
9119
9120
9121
9122
9123
9124
9125
9126
9127
  mp_clear(&M[1]);
  for (x = 1<<(winsize-1); x < (1 << winsize); x++) {
    mp_clear (&M[x]);
  }
  return err;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_s_mp_exptmod.c */

/* Start: bn_s_mp_mul_digs.c */
#include <tommath.h>
#ifdef BN_S_MP_MUL_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
8679
8680
8681
8682
8683
8684
8685
8686
8687
8688
8689
8690
8691
8692
8693
8694
8695
8696
8697
8698
8699
8700
8701
8702
8703
8704
8705
8706
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* multiplies |a| * |b| and only computes upto digs digits of result
 * HAC pp. 595, Algorithm 14.12  Modified so you can control how 
 * many digits of output are created.
 */
int s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
{
  mp_int  t;
  int     res, pa, pb, ix, iy;
  mp_digit u;
  mp_word r;
  mp_digit tmpx, *tmpt, *tmpy;

  /* can we use the fast multiplier? */
  if (((digs) < MP_WARRAY) &&
      MIN (a->used, b->used) < 
          (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) {
    return fast_s_mp_mul_digs (a, b, c, digs);
  }

  if ((res = mp_init_size (&t, digs)) != MP_OKAY) {
    return res;
  }







|












|







9136
9137
9138
9139
9140
9141
9142
9143
9144
9145
9146
9147
9148
9149
9150
9151
9152
9153
9154
9155
9156
9157
9158
9159
9160
9161
9162
9163
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */

/* multiplies |a| * |b| and only computes upto digs digits of result
 * HAC pp. 595, Algorithm 14.12  Modified so you can control how
 * many digits of output are created.
 */
int s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
{
  mp_int  t;
  int     res, pa, pb, ix, iy;
  mp_digit u;
  mp_word r;
  mp_digit tmpx, *tmpt, *tmpy;

  /* can we use the fast multiplier? */
  if (((digs) < MP_WARRAY) &&
      MIN (a->used, b->used) <
          (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) {
    return fast_s_mp_mul_digs (a, b, c, digs);
  }

  if ((res = mp_init_size (&t, digs)) != MP_OKAY) {
    return res;
  }
8714
8715
8716
8717
8718
8719
8720
8721
8722
8723
8724
8725
8726
8727
8728
8729
8730
8731

    /* limit ourselves to making digs digits of output */
    pb = MIN (b->used, digs - ix);

    /* setup some aliases */
    /* copy of the digit from a used within the nested loop */
    tmpx = a->dp[ix];
    
    /* an alias for the destination shifted ix places */
    tmpt = t.dp + ix;
    
    /* an alias for the digits of b */
    tmpy = b->dp;

    /* compute the columns of the output and propagate the carry */
    for (iy = 0; iy < pb; iy++) {
      /* compute the column as a mp_word */
      r       = ((mp_word)*tmpt) +







|


|







9171
9172
9173
9174
9175
9176
9177
9178
9179
9180
9181
9182
9183
9184
9185
9186
9187
9188

    /* limit ourselves to making digs digits of output */
    pb = MIN (b->used, digs - ix);

    /* setup some aliases */
    /* copy of the digit from a used within the nested loop */
    tmpx = a->dp[ix];

    /* an alias for the destination shifted ix places */
    tmpt = t.dp + ix;

    /* an alias for the digits of b */
    tmpy = b->dp;

    /* compute the columns of the output and propagate the carry */
    for (iy = 0; iy < pb; iy++) {
      /* compute the column as a mp_word */
      r       = ((mp_word)*tmpt) +
8747
8748
8749
8750
8751
8752
8753




8754
8755
8756
8757
8758
8759
8760
  mp_clamp (&t);
  mp_exch (&t, c);

  mp_clear (&t);
  return MP_OKAY;
}
#endif





/* End: bn_s_mp_mul_digs.c */

/* Start: bn_s_mp_mul_high_digs.c */
#include <tommath.h>
#ifdef BN_S_MP_MUL_HIGH_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







9204
9205
9206
9207
9208
9209
9210
9211
9212
9213
9214
9215
9216
9217
9218
9219
9220
9221
  mp_clamp (&t);
  mp_exch (&t, c);

  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_s_mp_mul_digs.c */

/* Start: bn_s_mp_mul_high_digs.c */
#include <tommath.h>
#ifdef BN_S_MP_MUL_HIGH_DIGS_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
8828
8829
8830
8831
8832
8833
8834




8835
8836
8837
8838
8839
8840
8841
  }
  mp_clamp (&t);
  mp_exch (&t, c);
  mp_clear (&t);
  return MP_OKAY;
}
#endif





/* End: bn_s_mp_mul_high_digs.c */

/* Start: bn_s_mp_sqr.c */
#include <tommath.h>
#ifdef BN_S_MP_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







9289
9290
9291
9292
9293
9294
9295
9296
9297
9298
9299
9300
9301
9302
9303
9304
9305
9306
  }
  mp_clamp (&t);
  mp_exch (&t, c);
  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_s_mp_mul_high_digs.c */

/* Start: bn_s_mp_sqr.c */
#include <tommath.h>
#ifdef BN_S_MP_SQR_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
8882
8883
8884
8885
8886
8887
8888
8889
8890
8891
8892
8893
8894
8895
8896
    u           = (mp_digit)(r >> ((mp_word) DIGIT_BIT));

    /* left hand side of A[ix] * A[iy] */
    tmpx        = a->dp[ix];

    /* alias for where to store the results */
    tmpt        = t.dp + (2*ix + 1);
    
    for (iy = ix + 1; iy < pa; iy++) {
      /* first calculate the product */
      r       = ((mp_word)tmpx) * ((mp_word)a->dp[iy]);

      /* now calculate the double precision result, note we use
       * addition instead of *2 since it's easier to optimize
       */







|







9347
9348
9349
9350
9351
9352
9353
9354
9355
9356
9357
9358
9359
9360
9361
    u           = (mp_digit)(r >> ((mp_word) DIGIT_BIT));

    /* left hand side of A[ix] * A[iy] */
    tmpx        = a->dp[ix];

    /* alias for where to store the results */
    tmpt        = t.dp + (2*ix + 1);

    for (iy = ix + 1; iy < pa; iy++) {
      /* first calculate the product */
      r       = ((mp_word)tmpx) * ((mp_word)a->dp[iy]);

      /* now calculate the double precision result, note we use
       * addition instead of *2 since it's easier to optimize
       */
8912
8913
8914
8915
8916
8917
8918




8919
8920
8921
8922
8923
8924
8925

  mp_clamp (&t);
  mp_exch (&t, b);
  mp_clear (&t);
  return MP_OKAY;
}
#endif





/* End: bn_s_mp_sqr.c */

/* Start: bn_s_mp_sub.c */
#include <tommath.h>
#ifdef BN_S_MP_SUB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







9377
9378
9379
9380
9381
9382
9383
9384
9385
9386
9387
9388
9389
9390
9391
9392
9393
9394

  mp_clamp (&t);
  mp_exch (&t, b);
  mp_clear (&t);
  return MP_OKAY;
}
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_s_mp_sqr.c */

/* Start: bn_s_mp_sub.c */
#include <tommath.h>
#ifdef BN_S_MP_SUB_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
9001
9002
9003
9004
9005
9006
9007




9008
9009
9010
9011
9012
9013
9014
  }

  mp_clamp (c);
  return MP_OKAY;
}

#endif





/* End: bn_s_mp_sub.c */

/* Start: bncore.c */
#include <tommath.h>
#ifdef BNCORE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis







>
>
>
>







9470
9471
9472
9473
9474
9475
9476
9477
9478
9479
9480
9481
9482
9483
9484
9485
9486
9487
  }

  mp_clamp (c);
  return MP_OKAY;
}

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bn_s_mp_sub.c */

/* Start: bncore.c */
#include <tommath.h>
#ifdef BNCORE_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
9028
9029
9030
9031
9032
9033
9034
9035
9036
9037
9038
9039
9040
9041
9042
9043




9044
9045
9046
9047
9048

/* Known optimal configurations

 CPU                    /Compiler     /MUL CUTOFF/SQR CUTOFF
-------------------------------------------------------------
 Intel P4 Northwood     /GCC v3.4.1   /        88/       128/LTM 0.32 ;-)
 AMD Athlon64           /GCC v3.4.4   /        80/       120/LTM 0.35
 
*/

int     KARATSUBA_MUL_CUTOFF = 80,      /* Min. number of digits before Karatsuba multiplication is used. */
        KARATSUBA_SQR_CUTOFF = 120,     /* Min. number of digits before Karatsuba squaring is used. */
        
        TOOM_MUL_CUTOFF      = 350,      /* no optimal values of these are known yet so set em high */
        TOOM_SQR_CUTOFF      = 400; 
#endif





/* End: bncore.c */


/* EOF */







|




|

|

>
>
>
>





9501
9502
9503
9504
9505
9506
9507
9508
9509
9510
9511
9512
9513
9514
9515
9516
9517
9518
9519
9520
9521
9522
9523
9524
9525

/* Known optimal configurations

 CPU                    /Compiler     /MUL CUTOFF/SQR CUTOFF
-------------------------------------------------------------
 Intel P4 Northwood     /GCC v3.4.1   /        88/       128/LTM 0.32 ;-)
 AMD Athlon64           /GCC v3.4.4   /        80/       120/LTM 0.35

*/

int     KARATSUBA_MUL_CUTOFF = 80,      /* Min. number of digits before Karatsuba multiplication is used. */
        KARATSUBA_SQR_CUTOFF = 120,     /* Min. number of digits before Karatsuba squaring is used. */

        TOOM_MUL_CUTOFF      = 350,      /* no optimal values of these are known yet so set em high */
        TOOM_SQR_CUTOFF      = 400;
#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */

/* End: bncore.c */


/* EOF */
Changes to libtommath/tommath.h.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65




66
67
68




69
70
71
72
73
74
75


76

77





78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123




124
125

126
127


128
129
130
131
132
133
134
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tom[email protected], http://math.libtomcrypt.com
 */
#ifndef BN_H_
#define BN_H_

#include <stdio.h>
#include <string.h>
#include <stdlib.h>
#include <ctype.h>
#include <limits.h>

#include <tommath_class.h>

#ifndef MIN
#   define MIN(x,y) ((x)<(y)?(x):(y))
#endif

#ifndef MAX
#   define MAX(x,y) ((x)>(y)?(x):(y))
#endif

#ifdef __cplusplus
extern "C" {

/* C++ compilers don't like assigning void * to mp_digit * */
#define  OPT_CAST(x)  (x *)

#else

/* C on the other hand doesn't care */
#define  OPT_CAST(x)

#endif


/* detect 64-bit mode if possible */
#if defined(__x86_64__) 
#   if !(defined(MP_64BIT) && defined(MP_16BIT) && defined(MP_8BIT))
#	define MP_64BIT
#   endif
#endif

/* some default configurations.
 *
 * A "mp_digit" must be able to hold DIGIT_BIT + 1 bits
 * A "mp_word" must be able to hold 2*DIGIT_BIT + 1 bits
 *
 * At the very least a mp_digit must be able to hold 7 bits
 * [any size beyond that is ok provided it doesn't overflow the data type]
 */
#ifdef MP_8BIT
   typedef unsigned char      mp_digit;
   typedef unsigned short     mp_word;




#elif defined(MP_16BIT)
   typedef unsigned short     mp_digit;
   typedef unsigned long      mp_word;




#elif defined(MP_64BIT)
   /* for GCC only on supported platforms */
#ifndef CRYPT
   typedef unsigned long long ulong64;
   typedef signed long long   long64;
#endif



   typedef unsigned long      mp_digit;

   typedef unsigned long      mp_word __attribute__ ((mode(TI)));






#  define DIGIT_BIT          60
#else
   /* this is the default case, 28-bit digits */
   
   /* this is to make porting into LibTomCrypt easier :-) */
#ifndef CRYPT
#  if defined(_MSC_VER) || defined(__BORLANDC__)
      typedef unsigned __int64   ulong64;
      typedef signed __int64     long64;
#  else
      typedef unsigned long long ulong64;
      typedef signed long long   long64;
#  endif
#endif

   typedef unsigned long      mp_digit;
   typedef ulong64            mp_word;

#ifdef MP_31BIT   
   /* this is an extension that uses 31-bit digits */
#  define DIGIT_BIT          31
#else
   /* default case is 28-bit digits, defines MP_28BIT as a handy macro to test */
#  define DIGIT_BIT          28
#  define MP_28BIT
#endif   
#endif

/* define heap macros */
#ifndef CRYPT
   /* default to libc stuff */
#  ifndef XMALLOC
#     define XMALLOC  malloc
#     define XFREE    free
#     define XREALLOC realloc
#     define XCALLOC  calloc
#  else
      /* prototypes for our heap functions */
      extern void *XMALLOC(size_t n);
      extern void *XREALLOC(void *p, size_t n);
      extern void *XCALLOC(size_t n, size_t s);
      extern void XFREE(void *p);
#  endif
#endif






/* otherwise the bits per digit is calculated automatically from the size of a mp_digit */

#ifndef DIGIT_BIT
#   define DIGIT_BIT     ((int)((CHAR_BIT * sizeof(mp_digit) - 1)))  /* bits per digit */


#endif

#define MP_DIGIT_BIT     DIGIT_BIT
#define MP_MASK          ((((mp_digit)1)<<((mp_digit)DIGIT_BIT))-((mp_digit)1))
#define MP_DIGIT_MAX     MP_MASK

/* equalities */












|





<

|




<
<
<
<
<
<
<
<


<
<
<
<
<
<
<
<
<


<

|
|
|
|











|
|
>
>
>
>

|
|
>
>
>
>



|
|


>
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/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, tstdenis82@gmail.com, http://math.libtomcrypt.com
 */
#ifndef BN_H_
#define BN_H_

#include <stdio.h>

#include <stdlib.h>
#include <stdint.h>
#include <limits.h>

#include <tommath_class.h>









#ifdef __cplusplus
extern "C" {









#endif


/* detect 64-bit mode if possible */
#if defined(__x86_64__)
   #if !(defined(MP_32BIT) || defined(MP_16BIT) || defined(MP_8BIT))
      #define MP_64BIT
   #endif
#endif

/* some default configurations.
 *
 * A "mp_digit" must be able to hold DIGIT_BIT + 1 bits
 * A "mp_word" must be able to hold 2*DIGIT_BIT + 1 bits
 *
 * At the very least a mp_digit must be able to hold 7 bits
 * [any size beyond that is ok provided it doesn't overflow the data type]
 */
#ifdef MP_8BIT
   typedef uint8_t              mp_digit;
   typedef uint16_t             mp_word;
#define MP_SIZEOF_MP_DIGIT      1
#ifdef DIGIT_BIT
#error You must not define DIGIT_BIT when using MP_8BIT
#endif
#elif defined(MP_16BIT)
   typedef uint16_t             mp_digit;
   typedef uint32_t             mp_word;
#define MP_SIZEOF_MP_DIGIT      2
#ifdef DIGIT_BIT
#error You must not define DIGIT_BIT when using MP_16BIT
#endif
#elif defined(MP_64BIT)
   /* for GCC only on supported platforms */
#ifndef CRYPT
   typedef unsigned long long   ulong64;
   typedef signed long long     long64;
#endif

   typedef uint64_t mp_digit;
#if defined(_WIN32)
   typedef unsigned __int128    mp_word;
#elif defined(__GNUC__)
   typedef unsigned long        mp_word __attribute__ ((mode(TI)));
#else
   /* it seems you have a problem
    * but we assume you can somewhere define your own uint128_t */
   typedef uint128_t            mp_word;
#endif

   #define DIGIT_BIT            60
#else
   /* this is the default case, 28-bit digits */

   /* this is to make porting into LibTomCrypt easier :-) */
#ifndef CRYPT




   typedef unsigned long long   ulong64;
   typedef signed long long     long64;

#endif

   typedef uint32_t             mp_digit;
   typedef uint64_t             mp_word;

#ifdef MP_31BIT
   /* this is an extension that uses 31-bit digits */
   #define DIGIT_BIT            31
#else
   /* default case is 28-bit digits, defines MP_28BIT as a handy macro to test */
   #define DIGIT_BIT            28
   #define MP_28BIT
#endif
#endif

/* otherwise the bits per digit is calculated automatically from the size of a mp_digit */
#ifndef DIGIT_BIT


   #define DIGIT_BIT     (((CHAR_BIT * MP_SIZEOF_MP_DIGIT) - 1))  /* bits per digit */
   typedef uint_least32_t mp_min_u32;


#else
   typedef mp_digit mp_min_u32;





#endif

/* platforms that can use a better rand function */
#if defined(__FreeBSD__) || defined(__OpenBSD__) || defined(__NetBSD__) || defined(__DragonFly__)
    #define MP_USE_ALT_RAND 1
#endif


/* use arc4random on platforms that support it */
#ifdef MP_USE_ALT_RAND
    #define MP_GEN_RANDOM()    arc4random()
#else
    #define MP_GEN_RANDOM()    rand()
#endif

#define MP_DIGIT_BIT     DIGIT_BIT
#define MP_MASK          ((((mp_digit)1)<<((mp_digit)DIGIT_BIT))-((mp_digit)1))
#define MP_DIGIT_MAX     MP_MASK

/* equalities */
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           TOOM_SQR_CUTOFF;

/* define this to use lower memory usage routines (exptmods mostly) */
/* #define MP_LOW_MEM */

/* default precision */
#ifndef MP_PREC
#  ifndef MP_LOW_MEM
#     define MP_PREC                 32     /* default digits of precision */
#  else
#     define MP_PREC                 8      /* default digits of precision */
#  endif
#endif

/* size of comba arrays, should be at least 2 * 2**(BITS_PER_WORD - BITS_PER_DIGIT*2) */
#define MP_WARRAY               (1 << (sizeof(mp_word) * CHAR_BIT - 2 * DIGIT_BIT + 1))

/* the infamous mp_int structure */
typedef struct  {
    int used, alloc, sign;
    mp_digit *dp;
} mp_int;

/* callback for mp_prime_random, should fill dst with random bytes and return how many read [upto len] */
typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat);


#define USED(m)    ((m)->used)
#define DIGIT(m,k) ((m)->dp[(k)])
#define SIGN(m)    ((m)->sign)

/* error code to char* string */
char *mp_error_to_string(int code);

/* ---> init and deinit bignum functions <--- */
/* init a bignum */
int mp_init(mp_int *a);

/* free a bignum */
void mp_clear(mp_int *a);







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           TOOM_SQR_CUTOFF;

/* define this to use lower memory usage routines (exptmods mostly) */
/* #define MP_LOW_MEM */

/* default precision */
#ifndef MP_PREC
   #ifndef MP_LOW_MEM
      #define MP_PREC                 32     /* default digits of precision */
   #else
      #define MP_PREC                 8      /* default digits of precision */
   #endif
#endif

/* size of comba arrays, should be at least 2 * 2**(BITS_PER_WORD - BITS_PER_DIGIT*2) */
#define MP_WARRAY               (1 << (((sizeof(mp_word) * CHAR_BIT) - (2 * DIGIT_BIT)) + 1))

/* the infamous mp_int structure */
typedef struct  {
    int used, alloc, sign;
    mp_digit *dp;
} mp_int;

/* callback for mp_prime_random, should fill dst with random bytes and return how many read [upto len] */
typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat);


#define USED(m)    ((m)->used)
#define DIGIT(m,k) ((m)->dp[(k)])
#define SIGN(m)    ((m)->sign)

/* error code to char* string */
const char *mp_error_to_string(int code);

/* ---> init and deinit bignum functions <--- */
/* init a bignum */
int mp_init(mp_int *a);

/* free a bignum */
void mp_clear(mp_int *a);
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int mp_grow(mp_int *a, int size);

/* init to a given number of digits */
int mp_init_size(mp_int *a, int size);

/* ---> Basic Manipulations <--- */
#define mp_iszero(a) (((a)->used == 0) ? MP_YES : MP_NO)
#define mp_iseven(a) (((a)->used == 0 || (((a)->dp[0] & 1) == 0)) ? MP_YES : MP_NO)
#define mp_isodd(a)  (((a)->used > 0 && (((a)->dp[0] & 1) == 1)) ? MP_YES : MP_NO)


/* set to zero */
void mp_zero(mp_int *a);

/* set to a digit */
void mp_set(mp_int *a, mp_digit b);

/* set a 32-bit const */
int mp_set_int(mp_int *a, unsigned long b);







/* get a 32-bit value */
unsigned long mp_get_int(mp_int * a);







/* initialize and set a digit */
int mp_init_set (mp_int * a, mp_digit b);

/* initialize and set 32-bit value */
int mp_init_set_int (mp_int * a, unsigned long b);

/* copy, b = a */
int mp_copy(const mp_int *a, mp_int *b);

/* inits and copies, a = b */
int mp_init_copy(mp_int *a, mp_int *b);

/* trim unused digits */
void mp_clamp(mp_int *a);







/* ---> digit manipulation <--- */

/* right shift by "b" digits */
void mp_rshd(mp_int *a, int b);

/* left shift by "b" digits */
int mp_lshd(mp_int *a, int b);

/* c = a / 2**b */
int mp_div_2d(const mp_int *a, int b, mp_int *c, mp_int *d);

/* b = a/2 */
int mp_div_2(mp_int *a, mp_int *b);

/* c = a * 2**b */
int mp_mul_2d(const mp_int *a, int b, mp_int *c);

/* b = a*2 */
int mp_mul_2(mp_int *a, mp_int *b);

/* c = a mod 2**d */
int mp_mod_2d(const mp_int *a, int b, mp_int *c);

/* computes a = 2**b */
int mp_2expt(mp_int *a, int b);

/* Counts the number of lsbs which are zero before the first zero bit */
int mp_cnt_lsb(mp_int *a);







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int mp_grow(mp_int *a, int size);

/* init to a given number of digits */
int mp_init_size(mp_int *a, int size);

/* ---> Basic Manipulations <--- */
#define mp_iszero(a) (((a)->used == 0) ? MP_YES : MP_NO)
#define mp_iseven(a) ((((a)->used > 0) && (((a)->dp[0] & 1u) == 0u)) ? MP_YES : MP_NO)
#define mp_isodd(a)  ((((a)->used > 0) && (((a)->dp[0] & 1u) == 1u)) ? MP_YES : MP_NO)
#define mp_isneg(a)  (((a)->sign != MP_ZPOS) ? MP_YES : MP_NO)

/* set to zero */
void mp_zero(mp_int *a);

/* set to a digit */
void mp_set(mp_int *a, mp_digit b);

/* set a 32-bit const */
int mp_set_int(mp_int *a, unsigned long b);

/* set a platform dependent unsigned long value */
int mp_set_long(mp_int *a, unsigned long b);

/* set a platform dependent unsigned long long value */
int mp_set_long_long(mp_int *a, unsigned long long b);

/* get a 32-bit value */
unsigned long mp_get_int(mp_int * a);

/* get a platform dependent unsigned long value */
unsigned long mp_get_long(mp_int * a);

/* get a platform dependent unsigned long long value */
unsigned long long mp_get_long_long(mp_int * a);

/* initialize and set a digit */
int mp_init_set (mp_int * a, mp_digit b);

/* initialize and set 32-bit value */
int mp_init_set_int (mp_int * a, unsigned long b);

/* copy, b = a */
int mp_copy(const mp_int *a, mp_int *b);

/* inits and copies, a = b */
int mp_init_copy(mp_int *a, mp_int *b);

/* trim unused digits */
void mp_clamp(mp_int *a);

/* import binary data */
int mp_import(mp_int* rop, size_t count, int order, size_t size, int endian, size_t nails, const void* op);

/* export binary data */
int mp_export(void* rop, size_t* countp, int order, size_t size, int endian, size_t nails, mp_int* op);

/* ---> digit manipulation <--- */

/* right shift by "b" digits */
void mp_rshd(mp_int *a, int b);

/* left shift by "b" digits */
int mp_lshd(mp_int *a, int b);

/* c = a / 2**b, implemented as c = a >> b */
int mp_div_2d(const mp_int *a, int b, mp_int *c, mp_int *d);

/* b = a/2 */
int mp_div_2(mp_int *a, mp_int *b);

/* c = a * 2**b, implemented as c = a << b */
int mp_mul_2d(const mp_int *a, int b, mp_int *c);

/* b = a*2 */
int mp_mul_2(mp_int *a, mp_int *b);

/* c = a mod 2**b */
int mp_mod_2d(const mp_int *a, int b, mp_int *c);

/* computes a = 2**b */
int mp_2expt(mp_int *a, int b);

/* Counts the number of lsbs which are zero before the first zero bit */
int mp_cnt_lsb(mp_int *a);
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int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d);

/* a/3 => 3c + d == a */
int mp_div_3(mp_int *a, mp_int *c, mp_digit *d);

/* c = a**b */
int mp_expt_d(mp_int *a, mp_digit b, mp_int *c);


/* c = a mod b, 0 <= c < b  */
int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c);

/* ---> number theory <--- */

/* d = a + b (mod c) */







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int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d);

/* a/3 => 3c + d == a */
int mp_div_3(mp_int *a, mp_int *c, mp_digit *d);

/* c = a**b */
int mp_expt_d(mp_int *a, mp_digit b, mp_int *c);
int mp_expt_d_ex (mp_int * a, mp_digit b, mp_int * c, int fast);

/* c = a mod b, 0 <= c < b  */
int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c);

/* ---> number theory <--- */

/* d = a + b (mod c) */
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int mp_lcm(mp_int *a, mp_int *b, mp_int *c);

/* finds one of the b'th root of a, such that |c|**b <= |a|
 *
 * returns error if a < 0 and b is even
 */
int mp_n_root(mp_int *a, mp_digit b, mp_int *c);


/* special sqrt algo */
int mp_sqrt(mp_int *arg, mp_int *ret);




/* is number a square? */
int mp_is_square(mp_int *arg, int *ret);

/* computes the jacobi c = (a | n) (or Legendre if b is prime)  */
int mp_jacobi(mp_int *a, mp_int *n, int *c);

/* used to setup the Barrett reduction for a given modulus b */







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int mp_lcm(mp_int *a, mp_int *b, mp_int *c);

/* finds one of the b'th root of a, such that |c|**b <= |a|
 *
 * returns error if a < 0 and b is even
 */
int mp_n_root(mp_int *a, mp_digit b, mp_int *c);
int mp_n_root_ex (mp_int * a, mp_digit b, mp_int * c, int fast);

/* special sqrt algo */
int mp_sqrt(mp_int *arg, mp_int *ret);

/* special sqrt (mod prime) */
int mp_sqrtmod_prime(mp_int *arg, mp_int *prime, mp_int *ret);

/* is number a square? */
int mp_is_square(mp_int *arg, int *ret);

/* computes the jacobi c = (a | n) (or Legendre if b is prime)  */
int mp_jacobi(mp_int *a, mp_int *n, int *c);

/* used to setup the Barrett reduction for a given modulus b */
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#ifdef MP_8BIT
#  define PRIME_SIZE      31
#else
#  define PRIME_SIZE      256
#endif

/* table of first PRIME_SIZE primes */
extern const mp_digit ltm_prime_tab[];

/* result=1 if a is divisible by one of the first PRIME_SIZE primes */
int mp_prime_is_divisible(mp_int *a, int *result);

/* performs one Fermat test of "a" using base "b".
 * Sets result to 0 if composite or 1 if probable prime
 */
int mp_prime_fermat(mp_int *a, mp_int *b, int *result);

/* performs one Miller-Rabin test of "a" using base "b".
 * Sets result to 0 if composite or 1 if probable prime
 */
int mp_prime_miller_rabin(mp_int *a, mp_int *b, int *result);

/* This gives [for a given bit size] the number of trials required
 * such that Miller-Rabin gives a prob of failure lower than 2^-96 
 */
int mp_prime_rabin_miller_trials(int size);

/* performs t rounds of Miller-Rabin on "a" using the first
 * t prime bases.  Also performs an initial sieve of trial
 * division.  Determines if "a" is prime with probability
 * of error no more than (1/4)**t.
 *
 * Sets result to 1 if probably prime, 0 otherwise
 */
int mp_prime_is_prime(mp_int *a, int t, int *result);

/* finds the next prime after the number "a" using "t" trials
 * of Miller-Rabin.
 *
 * bbs_style = 1 means the prime must be congruent to 3 mod 4
 */
int mp_prime_next_prime(mp_int *a, int t, int bbs_style);

/* makes a truly random prime of a given size (bytes),
 * call with bbs = 1 if you want it to be congruent to 3 mod 4 
 *
 * You have to supply a callback which fills in a buffer with random bytes.  "dat" is a parameter you can
 * have passed to the callback (e.g. a state or something).  This function doesn't use "dat" itself
 * so it can be NULL
 *
 * The prime generated will be larger than 2^(8*size).
 */
#define mp_prime_random(a, t, size, bbs, cb, dat) mp_prime_random_ex(a, t, ((size) * 8) + 1, (bbs==1)?LTM_PRIME_BBS:0, cb, dat)

/* makes a truly random prime of a given size (bits),
 *
 * Flags are as follows:
 * 
 *   LTM_PRIME_BBS      - make prime congruent to 3 mod 4
 *   LTM_PRIME_SAFE     - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS)
 *   LTM_PRIME_2MSB_OFF - make the 2nd highest bit zero
 *   LTM_PRIME_2MSB_ON  - make the 2nd highest bit one
 *
 * You have to supply a callback which fills in a buffer with random bytes.  "dat" is a parameter you can
 * have passed to the callback (e.g. a state or something).  This function doesn't use "dat" itself
 * so it can be NULL
 *
 */







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#ifdef MP_8BIT
#  define PRIME_SIZE      31
#else
#  define PRIME_SIZE      256
#endif

/* table of first PRIME_SIZE primes */
extern const mp_digit ltm_prime_tab[PRIME_SIZE];

/* result=1 if a is divisible by one of the first PRIME_SIZE primes */
int mp_prime_is_divisible(mp_int *a, int *result);

/* performs one Fermat test of "a" using base "b".
 * Sets result to 0 if composite or 1 if probable prime
 */
int mp_prime_fermat(mp_int *a, mp_int *b, int *result);

/* performs one Miller-Rabin test of "a" using base "b".
 * Sets result to 0 if composite or 1 if probable prime
 */
int mp_prime_miller_rabin(mp_int *a, mp_int *b, int *result);

/* This gives [for a given bit size] the number of trials required
 * such that Miller-Rabin gives a prob of failure lower than 2^-96
 */
int mp_prime_rabin_miller_trials(int size);

/* performs t rounds of Miller-Rabin on "a" using the first
 * t prime bases.  Also performs an initial sieve of trial
 * division.  Determines if "a" is prime with probability
 * of error no more than (1/4)**t.
 *
 * Sets result to 1 if probably prime, 0 otherwise
 */
int mp_prime_is_prime(mp_int *a, int t, int *result);

/* finds the next prime after the number "a" using "t" trials
 * of Miller-Rabin.
 *
 * bbs_style = 1 means the prime must be congruent to 3 mod 4
 */
int mp_prime_next_prime(mp_int *a, int t, int bbs_style);

/* makes a truly random prime of a given size (bytes),
 * call with bbs = 1 if you want it to be congruent to 3 mod 4
 *
 * You have to supply a callback which fills in a buffer with random bytes.  "dat" is a parameter you can
 * have passed to the callback (e.g. a state or something).  This function doesn't use "dat" itself
 * so it can be NULL
 *
 * The prime generated will be larger than 2^(8*size).
 */
#define mp_prime_random(a, t, size, bbs, cb, dat) mp_prime_random_ex(a, t, ((size) * 8) + 1, (bbs==1)?LTM_PRIME_BBS:0, cb, dat)

/* makes a truly random prime of a given size (bits),
 *
 * Flags are as follows:
 *
 *   LTM_PRIME_BBS      - make prime congruent to 3 mod 4
 *   LTM_PRIME_SAFE     - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS)

 *   LTM_PRIME_2MSB_ON  - make the 2nd highest bit one
 *
 * You have to supply a callback which fills in a buffer with random bytes.  "dat" is a parameter you can
 * have passed to the callback (e.g. a state or something).  This function doesn't use "dat" itself
 * so it can be NULL
 *
 */
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int mp_to_signed_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen);

int mp_read_radix(mp_int *a, const char *str, int radix);
int mp_toradix(mp_int *a, char *str, int radix);
int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen);
int mp_radix_size(mp_int *a, int radix, int *size);


int mp_fread(mp_int *a, int radix, FILE *stream);
int mp_fwrite(mp_int *a, int radix, FILE *stream);


#define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len))
#define mp_raw_size(mp)           mp_signed_bin_size(mp)
#define mp_toraw(mp, str)         mp_to_signed_bin((mp), (str))
#define mp_read_mag(mp, str, len) mp_read_unsigned_bin((mp), (str), (len))
#define mp_mag_size(mp)           mp_unsigned_bin_size(mp)
#define mp_tomag(mp, str)         mp_to_unsigned_bin((mp), (str))

#define mp_tobinary(M, S)  mp_toradix((M), (S), 2)
#define mp_tooctal(M, S)   mp_toradix((M), (S), 8)
#define mp_todecimal(M, S) mp_toradix((M), (S), 10)
#define mp_tohex(M, S)     mp_toradix((M), (S), 16)

/* lowlevel functions, do not call! */
int s_mp_add(mp_int *a, mp_int *b, mp_int *c);
int s_mp_sub(mp_int *a, mp_int *b, mp_int *c);
#define s_mp_mul(a, b, c) s_mp_mul_digs(a, b, c, (a)->used + (b)->used + 1)
int fast_s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
int s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
int fast_s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
int s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
int fast_s_mp_sqr(mp_int *a, mp_int *b);
int s_mp_sqr(mp_int *a, mp_int *b);
int mp_karatsuba_mul(mp_int *a, mp_int *b, mp_int *c);
int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c);
int mp_karatsuba_sqr(mp_int *a, mp_int *b);
int mp_toom_sqr(mp_int *a, mp_int *b);
int fast_mp_invmod(mp_int *a, mp_int *b, mp_int *c);
int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c);
int fast_mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
int mp_exptmod_fast(mp_int *G, mp_int *X, mp_int *P, mp_int *Y, int mode);
int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int mode);
void bn_reverse(unsigned char *s, int len);

extern const char *mp_s_rmap;

#ifdef __cplusplus
}
#endif

#endif












>


>













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int mp_to_signed_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen);

int mp_read_radix(mp_int *a, const char *str, int radix);
int mp_toradix(mp_int *a, char *str, int radix);
int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen);
int mp_radix_size(mp_int *a, int radix, int *size);

#ifndef LTM_NO_FILE
int mp_fread(mp_int *a, int radix, FILE *stream);
int mp_fwrite(mp_int *a, int radix, FILE *stream);
#endif

#define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len))
#define mp_raw_size(mp)           mp_signed_bin_size(mp)
#define mp_toraw(mp, str)         mp_to_signed_bin((mp), (str))
#define mp_read_mag(mp, str, len) mp_read_unsigned_bin((mp), (str), (len))
#define mp_mag_size(mp)           mp_unsigned_bin_size(mp)
#define mp_tomag(mp, str)         mp_to_unsigned_bin((mp), (str))

#define mp_tobinary(M, S)  mp_toradix((M), (S), 2)
#define mp_tooctal(M, S)   mp_toradix((M), (S), 8)
#define mp_todecimal(M, S) mp_toradix((M), (S), 10)
#define mp_tohex(M, S)     mp_toradix((M), (S), 16)
























#ifdef __cplusplus
}
#endif

#endif


/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/tommath.out.
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\BOOKMARK [0][-]{chapter.1}{Introduction}{}
\BOOKMARK [1][-]{section.1.1}{Multiple Precision Arithmetic}{chapter.1}
\BOOKMARK [2][-]{subsection.1.1.1}{What is Multiple Precision Arithmetic?}{section.1.1}
\BOOKMARK [2][-]{subsection.1.1.2}{The Need for Multiple Precision Arithmetic}{section.1.1}
\BOOKMARK [2][-]{subsection.1.1.3}{Benefits of Multiple Precision Arithmetic}{section.1.1}
\BOOKMARK [1][-]{section.1.2}{Purpose of This Text}{chapter.1}
\BOOKMARK [1][-]{section.1.3}{Discussion and Notation}{chapter.1}
\BOOKMARK [2][-]{subsection.1.3.1}{Notation}{section.1.3}
\BOOKMARK [2][-]{subsection.1.3.2}{Precision Notation}{section.1.3}
\BOOKMARK [2][-]{subsection.1.3.3}{Algorithm Inputs and Outputs}{section.1.3}
\BOOKMARK [2][-]{subsection.1.3.4}{Mathematical Expressions}{section.1.3}
\BOOKMARK [2][-]{subsection.1.3.5}{Work Effort}{section.1.3}
\BOOKMARK [1][-]{section.1.4}{Exercises}{chapter.1}
\BOOKMARK [1][-]{section.1.5}{Introduction to LibTomMath}{chapter.1}
\BOOKMARK [2][-]{subsection.1.5.1}{What is LibTomMath?}{section.1.5}
\BOOKMARK [2][-]{subsection.1.5.2}{Goals of LibTomMath}{section.1.5}
\BOOKMARK [1][-]{section.1.6}{Choice of LibTomMath}{chapter.1}
\BOOKMARK [2][-]{subsection.1.6.1}{Code Base}{section.1.6}
\BOOKMARK [2][-]{subsection.1.6.2}{API Simplicity}{section.1.6}
\BOOKMARK [2][-]{subsection.1.6.3}{Optimizations}{section.1.6}
\BOOKMARK [2][-]{subsection.1.6.4}{Portability and Stability}{section.1.6}
\BOOKMARK [2][-]{subsection.1.6.5}{Choice}{section.1.6}
\BOOKMARK [0][-]{chapter.2}{Getting Started}{}
\BOOKMARK [1][-]{section.2.1}{Library Basics}{chapter.2}
\BOOKMARK [1][-]{section.2.2}{What is a Multiple Precision Integer?}{chapter.2}
\BOOKMARK [2][-]{subsection.2.2.1}{The mp\137int Structure}{section.2.2}
\BOOKMARK [1][-]{section.2.3}{Argument Passing}{chapter.2}
\BOOKMARK [1][-]{section.2.4}{Return Values}{chapter.2}
\BOOKMARK [1][-]{section.2.5}{Initialization and Clearing}{chapter.2}
\BOOKMARK [2][-]{subsection.2.5.1}{Initializing an mp\137int}{section.2.5}
\BOOKMARK [2][-]{subsection.2.5.2}{Clearing an mp\137int}{section.2.5}
\BOOKMARK [1][-]{section.2.6}{Maintenance Algorithms}{chapter.2}
\BOOKMARK [2][-]{subsection.2.6.1}{Augmenting an mp\137int's Precision}{section.2.6}
\BOOKMARK [2][-]{subsection.2.6.2}{Initializing Variable Precision mp\137ints}{section.2.6}
\BOOKMARK [2][-]{subsection.2.6.3}{Multiple Integer Initializations and Clearings}{section.2.6}
\BOOKMARK [2][-]{subsection.2.6.4}{Clamping Excess Digits}{section.2.6}
\BOOKMARK [0][-]{chapter.3}{Basic Operations}{}
\BOOKMARK [1][-]{section.3.1}{Introduction}{chapter.3}
\BOOKMARK [1][-]{section.3.2}{Assigning Values to mp\137int Structures}{chapter.3}
\BOOKMARK [2][-]{subsection.3.2.1}{Copying an mp\137int}{section.3.2}
\BOOKMARK [2][-]{subsection.3.2.2}{Creating a Clone}{section.3.2}
\BOOKMARK [1][-]{section.3.3}{Zeroing an Integer}{chapter.3}
\BOOKMARK [1][-]{section.3.4}{Sign Manipulation}{chapter.3}
\BOOKMARK [2][-]{subsection.3.4.1}{Absolute Value}{section.3.4}
\BOOKMARK [2][-]{subsection.3.4.2}{Integer Negation}{section.3.4}
\BOOKMARK [1][-]{section.3.5}{Small Constants}{chapter.3}
\BOOKMARK [2][-]{subsection.3.5.1}{Setting Small Constants}{section.3.5}
\BOOKMARK [2][-]{subsection.3.5.2}{Setting Large Constants}{section.3.5}
\BOOKMARK [1][-]{section.3.6}{Comparisons}{chapter.3}
\BOOKMARK [2][-]{subsection.3.6.1}{Unsigned Comparisions}{section.3.6}
\BOOKMARK [2][-]{subsection.3.6.2}{Signed Comparisons}{section.3.6}
\BOOKMARK [0][-]{chapter.4}{Basic Arithmetic}{}
\BOOKMARK [1][-]{section.4.1}{Introduction}{chapter.4}
\BOOKMARK [1][-]{section.4.2}{Addition and Subtraction}{chapter.4}
\BOOKMARK [2][-]{subsection.4.2.1}{Low Level Addition}{section.4.2}
\BOOKMARK [2][-]{subsection.4.2.2}{Low Level Subtraction}{section.4.2}
\BOOKMARK [2][-]{subsection.4.2.3}{High Level Addition}{section.4.2}
\BOOKMARK [2][-]{subsection.4.2.4}{High Level Subtraction}{section.4.2}
\BOOKMARK [1][-]{section.4.3}{Bit and Digit Shifting}{chapter.4}
\BOOKMARK [2][-]{subsection.4.3.1}{Multiplication by Two}{section.4.3}
\BOOKMARK [2][-]{subsection.4.3.2}{Division by Two}{section.4.3}
\BOOKMARK [1][-]{section.4.4}{Polynomial Basis Operations}{chapter.4}
\BOOKMARK [2][-]{subsection.4.4.1}{Multiplication by x}{section.4.4}
\BOOKMARK [2][-]{subsection.4.4.2}{Division by x}{section.4.4}
\BOOKMARK [1][-]{section.4.5}{Powers of Two}{chapter.4}
\BOOKMARK [2][-]{subsection.4.5.1}{Multiplication by Power of Two}{section.4.5}
\BOOKMARK [2][-]{subsection.4.5.2}{Division by Power of Two}{section.4.5}
\BOOKMARK [2][-]{subsection.4.5.3}{Remainder of Division by Power of Two}{section.4.5}
\BOOKMARK [0][-]{chapter.5}{Multiplication and Squaring}{}
\BOOKMARK [1][-]{section.5.1}{The Multipliers}{chapter.5}
\BOOKMARK [1][-]{section.5.2}{Multiplication}{chapter.5}
\BOOKMARK [2][-]{subsection.5.2.1}{The Baseline Multiplication}{section.5.2}
\BOOKMARK [2][-]{subsection.5.2.2}{Faster Multiplication by the ``Comba'' Method}{section.5.2}
\BOOKMARK [2][-]{subsection.5.2.3}{Polynomial Basis Multiplication}{section.5.2}
\BOOKMARK [2][-]{subsection.5.2.4}{Karatsuba Multiplication}{section.5.2}
\BOOKMARK [2][-]{subsection.5.2.5}{Toom-Cook 3-Way Multiplication}{section.5.2}
\BOOKMARK [2][-]{subsection.5.2.6}{Signed Multiplication}{section.5.2}
\BOOKMARK [1][-]{section.5.3}{Squaring}{chapter.5}
\BOOKMARK [2][-]{subsection.5.3.1}{The Baseline Squaring Algorithm}{section.5.3}
\BOOKMARK [2][-]{subsection.5.3.2}{Faster Squaring by the ``Comba'' Method}{section.5.3}
\BOOKMARK [2][-]{subsection.5.3.3}{Polynomial Basis Squaring}{section.5.3}
\BOOKMARK [2][-]{subsection.5.3.4}{Karatsuba Squaring}{section.5.3}
\BOOKMARK [2][-]{subsection.5.3.5}{Toom-Cook Squaring}{section.5.3}
\BOOKMARK [2][-]{subsection.5.3.6}{High Level Squaring}{section.5.3}
\BOOKMARK [0][-]{chapter.6}{Modular Reduction}{}
\BOOKMARK [1][-]{section.6.1}{Basics of Modular Reduction}{chapter.6}
\BOOKMARK [1][-]{section.6.2}{The Barrett Reduction}{chapter.6}
\BOOKMARK [2][-]{subsection.6.2.1}{Fixed Point Arithmetic}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.2}{Choosing a Radix Point}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.3}{Trimming the Quotient}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.4}{Trimming the Residue}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.5}{The Barrett Algorithm}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.6}{The Barrett Setup Algorithm}{section.6.2}
\BOOKMARK [1][-]{section.6.3}{The Montgomery Reduction}{chapter.6}
\BOOKMARK [2][-]{subsection.6.3.1}{Digit Based Montgomery Reduction}{section.6.3}
\BOOKMARK [2][-]{subsection.6.3.2}{Baseline Montgomery Reduction}{section.6.3}
\BOOKMARK [2][-]{subsection.6.3.3}{Faster ``Comba'' Montgomery Reduction}{section.6.3}
\BOOKMARK [2][-]{subsection.6.3.4}{Montgomery Setup}{section.6.3}
\BOOKMARK [1][-]{section.6.4}{The Diminished Radix Algorithm}{chapter.6}
\BOOKMARK [2][-]{subsection.6.4.1}{Choice of Moduli}{section.6.4}
\BOOKMARK [2][-]{subsection.6.4.2}{Choice of k}{section.6.4}
\BOOKMARK [2][-]{subsection.6.4.3}{Restricted Diminished Radix Reduction}{section.6.4}
\BOOKMARK [2][-]{subsection.6.4.4}{Unrestricted Diminished Radix Reduction}{section.6.4}
\BOOKMARK [1][-]{section.6.5}{Algorithm Comparison}{chapter.6}
\BOOKMARK [0][-]{chapter.7}{Exponentiation}{}
\BOOKMARK [1][-]{section.7.1}{Exponentiation Basics}{chapter.7}
\BOOKMARK [2][-]{subsection.7.1.1}{Single Digit Exponentiation}{section.7.1}
\BOOKMARK [1][-]{section.7.2}{k-ary Exponentiation}{chapter.7}
\BOOKMARK [2][-]{subsection.7.2.1}{Optimal Values of k}{section.7.2}
\BOOKMARK [2][-]{subsection.7.2.2}{Sliding-Window Exponentiation}{section.7.2}
\BOOKMARK [1][-]{section.7.3}{Modular Exponentiation}{chapter.7}
\BOOKMARK [2][-]{subsection.7.3.1}{Barrett Modular Exponentiation}{section.7.3}
\BOOKMARK [1][-]{section.7.4}{Quick Power of Two}{chapter.7}
\BOOKMARK [0][-]{chapter.8}{Higher Level Algorithms}{}
\BOOKMARK [1][-]{section.8.1}{Integer Division with Remainder}{chapter.8}
\BOOKMARK [2][-]{subsection.8.1.1}{Quotient Estimation}{section.8.1}
\BOOKMARK [2][-]{subsection.8.1.2}{Normalized Integers}{section.8.1}
\BOOKMARK [2][-]{subsection.8.1.3}{Radix- Division with Remainder}{section.8.1}
\BOOKMARK [1][-]{section.8.2}{Single Digit Helpers}{chapter.8}
\BOOKMARK [2][-]{subsection.8.2.1}{Single Digit Addition and Subtraction}{section.8.2}
\BOOKMARK [2][-]{subsection.8.2.2}{Single Digit Multiplication}{section.8.2}
\BOOKMARK [2][-]{subsection.8.2.3}{Single Digit Division}{section.8.2}
\BOOKMARK [2][-]{subsection.8.2.4}{Single Digit Root Extraction}{section.8.2}
\BOOKMARK [1][-]{section.8.3}{Random Number Generation}{chapter.8}
\BOOKMARK [1][-]{section.8.4}{Formatted Representations}{chapter.8}
\BOOKMARK [2][-]{subsection.8.4.1}{Reading Radix-n Input}{section.8.4}
\BOOKMARK [2][-]{subsection.8.4.2}{Generating Radix-n Output}{section.8.4}
\BOOKMARK [0][-]{chapter.9}{Number Theoretic Algorithms}{}
\BOOKMARK [1][-]{section.9.1}{Greatest Common Divisor}{chapter.9}
\BOOKMARK [2][-]{subsection.9.1.1}{Complete Greatest Common Divisor}{section.9.1}
\BOOKMARK [1][-]{section.9.2}{Least Common Multiple}{chapter.9}
\BOOKMARK [1][-]{section.9.3}{Jacobi Symbol Computation}{chapter.9}
\BOOKMARK [2][-]{subsection.9.3.1}{Jacobi Symbol}{section.9.3}
\BOOKMARK [1][-]{section.9.4}{Modular Inverse}{chapter.9}
\BOOKMARK [2][-]{subsection.9.4.1}{General Case}{section.9.4}
\BOOKMARK [1][-]{section.9.5}{Primality Tests}{chapter.9}
\BOOKMARK [2][-]{subsection.9.5.1}{Trial Division}{section.9.5}
\BOOKMARK [2][-]{subsection.9.5.2}{The Fermat Test}{section.9.5}
\BOOKMARK [2][-]{subsection.9.5.3}{The Miller-Rabin Test}{section.9.5}
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\BOOKMARK [0][-]{chapter.1}{Introduction}{}% 1
\BOOKMARK [1][-]{section.1.1}{Multiple Precision Arithmetic}{chapter.1}% 2
\BOOKMARK [2][-]{subsection.1.1.1}{What is Multiple Precision Arithmetic?}{section.1.1}% 3
\BOOKMARK [2][-]{subsection.1.1.2}{The Need for Multiple Precision Arithmetic}{section.1.1}% 4
\BOOKMARK [2][-]{subsection.1.1.3}{Benefits of Multiple Precision Arithmetic}{section.1.1}% 5
\BOOKMARK [1][-]{section.1.2}{Purpose of This Text}{chapter.1}% 6
\BOOKMARK [1][-]{section.1.3}{Discussion and Notation}{chapter.1}% 7
\BOOKMARK [2][-]{subsection.1.3.1}{Notation}{section.1.3}% 8
\BOOKMARK [2][-]{subsection.1.3.2}{Precision Notation}{section.1.3}% 9
\BOOKMARK [2][-]{subsection.1.3.3}{Algorithm Inputs and Outputs}{section.1.3}% 10
\BOOKMARK [2][-]{subsection.1.3.4}{Mathematical Expressions}{section.1.3}% 11
\BOOKMARK [2][-]{subsection.1.3.5}{Work Effort}{section.1.3}% 12
\BOOKMARK [1][-]{section.1.4}{Exercises}{chapter.1}% 13
\BOOKMARK [1][-]{section.1.5}{Introduction to LibTomMath}{chapter.1}% 14
\BOOKMARK [2][-]{subsection.1.5.1}{What is LibTomMath?}{section.1.5}% 15
\BOOKMARK [2][-]{subsection.1.5.2}{Goals of LibTomMath}{section.1.5}% 16
\BOOKMARK [1][-]{section.1.6}{Choice of LibTomMath}{chapter.1}% 17
\BOOKMARK [2][-]{subsection.1.6.1}{Code Base}{section.1.6}% 18
\BOOKMARK [2][-]{subsection.1.6.2}{API Simplicity}{section.1.6}% 19
\BOOKMARK [2][-]{subsection.1.6.3}{Optimizations}{section.1.6}% 20
\BOOKMARK [2][-]{subsection.1.6.4}{Portability and Stability}{section.1.6}% 21
\BOOKMARK [2][-]{subsection.1.6.5}{Choice}{section.1.6}% 22
\BOOKMARK [0][-]{chapter.2}{Getting Started}{}% 23
\BOOKMARK [1][-]{section.2.1}{Library Basics}{chapter.2}% 24
\BOOKMARK [1][-]{section.2.2}{What is a Multiple Precision Integer?}{chapter.2}% 25
\BOOKMARK [2][-]{subsection.2.2.1}{The mp\137int Structure}{section.2.2}% 26
\BOOKMARK [1][-]{section.2.3}{Argument Passing}{chapter.2}% 27
\BOOKMARK [1][-]{section.2.4}{Return Values}{chapter.2}% 28
\BOOKMARK [1][-]{section.2.5}{Initialization and Clearing}{chapter.2}% 29
\BOOKMARK [2][-]{subsection.2.5.1}{Initializing an mp\137int}{section.2.5}% 30
\BOOKMARK [2][-]{subsection.2.5.2}{Clearing an mp\137int}{section.2.5}% 31
\BOOKMARK [1][-]{section.2.6}{Maintenance Algorithms}{chapter.2}% 32
\BOOKMARK [2][-]{subsection.2.6.1}{Augmenting an mp\137int's Precision}{section.2.6}% 33
\BOOKMARK [2][-]{subsection.2.6.2}{Initializing Variable Precision mp\137ints}{section.2.6}% 34
\BOOKMARK [2][-]{subsection.2.6.3}{Multiple Integer Initializations and Clearings}{section.2.6}% 35
\BOOKMARK [2][-]{subsection.2.6.4}{Clamping Excess Digits}{section.2.6}% 36
\BOOKMARK [0][-]{chapter.3}{Basic Operations}{}% 37
\BOOKMARK [1][-]{section.3.1}{Introduction}{chapter.3}% 38
\BOOKMARK [1][-]{section.3.2}{Assigning Values to mp\137int Structures}{chapter.3}% 39
\BOOKMARK [2][-]{subsection.3.2.1}{Copying an mp\137int}{section.3.2}% 40
\BOOKMARK [2][-]{subsection.3.2.2}{Creating a Clone}{section.3.2}% 41
\BOOKMARK [1][-]{section.3.3}{Zeroing an Integer}{chapter.3}% 42
\BOOKMARK [1][-]{section.3.4}{Sign Manipulation}{chapter.3}% 43
\BOOKMARK [2][-]{subsection.3.4.1}{Absolute Value}{section.3.4}% 44
\BOOKMARK [2][-]{subsection.3.4.2}{Integer Negation}{section.3.4}% 45
\BOOKMARK [1][-]{section.3.5}{Small Constants}{chapter.3}% 46
\BOOKMARK [2][-]{subsection.3.5.1}{Setting Small Constants}{section.3.5}% 47
\BOOKMARK [2][-]{subsection.3.5.2}{Setting Large Constants}{section.3.5}% 48
\BOOKMARK [1][-]{section.3.6}{Comparisons}{chapter.3}% 49
\BOOKMARK [2][-]{subsection.3.6.1}{Unsigned Comparisions}{section.3.6}% 50
\BOOKMARK [2][-]{subsection.3.6.2}{Signed Comparisons}{section.3.6}% 51
\BOOKMARK [0][-]{chapter.4}{Basic Arithmetic}{}% 52
\BOOKMARK [1][-]{section.4.1}{Introduction}{chapter.4}% 53
\BOOKMARK [1][-]{section.4.2}{Addition and Subtraction}{chapter.4}% 54
\BOOKMARK [2][-]{subsection.4.2.1}{Low Level Addition}{section.4.2}% 55
\BOOKMARK [2][-]{subsection.4.2.2}{Low Level Subtraction}{section.4.2}% 56
\BOOKMARK [2][-]{subsection.4.2.3}{High Level Addition}{section.4.2}% 57
\BOOKMARK [2][-]{subsection.4.2.4}{High Level Subtraction}{section.4.2}% 58
\BOOKMARK [1][-]{section.4.3}{Bit and Digit Shifting}{chapter.4}% 59
\BOOKMARK [2][-]{subsection.4.3.1}{Multiplication by Two}{section.4.3}% 60
\BOOKMARK [2][-]{subsection.4.3.2}{Division by Two}{section.4.3}% 61
\BOOKMARK [1][-]{section.4.4}{Polynomial Basis Operations}{chapter.4}% 62
\BOOKMARK [2][-]{subsection.4.4.1}{Multiplication by x}{section.4.4}% 63
\BOOKMARK [2][-]{subsection.4.4.2}{Division by x}{section.4.4}% 64
\BOOKMARK [1][-]{section.4.5}{Powers of Two}{chapter.4}% 65
\BOOKMARK [2][-]{subsection.4.5.1}{Multiplication by Power of Two}{section.4.5}% 66
\BOOKMARK [2][-]{subsection.4.5.2}{Division by Power of Two}{section.4.5}% 67
\BOOKMARK [2][-]{subsection.4.5.3}{Remainder of Division by Power of Two}{section.4.5}% 68
\BOOKMARK [0][-]{chapter.5}{Multiplication and Squaring}{}% 69
\BOOKMARK [1][-]{section.5.1}{The Multipliers}{chapter.5}% 70
\BOOKMARK [1][-]{section.5.2}{Multiplication}{chapter.5}% 71
\BOOKMARK [2][-]{subsection.5.2.1}{The Baseline Multiplication}{section.5.2}% 72
\BOOKMARK [2][-]{subsection.5.2.2}{Faster Multiplication by the ``Comba'' Method}{section.5.2}% 73
\BOOKMARK [2][-]{subsection.5.2.3}{Polynomial Basis Multiplication}{section.5.2}% 74
\BOOKMARK [2][-]{subsection.5.2.4}{Karatsuba Multiplication}{section.5.2}% 75
\BOOKMARK [2][-]{subsection.5.2.5}{Toom-Cook 3-Way Multiplication}{section.5.2}% 76
\BOOKMARK [2][-]{subsection.5.2.6}{Signed Multiplication}{section.5.2}% 77
\BOOKMARK [1][-]{section.5.3}{Squaring}{chapter.5}% 78
\BOOKMARK [2][-]{subsection.5.3.1}{The Baseline Squaring Algorithm}{section.5.3}% 79
\BOOKMARK [2][-]{subsection.5.3.2}{Faster Squaring by the ``Comba'' Method}{section.5.3}% 80
\BOOKMARK [2][-]{subsection.5.3.3}{Polynomial Basis Squaring}{section.5.3}% 81
\BOOKMARK [2][-]{subsection.5.3.4}{Karatsuba Squaring}{section.5.3}% 82
\BOOKMARK [2][-]{subsection.5.3.5}{Toom-Cook Squaring}{section.5.3}% 83
\BOOKMARK [2][-]{subsection.5.3.6}{High Level Squaring}{section.5.3}% 84
\BOOKMARK [0][-]{chapter.6}{Modular Reduction}{}% 85
\BOOKMARK [1][-]{section.6.1}{Basics of Modular Reduction}{chapter.6}% 86
\BOOKMARK [1][-]{section.6.2}{The Barrett Reduction}{chapter.6}% 87
\BOOKMARK [2][-]{subsection.6.2.1}{Fixed Point Arithmetic}{section.6.2}% 88
\BOOKMARK [2][-]{subsection.6.2.2}{Choosing a Radix Point}{section.6.2}% 89
\BOOKMARK [2][-]{subsection.6.2.3}{Trimming the Quotient}{section.6.2}% 90
\BOOKMARK [2][-]{subsection.6.2.4}{Trimming the Residue}{section.6.2}% 91
\BOOKMARK [2][-]{subsection.6.2.5}{The Barrett Algorithm}{section.6.2}% 92
\BOOKMARK [2][-]{subsection.6.2.6}{The Barrett Setup Algorithm}{section.6.2}% 93
\BOOKMARK [1][-]{section.6.3}{The Montgomery Reduction}{chapter.6}% 94
\BOOKMARK [2][-]{subsection.6.3.1}{Digit Based Montgomery Reduction}{section.6.3}% 95
\BOOKMARK [2][-]{subsection.6.3.2}{Baseline Montgomery Reduction}{section.6.3}% 96
\BOOKMARK [2][-]{subsection.6.3.3}{Faster ``Comba'' Montgomery Reduction}{section.6.3}% 97
\BOOKMARK [2][-]{subsection.6.3.4}{Montgomery Setup}{section.6.3}% 98
\BOOKMARK [1][-]{section.6.4}{The Diminished Radix Algorithm}{chapter.6}% 99
\BOOKMARK [2][-]{subsection.6.4.1}{Choice of Moduli}{section.6.4}% 100
\BOOKMARK [2][-]{subsection.6.4.2}{Choice of k}{section.6.4}% 101
\BOOKMARK [2][-]{subsection.6.4.3}{Restricted Diminished Radix Reduction}{section.6.4}% 102
\BOOKMARK [2][-]{subsection.6.4.4}{Unrestricted Diminished Radix Reduction}{section.6.4}% 103
\BOOKMARK [1][-]{section.6.5}{Algorithm Comparison}{chapter.6}% 104
\BOOKMARK [0][-]{chapter.7}{Exponentiation}{}% 105
\BOOKMARK [1][-]{section.7.1}{Exponentiation Basics}{chapter.7}% 106
\BOOKMARK [2][-]{subsection.7.1.1}{Single Digit Exponentiation}{section.7.1}% 107
\BOOKMARK [1][-]{section.7.2}{k-ary Exponentiation}{chapter.7}% 108
\BOOKMARK [2][-]{subsection.7.2.1}{Optimal Values of k}{section.7.2}% 109
\BOOKMARK [2][-]{subsection.7.2.2}{Sliding-Window Exponentiation}{section.7.2}% 110
\BOOKMARK [1][-]{section.7.3}{Modular Exponentiation}{chapter.7}% 111
\BOOKMARK [2][-]{subsection.7.3.1}{Barrett Modular Exponentiation}{section.7.3}% 112
\BOOKMARK [1][-]{section.7.4}{Quick Power of Two}{chapter.7}% 113
\BOOKMARK [0][-]{chapter.8}{Higher Level Algorithms}{}% 114
\BOOKMARK [1][-]{section.8.1}{Integer Division with Remainder}{chapter.8}% 115
\BOOKMARK [2][-]{subsection.8.1.1}{Quotient Estimation}{section.8.1}% 116
\BOOKMARK [2][-]{subsection.8.1.2}{Normalized Integers}{section.8.1}% 117
\BOOKMARK [2][-]{subsection.8.1.3}{Radix- Division with Remainder}{section.8.1}% 118
\BOOKMARK [1][-]{section.8.2}{Single Digit Helpers}{chapter.8}% 119
\BOOKMARK [2][-]{subsection.8.2.1}{Single Digit Addition and Subtraction}{section.8.2}% 120
\BOOKMARK [2][-]{subsection.8.2.2}{Single Digit Multiplication}{section.8.2}% 121
\BOOKMARK [2][-]{subsection.8.2.3}{Single Digit Division}{section.8.2}% 122
\BOOKMARK [2][-]{subsection.8.2.4}{Single Digit Root Extraction}{section.8.2}% 123
\BOOKMARK [1][-]{section.8.3}{Random Number Generation}{chapter.8}% 124
\BOOKMARK [1][-]{section.8.4}{Formatted Representations}{chapter.8}% 125
\BOOKMARK [2][-]{subsection.8.4.1}{Reading Radix-n Input}{section.8.4}% 126
\BOOKMARK [2][-]{subsection.8.4.2}{Generating Radix-n Output}{section.8.4}% 127
\BOOKMARK [0][-]{chapter.9}{Number Theoretic Algorithms}{}% 128
\BOOKMARK [1][-]{section.9.1}{Greatest Common Divisor}{chapter.9}% 129
\BOOKMARK [2][-]{subsection.9.1.1}{Complete Greatest Common Divisor}{section.9.1}% 130
\BOOKMARK [1][-]{section.9.2}{Least Common Multiple}{chapter.9}% 131
\BOOKMARK [1][-]{section.9.3}{Jacobi Symbol Computation}{chapter.9}% 132
\BOOKMARK [2][-]{subsection.9.3.1}{Jacobi Symbol}{section.9.3}% 133
\BOOKMARK [1][-]{section.9.4}{Modular Inverse}{chapter.9}% 134
\BOOKMARK [2][-]{subsection.9.4.1}{General Case}{section.9.4}% 135
\BOOKMARK [1][-]{section.9.5}{Primality Tests}{chapter.9}% 136
\BOOKMARK [2][-]{subsection.9.5.1}{Trial Division}{section.9.5}% 137
\BOOKMARK [2][-]{subsection.9.5.2}{The Fermat Test}{section.9.5}% 138
\BOOKMARK [2][-]{subsection.9.5.3}{The Miller-Rabin Test}{section.9.5}% 139
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Greg Rose \\
QUALCOMM Australia \\
\end{tabular}
%\end{small}
}
}
\maketitle
This text has been placed in the public domain.  This text corresponds to the v0.39 release of the 
LibTomMath project.

\begin{alltt}
Tom St Denis
111 Banning Rd
Ottawa, Ontario
K2L 1C3
Canada

Phone: 1-613-836-3160
Email: [email protected]
\end{alltt}

This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{} 
{\em book} macro package and the Perl {\em booker} package.

\tableofcontents
\listoffigures
\chapter*{Prefaces}
When I tell people about my LibTom projects and that I release them as public domain they are often puzzled.  
They ask why I did it and especially why I continue to work on them for free.  The best I can explain it is ``Because I can.''  
Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which 
perhaps explains it better.  I am the first to admit there is not anything that special with what I have done.  Perhaps
others can see that too and then we would have a society to be proud of.  My LibTom projects are what I am doing to give 
back to society in the form of tools and knowledge that can help others in their endeavours.

I started writing this book because it was the most logical task to further my goal of open academia.  The LibTomMath source
code itself was written to be easy to follow and learn from.  There are times, however, where pure C source code does not
explain the algorithms properly.  Hence this book.  The book literally starts with the foundation of the library and works
itself outwards to the more complicated algorithms.  The use of both pseudo--code and verbatim source code provides a duality
of ``theory'' and ``practice'' that the computer science students of the world shall appreciate.  I never deviate too far
from relatively straightforward algebra and I hope that this book can be a valuable learning asset.

This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora
of kind people donating their time, resources and kind words to help support my work.  Writing a text of significant
length (along with the source code) is a tiresome and lengthy process.  Currently the LibTom project is four years old,
comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material.  People like Mads and Greg 
were there at the beginning to encourage me to work well.  It is amazing how timely validation from others can boost morale to 
continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003.  

To my many friends whom I have met through the years I thank you for the good times and the words of encouragement.  I hope I
honour your kind gestures with this project.

Open Source.  Open Academia.  Open Minds.

\begin{flushright} Tom St Denis \end{flushright}

\newpage
I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also 
contribute to educate others facing the problem of having to handle big number mathematical calculations.

This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of 
how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about 
the layout and language used.

I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the 
practical aspects of cryptography. 

Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a 
great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up 
multiple precision calculations is often very important since we deal with outdated machine architecture where modular 
reductions, for example, become painfully slow.

This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks 
themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?''

\begin{flushright}
Mads Rasmussen

S\~{a}o Paulo - SP

Brazil
\end{flushright}

\newpage
It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about 
Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not 
really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once.

At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the 
sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real
contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity. 
Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake.

When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully, 
and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close 
friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort, 
and I'm pleased to be involved with it.

\begin{flushright}
Greg Rose, Sydney, Australia, June 2003. 
\end{flushright}

\mainmatter
\pagestyle{headings}
\chapter{Introduction}
\section{Multiple Precision Arithmetic}

\subsection{What is Multiple Precision Arithmetic?}
When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively
raise or lower the precision of the numbers we are dealing with.  For example, in decimal we almost immediate can 
reason that $7$ times $6$ is $42$.  However, $42$ has two digits of precision as opposed to one digit we started with.  
Further multiplications of say $3$ result in a larger precision result $126$.  In these few examples we have multiple 
precisions for the numbers we are working with.  Despite the various levels of precision a single subset\footnote{With the occasional optimization.}
 of algorithms can be designed to accomodate them.  

By way of comparison a fixed or single precision operation would lose precision on various operations.  For example, in
the decimal system with fixed precision $6 \cdot 7 = 2$.

Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in
schools to manually add, subtract, multiply and divide.  

\subsection{The Need for Multiple Precision Arithmetic}
The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation
of public-key cryptography algorithms.   Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require 
integers of significant magnitude to resist known cryptanalytic attacks.  For example, at the time of this writing a 
typical RSA modulus would be at least greater than $10^{309}$.  However, modern programming languages such as ISO C \cite{ISOC} and 
Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision.

\begin{figure}[!here]
\begin{center}
\begin{tabular}{|r|c|}
\hline \textbf{Data Type} & \textbf{Range} \\
\hline char  & $-128 \ldots 127$ \\
\hline short & $-32768 \ldots 32767$ \\
\hline long  & $-2147483648 \ldots 2147483647$ \\
\hline long long & $-9223372036854775808 \ldots 9223372036854775807$ \\
\hline
\end{tabular}
\end{center}
\caption{Typical Data Types for the C Programming Language}
\label{fig:ISOC}
\end{figure}

The largest data type guaranteed to be provided by the ISO C programming 
language\footnote{As per the ISO C standard.  However, each compiler vendor is allowed to augment the precision as they 
see fit.}  can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is 
insufficient to accomodate the magnitude required for the problem at hand.  An RSA modulus of magnitude $10^{19}$ could be 
trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, 
rendering any protocol based on the algorithm insecure.  Multiple precision algorithms solve this very problem by 
extending the range of representable integers while using single precision data types.

Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic 
primitives.  Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in 
various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient.  In fact, several 
major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and 
deployment of efficient algorithms.

However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines.  
Another auxiliary use of multiple precision integers is high precision floating point data types.  
The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$.  
Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE.  Since IEEE 
floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small 
(\textit{23, 48 and 64 bits}).  The mantissa is merely an integer and a multiple precision integer could be used to create
a mantissa of much larger precision than hardware alone can efficiently support.  This approach could be useful where 
scientific applications must minimize the total output error over long calculations.

Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$).
In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}.

\subsection{Benefits of Multiple Precision Arithmetic}
\index{precision}
The benefit of multiple precision representations over single or fixed precision representations is that 
no precision is lost while representing the result of an operation which requires excess precision.  For example, 
the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully.  A multiple 
precision algorithm would augment the precision of the destination to accomodate the result while a single precision system 
would truncate excess bits to maintain a fixed level of precision.

It is possible to implement algorithms which require large integers with fixed precision algorithms.  For example, elliptic
curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum 
size the system will ever need.  Such an approach can lead to vastly simpler algorithms which can accomodate the 
integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard 
processor has an 8 bit accumulator.}.  However, as efficient as such an approach may be, the resulting source code is not
normally very flexible.  It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated.

Multiple precision algorithms have the most overhead of any style of arithmetic.  For the the most part the 
overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved
platforms.  However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the 
inputs.  That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input 
without the designer's explicit forethought.  This leads to lower cost of ownership for the code as it only has to 
be written and tested once.

\section{Purpose of This Text}
The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms.  
That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' 
elements that are neglected by authors of other texts on the subject.  Several well reknowned texts \cite{TAOCPV2,HAC} 
give considerably detailed explanations of the theoretical aspects of algorithms and often very little information 
regarding the practical implementation aspects.  

In most cases how an algorithm is explained and how it is actually implemented are two very different concepts.  For 
example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple 
algorithm for performing multiple precision integer addition.  However, the description lacks any discussion concerning 
the fact that the two integer inputs may be of differing magnitudes.  As a result the implementation is not as simple
as the text would lead people to believe.  Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not 
discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}).

Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers 
and fast modular inversion, which we consider practical oversights.  These optimal algorithms are vital to achieve 
any form of useful performance in non-trivial applications.  

To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer
package.  As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.com}} package is used 
to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field 
tested and work very well.  The LibTomMath library is freely available on the Internet for all uses and this text 
discusses a very large portion of the inner workings of the library.

The algorithms that are presented will always include at least one ``pseudo-code'' description followed 
by the actual C source code that implements the algorithm.  The pseudo-code can be used to implement the same 
algorithm in other programming languages as the reader sees fit.  

This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch.  Showing
the reader how the algorithms fit together as well as where to start on various taskings.  

\section{Discussion and Notation}
\subsection{Notation}
A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent
the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$.  The elements of the array $x$ are said to be the radix $\beta$ digits 
of the integer.  For example, $x = (1,2,3)_{10}$ would represent the integer 
$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$.  

\index{mp\_int}
The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well 
as auxilary data required to manipulate the data.  These additional members are discussed further in section 
\ref{sec:MPINT}.  For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be 
synonymous.  When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members 
are present as well.  An expression of the type \textit{variablename.item} implies that it should evaluate to the 
member named ``item'' of the variable.  For example, a string of characters may have a member ``length'' which would 
evaluate to the number of characters in the string.  If the string $a$ equals ``hello'' then it follows that 
$a.length = 5$.  

For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used
to solve a given problem.  When an algorithm is described as accepting an integer input it is assumed the input is 
a plain integer with no additional multiple-precision members.  That is, algorithms that use integers as opposed to 
mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management.  These 
algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple
precision algorithm to solve the same problem.  

\subsection{Precision Notation}
The variable $\beta$ represents the radix of a single digit of a multiple precision integer and 
must be of the form $q^p$ for $q, p \in \Z^+$.  A single precision variable must be able to represent integers in 
the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range 
$0 \le x < q \beta^2$.  The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the 
carry.  Since all modern computers are binary, it is assumed that $q$ is two.

\index{mp\_digit} \index{mp\_word}
Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent 
a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type.  In 
several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words.  
For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to 
the $j$'th digit of a double precision array.  Whenever an expression is to be assigned to a double precision
variable it is assumed that all single precision variables are promoted to double precision during the evaluation.  
Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single
precision data type.

For example, if $\beta = 10^2$ a single precision data type may represent a value in the 
range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$.  Let
$a = 23$ and $b = 49$ represent two single precision variables.  The single precision product shall be written
as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$.
In this particular case, $\hat c = 1127$ and $c = 127$.  The most significant digit of the product would not fit 
in a single precision data type and as a result $c \ne \hat c$.  

\subsection{Algorithm Inputs and Outputs}
Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision
as indicated.  The only exception to this rule is when variables have been indicated to be of type mp\_int.  This 
distinction is important as scalars are often used as array indicies and various other counters.  

\subsection{Mathematical Expressions}
The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression 
itself.  For example, $\lfloor 5.7 \rfloor = 5$.  Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression
rounded to an integer not less than the expression itself.  For example, $\lceil 5.1 \rceil = 6$.  Typically when 
the $/$ division symbol is used the intention is to perform an integer division with truncation.  For example, 
$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity.  When an expression is written as a 
fraction a real value division is implied, for example ${5 \over 2} = 2.5$.  

The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation
of the integer.  For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$.  

\subsection{Work Effort}
\index{big-Oh}
To measure the efficiency of the specified algorithms, a modified big-Oh notation is used.  In this system all 
single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}.  
That is a single precision addition, multiplication and division are assumed to take the same time to 
complete.  While this is generally not true in practice, it will simplify the discussions considerably.

Some algorithms have slight advantages over others which is why some constants will not be removed in 
the notation.  For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a 
baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work.  In standard big-Oh notation these 
would both be said to be equivalent to $O(n^2)$.  However, 
in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small.  As a 
result small constant factors in the work effort will make an observable difference in algorithm efficiency.

All of the algorithms presented in this text have a polynomial time work level.  That is, of the form 
$O(n^k)$ for $n, k \in \Z^{+}$.  This will help make useful comparisons in terms of the speed of the algorithms and how 
various optimizations will help pay off in the long run.

\section{Exercises}
Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to
the discussion at hand.  These exercises are not designed to be prize winning problems, but instead to be thought 
provoking.  Wherever possible the problems are forward minded, stating problems that will be answered in subsequent 
chapters.  The reader is encouraged to finish the exercises as they appear to get a better understanding of the 
subject material.  

That being said, the problems are designed to affirm knowledge of a particular subject matter.  Students in particular
are encouraged to verify they can answer the problems correctly before moving on.

Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of
the problem.  However, unlike \cite{TAOCPV2} the problems do not get nearly as hard.  The scoring of these 
exercises ranges from one (the easiest) to five (the hardest).  The following table sumarizes the 
scoring system used.

\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|c|l|}
\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\







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Greg Rose \\
QUALCOMM Australia \\
\end{tabular}
%\end{small}
}
}
\maketitle
This text has been placed in the public domain.  This text corresponds to the v0.39 release of the
LibTomMath project.












This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{}
{\em book} macro package and the Perl {\em booker} package.

\tableofcontents
\listoffigures
\chapter*{Prefaces}
When I tell people about my LibTom projects and that I release them as public domain they are often puzzled.
They ask why I did it and especially why I continue to work on them for free.  The best I can explain it is ``Because I can.''
Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which
perhaps explains it better.  I am the first to admit there is not anything that special with what I have done.  Perhaps
others can see that too and then we would have a society to be proud of.  My LibTom projects are what I am doing to give
back to society in the form of tools and knowledge that can help others in their endeavours.

I started writing this book because it was the most logical task to further my goal of open academia.  The LibTomMath source
code itself was written to be easy to follow and learn from.  There are times, however, where pure C source code does not
explain the algorithms properly.  Hence this book.  The book literally starts with the foundation of the library and works
itself outwards to the more complicated algorithms.  The use of both pseudo--code and verbatim source code provides a duality
of ``theory'' and ``practice'' that the computer science students of the world shall appreciate.  I never deviate too far
from relatively straightforward algebra and I hope that this book can be a valuable learning asset.

This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora
of kind people donating their time, resources and kind words to help support my work.  Writing a text of significant
length (along with the source code) is a tiresome and lengthy process.  Currently the LibTom project is four years old,
comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material.  People like Mads and Greg
were there at the beginning to encourage me to work well.  It is amazing how timely validation from others can boost morale to
continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003.

To my many friends whom I have met through the years I thank you for the good times and the words of encouragement.  I hope I
honour your kind gestures with this project.

Open Source.  Open Academia.  Open Minds.

\begin{flushright} Tom St Denis \end{flushright}

\newpage
I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also
contribute to educate others facing the problem of having to handle big number mathematical calculations.

This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of
how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about
the layout and language used.

I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the
practical aspects of cryptography.

Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a
great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up
multiple precision calculations is often very important since we deal with outdated machine architecture where modular
reductions, for example, become painfully slow.

This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks
themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?''

\begin{flushright}
Mads Rasmussen

S\~{a}o Paulo - SP

Brazil
\end{flushright}

\newpage
It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about
Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not
really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once.

At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the
sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real
contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity.
Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake.

When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully,
and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close
friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort,
and I'm pleased to be involved with it.

\begin{flushright}
Greg Rose, Sydney, Australia, June 2003.
\end{flushright}

\mainmatter
\pagestyle{headings}
\chapter{Introduction}
\section{Multiple Precision Arithmetic}

\subsection{What is Multiple Precision Arithmetic?}
When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively
raise or lower the precision of the numbers we are dealing with.  For example, in decimal we almost immediate can
reason that $7$ times $6$ is $42$.  However, $42$ has two digits of precision as opposed to one digit we started with.
Further multiplications of say $3$ result in a larger precision result $126$.  In these few examples we have multiple
precisions for the numbers we are working with.  Despite the various levels of precision a single subset\footnote{With the occasional optimization.}
 of algorithms can be designed to accomodate them.

By way of comparison a fixed or single precision operation would lose precision on various operations.  For example, in
the decimal system with fixed precision $6 \cdot 7 = 2$.

Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in
schools to manually add, subtract, multiply and divide.

\subsection{The Need for Multiple Precision Arithmetic}
The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation
of public-key cryptography algorithms.   Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require
integers of significant magnitude to resist known cryptanalytic attacks.  For example, at the time of this writing a
typical RSA modulus would be at least greater than $10^{309}$.  However, modern programming languages such as ISO C \cite{ISOC} and
Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision.

\begin{figure}[!here]
\begin{center}
\begin{tabular}{|r|c|}
\hline \textbf{Data Type} & \textbf{Range} \\
\hline char  & $-128 \ldots 127$ \\
\hline short & $-32768 \ldots 32767$ \\
\hline long  & $-2147483648 \ldots 2147483647$ \\
\hline long long & $-9223372036854775808 \ldots 9223372036854775807$ \\
\hline
\end{tabular}
\end{center}
\caption{Typical Data Types for the C Programming Language}
\label{fig:ISOC}
\end{figure}

The largest data type guaranteed to be provided by the ISO C programming
language\footnote{As per the ISO C standard.  However, each compiler vendor is allowed to augment the precision as they
see fit.}  can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is
insufficient to accomodate the magnitude required for the problem at hand.  An RSA modulus of magnitude $10^{19}$ could be
trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer,
rendering any protocol based on the algorithm insecure.  Multiple precision algorithms solve this very problem by
extending the range of representable integers while using single precision data types.

Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic
primitives.  Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in
various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient.  In fact, several
major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and
deployment of efficient algorithms.

However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines.
Another auxiliary use of multiple precision integers is high precision floating point data types.
The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$.
Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE.  Since IEEE
floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small
(\textit{23, 48 and 64 bits}).  The mantissa is merely an integer and a multiple precision integer could be used to create
a mantissa of much larger precision than hardware alone can efficiently support.  This approach could be useful where
scientific applications must minimize the total output error over long calculations.

Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$).
In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}.

\subsection{Benefits of Multiple Precision Arithmetic}
\index{precision}
The benefit of multiple precision representations over single or fixed precision representations is that
no precision is lost while representing the result of an operation which requires excess precision.  For example,
the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully.  A multiple
precision algorithm would augment the precision of the destination to accomodate the result while a single precision system
would truncate excess bits to maintain a fixed level of precision.

It is possible to implement algorithms which require large integers with fixed precision algorithms.  For example, elliptic
curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum
size the system will ever need.  Such an approach can lead to vastly simpler algorithms which can accomodate the
integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard
processor has an 8 bit accumulator.}.  However, as efficient as such an approach may be, the resulting source code is not
normally very flexible.  It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated.

Multiple precision algorithms have the most overhead of any style of arithmetic.  For the the most part the
overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved
platforms.  However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the
inputs.  That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input
without the designer's explicit forethought.  This leads to lower cost of ownership for the code as it only has to
be written and tested once.

\section{Purpose of This Text}
The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms.
That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping''
elements that are neglected by authors of other texts on the subject.  Several well reknowned texts \cite{TAOCPV2,HAC}
give considerably detailed explanations of the theoretical aspects of algorithms and often very little information
regarding the practical implementation aspects.

In most cases how an algorithm is explained and how it is actually implemented are two very different concepts.  For
example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple
algorithm for performing multiple precision integer addition.  However, the description lacks any discussion concerning
the fact that the two integer inputs may be of differing magnitudes.  As a result the implementation is not as simple
as the text would lead people to believe.  Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not
discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}).

Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers
and fast modular inversion, which we consider practical oversights.  These optimal algorithms are vital to achieve
any form of useful performance in non-trivial applications.

To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer
package.  As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.com}} package is used
to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field
tested and work very well.  The LibTomMath library is freely available on the Internet for all uses and this text
discusses a very large portion of the inner workings of the library.

The algorithms that are presented will always include at least one ``pseudo-code'' description followed
by the actual C source code that implements the algorithm.  The pseudo-code can be used to implement the same
algorithm in other programming languages as the reader sees fit.

This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch.  Showing
the reader how the algorithms fit together as well as where to start on various taskings.

\section{Discussion and Notation}
\subsection{Notation}
A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent
the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$.  The elements of the array $x$ are said to be the radix $\beta$ digits
of the integer.  For example, $x = (1,2,3)_{10}$ would represent the integer
$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$.

\index{mp\_int}
The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well
as auxilary data required to manipulate the data.  These additional members are discussed further in section
\ref{sec:MPINT}.  For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be
synonymous.  When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members
are present as well.  An expression of the type \textit{variablename.item} implies that it should evaluate to the
member named ``item'' of the variable.  For example, a string of characters may have a member ``length'' which would
evaluate to the number of characters in the string.  If the string $a$ equals ``hello'' then it follows that
$a.length = 5$.

For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used
to solve a given problem.  When an algorithm is described as accepting an integer input it is assumed the input is
a plain integer with no additional multiple-precision members.  That is, algorithms that use integers as opposed to
mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management.  These
algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple
precision algorithm to solve the same problem.

\subsection{Precision Notation}
The variable $\beta$ represents the radix of a single digit of a multiple precision integer and
must be of the form $q^p$ for $q, p \in \Z^+$.  A single precision variable must be able to represent integers in
the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range
$0 \le x < q \beta^2$.  The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the
carry.  Since all modern computers are binary, it is assumed that $q$ is two.

\index{mp\_digit} \index{mp\_word}
Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent
a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type.  In
several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words.
For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to
the $j$'th digit of a double precision array.  Whenever an expression is to be assigned to a double precision
variable it is assumed that all single precision variables are promoted to double precision during the evaluation.
Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single
precision data type.

For example, if $\beta = 10^2$ a single precision data type may represent a value in the
range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$.  Let
$a = 23$ and $b = 49$ represent two single precision variables.  The single precision product shall be written
as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$.
In this particular case, $\hat c = 1127$ and $c = 127$.  The most significant digit of the product would not fit
in a single precision data type and as a result $c \ne \hat c$.

\subsection{Algorithm Inputs and Outputs}
Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision
as indicated.  The only exception to this rule is when variables have been indicated to be of type mp\_int.  This
distinction is important as scalars are often used as array indicies and various other counters.

\subsection{Mathematical Expressions}
The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression
itself.  For example, $\lfloor 5.7 \rfloor = 5$.  Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression
rounded to an integer not less than the expression itself.  For example, $\lceil 5.1 \rceil = 6$.  Typically when
the $/$ division symbol is used the intention is to perform an integer division with truncation.  For example,
$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity.  When an expression is written as a
fraction a real value division is implied, for example ${5 \over 2} = 2.5$.

The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation
of the integer.  For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$.

\subsection{Work Effort}
\index{big-Oh}
To measure the efficiency of the specified algorithms, a modified big-Oh notation is used.  In this system all
single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}.
That is a single precision addition, multiplication and division are assumed to take the same time to
complete.  While this is generally not true in practice, it will simplify the discussions considerably.

Some algorithms have slight advantages over others which is why some constants will not be removed in
the notation.  For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a
baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work.  In standard big-Oh notation these
would both be said to be equivalent to $O(n^2)$.  However,
in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small.  As a
result small constant factors in the work effort will make an observable difference in algorithm efficiency.

All of the algorithms presented in this text have a polynomial time work level.  That is, of the form
$O(n^k)$ for $n, k \in \Z^{+}$.  This will help make useful comparisons in terms of the speed of the algorithms and how
various optimizations will help pay off in the long run.

\section{Exercises}
Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to
the discussion at hand.  These exercises are not designed to be prize winning problems, but instead to be thought
provoking.  Wherever possible the problems are forward minded, stating problems that will be answered in subsequent
chapters.  The reader is encouraged to finish the exercises as they appear to get a better understanding of the
subject material.

That being said, the problems are designed to affirm knowledge of a particular subject matter.  Students in particular
are encouraged to verify they can answer the problems correctly before moving on.

Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of
the problem.  However, unlike \cite{TAOCPV2} the problems do not get nearly as hard.  The scoring of these
exercises ranges from one (the easiest) to five (the hardest).  The following table sumarizes the
scoring system used.

\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|c|l|}
\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\
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\end{tabular}
\end{small}
\end{center}
\caption{Exercise Scoring System}
\end{figure}

Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or
devising new theory.  These problems are quick tests to see if the material is understood.  Problems at the second level 
are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer.  These
two levels are essentially entry level questions.  

Problems at the third level are meant to be a bit more difficult than the first two levels.  The answer is often 
fairly obvious but arriving at an exacting solution requires some thought and skill.  These problems will almost always 
involve devising a new algorithm or implementing a variation of another algorithm previously presented.  Readers who can
answer these questions will feel comfortable with the concepts behind the topic at hand.

Problems at the fourth level are meant to be similar to those of the level three questions except they will require 
additional research to be completed.  The reader will most likely not know the answer right away, nor will the text provide 
the exact details of the answer until a subsequent chapter.  

Problems at the fifth level are meant to be the hardest 
problems relative to all the other problems in the chapter.  People who can correctly answer fifth level problems have a 
mastery of the subject matter at hand.

Often problems will be tied together.  The purpose of this is to start a chain of thought that will be discussed in future chapters.  The reader
is encouraged to answer the follow-up problems and try to draw the relevance of problems.

\section{Introduction to LibTomMath}

\subsection{What is LibTomMath?}
LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C.  By portable it 
is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on 
any given platform.  

The library has been successfully tested under numerous operating systems including Unix\footnote{All of these
trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such 
as the Gameboy Advance.  The library is designed to contain enough functionality to be able to develop applications such 
as public key cryptosystems and still maintain a relatively small footprint.

\subsection{Goals of LibTomMath}

Libraries which obtain the most efficiency are rarely written in a high level programming language such as C.  However, 
even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the 
library.  Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM 
processors.  Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window 
exponentiation and Montgomery reduction have been provided to make the library more efficient.  

Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface 
(\textit{API}) has been kept as simple as possible.  Often generic place holder routines will make use of specialized 
algorithms automatically without the developer's specific attention.  One such example is the generic multiplication 
algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication 
based on the magnitude of the inputs and the configuration of the library.  

Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project.  Ideally the library should 
be source compatible with another popular library which makes it more attractive for developers to use.  In this case the
MPI library was used as a API template for all the basic functions.  MPI was chosen because it is another library that fits 
in the same niche as LibTomMath.  Even though LibTomMath uses MPI as the template for the function names and argument 
passing conventions, it has been written from scratch by Tom St Denis.

The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' 
library exists which can be used to teach computer science students how to perform fast and reliable multiple precision 
integer arithmetic.  To this end the source code has been given quite a few comments and algorithm discussion points.  

\section{Choice of LibTomMath}
LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but
for more worthy reasons.  Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL 
\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for 
reasons that will be explained in the following sub-sections.

\subsection{Code Base}
The LibTomMath code base is all portable ISO C source code.  This means that there are no platform dependent conditional
segments of code littered throughout the source.  This clean and uncluttered approach to the library means that a
developer can more readily discern the true intent of a given section of source code without trying to keep track of
what conditional code will be used.

The code base of LibTomMath is well organized.  Each function is in its own separate source code file 
which allows the reader to find a given function very quickly.  On average there are $76$ lines of code per source
file which makes the source very easily to follow.  By comparison MPI and LIP are single file projects making code tracing
very hard.  GMP has many conditional code segments which also hinder tracing.  

When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.}
 which is fairly small compared to GMP (over $250$KiB).  LibTomMath is slightly larger than MPI (which compiles to about 
$50$KiB) but LibTomMath is also much faster and more complete than MPI.

\subsection{API Simplicity}
LibTomMath is designed after the MPI library and shares the API design.  Quite often programs that use MPI will build 
with LibTomMath without change. The function names correlate directly to the action they perform.  Almost all of the 
functions share the same parameter passing convention.  The learning curve is fairly shallow with the API provided 
which is an extremely valuable benefit for the student and developer alike.  

The LIP library is an example of a library with an API that is awkward to work with.  LIP uses function names that are often ``compressed'' to 
illegible short hand.  LibTomMath does not share this characteristic.  

The GMP library also does not return error codes.  Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors
are signaled to the host application.  This happens to be the fastest approach but definitely not the most versatile.  In
effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely 
undersireable in many situations.

\subsection{Optimizations}
While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does
feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring.  GMP 
and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations.  GMP lacks a few
of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP
only had Barrett and Montgomery modular reduction algorithms.}.  

LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular
exponentiation.  In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually  
slower than the best libraries such as GMP and OpenSSL by only a small factor.

\subsection{Portability and Stability}
LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler 
(\textit{GCC}).  This means that without changes the library will build without configuration or setting up any 
variables.  LIP and MPI will build ``out of the box'' as well but have numerous known bugs.  Most notably the author of 
MPI has recently stopped working on his library and LIP has long since been discontinued.  

GMP requires a configuration script to run and will not build out of the box.   GMP and LibTomMath are still in active
development and are very stable across a variety of platforms.

\subsection{Choice}
LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for
the case study of this text.  Various source files from the LibTomMath project will be included within the text.  However, 
the reader is encouraged to download their own copy of the library to actually be able to work with the library.  

\chapter{Getting Started}
\section{Library Basics}
The trick to writing any useful library of source code is to build a solid foundation and work outwards from it.  First, 
a problem along with allowable solution parameters should be identified and analyzed.  In this particular case the 
inability to accomodate multiple precision integers is the problem.  Futhermore, the solution must be written
as portable source code that is reasonably efficient across several different computer platforms.

After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion.  
That is, to implement the lowest level dependencies first and work towards the most abstract functions last.  For example, 
before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm.
By building outwards from a base foundation instead of using a parallel design methodology the resulting project is 
highly modular.  Being highly modular is a desirable property of any project as it often means the resulting product
has a small footprint and updates are easy to perform.  

Usually when I start a project I will begin with the header files.  I define the data types I think I will need and 
prototype the initial functions that are not dependent on other functions (within the library).  After I 
implement these base functions I prototype more dependent functions and implement them.   The process repeats until
I implement all of the functions I require.  For example, in the case of LibTomMath I implemented functions such as 
mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod().  As an example as to 
why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the 
dependent function mp\_exptmod() was written.  Adding the new multiplication algorithms did not require changes to the 
mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development 
for new algorithms.  This methodology allows new algorithms to be tested in a complete framework with relative ease.

FIGU,design_process,Design Flow of the First Few Original LibTomMath Functions.

Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing
the source code.  For example, one day I may audit the multipliers and the next day the polynomial basis functions.  

It only makes sense to begin the text with the preliminary data types and support algorithms required as well.  
This chapter discusses the core algorithms of the library which are the dependents for every other algorithm.

\section{What is a Multiple Precision Integer?}
Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot 
be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is 
to use fixed precision data types to create and manipulate multiple precision integers which may represent values 
that are very large.  

As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits.  In the decimal system
the largest single digit value is $9$.  However, by concatenating digits together larger numbers may be represented.  Newly prepended digits 
(\textit{to the left}) are said to be in a different power of ten column.  That is, the number $123$ can be described as having a $1$ in the hundreds 
column, $2$ in the tens column and $3$ in the ones column.  Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$.  Computer based 
multiple precision arithmetic is essentially the same concept.  Larger integers are represented by adjoining fixed 
precision computer words with the exception that a different radix is used.

What most people probably do not think about explicitly are the various other attributes that describe a multiple precision 
integer.  For example, the integer $154_{10}$ has two immediately obvious properties.  First, the integer is positive, 
that is the sign of this particular integer is positive as opposed to negative.  Second, the integer has three digits in 
its representation.  There is an additional property that the integer posesses that does not concern pencil-and-paper 
arithmetic.  The third property is how many digits placeholders are available to hold the integer.  

The human analogy of this third property is ensuring there is enough space on the paper to write the integer.  For example,
if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left.  
Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer
will not exceed the allowed boundaries.  These three properties make up what is known as a multiple precision 
integer or mp\_int for short.  

\subsection{The mp\_int Structure}
\label{sec:MPINT}
The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer.  The ISO C standard does not provide for 
any such data type but it does provide for making composite data types known as structures.  The following is the structure definition 
used within LibTomMath.

\index{mp\_int}
\begin{figure}[here]
\begin{center}
\begin{small}
%\begin{verbatim}







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\end{tabular}
\end{small}
\end{center}
\caption{Exercise Scoring System}
\end{figure}

Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or
devising new theory.  These problems are quick tests to see if the material is understood.  Problems at the second level
are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer.  These
two levels are essentially entry level questions.

Problems at the third level are meant to be a bit more difficult than the first two levels.  The answer is often
fairly obvious but arriving at an exacting solution requires some thought and skill.  These problems will almost always
involve devising a new algorithm or implementing a variation of another algorithm previously presented.  Readers who can
answer these questions will feel comfortable with the concepts behind the topic at hand.

Problems at the fourth level are meant to be similar to those of the level three questions except they will require
additional research to be completed.  The reader will most likely not know the answer right away, nor will the text provide
the exact details of the answer until a subsequent chapter.

Problems at the fifth level are meant to be the hardest
problems relative to all the other problems in the chapter.  People who can correctly answer fifth level problems have a
mastery of the subject matter at hand.

Often problems will be tied together.  The purpose of this is to start a chain of thought that will be discussed in future chapters.  The reader
is encouraged to answer the follow-up problems and try to draw the relevance of problems.

\section{Introduction to LibTomMath}

\subsection{What is LibTomMath?}
LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C.  By portable it
is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on
any given platform.

The library has been successfully tested under numerous operating systems including Unix\footnote{All of these
trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such
as the Gameboy Advance.  The library is designed to contain enough functionality to be able to develop applications such
as public key cryptosystems and still maintain a relatively small footprint.

\subsection{Goals of LibTomMath}

Libraries which obtain the most efficiency are rarely written in a high level programming language such as C.  However,
even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the
library.  Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM
processors.  Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window
exponentiation and Montgomery reduction have been provided to make the library more efficient.

Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface
(\textit{API}) has been kept as simple as possible.  Often generic place holder routines will make use of specialized
algorithms automatically without the developer's specific attention.  One such example is the generic multiplication
algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication
based on the magnitude of the inputs and the configuration of the library.

Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project.  Ideally the library should
be source compatible with another popular library which makes it more attractive for developers to use.  In this case the
MPI library was used as a API template for all the basic functions.  MPI was chosen because it is another library that fits
in the same niche as LibTomMath.  Even though LibTomMath uses MPI as the template for the function names and argument
passing conventions, it has been written from scratch by Tom St Denis.

The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum''
library exists which can be used to teach computer science students how to perform fast and reliable multiple precision
integer arithmetic.  To this end the source code has been given quite a few comments and algorithm discussion points.

\section{Choice of LibTomMath}
LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but
for more worthy reasons.  Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL
\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for
reasons that will be explained in the following sub-sections.

\subsection{Code Base}
The LibTomMath code base is all portable ISO C source code.  This means that there are no platform dependent conditional
segments of code littered throughout the source.  This clean and uncluttered approach to the library means that a
developer can more readily discern the true intent of a given section of source code without trying to keep track of
what conditional code will be used.

The code base of LibTomMath is well organized.  Each function is in its own separate source code file
which allows the reader to find a given function very quickly.  On average there are $76$ lines of code per source
file which makes the source very easily to follow.  By comparison MPI and LIP are single file projects making code tracing
very hard.  GMP has many conditional code segments which also hinder tracing.

When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.}
 which is fairly small compared to GMP (over $250$KiB).  LibTomMath is slightly larger than MPI (which compiles to about
$50$KiB) but LibTomMath is also much faster and more complete than MPI.

\subsection{API Simplicity}
LibTomMath is designed after the MPI library and shares the API design.  Quite often programs that use MPI will build
with LibTomMath without change. The function names correlate directly to the action they perform.  Almost all of the
functions share the same parameter passing convention.  The learning curve is fairly shallow with the API provided
which is an extremely valuable benefit for the student and developer alike.

The LIP library is an example of a library with an API that is awkward to work with.  LIP uses function names that are often ``compressed'' to
illegible short hand.  LibTomMath does not share this characteristic.

The GMP library also does not return error codes.  Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors
are signaled to the host application.  This happens to be the fastest approach but definitely not the most versatile.  In
effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely
undersireable in many situations.

\subsection{Optimizations}
While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does
feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring.  GMP
and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations.  GMP lacks a few
of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP
only had Barrett and Montgomery modular reduction algorithms.}.

LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular
exponentiation.  In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually
slower than the best libraries such as GMP and OpenSSL by only a small factor.

\subsection{Portability and Stability}
LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler
(\textit{GCC}).  This means that without changes the library will build without configuration or setting up any
variables.  LIP and MPI will build ``out of the box'' as well but have numerous known bugs.  Most notably the author of
MPI has recently stopped working on his library and LIP has long since been discontinued.

GMP requires a configuration script to run and will not build out of the box.   GMP and LibTomMath are still in active
development and are very stable across a variety of platforms.

\subsection{Choice}
LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for
the case study of this text.  Various source files from the LibTomMath project will be included within the text.  However,
the reader is encouraged to download their own copy of the library to actually be able to work with the library.

\chapter{Getting Started}
\section{Library Basics}
The trick to writing any useful library of source code is to build a solid foundation and work outwards from it.  First,
a problem along with allowable solution parameters should be identified and analyzed.  In this particular case the
inability to accomodate multiple precision integers is the problem.  Futhermore, the solution must be written
as portable source code that is reasonably efficient across several different computer platforms.

After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion.
That is, to implement the lowest level dependencies first and work towards the most abstract functions last.  For example,
before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm.
By building outwards from a base foundation instead of using a parallel design methodology the resulting project is
highly modular.  Being highly modular is a desirable property of any project as it often means the resulting product
has a small footprint and updates are easy to perform.

Usually when I start a project I will begin with the header files.  I define the data types I think I will need and
prototype the initial functions that are not dependent on other functions (within the library).  After I
implement these base functions I prototype more dependent functions and implement them.   The process repeats until
I implement all of the functions I require.  For example, in the case of LibTomMath I implemented functions such as
mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod().  As an example as to
why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the
dependent function mp\_exptmod() was written.  Adding the new multiplication algorithms did not require changes to the
mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development
for new algorithms.  This methodology allows new algorithms to be tested in a complete framework with relative ease.

FIGU,design_process,Design Flow of the First Few Original LibTomMath Functions.

Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing
the source code.  For example, one day I may audit the multipliers and the next day the polynomial basis functions.

It only makes sense to begin the text with the preliminary data types and support algorithms required as well.
This chapter discusses the core algorithms of the library which are the dependents for every other algorithm.

\section{What is a Multiple Precision Integer?}
Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot
be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is
to use fixed precision data types to create and manipulate multiple precision integers which may represent values
that are very large.

As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits.  In the decimal system
the largest single digit value is $9$.  However, by concatenating digits together larger numbers may be represented.  Newly prepended digits
(\textit{to the left}) are said to be in a different power of ten column.  That is, the number $123$ can be described as having a $1$ in the hundreds
column, $2$ in the tens column and $3$ in the ones column.  Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$.  Computer based
multiple precision arithmetic is essentially the same concept.  Larger integers are represented by adjoining fixed
precision computer words with the exception that a different radix is used.

What most people probably do not think about explicitly are the various other attributes that describe a multiple precision
integer.  For example, the integer $154_{10}$ has two immediately obvious properties.  First, the integer is positive,
that is the sign of this particular integer is positive as opposed to negative.  Second, the integer has three digits in
its representation.  There is an additional property that the integer posesses that does not concern pencil-and-paper
arithmetic.  The third property is how many digits placeholders are available to hold the integer.

The human analogy of this third property is ensuring there is enough space on the paper to write the integer.  For example,
if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left.
Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer
will not exceed the allowed boundaries.  These three properties make up what is known as a multiple precision
integer or mp\_int for short.

\subsection{The mp\_int Structure}
\label{sec:MPINT}
The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer.  The ISO C standard does not provide for
any such data type but it does provide for making composite data types known as structures.  The following is the structure definition
used within LibTomMath.

\index{mp\_int}
\begin{figure}[here]
\begin{center}
\begin{small}
%\begin{verbatim}
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\end{center}
\end{figure}

The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows.

\begin{enumerate}
\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent
a given integer.  The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count.  

\item The \textbf{alloc} parameter denotes how 
many digits are available in the array to use by functions before it has to increase in size.  When the \textbf{used} count 
of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the 
array to accommodate the precision of the result.  

\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple 
precision integer.  It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits.  The array is maintained in a least 
significant digit order.  As a pencil and paper analogy the array is organized such that the right most digits are stored
first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array.  For example, 
if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then 
it would represent the integer $a + b\beta + c\beta^2 + \ldots$  

\index{MP\_ZPOS} \index{MP\_NEG}
\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}).  
\end{enumerate}

\subsubsection{Valid mp\_int Structures}
Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency.  
The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy().

\begin{enumerate}
\item The value of \textbf{alloc} may not be less than one.  That is \textbf{dp} always points to a previously allocated
array of digits.
\item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero.
\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero.  That is, 
leading zero digits in the most significant positions must be trimmed.
   \begin{enumerate}
   \item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero.
   \end{enumerate}
\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero; 
this represents the mp\_int value of zero.
\end{enumerate}

\section{Argument Passing}
A convention of argument passing must be adopted early on in the development of any library.  Making the function 
prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity.  
In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int 
structures.  That means that the source (input) operands are placed on the left and the destination (output) on the right.   
Consider the following examples.

\begin{verbatim}
   mp_mul(&a, &b, &c);   /* c = a * b */
   mp_add(&a, &b, &a);   /* a = a + b */
   mp_sqr(&a, &b);       /* b = a * a */
\end{verbatim}

The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the
functions and make sense of them.  For example, the first function would read ``multiply a and b and store in c''.

Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order
of assignment expressions.  That is, the destination (output) is on the left and arguments (inputs) are on the right.  In 
truth, it is entirely a matter of preference.  In the case of LibTomMath the convention from the MPI library has been 
adopted.  

Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a 
destination.  For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$.  This is an important 
feature to implement since it allows the calling functions to cut down on the number of variables it must maintain.  
However, to implement this feature specific care has to be given to ensure the destination is not modified before the 
source is fully read.

\section{Return Values}
A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them 
to the caller.  By catching runtime errors a library can be guaranteed to prevent undefined behaviour.  However, the end 
developer can still manage to cause a library to crash.  For example, by passing an invalid pointer an application may
fault by dereferencing memory not owned by the application.

In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for 
instance) and memory allocation errors.  It will not check that the mp\_int passed to any function is valid nor 
will it check pointers for validity.  Any function that can cause a runtime error will return an error code as an 
\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}).

\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM}
\begin{figure}[here]
\begin{center}
\begin{tabular}{|l|l|}
\hline \textbf{Value} & \textbf{Meaning} \\
\hline \textbf{MP\_OKAY} & The function was successful \\
\hline \textbf{MP\_VAL}  & One of the input value(s) was invalid \\
\hline \textbf{MP\_MEM}  & The function ran out of heap memory \\
\hline
\end{tabular}
\end{center}
\caption{LibTomMath Error Codes}
\label{fig:errcodes}
\end{figure}

When an error is detected within a function it should free any memory it allocated, often during the initialization of
temporary mp\_ints, and return as soon as possible.  The goal is to leave the system in the same state it was when the 
function was called.  Error checking with this style of API is fairly simple.

\begin{verbatim}
   int err;
   if ((err = mp_add(&a, &b, &c)) != MP_OKAY) {
      printf("Error: %s\n", mp_error_to_string(err));
      exit(EXIT_FAILURE);
   }
\end{verbatim}

The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use.  Not all errors are fatal 
and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases.

\section{Initialization and Clearing}
The logical starting point when actually writing multiple precision integer functions is the initialization and 
clearing of the mp\_int structures.  These two algorithms will be used by the majority of the higher level algorithms.

Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of
the integer.  Often it is optimal to allocate a sufficiently large pre-set number of digits even though
the initial integer will represent zero.  If only a single digit were allocated quite a few subsequent re-allocations
would occur when operations are performed on the integers.  There is a tradeoff between how many default digits to allocate
and how many re-allocations are tolerable.  Obviously allocating an excessive amount of digits initially will waste 
memory and become unmanageable.  

If the memory for the digits has been successfully allocated then the rest of the members of the structure must
be initialized.  Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set
to zero.  The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}.

\subsection{Initializing an mp\_int}
An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the







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\end{center}
\end{figure}

The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows.

\begin{enumerate}
\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent
a given integer.  The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count.

\item The \textbf{alloc} parameter denotes how
many digits are available in the array to use by functions before it has to increase in size.  When the \textbf{used} count
of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the
array to accommodate the precision of the result.

\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple
precision integer.  It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits.  The array is maintained in a least
significant digit order.  As a pencil and paper analogy the array is organized such that the right most digits are stored
first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array.  For example,
if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then
it would represent the integer $a + b\beta + c\beta^2 + \ldots$

\index{MP\_ZPOS} \index{MP\_NEG}
\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}).
\end{enumerate}

\subsubsection{Valid mp\_int Structures}
Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency.
The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy().

\begin{enumerate}
\item The value of \textbf{alloc} may not be less than one.  That is \textbf{dp} always points to a previously allocated
array of digits.
\item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero.
\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero.  That is,
leading zero digits in the most significant positions must be trimmed.
   \begin{enumerate}
   \item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero.
   \end{enumerate}
\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero;
this represents the mp\_int value of zero.
\end{enumerate}

\section{Argument Passing}
A convention of argument passing must be adopted early on in the development of any library.  Making the function
prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity.
In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int
structures.  That means that the source (input) operands are placed on the left and the destination (output) on the right.
Consider the following examples.

\begin{verbatim}
   mp_mul(&a, &b, &c);   /* c = a * b */
   mp_add(&a, &b, &a);   /* a = a + b */
   mp_sqr(&a, &b);       /* b = a * a */
\end{verbatim}

The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the
functions and make sense of them.  For example, the first function would read ``multiply a and b and store in c''.

Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order
of assignment expressions.  That is, the destination (output) is on the left and arguments (inputs) are on the right.  In
truth, it is entirely a matter of preference.  In the case of LibTomMath the convention from the MPI library has been
adopted.

Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a
destination.  For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$.  This is an important
feature to implement since it allows the calling functions to cut down on the number of variables it must maintain.
However, to implement this feature specific care has to be given to ensure the destination is not modified before the
source is fully read.

\section{Return Values}
A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them
to the caller.  By catching runtime errors a library can be guaranteed to prevent undefined behaviour.  However, the end
developer can still manage to cause a library to crash.  For example, by passing an invalid pointer an application may
fault by dereferencing memory not owned by the application.

In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for
instance) and memory allocation errors.  It will not check that the mp\_int passed to any function is valid nor
will it check pointers for validity.  Any function that can cause a runtime error will return an error code as an
\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}).

\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM}
\begin{figure}[here]
\begin{center}
\begin{tabular}{|l|l|}
\hline \textbf{Value} & \textbf{Meaning} \\
\hline \textbf{MP\_OKAY} & The function was successful \\
\hline \textbf{MP\_VAL}  & One of the input value(s) was invalid \\
\hline \textbf{MP\_MEM}  & The function ran out of heap memory \\
\hline
\end{tabular}
\end{center}
\caption{LibTomMath Error Codes}
\label{fig:errcodes}
\end{figure}

When an error is detected within a function it should free any memory it allocated, often during the initialization of
temporary mp\_ints, and return as soon as possible.  The goal is to leave the system in the same state it was when the
function was called.  Error checking with this style of API is fairly simple.

\begin{verbatim}
   int err;
   if ((err = mp_add(&a, &b, &c)) != MP_OKAY) {
      printf("Error: %s\n", mp_error_to_string(err));
      exit(EXIT_FAILURE);
   }
\end{verbatim}

The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use.  Not all errors are fatal
and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases.

\section{Initialization and Clearing}
The logical starting point when actually writing multiple precision integer functions is the initialization and
clearing of the mp\_int structures.  These two algorithms will be used by the majority of the higher level algorithms.

Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of
the integer.  Often it is optimal to allocate a sufficiently large pre-set number of digits even though
the initial integer will represent zero.  If only a single digit were allocated quite a few subsequent re-allocations
would occur when operations are performed on the integers.  There is a tradeoff between how many default digits to allocate
and how many re-allocations are tolerable.  Obviously allocating an excessive amount of digits initially will waste
memory and become unmanageable.

If the memory for the digits has been successfully allocated then the rest of the members of the structure must
be initialized.  Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set
to zero.  The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}.

\subsection{Initializing an mp\_int}
An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the
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\end{center}
\caption{Algorithm mp\_init}
\end{figure}

\textbf{Algorithm mp\_init.}
The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly
manipulte it.  It is assumed that the input may not have had any of its members previously initialized which is certainly
a valid assumption if the input resides on the stack.  

Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for
the digits is allocated.  If this fails the function returns before setting any of the other members.  The \textbf{MP\_PREC} 
name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} 
used to dictate the minimum precision of newly initialized mp\_int integers.  Ideally, it is at least equal to the smallest
precision number you'll be working with.

Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow
heap operations later functions will have to perform in the future.  If \textbf{MP\_PREC} is set correctly the slack 
memory and the number of heap operations will be trivial.

Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and
\textbf{alloc} members initialized.  This ensures that the mp\_int will always represent the default state of zero regardless
of the original condition of the input.

\textbf{Remark.}
This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally
when the ``to'' keyword is placed between two expressions.  For example, ``for $a$ from $b$ to $c$ do'' means that
a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$.  In each
iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$.  If $b > c$ occured
the loop would not iterate.  By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate 
decrementally.

EXAM,bn_mp_init.c

One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure.  It 
is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack.  The 
call to mp\_init() is used only to initialize the members of the structure to a known default state.  

Here we see (line @23,XMALLOC@) the memory allocation is performed first.  This allows us to exit cleanly and quickly
if there is an error.  If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there
was a memory error.  The function XMALLOC is what actually allocates the memory.  Technically XMALLOC is not a function
but a macro defined in ``tommath.h``.  By default, XMALLOC will evaluate to malloc() which is the C library's built--in
memory allocation routine.

In order to assure the mp\_int is in a known state the digits must be set to zero.  On most platforms this could have been
accomplished by using calloc() instead of malloc().  However,  to correctly initialize a integer type to a given value in a 
portable fashion you have to actually assign the value.  The for loop (line @28,for@) performs this required
operation.

After the memory has been successfully initialized the remainder of the members are initialized 
(lines @29,used@ through @31,sign@) to their respective default states.  At this point the algorithm has succeeded and
a success code is returned to the calling function.  If this function returns \textbf{MP\_OKAY} it is safe to assume the 
mp\_int structure has been properly initialized and is safe to use with other functions within the library.  

\subsection{Clearing an mp\_int}
When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be 
returned to the application's memory pool with the mp\_clear algorithm.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clear}. \\
\textbf{Input}.   An mp\_int $a$ \\







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\end{center}
\caption{Algorithm mp\_init}
\end{figure}

\textbf{Algorithm mp\_init.}
The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly
manipulte it.  It is assumed that the input may not have had any of its members previously initialized which is certainly
a valid assumption if the input resides on the stack.

Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for
the digits is allocated.  If this fails the function returns before setting any of the other members.  The \textbf{MP\_PREC}
name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.}
used to dictate the minimum precision of newly initialized mp\_int integers.  Ideally, it is at least equal to the smallest
precision number you'll be working with.

Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow
heap operations later functions will have to perform in the future.  If \textbf{MP\_PREC} is set correctly the slack
memory and the number of heap operations will be trivial.

Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and
\textbf{alloc} members initialized.  This ensures that the mp\_int will always represent the default state of zero regardless
of the original condition of the input.

\textbf{Remark.}
This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally
when the ``to'' keyword is placed between two expressions.  For example, ``for $a$ from $b$ to $c$ do'' means that
a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$.  In each
iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$.  If $b > c$ occured
the loop would not iterate.  By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate
decrementally.

EXAM,bn_mp_init.c

One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure.  It
is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack.  The
call to mp\_init() is used only to initialize the members of the structure to a known default state.

Here we see (line @23,XMALLOC@) the memory allocation is performed first.  This allows us to exit cleanly and quickly
if there is an error.  If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there
was a memory error.  The function XMALLOC is what actually allocates the memory.  Technically XMALLOC is not a function
but a macro defined in ``tommath.h``.  By default, XMALLOC will evaluate to malloc() which is the C library's built--in
memory allocation routine.

In order to assure the mp\_int is in a known state the digits must be set to zero.  On most platforms this could have been
accomplished by using calloc() instead of malloc().  However,  to correctly initialize a integer type to a given value in a
portable fashion you have to actually assign the value.  The for loop (line @28,for@) performs this required
operation.

After the memory has been successfully initialized the remainder of the members are initialized
(lines @29,used@ through @31,sign@) to their respective default states.  At this point the algorithm has succeeded and
a success code is returned to the calling function.  If this function returns \textbf{MP\_OKAY} it is safe to assume the
mp\_int structure has been properly initialized and is safe to use with other functions within the library.

\subsection{Clearing an mp\_int}
When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be
returned to the application's memory pool with the mp\_clear algorithm.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clear}. \\
\textbf{Input}.   An mp\_int $a$ \\
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_clear}
\end{figure}

\textbf{Algorithm mp\_clear.}
This algorithm accomplishes two goals.  First, it clears the digits and the other mp\_int members.  This ensures that 
if a developer accidentally re-uses a cleared structure it is less likely to cause problems.  The second goal
is to free the allocated memory.

The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this
algorithm will not try to free the memory multiple times.  Cleared mp\_ints are detectable by having a pre-defined invalid 
digit pointer \textbf{dp} setting.

Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm
with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear.

EXAM,bn_mp_clear.c

The algorithm only operates on the mp\_int if it hasn't been previously cleared.  The if statement (line @23,a->dp != NULL@)
checks to see if the \textbf{dp} member is not \textbf{NULL}.  If the mp\_int is a valid mp\_int then \textbf{dp} cannot be
\textbf{NULL} in which case the if statement will evaluate to true.

The digits of the mp\_int are cleared by the for loop (line @25,for@) which assigns a zero to every digit.  Similar to mp\_init()
the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable.  

The digits are deallocated off the heap via the XFREE macro.  Similar to XMALLOC the XFREE macro actually evaluates to
a standard C library function.  In this case the free() function.  Since free() only deallocates the memory the pointer
still has to be reset to \textbf{NULL} manually (line @33,NULL@).  

Now that the digits have been cleared and deallocated the other members are set to their final values (lines @34,= 0@ and @35,ZPOS@).

\section{Maintenance Algorithms}

The previous sections describes how to initialize and clear an mp\_int structure.  To further support operations
that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be
able to augment the precision of an mp\_int and 
initialize mp\_ints with differing initial conditions.  

These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level
algorithms such as addition, multiplication and modular exponentiation.

\subsection{Augmenting an mp\_int's Precision}
When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire 
result of an operation without loss of precision.  Quite often the size of the array given by the \textbf{alloc} member 
is large enough to simply increase the \textbf{used} digit count.  However, when the size of the array is too small it 
must be re-sized appropriately to accomodate the result.  The mp\_grow algorithm will provide this functionality.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_grow}. \\
\textbf{Input}.   An mp\_int $a$ and an integer $b$. \\







|




|












|



|







|
|





|
|
|







811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_clear}
\end{figure}

\textbf{Algorithm mp\_clear.}
This algorithm accomplishes two goals.  First, it clears the digits and the other mp\_int members.  This ensures that
if a developer accidentally re-uses a cleared structure it is less likely to cause problems.  The second goal
is to free the allocated memory.

The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this
algorithm will not try to free the memory multiple times.  Cleared mp\_ints are detectable by having a pre-defined invalid
digit pointer \textbf{dp} setting.

Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm
with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear.

EXAM,bn_mp_clear.c

The algorithm only operates on the mp\_int if it hasn't been previously cleared.  The if statement (line @23,a->dp != NULL@)
checks to see if the \textbf{dp} member is not \textbf{NULL}.  If the mp\_int is a valid mp\_int then \textbf{dp} cannot be
\textbf{NULL} in which case the if statement will evaluate to true.

The digits of the mp\_int are cleared by the for loop (line @25,for@) which assigns a zero to every digit.  Similar to mp\_init()
the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable.

The digits are deallocated off the heap via the XFREE macro.  Similar to XMALLOC the XFREE macro actually evaluates to
a standard C library function.  In this case the free() function.  Since free() only deallocates the memory the pointer
still has to be reset to \textbf{NULL} manually (line @33,NULL@).

Now that the digits have been cleared and deallocated the other members are set to their final values (lines @34,= 0@ and @35,ZPOS@).

\section{Maintenance Algorithms}

The previous sections describes how to initialize and clear an mp\_int structure.  To further support operations
that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be
able to augment the precision of an mp\_int and
initialize mp\_ints with differing initial conditions.

These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level
algorithms such as addition, multiplication and modular exponentiation.

\subsection{Augmenting an mp\_int's Precision}
When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire
result of an operation without loss of precision.  Quite often the size of the array given by the \textbf{alloc} member
is large enough to simply increase the \textbf{used} digit count.  However, when the size of the array is too small it
must be re-sized appropriately to accomodate the result.  The mp\_grow algorithm will provide this functionality.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_grow}. \\
\textbf{Input}.   An mp\_int $a$ and an integer $b$. \\
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_grow}
\end{figure}

\textbf{Algorithm mp\_grow.}
It is ideal to prevent re-allocations from being performed if they are not required (step one).  This is useful to 
prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow.  

The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three).  
This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values.  

It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact.  This is much 
akin to how the \textit{realloc} function from the standard C library works.  Since the newly allocated digits are 
assumed to contain undefined values they are initially set to zero.

EXAM,bn_mp_grow.c

A quick optimization is to first determine if a memory re-allocation is required at all.  The if statement (line @24,alloc@) checks
if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count.  If the count is not larger than \textbf{alloc}
the function skips the re-allocation part thus saving time.

When a re-allocation is performed it is turned into an optimal request to save time in the future.  The requested digit count is
padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line @25, size@).  The XREALLOC function is used
to re-allocate the memory.  As per the other functions XREALLOC is actually a macro which evaluates to realloc by default.  The realloc
function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before
the re-allocation.  All	that is left is to clear the newly allocated digits and return.

Note that the re-allocation result is actually stored in a temporary pointer $tmp$.  This is to allow this function to return
an error with a valid pointer.  Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$.  That would
result in a memory leak if XREALLOC ever failed.  

\subsection{Initializing Variable Precision mp\_ints}
Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size 
of input mp\_ints to a given algorithm.  The purpose of algorithm mp\_init\_size is similar to mp\_init except that it 
will allocate \textit{at least} a specified number of digits.  

\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_size}. \\
\textbf{Input}.   An mp\_int $a$ and the requested number of digits $b$. \\







|
|

|
|

|
|
















|


|
|
|







876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_grow}
\end{figure}

\textbf{Algorithm mp\_grow.}
It is ideal to prevent re-allocations from being performed if they are not required (step one).  This is useful to
prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow.

The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three).
This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values.

It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact.  This is much
akin to how the \textit{realloc} function from the standard C library works.  Since the newly allocated digits are
assumed to contain undefined values they are initially set to zero.

EXAM,bn_mp_grow.c

A quick optimization is to first determine if a memory re-allocation is required at all.  The if statement (line @24,alloc@) checks
if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count.  If the count is not larger than \textbf{alloc}
the function skips the re-allocation part thus saving time.

When a re-allocation is performed it is turned into an optimal request to save time in the future.  The requested digit count is
padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line @25, size@).  The XREALLOC function is used
to re-allocate the memory.  As per the other functions XREALLOC is actually a macro which evaluates to realloc by default.  The realloc
function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before
the re-allocation.  All	that is left is to clear the newly allocated digits and return.

Note that the re-allocation result is actually stored in a temporary pointer $tmp$.  This is to allow this function to return
an error with a valid pointer.  Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$.  That would
result in a memory leak if XREALLOC ever failed.

\subsection{Initializing Variable Precision mp\_ints}
Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size
of input mp\_ints to a given algorithm.  The purpose of algorithm mp\_init\_size is similar to mp\_init except that it
will allocate \textit{at least} a specified number of digits.

\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_size}. \\
\textbf{Input}.   An mp\_int $a$ and the requested number of digits $b$. \\
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_init\_size}
\end{figure}

\textbf{Algorithm mp\_init\_size.}
This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of 
digits allocated can be controlled by the second input argument $b$.  The input size is padded upwards so it is a 
multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits.  This padding is used to prevent trivial 
allocations from becoming a bottleneck in the rest of the algorithms.

Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero.  This 
particular algorithm is useful if it is known ahead of time the approximate size of the input.  If the approximation is
correct no further memory re-allocations are required to work with the mp\_int.

EXAM,bn_mp_init_size.c

The number of digits $b$ requested is padded (line @22,MP_PREC@) by first augmenting it to the next multiple of 
\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result.  If the memory can be successfully allocated the 
mp\_int is placed in a default state representing the integer zero.  Otherwise, the error code \textbf{MP\_MEM} will be 
returned (line @27,return@).  

The digits are allocated and set to zero at the same time with the calloc() function (line @25,XCALLOC@).  The 
\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set 
to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines @29,used@, @30,alloc@ and @31,sign@).  If the function 
returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the 
functions to work with.

\subsection{Multiple Integer Initializations and Clearings}
Occasionally a function will require a series of mp\_int data types to be made available simultaneously.  
The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single
statement.  It is essentially a shortcut to multiple initializations.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_multi}. \\







|
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|





|
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|

|
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|



|







932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_init\_size}
\end{figure}

\textbf{Algorithm mp\_init\_size.}
This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of
digits allocated can be controlled by the second input argument $b$.  The input size is padded upwards so it is a
multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits.  This padding is used to prevent trivial
allocations from becoming a bottleneck in the rest of the algorithms.

Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero.  This
particular algorithm is useful if it is known ahead of time the approximate size of the input.  If the approximation is
correct no further memory re-allocations are required to work with the mp\_int.

EXAM,bn_mp_init_size.c

The number of digits $b$ requested is padded (line @22,MP_PREC@) by first augmenting it to the next multiple of
\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result.  If the memory can be successfully allocated the
mp\_int is placed in a default state representing the integer zero.  Otherwise, the error code \textbf{MP\_MEM} will be
returned (line @27,return@).

The digits are allocated with the malloc() function (line @27,XMALLOC@) and set to zero afterwards (line @38,for@).  The
\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set
to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines @29,used@, @30,alloc@ and @31,sign@).  If the function
returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the
functions to work with.

\subsection{Multiple Integer Initializations and Clearings}
Occasionally a function will require a series of mp\_int data types to be made available simultaneously.
The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single
statement.  It is essentially a shortcut to multiple initializations.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_multi}. \\
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init\_multi}
\end{figure}

\textbf{Algorithm mp\_init\_multi.}
The algorithm will initialize the array of mp\_int variables one at a time.  If a runtime error has been detected 
(\textit{step 1.2}) all of the previously initialized variables are cleared.  The goal is an ``all or nothing'' 
initialization which allows for quick recovery from runtime errors.

EXAM,bn_mp_init_multi.c

This function intializes a variable length list of mp\_int structure pointers.  However, instead of having the mp\_int
structures in an actual C array they are simply passed as arguments to the function.  This function makes use of the 
``...'' argument syntax of the C programming language.  The list is terminated with a final \textbf{NULL} argument 
appended on the right.  

The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function.  A count
$n$ of succesfully initialized mp\_int structures is maintained (line @47,n++@) such that if a failure does occur,
the algorithm can backtrack and free the previously initialized structures (lines @27,if@ to @46,}@).  


\subsection{Clamping Excess Digits}
When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of 
the function instead of checking during the computation.  For example, a multiplication of a $i$ digit number by a 
$j$ digit produces a result of at most $i + j$ digits.  It is entirely possible that the result is $i + j - 1$ 
though, with no final carry into the last position.  However, suppose the destination had to be first expanded 
(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry.  
That would be a considerable waste of time since heap operations are relatively slow.

The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function
terminates.  This way a single heap operation (\textit{at most}) is required.  However, if the result was not checked
there would be an excess high order zero digit.  

For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$.  The leading zero digit 
will not contribute to the precision of the result.  In fact, through subsequent operations more leading zero digits would
accumulate to the point the size of the integer would be prohibitive.  As a result even though the precision is very 
low the representation is excessively large.  

The mp\_clamp algorithm is designed to solve this very problem.  It will trim high-order zeros by decrementing the 
\textbf{used} count until a non-zero most significant digit is found.  Also in this system, zero is considered to be a 
positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to 
\textbf{MP\_ZPOS}.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clamp}. \\
\textbf{Input}.   An mp\_int $a$ \\







|
|





|
|
|



|



|
|
|
|
|




|

|

|
|

|
|
|







980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init\_multi}
\end{figure}

\textbf{Algorithm mp\_init\_multi.}
The algorithm will initialize the array of mp\_int variables one at a time.  If a runtime error has been detected
(\textit{step 1.2}) all of the previously initialized variables are cleared.  The goal is an ``all or nothing''
initialization which allows for quick recovery from runtime errors.

EXAM,bn_mp_init_multi.c

This function intializes a variable length list of mp\_int structure pointers.  However, instead of having the mp\_int
structures in an actual C array they are simply passed as arguments to the function.  This function makes use of the
``...'' argument syntax of the C programming language.  The list is terminated with a final \textbf{NULL} argument
appended on the right.

The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function.  A count
$n$ of succesfully initialized mp\_int structures is maintained (line @47,n++@) such that if a failure does occur,
the algorithm can backtrack and free the previously initialized structures (lines @27,if@ to @46,}@).


\subsection{Clamping Excess Digits}
When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of
the function instead of checking during the computation.  For example, a multiplication of a $i$ digit number by a
$j$ digit produces a result of at most $i + j$ digits.  It is entirely possible that the result is $i + j - 1$
though, with no final carry into the last position.  However, suppose the destination had to be first expanded
(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry.
That would be a considerable waste of time since heap operations are relatively slow.

The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function
terminates.  This way a single heap operation (\textit{at most}) is required.  However, if the result was not checked
there would be an excess high order zero digit.

For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$.  The leading zero digit
will not contribute to the precision of the result.  In fact, through subsequent operations more leading zero digits would
accumulate to the point the size of the integer would be prohibitive.  As a result even though the precision is very
low the representation is excessively large.

The mp\_clamp algorithm is designed to solve this very problem.  It will trim high-order zeros by decrementing the
\textbf{used} count until a non-zero most significant digit is found.  Also in this system, zero is considered to be a
positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to
\textbf{MP\_ZPOS}.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clamp}. \\
\textbf{Input}.   An mp\_int $a$ \\
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
\end{tabular}
\end{center}
\caption{Algorithm mp\_clamp}
\end{figure}

\textbf{Algorithm mp\_clamp.}
As can be expected this algorithm is very simple.  The loop on step one is expected to iterate only once or twice at
the most.  For example, this will happen in cases where there is not a carry to fill the last position.  Step two fixes the sign for 
when all of the digits are zero to ensure that the mp\_int is valid at all times.

EXAM,bn_mp_clamp.c

Note on line @27,while@ how to test for the \textbf{used} count is made on the left of the \&\& operator.  In the C programming
language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails.  This is 
important since if the \textbf{used} is zero the test on the right would fetch below the array.  That is obviously 
undesirable.  The parenthesis on line @28,a->used@ is used to make sure the \textbf{used} count is decremented and not
the pointer ``a''.  

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\
                     & \\
$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations.  \\
                     & \\







|





|
|

|







1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
\end{tabular}
\end{center}
\caption{Algorithm mp\_clamp}
\end{figure}

\textbf{Algorithm mp\_clamp.}
As can be expected this algorithm is very simple.  The loop on step one is expected to iterate only once or twice at
the most.  For example, this will happen in cases where there is not a carry to fill the last position.  Step two fixes the sign for
when all of the digits are zero to ensure that the mp\_int is valid at all times.

EXAM,bn_mp_clamp.c

Note on line @27,while@ how to test for the \textbf{used} count is made on the left of the \&\& operator.  In the C programming
language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails.  This is
important since if the \textbf{used} is zero the test on the right would fetch below the array.  That is obviously
undesirable.  The parenthesis on line @28,a->used@ is used to make sure the \textbf{used} count is decremented and not
the pointer ``a''.

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\
                     & \\
$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations.  \\
                     & \\
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
% CHAPTER FOUR
%%%

\chapter{Basic Operations}

\section{Introduction}
In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining
mp\_int structures.  This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low 
level basis of the entire library.  While these algorithm are relatively trivial it is important to understand how they
work before proceeding since these algorithms will be used almost intrinsically in the following chapters.

The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of
mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures
represent.   

\section{Assigning Values to mp\_int Structures}
\subsection{Copying an mp\_int}
Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making
a copy for the purposes of this text.  The copy of the mp\_int will be a separate entity that represents the same
value as the mp\_int it was copied from.  The mp\_copy algorithm provides this functionality. 

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_copy}. \\
\textbf{Input}.  An mp\_int $a$ and $b$. \\
\textbf{Output}.  Store a copy of $a$ in $b$. \\







|





|





|







1072
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1076
1077
1078
1079
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1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
% CHAPTER FOUR
%%%

\chapter{Basic Operations}

\section{Introduction}
In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining
mp\_int structures.  This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low
level basis of the entire library.  While these algorithm are relatively trivial it is important to understand how they
work before proceeding since these algorithms will be used almost intrinsically in the following chapters.

The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of
mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures
represent.

\section{Assigning Values to mp\_int Structures}
\subsection{Copying an mp\_int}
Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making
a copy for the purposes of this text.  The copy of the mp\_int will be a separate entity that represents the same
value as the mp\_int it was copied from.  The mp\_copy algorithm provides this functionality.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_copy}. \\
\textbf{Input}.  An mp\_int $a$ and $b$. \\
\textbf{Output}.  Store a copy of $a$ in $b$. \\
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
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1134
1135
1136
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1138
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1140
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1159
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1177
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1179
1180
1181
1182
1183
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1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
\end{tabular}
\end{center}
\caption{Algorithm mp\_copy}
\end{figure}

\textbf{Algorithm mp\_copy.}
This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will
represent the same integer as the mp\_int $a$.  The mp\_int $b$ shall be a complete and distinct copy of the 
mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$.

If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow 
algorithm.  The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two
and three).  The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of
$b$.

\textbf{Remark.}  This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the
text.  The error return codes of other algorithms are not explicitly checked in the pseudo-code presented.  For example, in 
step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded.  Text space is 
limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return
the error code itself.  However, the C code presented will demonstrate all of the error handling logic required to 
implement the pseudo-code.

EXAM,bn_mp_copy.c

Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output
mp\_int structures passed to a function are one and the same.  For this case it is optimal to return immediately without 
copying digits (line @24,a == b@).  

The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$.  If $b.alloc$ is less than
$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines @29,alloc@ to @33,}@).  In order to
simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits
of the mp\_ints $a$ and $b$ respectively.  These aliases (lines @42,tmpa@ and @45,tmpb@) allow the compiler to access the digits without first dereferencing the
mp\_int pointers and then subsequently the pointer to the digits.  

After the aliases are established the digits from $a$ are copied into $b$ (lines @48,for@ to @50,}@) and then the excess 
digits of $b$ are set to zero (lines @53,for@ to @55,}@).  Both ``for'' loops make use of the pointer aliases and in 
fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits.  This optimization 
allows the alias to stay in a machine register fairly easy between the two loops.

\textbf{Remarks.}  The use of pointer aliases is an implementation methodology first introduced in this function that will
be used considerably in other functions.  Technically, a pointer alias is simply a short hand alias used to lower the 
number of pointer dereferencing operations required to access data.  For example, a for loop may resemble

\begin{alltt}
for (x = 0; x < 100; x++) \{
    a->num[4]->dp[x] = 0;
\}
\end{alltt}

This could be re-written using aliases as 

\begin{alltt}
mp_digit *tmpa;
a = a->num[4]->dp;
for (x = 0; x < 100; x++) \{
    *a++ = 0;
\}
\end{alltt}

In this case an alias is used to access the 
array of digits within an mp\_int structure directly.  It may seem that a pointer alias is strictly not required 
as a compiler may optimize out the redundant pointer operations.  However, there are two dominant reasons to use aliases.

The first reason is that most compilers will not effectively optimize pointer arithmetic.  For example, some optimizations 
may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC).  Also some optimizations may 
work for GCC and not MSVC.  As such it is ideal to find a common ground for as many compilers as possible.  Pointer 
aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code 
stands a better chance of being faster.

The second reason is that pointer aliases often can make an algorithm simpler to read.  Consider the first ``for'' 
loop of the function mp\_copy() re-written to not use pointer aliases.

\begin{alltt}
    /* copy all the digits */
    for (n = 0; n < a->used; n++) \{
      b->dp[n] = a->dp[n];
    \}
\end{alltt}

Whether this code is harder to read depends strongly on the individual.  However, it is quantifiably slightly more 
complicated as there are four variables within the statement instead of just two.

\subsubsection{Nested Statements}
Another commonly used technique in the source routines is that certain sections of code are nested.  This is used in
particular with the pointer aliases to highlight code phases.  For example, a Comba multiplier (discussed in chapter six)
will typically have three different phases.  First the temporaries are initialized, then the columns calculated and 
finally the carries are propagated.  In this example the middle column production phase will typically be nested as it
uses temporary variables and aliases the most.

The nesting also simplies the source code as variables that are nested are only valid for their scope.  As a result
the various temporary variables required do not propagate into other sections of code.


\subsection{Creating a Clone}
Another common operation is to make a local temporary copy of an mp\_int argument.  To initialize an mp\_int 
and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone.  This is 
useful within functions that need to modify an argument but do not wish to actually modify the original copy.  The 
mp\_init\_copy algorithm has been designed to help perform this task.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_copy}. \\
\textbf{Input}.   An mp\_int $a$ and $b$\\
\textbf{Output}.  $a$ is initialized to be a copy of $b$. \\
\hline \\
1.  Init $a$.  (\textit{mp\_init}) \\
2.  Copy $b$ to $a$.  (\textit{mp\_copy}) \\
3.  Return the status of the copy operation. \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init\_copy}
\end{figure}

\textbf{Algorithm mp\_init\_copy.}
This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it.  As 
such this algorithm will perform two operations in one step.  

EXAM,bn_mp_init_copy.c

This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}.  Note that 
\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call
and \textbf{a} will be left intact.  

\section{Zeroing an Integer}
Reseting an mp\_int to the default state is a common step in many algorithms.  The mp\_zero algorithm will be the algorithm used to
perform this task.

\begin{figure}[here]
\begin{center}







|


|





|
|

|





|
|





|

|
|
|



|








|









|
|


|
|
|
|


|









|





|








|
|
|



















|
|



|

|







1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
\end{tabular}
\end{center}
\caption{Algorithm mp\_copy}
\end{figure}

\textbf{Algorithm mp\_copy.}
This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will
represent the same integer as the mp\_int $a$.  The mp\_int $b$ shall be a complete and distinct copy of the
mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$.

If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow
algorithm.  The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two
and three).  The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of
$b$.

\textbf{Remark.}  This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the
text.  The error return codes of other algorithms are not explicitly checked in the pseudo-code presented.  For example, in
step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded.  Text space is
limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return
the error code itself.  However, the C code presented will demonstrate all of the error handling logic required to
implement the pseudo-code.

EXAM,bn_mp_copy.c

Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output
mp\_int structures passed to a function are one and the same.  For this case it is optimal to return immediately without
copying digits (line @24,a == b@).

The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$.  If $b.alloc$ is less than
$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines @29,alloc@ to @33,}@).  In order to
simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits
of the mp\_ints $a$ and $b$ respectively.  These aliases (lines @42,tmpa@ and @45,tmpb@) allow the compiler to access the digits without first dereferencing the
mp\_int pointers and then subsequently the pointer to the digits.

After the aliases are established the digits from $a$ are copied into $b$ (lines @48,for@ to @50,}@) and then the excess
digits of $b$ are set to zero (lines @53,for@ to @55,}@).  Both ``for'' loops make use of the pointer aliases and in
fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits.  This optimization
allows the alias to stay in a machine register fairly easy between the two loops.

\textbf{Remarks.}  The use of pointer aliases is an implementation methodology first introduced in this function that will
be used considerably in other functions.  Technically, a pointer alias is simply a short hand alias used to lower the
number of pointer dereferencing operations required to access data.  For example, a for loop may resemble

\begin{alltt}
for (x = 0; x < 100; x++) \{
    a->num[4]->dp[x] = 0;
\}
\end{alltt}

This could be re-written using aliases as

\begin{alltt}
mp_digit *tmpa;
a = a->num[4]->dp;
for (x = 0; x < 100; x++) \{
    *a++ = 0;
\}
\end{alltt}

In this case an alias is used to access the
array of digits within an mp\_int structure directly.  It may seem that a pointer alias is strictly not required
as a compiler may optimize out the redundant pointer operations.  However, there are two dominant reasons to use aliases.

The first reason is that most compilers will not effectively optimize pointer arithmetic.  For example, some optimizations
may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC).  Also some optimizations may
work for GCC and not MSVC.  As such it is ideal to find a common ground for as many compilers as possible.  Pointer
aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code
stands a better chance of being faster.

The second reason is that pointer aliases often can make an algorithm simpler to read.  Consider the first ``for''
loop of the function mp\_copy() re-written to not use pointer aliases.

\begin{alltt}
    /* copy all the digits */
    for (n = 0; n < a->used; n++) \{
      b->dp[n] = a->dp[n];
    \}
\end{alltt}

Whether this code is harder to read depends strongly on the individual.  However, it is quantifiably slightly more
complicated as there are four variables within the statement instead of just two.

\subsubsection{Nested Statements}
Another commonly used technique in the source routines is that certain sections of code are nested.  This is used in
particular with the pointer aliases to highlight code phases.  For example, a Comba multiplier (discussed in chapter six)
will typically have three different phases.  First the temporaries are initialized, then the columns calculated and
finally the carries are propagated.  In this example the middle column production phase will typically be nested as it
uses temporary variables and aliases the most.

The nesting also simplies the source code as variables that are nested are only valid for their scope.  As a result
the various temporary variables required do not propagate into other sections of code.


\subsection{Creating a Clone}
Another common operation is to make a local temporary copy of an mp\_int argument.  To initialize an mp\_int
and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone.  This is
useful within functions that need to modify an argument but do not wish to actually modify the original copy.  The
mp\_init\_copy algorithm has been designed to help perform this task.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_copy}. \\
\textbf{Input}.   An mp\_int $a$ and $b$\\
\textbf{Output}.  $a$ is initialized to be a copy of $b$. \\
\hline \\
1.  Init $a$.  (\textit{mp\_init}) \\
2.  Copy $b$ to $a$.  (\textit{mp\_copy}) \\
3.  Return the status of the copy operation. \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init\_copy}
\end{figure}

\textbf{Algorithm mp\_init\_copy.}
This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it.  As
such this algorithm will perform two operations in one step.

EXAM,bn_mp_init_copy.c

This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}.  Note that
\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call
and \textbf{a} will be left intact.

\section{Zeroing an Integer}
Reseting an mp\_int to the default state is a common step in many algorithms.  The mp\_zero algorithm will be the algorithm used to
perform this task.

\begin{figure}[here]
\begin{center}
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_zero}
\end{figure}

\textbf{Algorithm mp\_zero.}
This algorithm simply resets a mp\_int to the default state.  

EXAM,bn_mp_zero.c

After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the 
\textbf{sign} variable is set to \textbf{MP\_ZPOS}.

\section{Sign Manipulation}
\subsection{Absolute Value}
With the mp\_int representation of an integer, calculating the absolute value is trivial.  The mp\_abs algorithm will compute
the absolute value of an mp\_int.








|



|







1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_zero}
\end{figure}

\textbf{Algorithm mp\_zero.}
This algorithm simply resets a mp\_int to the default state.

EXAM,bn_mp_zero.c

After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the
\textbf{sign} variable is set to \textbf{MP\_ZPOS}.

\section{Sign Manipulation}
\subsection{Absolute Value}
With the mp\_int representation of an integer, calculating the absolute value is trivial.  The mp\_abs algorithm will compute
the absolute value of an mp\_int.

1292
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1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
\end{center}
\caption{Algorithm mp\_abs}
\end{figure}

\textbf{Algorithm mp\_abs.}
This algorithm computes the absolute of an mp\_int input.  First it copies $a$ over $b$.  This is an example of an
algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful.  This allows,
for instance, the developer to pass the same mp\_int as the source and destination to this function without addition 
logic to handle it.

EXAM,bn_mp_abs.c

This fairly trivial algorithm first eliminates non--required duplications (line @27,a != b@) and then sets the
\textbf{sign} flag to \textbf{MP\_ZPOS}.








|







1281
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1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
\end{center}
\caption{Algorithm mp\_abs}
\end{figure}

\textbf{Algorithm mp\_abs.}
This algorithm computes the absolute of an mp\_int input.  First it copies $a$ over $b$.  This is an example of an
algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful.  This allows,
for instance, the developer to pass the same mp\_int as the source and destination to this function without addition
logic to handle it.

EXAM,bn_mp_abs.c

This fairly trivial algorithm first eliminates non--required duplications (line @27,a != b@) and then sets the
\textbf{sign} flag to \textbf{MP\_ZPOS}.

1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
\end{tabular}
\end{center}
\caption{Algorithm mp\_neg}
\end{figure}

\textbf{Algorithm mp\_neg.}
This algorithm computes the negation of an input.  First it copies $a$ over $b$.  If $a$ has no used digits then
the algorithm returns immediately.  Otherwise it flips the sign flag and stores the result in $b$.  Note that if 
$a$ had no digits then it must be positive by definition.  Had step three been omitted then the algorithm would return
zero as negative.

EXAM,bn_mp_neg.c

Like mp\_abs() this function avoids non--required duplications (line @21,a != b@) and then sets the sign.  We
have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}.  If the mp\_int is zero







|







1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
\end{tabular}
\end{center}
\caption{Algorithm mp\_neg}
\end{figure}

\textbf{Algorithm mp\_neg.}
This algorithm computes the negation of an input.  First it copies $a$ over $b$.  If $a$ has no used digits then
the algorithm returns immediately.  Otherwise it flips the sign flag and stores the result in $b$.  Note that if
$a$ had no digits then it must be positive by definition.  Had step three been omitted then the algorithm would return
zero as negative.

EXAM,bn_mp_neg.c

Like mp\_abs() this function avoids non--required duplications (line @21,a != b@) and then sets the sign.  We
have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}.  If the mp\_int is zero
1352
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1356
1357
1358
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\textbf{Input}.   An mp\_int $a$ and a digit $b$ \\
\textbf{Output}.  Make $a$ equivalent to $b$ \\
\hline \\
1.  Zero $a$ (\textit{mp\_zero}). \\
2.  $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\
3.  $a.used \leftarrow  \left \lbrace \begin{array}{ll}
                              1 &  \mbox{if }a_0 > 0 \\
                              0 &  \mbox{if }a_0 = 0 
                              \end{array} \right .$ \\
\hline                              
\end{tabular}
\end{center}
\caption{Algorithm mp\_set}
\end{figure}

\textbf{Algorithm mp\_set.}
This algorithm sets a mp\_int to a small single digit value.  Step number 1 ensures that the integer is reset to the default state.  The
single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly.

EXAM,bn_mp_set.c

First we zero (line @21,mp_zero@) the mp\_int to make sure that the other members are initialized for a 
small positive constant.  mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count
is zero.  Next we set the digit and reduce it modulo $\beta$ (line @22,MP_MASK@).  After this step we have to 
check if the resulting digit is zero or not.  If it is not then we set the \textbf{used} count to one, otherwise
to zero.

We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with 
$2^k - 1$ will perform the same operation.

One important limitation of this function is that it will only set one digit.  The size of a digit is not fixed, meaning source that uses 
this function should take that into account.  Only trivially small constants can be set using this function.

\subsection{Setting Large Constants}
To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal.  It accepts a ``long''
data type as input and will always treat it as a 32-bit integer.

\begin{figure}[here]







|

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\textbf{Input}.   An mp\_int $a$ and a digit $b$ \\
\textbf{Output}.  Make $a$ equivalent to $b$ \\
\hline \\
1.  Zero $a$ (\textit{mp\_zero}). \\
2.  $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\
3.  $a.used \leftarrow  \left \lbrace \begin{array}{ll}
                              1 &  \mbox{if }a_0 > 0 \\
                              0 &  \mbox{if }a_0 = 0
                              \end{array} \right .$ \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_set}
\end{figure}

\textbf{Algorithm mp\_set.}
This algorithm sets a mp\_int to a small single digit value.  Step number 1 ensures that the integer is reset to the default state.  The
single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly.

EXAM,bn_mp_set.c

First we zero (line @21,mp_zero@) the mp\_int to make sure that the other members are initialized for a
small positive constant.  mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count
is zero.  Next we set the digit and reduce it modulo $\beta$ (line @22,MP_MASK@).  After this step we have to
check if the resulting digit is zero or not.  If it is not then we set the \textbf{used} count to one, otherwise
to zero.

We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with
$2^k - 1$ will perform the same operation.

One important limitation of this function is that it will only set one digit.  The size of a digit is not fixed, meaning source that uses
this function should take that into account.  Only trivially small constants can be set using this function.

\subsection{Setting Large Constants}
To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal.  It accepts a ``long''
data type as input and will always treat it as a 32-bit integer.

\begin{figure}[here]
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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_set\_int}
\end{figure}

\textbf{Algorithm mp\_set\_int.}
The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the 
mp\_int.  Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions.  In step 2.2 the
next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is 
incremented to reflect the addition.  The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have
zero digits used and the newly added four bits would be ignored.

Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp.

EXAM,bn_mp_set_int.c

This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes.  The weird
addition on line @38,a->used@ ensures that the newly added in bits are added to the number of digits.  While it may not 
seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line @27,mp_mul_2d@ 
as well as the  call to mp\_clamp() on line @40,mp_clamp@.  Both functions will clamp excess leading digits which keeps 
the number of used digits low.

\section{Comparisons}
\subsection{Unsigned Comparisions}
Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers.  For example,
to compare $1,234$ to $1,264$ the digits are extracted by their positions.  That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$
to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude 
positions.  If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater.  

The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two
mp\_int variables alone.  It will ignore the sign of the two inputs.  Such a function is useful when an absolute comparison is required or if the 
signs are known to agree in advance.

To facilitate working with the results of the comparison functions three constants are required.  

\begin{figure}[here]
\begin{center}
\begin{tabular}{|r|l|}
\hline \textbf{Constant} & \textbf{Meaning} \\
\hline \textbf{MP\_GT} & Greater Than \\
\hline \textbf{MP\_EQ} & Equal To \\







|

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|







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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_set\_int}
\end{figure}

\textbf{Algorithm mp\_set\_int.}
The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the
mp\_int.  Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions.  In step 2.2 the
next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is
incremented to reflect the addition.  The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have
zero digits used and the newly added four bits would be ignored.

Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp.

EXAM,bn_mp_set_int.c

This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes.  The weird
addition on line @38,a->used@ ensures that the newly added in bits are added to the number of digits.  While it may not
seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line @27,mp_mul_2d@
as well as the  call to mp\_clamp() on line @40,mp_clamp@.  Both functions will clamp excess leading digits which keeps
the number of used digits low.

\section{Comparisons}
\subsection{Unsigned Comparisions}
Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers.  For example,
to compare $1,234$ to $1,264$ the digits are extracted by their positions.  That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$
to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude
positions.  If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater.

The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two
mp\_int variables alone.  It will ignore the sign of the two inputs.  Such a function is useful when an absolute comparison is required or if the
signs are known to agree in advance.

To facilitate working with the results of the comparison functions three constants are required.

\begin{figure}[here]
\begin{center}
\begin{tabular}{|r|l|}
\hline \textbf{Constant} & \textbf{Meaning} \\
\hline \textbf{MP\_GT} & Greater Than \\
\hline \textbf{MP\_EQ} & Equal To \\
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1471
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\end{tabular}
\end{center}
\caption{Algorithm mp\_cmp\_mag}
\end{figure}

\textbf{Algorithm mp\_cmp\_mag.}
By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return
\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$.  The first two steps compare the number of digits used in both $a$ and $b$.  
Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is.  
If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit.  

By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to
the zero'th digit.  If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}.

EXAM,bn_mp_cmp_mag.c

The two if statements (lines @24,if@ and @28,if@) compare the number of digits in the two inputs.  These two are 
performed before all of the digits are compared since it is a very cheap test to perform and can potentially save 
considerable time.  The implementation given is also not valid without those two statements.  $b.alloc$ may be 
smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits.



\subsection{Signed Comparisons}
Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}).  Based on an unsigned magnitude 
comparison a trivial signed comparison algorithm can be written.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_cmp}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$ \\







|
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|





|







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\end{tabular}
\end{center}
\caption{Algorithm mp\_cmp\_mag}
\end{figure}

\textbf{Algorithm mp\_cmp\_mag.}
By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return
\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$.  The first two steps compare the number of digits used in both $a$ and $b$.
Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is.
If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit.

By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to
the zero'th digit.  If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}.

EXAM,bn_mp_cmp_mag.c

The two if statements (lines @24,if@ and @28,if@) compare the number of digits in the two inputs.  These two are
performed before all of the digits are compared since it is a very cheap test to perform and can potentially save
considerable time.  The implementation given is also not valid without those two statements.  $b.alloc$ may be
smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits.



\subsection{Signed Comparisons}
Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}).  Based on an unsigned magnitude
comparison a trivial signed comparison algorithm can be written.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_cmp}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$ \\
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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_cmp}
\end{figure}

\textbf{Algorithm mp\_cmp.}
The first two steps compare the signs of the two inputs.  If the signs do not agree then it can return right away with the appropriate 
comparison code.  When the signs are equal the digits of the inputs must be compared to determine the correct result.  In step 
three the unsigned comparision flips the order of the arguments since they are both negative.  For instance, if $-a > -b$ then 
$\vert a \vert < \vert b \vert$.  Step number four will compare the two when they are both positive.

EXAM,bn_mp_cmp.c

The two if statements (lines @22,if@ and @26,if@) perform the initial sign comparison.  If the signs are not the equal then which ever
has the positive sign is larger.   The inputs are compared (line @30,if@) based on magnitudes.  If the signs were both 
negative then the unsigned comparison is performed in the opposite direction (line @31,mp_cmp_mag@).  Otherwise, the signs are assumed to 
be both positive and a forward direction unsigned comparison is performed.

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\
                     & \\
$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits  \\
                     & of two random digits (of equal magnitude) before a difference is found. \\
                     & \\
$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based  \\
                     & on the observations made in the previous problem. \\
                     &
\end{tabular}

\chapter{Basic Arithmetic}
\section{Introduction}
At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been 
established.  The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms.  These 
algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms.  It is very important 
that these algorithms are highly optimized.  On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms 
which easily places them at $O(n^2)$ or even $O(n^3)$ work levels.  

MARK,SHIFTS
All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right 
logical shifts respectively.  A logical shift is analogous to sliding the decimal point of radix-10 representations.  For example, the real 
number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}).  
Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two.  
For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$.

One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed
from the number.  For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$.  However, with a logical shift the 
result is $110_2$.  

\section{Addition and Subtraction}
In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus.  For example, with 32-bit integers
$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$  since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$.  
As a result subtraction can be performed with a trivial series of logical operations and an addition.

However, in multiple precision arithmetic negative numbers are not represented in the same way.  Instead a sign flag is used to keep track of the
sign of the integer.  As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or 
subtraction algorithms with the sign fixed up appropriately.

The lower level algorithms will add or subtract integers without regard to the sign flag.  That is they will add or subtract the magnitude of
the integers respectively.

\subsection{Low Level Addition}
An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers.  That is to add the 
trailing digits first and propagate the resulting carry upwards.  Since this is a lower level algorithm the name will have a ``s\_'' prefix.  
Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely.

\newpage
\begin{figure}[!here]
\begin{center}
\begin{small}
\begin{tabular}{l}







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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_cmp}
\end{figure}

\textbf{Algorithm mp\_cmp.}
The first two steps compare the signs of the two inputs.  If the signs do not agree then it can return right away with the appropriate
comparison code.  When the signs are equal the digits of the inputs must be compared to determine the correct result.  In step
three the unsigned comparision flips the order of the arguments since they are both negative.  For instance, if $-a > -b$ then
$\vert a \vert < \vert b \vert$.  Step number four will compare the two when they are both positive.

EXAM,bn_mp_cmp.c

The two if statements (lines @22,if@ and @26,if@) perform the initial sign comparison.  If the signs are not the equal then which ever
has the positive sign is larger.   The inputs are compared (line @30,if@) based on magnitudes.  If the signs were both
negative then the unsigned comparison is performed in the opposite direction (line @31,mp_cmp_mag@).  Otherwise, the signs are assumed to
be both positive and a forward direction unsigned comparison is performed.

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\
                     & \\
$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits  \\
                     & of two random digits (of equal magnitude) before a difference is found. \\
                     & \\
$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based  \\
                     & on the observations made in the previous problem. \\
                     &
\end{tabular}

\chapter{Basic Arithmetic}
\section{Introduction}
At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been
established.  The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms.  These
algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms.  It is very important
that these algorithms are highly optimized.  On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms
which easily places them at $O(n^2)$ or even $O(n^3)$ work levels.

MARK,SHIFTS
All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right
logical shifts respectively.  A logical shift is analogous to sliding the decimal point of radix-10 representations.  For example, the real
number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}).
Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two.
For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$.

One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed
from the number.  For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$.  However, with a logical shift the
result is $110_2$.

\section{Addition and Subtraction}
In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus.  For example, with 32-bit integers
$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$  since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$.
As a result subtraction can be performed with a trivial series of logical operations and an addition.

However, in multiple precision arithmetic negative numbers are not represented in the same way.  Instead a sign flag is used to keep track of the
sign of the integer.  As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or
subtraction algorithms with the sign fixed up appropriately.

The lower level algorithms will add or subtract integers without regard to the sign flag.  That is they will add or subtract the magnitude of
the integers respectively.

\subsection{Low Level Addition}
An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers.  That is to add the
trailing digits first and propagate the resulting carry upwards.  Since this is a lower level algorithm the name will have a ``s\_'' prefix.
Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely.

\newpage
\begin{figure}[!here]
\begin{center}
\begin{small}
\begin{tabular}{l}
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\end{tabular}
\end{small}
\end{center}
\caption{Algorithm s\_mp\_add}
\end{figure}

\textbf{Algorithm s\_mp\_add.}
This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes.  
Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}.  Even the 
MIX pseudo  machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes.

The first thing that has to be accomplished is to sort out which of the two inputs is the largest.  The addition logic
will simply add all of the smallest input to the largest input and store that first part of the result in the
destination.  Then it will apply a simpler addition loop to excess digits of the larger input.

The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two 
inputs.  The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the
same number of digits.  After the inputs are sorted the destination $c$ is grown as required to accomodate the sum 
of the two inputs.  The original \textbf{used} count of $c$ is copied and set to the new used count.  

At this point the first addition loop will go through as many digit positions that both inputs have.  The carry
variable $\mu$ is set to zero outside the loop.  Inside the loop an ``addition'' step requires three statements to produce
one digit of the summand.  First
two digits from $a$ and $b$ are added together along with the carry $\mu$.  The carry of this step is extracted and stored
in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$.

Now all of the digit positions that both inputs have in common have been exhausted.  If $min \ne max$ then $x$ is an alias
for one of the inputs that has more digits.  A simplified addition loop is then used to essentially copy the remaining digits
and the carry to the destination.

The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are zeroed which completes the addition.


EXAM,bn_s_mp_add.c

We first sort (lines @27,if@ to @35,}@) the inputs based on magnitude and determine the $min$ and $max$ variables.
Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias.  Next we
grow the destination (@37,init@ to @42,}@) ensure that it can accomodate the result of the addition. 

Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style.  The three aliases that are on 
lines @56,tmpa@, @59,tmpb@ and @62,tmpc@ represent the two inputs and destination variables respectively.  These aliases are used to ensure the
compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int.

The initial carry $u$ will be cleared (line @65,u = 0@), note that $u$ is of type mp\_digit which ensures type 
compatibility within the implementation.  The initial addition (line @66,for@ to @75,}@) adds digits from
both inputs until the smallest input runs out of digits.  Similarly the conditional addition loop
(line @81,for@ to @90,}@) adds the remaining digits from the larger of the two inputs.  The addition is finished 
with the final carry being stored in $tmpc$ (line @94,tmpc++@).  Note the ``++'' operator within the same expression.
After line @94,tmpc++@, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$.  This is useful
for the next loop (line @97,for@ to @99,}@) which set any old upper digits to zero.

\subsection{Low Level Subtraction}
The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm.  The principle difference is that the
unsigned subtraction algorithm requires the result to be positive.  That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must 
be met for this algorithm to function properly.  Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly.  
This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms.

MARK,GAMMA

For this algorithm a new variable is required to make the description simpler.  Recall from section 1.3.1 that a mp\_digit must be able to represent
the range $0 \le x < 2\beta$ for the algorithms to work correctly.  However, it is allowable that a mp\_digit represent a larger range of values.  For 
this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a 
mp\_digit (\textit{this implies $2^{\gamma} > \beta$}).  

For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$.  In ISO C an ``unsigned long''
data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$.

\newpage\begin{figure}[!here]
\begin{center}
\begin{small}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_sub}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\
\textbf{Output}.  The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\
\hline \\
1.  $min \leftarrow b.used$ \\
2.  $max \leftarrow a.used$ \\
3.  If $c.alloc < max$ then grow $c$ to hold at least $max$ digits.  (\textit{mp\_grow}) \\
4.  $oldused \leftarrow c.used$ \\ 
5.  $c.used \leftarrow max$ \\
6.  $u \leftarrow 0$ \\
7.  for $n$ from $0$ to $min - 1$ do \\
\hspace{3mm}7.1  $c_n \leftarrow a_n - b_n - u$ \\
\hspace{3mm}7.2  $u   \leftarrow c_n >> (\gamma - 1)$ \\
\hspace{3mm}7.3  $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
8.  if $min < max$ then do \\







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\end{tabular}
\end{small}
\end{center}
\caption{Algorithm s\_mp\_add}
\end{figure}

\textbf{Algorithm s\_mp\_add.}
This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes.
Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}.  Even the
MIX pseudo  machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes.

The first thing that has to be accomplished is to sort out which of the two inputs is the largest.  The addition logic
will simply add all of the smallest input to the largest input and store that first part of the result in the
destination.  Then it will apply a simpler addition loop to excess digits of the larger input.

The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two
inputs.  The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the
same number of digits.  After the inputs are sorted the destination $c$ is grown as required to accomodate the sum
of the two inputs.  The original \textbf{used} count of $c$ is copied and set to the new used count.

At this point the first addition loop will go through as many digit positions that both inputs have.  The carry
variable $\mu$ is set to zero outside the loop.  Inside the loop an ``addition'' step requires three statements to produce
one digit of the summand.  First
two digits from $a$ and $b$ are added together along with the carry $\mu$.  The carry of this step is extracted and stored
in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$.

Now all of the digit positions that both inputs have in common have been exhausted.  If $min \ne max$ then $x$ is an alias
for one of the inputs that has more digits.  A simplified addition loop is then used to essentially copy the remaining digits
and the carry to the destination.

The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are zeroed which completes the addition.


EXAM,bn_s_mp_add.c

We first sort (lines @27,if@ to @35,}@) the inputs based on magnitude and determine the $min$ and $max$ variables.
Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias.  Next we
grow the destination (@37,init@ to @42,}@) ensure that it can accomodate the result of the addition.

Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style.  The three aliases that are on
lines @56,tmpa@, @59,tmpb@ and @62,tmpc@ represent the two inputs and destination variables respectively.  These aliases are used to ensure the
compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int.

The initial carry $u$ will be cleared (line @65,u = 0@), note that $u$ is of type mp\_digit which ensures type
compatibility within the implementation.  The initial addition (line @66,for@ to @75,}@) adds digits from
both inputs until the smallest input runs out of digits.  Similarly the conditional addition loop
(line @81,for@ to @90,}@) adds the remaining digits from the larger of the two inputs.  The addition is finished
with the final carry being stored in $tmpc$ (line @94,tmpc++@).  Note the ``++'' operator within the same expression.
After line @94,tmpc++@, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$.  This is useful
for the next loop (line @97,for@ to @99,}@) which set any old upper digits to zero.

\subsection{Low Level Subtraction}
The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm.  The principle difference is that the
unsigned subtraction algorithm requires the result to be positive.  That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must
be met for this algorithm to function properly.  Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly.
This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms.

MARK,GAMMA

For this algorithm a new variable is required to make the description simpler.  Recall from section 1.3.1 that a mp\_digit must be able to represent
the range $0 \le x < 2\beta$ for the algorithms to work correctly.  However, it is allowable that a mp\_digit represent a larger range of values.  For
this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a
mp\_digit (\textit{this implies $2^{\gamma} > \beta$}).

For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$.  In ISO C an ``unsigned long''
data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$.

\newpage\begin{figure}[!here]
\begin{center}
\begin{small}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_sub}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\
\textbf{Output}.  The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\
\hline \\
1.  $min \leftarrow b.used$ \\
2.  $max \leftarrow a.used$ \\
3.  If $c.alloc < max$ then grow $c$ to hold at least $max$ digits.  (\textit{mp\_grow}) \\
4.  $oldused \leftarrow c.used$ \\
5.  $c.used \leftarrow max$ \\
6.  $u \leftarrow 0$ \\
7.  for $n$ from $0$ to $min - 1$ do \\
\hspace{3mm}7.1  $c_n \leftarrow a_n - b_n - u$ \\
\hspace{3mm}7.2  $u   \leftarrow c_n >> (\gamma - 1)$ \\
\hspace{3mm}7.3  $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
8.  if $min < max$ then do \\
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1768
1769
1770
1771
1772

\textbf{Algorithm s\_mp\_sub.}
This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive.  That is when
passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly.  This
algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well.  As was the case
of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude.

The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$.  Steps 1 and 2 
set the $min$ and $max$ variables.  Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at 
most $max$ digits in length as opposed to $max + 1$.  Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and 
set to the maximal count for the operation.

The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision 
subtraction is used instead.  Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction 
loops.  Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry.  

For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$.  The least significant bit will force a carry upwards to 
the third bit which will be set to zero after the borrow.  After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain,  When the 
third bit of $0101_2$ is subtracted from the result it will cause another carry.  In this case though the carry will be forced to propagate all the 
way to the most significant bit.  

Recall that $\beta < 2^{\gamma}$.  This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most 
significant bit.  Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that
is needed is a single zero or one bit for the carry.  Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the 
carry.  This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed.  

If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$.  Step
10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed.

EXAM,bn_s_mp_sub.c

Like low level addition we ``sort'' the inputs.  Except in this case the sorting is hardcoded 
(lines @24,min@ and @25,max@).  In reality the $min$ and $max$ variables are only aliases and are only 
used to make the source code easier to read.  Again the pointer alias optimization is used 
within this algorithm.  The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized
(lines @42,tmpa@, @43,tmpb@ and @44,tmpc@) for $a$, $b$ and $c$ respectively.

The first subtraction loop (lines @47,u = 0@ through @61,}@) subtract digits from both inputs until the smaller of
the two inputs has been exhausted.  As remarked earlier there is an implementation reason for using the ``awkward'' 
method of extracting the carry (line @57, >>@).  The traditional method for extracting the carry would be to shift 
by $lg(\beta)$ positions and logically AND the least significant bit.  The AND operation is required because all of 
the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction.  This carry 
extraction requires two relatively cheap operations to extract the carry.  The other method is to simply shift the 
most significant bit to the least significant bit thus extracting the carry with a single cheap operation.  This 
optimization only works on twos compliment machines which is a safe assumption to make.

If $a$ has a larger magnitude than $b$ an additional loop (lines @64,for@ through @73,}@) is required to propagate 
the carry through $a$ and copy the result to $c$.  

\subsection{High Level Addition}
Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be
established.  This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data 
types.  

Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} 
flag.  A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases.

\begin{figure}[!here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_add}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$  \\







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1700
1701
1702
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1704
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1749
1750
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1752
1753
1754
1755
1756
1757
1758
1759
1760
1761

\textbf{Algorithm s\_mp\_sub.}
This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive.  That is when
passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly.  This
algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well.  As was the case
of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude.

The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$.  Steps 1 and 2
set the $min$ and $max$ variables.  Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at
most $max$ digits in length as opposed to $max + 1$.  Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and
set to the maximal count for the operation.

The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision
subtraction is used instead.  Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction
loops.  Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry.

For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$.  The least significant bit will force a carry upwards to
the third bit which will be set to zero after the borrow.  After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain,  When the
third bit of $0101_2$ is subtracted from the result it will cause another carry.  In this case though the carry will be forced to propagate all the
way to the most significant bit.

Recall that $\beta < 2^{\gamma}$.  This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most
significant bit.  Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that
is needed is a single zero or one bit for the carry.  Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the
carry.  This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed.

If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$.  Step
10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed.

EXAM,bn_s_mp_sub.c

Like low level addition we ``sort'' the inputs.  Except in this case the sorting is hardcoded
(lines @24,min@ and @25,max@).  In reality the $min$ and $max$ variables are only aliases and are only
used to make the source code easier to read.  Again the pointer alias optimization is used
within this algorithm.  The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized
(lines @42,tmpa@, @43,tmpb@ and @44,tmpc@) for $a$, $b$ and $c$ respectively.

The first subtraction loop (lines @47,u = 0@ through @61,}@) subtract digits from both inputs until the smaller of
the two inputs has been exhausted.  As remarked earlier there is an implementation reason for using the ``awkward''
method of extracting the carry (line @57, >>@).  The traditional method for extracting the carry would be to shift
by $lg(\beta)$ positions and logically AND the least significant bit.  The AND operation is required because all of
the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction.  This carry
extraction requires two relatively cheap operations to extract the carry.  The other method is to simply shift the
most significant bit to the least significant bit thus extracting the carry with a single cheap operation.  This
optimization only works on twos compliment machines which is a safe assumption to make.

If $a$ has a larger magnitude than $b$ an additional loop (lines @64,for@ through @73,}@) is required to propagate
the carry through $a$ and copy the result to $c$.

\subsection{High Level Addition}
Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be
established.  This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data
types.

Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign}
flag.  A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases.

\begin{figure}[!here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_add}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$  \\
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1793
1794
1795
1796
1797
1798
1799
1800
1801
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_add}
\end{figure}

\textbf{Algorithm mp\_add.}
This algorithm performs the signed addition of two mp\_int variables.  There is no reference algorithm to draw upon from 
either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations.  The algorithm is fairly 
straightforward but restricted since subtraction can only produce positive results.

\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\







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1783
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1788
1789
1790
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_add}
\end{figure}

\textbf{Algorithm mp\_add.}
This algorithm performs the signed addition of two mp\_int variables.  There is no reference algorithm to draw upon from
either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations.  The algorithm is fairly
straightforward but restricted since subtraction can only produce positive results.

\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\
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1844
1845
1846
1847
1848
1849
1850
1851
1852
\end{tabular}
\end{center}
\end{small}
\caption{Addition Guide Chart}
\label{fig:AddChart}
\end{figure}

Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three 
specific cases need to be handled.  The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are 
forwarded to step three to check for errors.  This simplifies the description of the algorithm considerably and best 
follows how the implementation actually was achieved.

Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed.  Recall from the descriptions of algorithms
s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits.  The mp\_clamp algorithm will set the \textbf{sign}
to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero.

For example, consider performing $-a + a$ with algorithm mp\_add.  By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would
produce a result of $-0$.  However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp 
within algorithm s\_mp\_add will force $-0$ to become $0$.  

EXAM,bn_mp_add.c

The source code follows the algorithm fairly closely.  The most notable new source code addition is the usage of the $res$ integer variable which
is used to pass result of the unsigned operations forward.  Unlike in the algorithm, the variable $res$ is merely returned as is without
explicitly checking it and returning the constant \textbf{MP\_OKAY}.  The observation is this algorithm will succeed or fail only if the lower
level functions do so.  Returning their return code is sufficient.

\subsection{High Level Subtraction}
The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm.  

\newpage\begin{figure}[!here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_sub}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$  \\
\textbf{Output}.  The signed subtraction $c = a - b$. \\







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1841
\end{tabular}
\end{center}
\end{small}
\caption{Addition Guide Chart}
\label{fig:AddChart}
\end{figure}

Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three
specific cases need to be handled.  The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are
forwarded to step three to check for errors.  This simplifies the description of the algorithm considerably and best
follows how the implementation actually was achieved.

Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed.  Recall from the descriptions of algorithms
s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits.  The mp\_clamp algorithm will set the \textbf{sign}
to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero.

For example, consider performing $-a + a$ with algorithm mp\_add.  By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would
produce a result of $-0$.  However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp
within algorithm s\_mp\_add will force $-0$ to become $0$.

EXAM,bn_mp_add.c

The source code follows the algorithm fairly closely.  The most notable new source code addition is the usage of the $res$ integer variable which
is used to pass result of the unsigned operations forward.  Unlike in the algorithm, the variable $res$ is merely returned as is without
explicitly checking it and returning the constant \textbf{MP\_OKAY}.  The observation is this algorithm will succeed or fail only if the lower
level functions do so.  Returning their return code is sufficient.

\subsection{High Level Subtraction}
The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm.

\newpage\begin{figure}[!here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_sub}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$  \\
\textbf{Output}.  The signed subtraction $c = a - b$. \\
1868
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1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_sub}
\end{figure}

\textbf{Algorithm mp\_sub.}
This algorithm performs the signed subtraction of two inputs.  Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or 
\cite{HAC}.  Also this algorithm is restricted by algorithm s\_mp\_sub.  Chart \ref{fig:SubChart} lists the eight possible inputs and
the operations required.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}







|







1857
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1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_sub}
\end{figure}

\textbf{Algorithm mp\_sub.}
This algorithm performs the signed subtraction of two inputs.  Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or
\cite{HAC}.  Also this algorithm is restricted by algorithm s\_mp\_sub.  Chart \ref{fig:SubChart} lists the eight possible inputs and
the operations required.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
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1918
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1930
\end{tabular}
\end{center}
\end{small}
\caption{Subtraction Guide Chart}
\label{fig:SubChart}
\end{figure}

Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction.  That is to prevent the 
algorithm from producing $-a - -a = -0$ as a result.  

EXAM,bn_mp_sub.c

Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations
and forward it to the end of the function.  On line @38, != MP_LT@ the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a 
``greater than or equal to'' comparison.  

\section{Bit and Digit Shifting}
MARK,POLY
It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$.  
This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring.  

In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established.  That is to shift
the digits left or right as well to shift individual bits of the digits left and right.  It is important to note that not all ``shift'' operations
are on radix-$\beta$ digits.  

\subsection{Multiplication by Two}

In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient 
operation to perform.  A single precision logical shift left is sufficient to multiply a single digit by two.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mul\_2}. \\
\textbf{Input}.   One mp\_int $a$ \\







|
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|
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1884
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\end{tabular}
\end{center}
\end{small}
\caption{Subtraction Guide Chart}
\label{fig:SubChart}
\end{figure}

Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction.  That is to prevent the
algorithm from producing $-a - -a = -0$ as a result.

EXAM,bn_mp_sub.c

Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations
and forward it to the end of the function.  On line @38, != MP_LT@ the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a
``greater than or equal to'' comparison.

\section{Bit and Digit Shifting}
MARK,POLY
It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$.
This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring.

In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established.  That is to shift
the digits left or right as well to shift individual bits of the digits left and right.  It is important to note that not all ``shift'' operations
are on radix-$\beta$ digits.

\subsection{Multiplication by Two}

In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient
operation to perform.  A single precision logical shift left is sufficient to multiply a single digit by two.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mul\_2}. \\
\textbf{Input}.   One mp\_int $a$ \\
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
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1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul\_2}
\end{figure}

\textbf{Algorithm mp\_mul\_2.}
This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two.  Neither \cite{TAOCPV2} nor \cite{HAC} describe such 
an algorithm despite the fact it arises often in other algorithms.  The algorithm is setup much like the lower level algorithm s\_mp\_add since 
it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$.  

Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result.  The initial \textbf{used} count
is set to $a.used$ at step 4.  Only if there is a final carry will the \textbf{used} count require adjustment.

Step 6 is an optimization implementation of the addition loop for this specific case.  That is since the two values being added together 
are the same there is no need to perform two reads from the digits of $a$.  Step 6.1 performs a single precision shift on the current digit $a_n$ to
obtain what will be the carry for the next iteration.  Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus
the previous carry.  Recall from ~SHIFTS~ that $a_n << 1$ is equivalent to $a_n \cdot 2$.  An iteration of the addition loop is finished with 
forwarding the carry to the next iteration.

Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$.  
Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$.

EXAM,bn_mp_mul_2.c

This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input.  The only noteworthy difference
is the use of the logical shift operator on line @52,<<@ to perform a single precision doubling.  

\subsection{Division by Two}
A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}







|
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|


|


|





|







1939
1940
1941
1942
1943
1944
1945
1946
1947
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1961
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1963
1964
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1968
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1970
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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul\_2}
\end{figure}

\textbf{Algorithm mp\_mul\_2.}
This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two.  Neither \cite{TAOCPV2} nor \cite{HAC} describe such
an algorithm despite the fact it arises often in other algorithms.  The algorithm is setup much like the lower level algorithm s\_mp\_add since
it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$.

Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result.  The initial \textbf{used} count
is set to $a.used$ at step 4.  Only if there is a final carry will the \textbf{used} count require adjustment.

Step 6 is an optimization implementation of the addition loop for this specific case.  That is since the two values being added together
are the same there is no need to perform two reads from the digits of $a$.  Step 6.1 performs a single precision shift on the current digit $a_n$ to
obtain what will be the carry for the next iteration.  Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus
the previous carry.  Recall from ~SHIFTS~ that $a_n << 1$ is equivalent to $a_n \cdot 2$.  An iteration of the addition loop is finished with
forwarding the carry to the next iteration.

Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$.
Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$.

EXAM,bn_mp_mul_2.c

This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input.  The only noteworthy difference
is the use of the logical shift operator on line @52,<<@ to perform a single precision doubling.

\subsection{Division by Two}
A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
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2043

\textbf{Algorithm mp\_div\_2.}
This algorithm will divide an mp\_int by two using logical shifts to the right.  Like mp\_mul\_2 it uses a modified low level addition
core as the basis of the algorithm.  Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit.  The algorithm
could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent
reading past the end of the array of digits.

Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the 
least significant bit not the most significant bit.  

EXAM,bn_mp_div_2.c

\section{Polynomial Basis Operations}
Recall from ~POLY~ that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$.  Such a representation is also known as
the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single 
place.  The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer
division and Karatsuba multiplication.  

Converting from an array of digits to polynomial basis is very simple.  Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that
$y = \sum_{i=0}^{2} a_i \beta^i$.  Simply replace $\beta$ with $x$ and the expression is in polynomial basis.  For example, $f(x) = 8x + 9$ is the
polynomial basis representation for $89$ using radix ten.  That is, $f(10) = 8(10) + 9 = 89$.  

\subsection{Multiplication by $x$}

Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one 
degree.  In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$.  From a scalar basis point of view multiplying by $x$ is equivalent to
multiplying by the integer $\beta$.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_lshd}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\







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|







1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
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2011
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2017
2018
2019
2020
2021
2022
2023
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2027
2028
2029
2030
2031
2032

\textbf{Algorithm mp\_div\_2.}
This algorithm will divide an mp\_int by two using logical shifts to the right.  Like mp\_mul\_2 it uses a modified low level addition
core as the basis of the algorithm.  Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit.  The algorithm
could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent
reading past the end of the array of digits.

Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the
least significant bit not the most significant bit.

EXAM,bn_mp_div_2.c

\section{Polynomial Basis Operations}
Recall from ~POLY~ that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$.  Such a representation is also known as
the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single
place.  The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer
division and Karatsuba multiplication.

Converting from an array of digits to polynomial basis is very simple.  Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that
$y = \sum_{i=0}^{2} a_i \beta^i$.  Simply replace $\beta$ with $x$ and the expression is in polynomial basis.  For example, $f(x) = 8x + 9$ is the
polynomial basis representation for $89$ using radix ten.  That is, $f(10) = 8(10) + 9 = 89$.

\subsection{Multiplication by $x$}

Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one
degree.  In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$.  From a scalar basis point of view multiplying by $x$ is equivalent to
multiplying by the integer $\beta$.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_lshd}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\
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2070
2071
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2074
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2079
2080
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2090
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2093
2094
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2096
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2098
2099
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_lshd}
\end{figure}

\textbf{Algorithm mp\_lshd.}
This algorithm multiplies an mp\_int by the $b$'th power of $x$.  This is equivalent to multiplying by $\beta^b$.  The algorithm differs 
from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location.  The
motivation behind this change is due to the way this function is typically used.  Algorithms such as mp\_add store the result in an optionally
different third mp\_int because the original inputs are often still required.  Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is
typically used on values where the original value is no longer required.  The algorithm will return success immediately if 
$b \le 0$ since the rest of algorithm is only valid when $b > 0$.  

First the destination $a$ is grown as required to accomodate the result.  The counters $i$ and $j$ are used to form a \textit{sliding window} over
the digits of $a$ of length $b$.  The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}).  
The loop on step 7 copies the digit from the tail to the head.  In each iteration the window is moved down one digit.   The last loop on 
step 8 sets the lower $b$ digits to zero.

\newpage
FIGU,sliding_window,Sliding Window Movement

EXAM,bn_mp_lshd.c

The if statement (line @24,if@) ensures that the $b$ variable is greater than zero since we do not interpret negative
shift counts properly.  The \textbf{used} count is incremented by $b$ before the copy loop begins.  This elminates 
the need for an additional variable in the for loop.  The variable $top$ (line @42,top@) is an alias
for the leading digit while $bottom$ (line @45,bottom@) is an alias for the trailing edge.  The aliases form a 
window of exactly $b$ digits over the input.  

\subsection{Division by $x$}

Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_rshd}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\







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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_lshd}
\end{figure}

\textbf{Algorithm mp\_lshd.}
This algorithm multiplies an mp\_int by the $b$'th power of $x$.  This is equivalent to multiplying by $\beta^b$.  The algorithm differs
from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location.  The
motivation behind this change is due to the way this function is typically used.  Algorithms such as mp\_add store the result in an optionally
different third mp\_int because the original inputs are often still required.  Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is
typically used on values where the original value is no longer required.  The algorithm will return success immediately if
$b \le 0$ since the rest of algorithm is only valid when $b > 0$.

First the destination $a$ is grown as required to accomodate the result.  The counters $i$ and $j$ are used to form a \textit{sliding window} over
the digits of $a$ of length $b$.  The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}).
The loop on step 7 copies the digit from the tail to the head.  In each iteration the window is moved down one digit.   The last loop on
step 8 sets the lower $b$ digits to zero.

\newpage
FIGU,sliding_window,Sliding Window Movement

EXAM,bn_mp_lshd.c

The if statement (line @24,if@) ensures that the $b$ variable is greater than zero since we do not interpret negative
shift counts properly.  The \textbf{used} count is incremented by $b$ before the copy loop begins.  This elminates
the need for an additional variable in the for loop.  The variable $top$ (line @42,top@) is an alias
for the leading digit while $bottom$ (line @45,bottom@) is an alias for the trailing edge.  The aliases form a
window of exactly $b$ digits over the input.

\subsection{Division by $x$}

Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_rshd}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\
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2151
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2153
\end{center}
\end{small}
\caption{Algorithm mp\_rshd}
\end{figure}

\textbf{Algorithm mp\_rshd.}
This algorithm divides the input in place by the $b$'th power of $x$.  It is analogous to dividing by a $\beta^b$ but much quicker since
it does not require single precision division.  This algorithm does not actually return an error code as it cannot fail.  

If the input $b$ is less than one the algorithm quickly returns without performing any work.  If the \textbf{used} count is less than or equal
to the shift count $b$ then it will simply zero the input and return.

After the trivial cases of inputs have been handled the sliding window is setup.  Much like the case of algorithm mp\_lshd a sliding window that
is $b$ digits wide is used to copy the digits.  Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit.  
Also the digits are copied from the leading to the trailing edge.

Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented.

EXAM,bn_mp_rshd.c

The only noteworthy element of this routine is the lack of a return type since it cannot fail.  Like mp\_lshd() we
form a sliding window except we copy in the other direction.  After the window (line @59,for (;@) we then zero
the upper digits of the input to make sure the result is correct.

\section{Powers of Two}

Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required.  For 
example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful.  Instead of performing single
shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed.  

\subsection{Multiplication by Power of Two}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}







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\end{center}
\end{small}
\caption{Algorithm mp\_rshd}
\end{figure}

\textbf{Algorithm mp\_rshd.}
This algorithm divides the input in place by the $b$'th power of $x$.  It is analogous to dividing by a $\beta^b$ but much quicker since
it does not require single precision division.  This algorithm does not actually return an error code as it cannot fail.

If the input $b$ is less than one the algorithm quickly returns without performing any work.  If the \textbf{used} count is less than or equal
to the shift count $b$ then it will simply zero the input and return.

After the trivial cases of inputs have been handled the sliding window is setup.  Much like the case of algorithm mp\_lshd a sliding window that
is $b$ digits wide is used to copy the digits.  Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit.
Also the digits are copied from the leading to the trailing edge.

Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented.

EXAM,bn_mp_rshd.c

The only noteworthy element of this routine is the lack of a return type since it cannot fail.  Like mp\_lshd() we
form a sliding window except we copy in the other direction.  After the window (line @59,for (;@) we then zero
the upper digits of the input to make sure the result is correct.

\section{Powers of Two}

Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required.  For
example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful.  Instead of performing single
shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed.

\subsection{Multiplication by Power of Two}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
\caption{Algorithm mp\_mul\_2d}
\end{figure}

\textbf{Algorithm mp\_mul\_2d.}
This algorithm multiplies $a$ by $2^b$ and stores the result in $c$.  The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to
quickly compute the product.

First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than 
$\beta$.  For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ 
left.

After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform.  Step 5 calculates the number of remaining shifts 
required.  If it is non-zero a modified shift loop is used to calculate the remaining product.  
Essentially the loop is a generic version of algorithm mp\_mul\_2 designed to handle any shift count in the range $1 \le x < lg(\beta)$.  The $mask$
variable is used to extract the upper $d$ bits to form the carry for the next iteration.  

This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to 
complete.  It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow.

EXAM,bn_mp_mul_2d.c

The shifting is performed in--place which means the first step (line @24,a != c@) is to copy the input to the 
destination.  We avoid calling mp\_copy() by making sure the mp\_ints are different.  The destination then
has to be grown (line @31,grow@) to accomodate the result.

If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples 
of $lg(\beta)$.  Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left.  Inside the actual shift 
loop (lines @45,if@ to @76,}@) we make use of pre--computed values $shift$ and $mask$.   These are used to
extract the carry bit(s) to pass into the next iteration of the loop.  The $r$ and $rr$ variables form a 
chain between consecutive iterations to propagate the carry.  

\subsection{Division by Power of Two}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}







|
|


|
|

|

|




|



|
|

|
|







2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
\caption{Algorithm mp\_mul\_2d}
\end{figure}

\textbf{Algorithm mp\_mul\_2d.}
This algorithm multiplies $a$ by $2^b$ and stores the result in $c$.  The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to
quickly compute the product.

First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than
$\beta$.  For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$
left.

After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform.  Step 5 calculates the number of remaining shifts
required.  If it is non-zero a modified shift loop is used to calculate the remaining product.
Essentially the loop is a generic version of algorithm mp\_mul\_2 designed to handle any shift count in the range $1 \le x < lg(\beta)$.  The $mask$
variable is used to extract the upper $d$ bits to form the carry for the next iteration.

This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to
complete.  It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow.

EXAM,bn_mp_mul_2d.c

The shifting is performed in--place which means the first step (line @24,a != c@) is to copy the input to the
destination.  We avoid calling mp\_copy() by making sure the mp\_ints are different.  The destination then
has to be grown (line @31,grow@) to accomodate the result.

If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples
of $lg(\beta)$.  Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left.  Inside the actual shift
loop (lines @45,if@ to @76,}@) we make use of pre--computed values $shift$ and $mask$.   These are used to
extract the carry bit(s) to pass into the next iteration of the loop.  The $r$ and $rr$ variables form a
chain between consecutive iterations to propagate the carry.

\subsection{Division by Power of Two}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_div\_2d}
\end{figure}

\textbf{Algorithm mp\_div\_2d.}
This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder.  The algorithm is designed much like algorithm 
mp\_mul\_2d by first using whole digit shifts then single precision shifts.  This algorithm will also produce the remainder of the division
by using algorithm mp\_mod\_2d.

EXAM,bn_mp_div_2d.c

The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies.  The remainder $d$ may be optionally 
ignored by passing \textbf{NULL} as the pointer to the mp\_int variable.    The temporary mp\_int variable $t$ is used to hold the 
result of the remainder operation until the end.  This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before
the quotient is obtained.

The remainder of the source code is essentially the same as the source code for mp\_mul\_2d.  The only significant difference is
the direction of the shifts.

\subsection{Remainder of Division by Power of Two}

The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$.  This
algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mod\_2d}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\







|





|
|









|







2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_div\_2d}
\end{figure}

\textbf{Algorithm mp\_div\_2d.}
This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder.  The algorithm is designed much like algorithm
mp\_mul\_2d by first using whole digit shifts then single precision shifts.  This algorithm will also produce the remainder of the division
by using algorithm mp\_mod\_2d.

EXAM,bn_mp_div_2d.c

The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies.  The remainder $d$ may be optionally
ignored by passing \textbf{NULL} as the pointer to the mp\_int variable.    The temporary mp\_int variable $t$ is used to hold the
result of the remainder operation until the end.  This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before
the quotient is obtained.

The remainder of the source code is essentially the same as the source code for mp\_mul\_2d.  The only significant difference is
the direction of the shifts.

\subsection{Remainder of Division by Power of Two}

The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$.  This
algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mod\_2d}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mod\_2d}
\end{figure}

\textbf{Algorithm mp\_mod\_2d.}
This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$.  First if $b$ is less than or equal to zero the 
result is set to zero.  If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns.  Otherwise, $a$ 
is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count.

EXAM,bn_mp_mod_2d.c

We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases.  Next if $2^b$ is larger
than the input we just mp\_copy() the input and return right away.  After this point we know we must actually
perform some work to produce the remainder.

Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce 
the number.  First we zero any digits above the last digit in $2^b$ (line @41,for@).  Next we reduce the 
leading digit of both (line @45,&=@) and then mp\_clamp().

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\
                      & in $O(n)$ time. \\
                      &\\







|
|








|
|







2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mod\_2d}
\end{figure}

\textbf{Algorithm mp\_mod\_2d.}
This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$.  First if $b$ is less than or equal to zero the
result is set to zero.  If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns.  Otherwise, $a$
is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count.

EXAM,bn_mp_mod_2d.c

We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases.  Next if $2^b$ is larger
than the input we just mp\_copy() the input and return right away.  After this point we know we must actually
perform some work to produce the remainder.

Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce
the number.  First we zero any digits above the last digit in $2^b$ (line @41,for@).  Next we reduce the
leading digit of both (line @45,&=@) and then mp\_clamp().

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\
                      & in $O(n)$ time. \\
                      &\\
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
2375
2376
2377
2378
2379
2380
2381
$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\
                      & calculating the result of a signed comparison. \\
                      &
\end{tabular}

\chapter{Multiplication and Squaring}
\section{The Multipliers}
For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of 
algorithms of any multiple precision integer package.  The set of multiplier algorithms include integer multiplication, squaring and modular reduction 
where in each of the algorithms single precision multiplication is the dominant operation performed.  This chapter will discuss integer multiplication 
and squaring, leaving modular reductions for the subsequent chapter.  

The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular 
exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$.  During a modular
exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions, 
35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision 
multiplications.

For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied 
against every digit of the other multiplicand.  Traditional long-hand multiplication is based on this process;  while the techniques can differ the 
overall algorithm used is essentially the same.  Only ``recently'' have faster algorithms been studied.  First Karatsuba multiplication was discovered in 
1962.  This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach.  
This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions.  

\section{Multiplication}
\subsection{The Baseline Multiplication}
\label{sec:basemult}
\index{baseline multiplication}
Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication
algorithm that school children are taught.  The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision 
multiplications are required.  More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required.  To 
simplify most discussions, it will be assumed that the inputs have comparable number of digits.  

The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be 
used.  This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible.    One important 
facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution.  The importance of this 
modification will become evident during the discussion of Barrett modular reduction.  Recall that for a $n$ and $m$ digit input the product 
will be at most $n + m$ digits.  Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product.  

Recall from ~GAMMA~ the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}.  We shall now extend the variable set to 
include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}.  This implies that $2^{\alpha} > 2 \cdot \beta^2$.  The 
constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see ~COMBA~ for more information}).

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\







|
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|


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2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\
                      & calculating the result of a signed comparison. \\
                      &
\end{tabular}

\chapter{Multiplication and Squaring}
\section{The Multipliers}
For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of
algorithms of any multiple precision integer package.  The set of multiplier algorithms include integer multiplication, squaring and modular reduction
where in each of the algorithms single precision multiplication is the dominant operation performed.  This chapter will discuss integer multiplication
and squaring, leaving modular reductions for the subsequent chapter.

The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular
exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$.  During a modular
exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions,
35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision
multiplications.

For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied
against every digit of the other multiplicand.  Traditional long-hand multiplication is based on this process;  while the techniques can differ the
overall algorithm used is essentially the same.  Only ``recently'' have faster algorithms been studied.  First Karatsuba multiplication was discovered in
1962.  This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach.
This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions.

\section{Multiplication}
\subsection{The Baseline Multiplication}
\label{sec:basemult}
\index{baseline multiplication}
Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication
algorithm that school children are taught.  The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision
multiplications are required.  More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required.  To
simplify most discussions, it will be assumed that the inputs have comparable number of digits.

The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be
used.  This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible.    One important
facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution.  The importance of this
modification will become evident during the discussion of Barrett modular reduction.  Recall that for a $n$ and $m$ digit input the product
will be at most $n + m$ digits.  Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product.

Recall from ~GAMMA~ the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}.  We shall now extend the variable set to
include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}.  This implies that $2^{\alpha} > 2 \cdot \beta^2$.  The
constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see ~COMBA~ for more information}).

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490
2491
2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
2510
2511
2512
2513
2514
2515
2516
2517
2518
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
\end{center}
\end{small}
\caption{Algorithm s\_mp\_mul\_digs}
\end{figure}

\textbf{Algorithm s\_mp\_mul\_digs.}
This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits.  While it may seem
a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent 
algorithm.  The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}.  
Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the 
inputs.

The first thing this algorithm checks for is whether a Comba multiplier can be used instead.   If the minimum digit count of either
input is less than $\delta$, then the Comba method may be used instead.    After the Comba method is ruled out, the baseline algorithm begins.  A 
temporary mp\_int variable $t$ is used to hold the intermediate result of the product.  This allows the algorithm to be used to 
compute products when either $a = c$ or $b = c$ without overwriting the inputs.  

All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output.  The $pb$ variable
is given the count of digits to read from $b$ inside the nested loop.  If $pb \le 1$ then no more output digits can be produced and the algorithm
will exit the loop.  The best way to think of the loops are as a series of $pb \times 1$ multiplications.    That is, in each pass of the 
innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$.  

For example, consider multiplying $576$ by $241$.  That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best
visualized in the following table.

\begin{figure}[here]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|l|}
\hline   &&          & 5 & 7 & 6 & \\
\hline   $\times$&&  & 2 & 4 & 1 & \\
\hline &&&&&&\\
  &&          & 5 & 7 & 6 & $10^0(1)(576)$ \\
  &2 &   3    & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\
  1 & 3 & 8 & 8 & 1 & 6 &   $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\
\hline  
\end{tabular}
\end{center}
\caption{Long-Hand Multiplication Diagram}
\end{figure}

Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate 
count.  That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult.

Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable.  The multiplication on that step
is assumed to be a double wide output single precision multiplication.  That is, two single precision variables are multiplied to produce a
double precision result.  The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step
5.4.1 is propagated through the nested loop.  If the carry was not propagated immediately it would overflow the single precision digit 
$t_{ix+iy}$ and the result would be lost.  

At step 5.5 the nested loop is finished and any carry that was left over should be forwarded.  The carry does not have to be added to the $ix+pb$'th
digit since that digit is assumed to be zero at this point.  However, if $ix + pb \ge digs$ the carry is not set as it would make the result
exceed the precision requested.

EXAM,bn_s_mp_mul_digs.c

First we determine (line @30,if@) if the Comba method can be used first since it's faster.  The conditions for 
sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than 
\textbf{MP\_WARRAY}.  This new constant is used to control the stack usage in the Comba routines.  By default it is 
set to $\delta$ but can be reduced when memory is at a premium.

If we cannot use the Comba method we proceed to setup the baseline routine.  We allocate the the destination mp\_int
$t$ (line @36,init@) to the exact size of the output to avoid further re--allocations.  At this point we now 
begin the $O(n^2)$ loop.

This implementation of multiplication has the caveat that it can be trimmed to only produce a variable number of
digits as output.  In each iteration of the outer loop the $pb$ variable is set (line @48,MIN@) to the maximum 
number of inner loop iterations.  

Inside the inner loop we calculate $\hat r$ as the mp\_word product of the two mp\_digits and the addition of the
carry from the previous iteration.  A particularly important observation is that most modern optimizing 
C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that 
is required for the product.  In x86 terms for example, this means using the MUL instruction.

Each digit of the product is stored in turn (line @68,tmpt@) and the carry propagated (line @71,>>@) to the 
next iteration.

\subsection{Faster Multiplication by the ``Comba'' Method}
MARK,COMBA

One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be 
computed and propagated upwards.  This makes the nested loop very sequential and hard to unroll and implement 
in parallel.  The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G. 
Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations.  As an 
interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written 
five years before.

At the heart of the Comba technique is once again the long-hand algorithm.  Except in this case a slight 
twist is placed on how the columns of the result are produced.  In the standard long-hand algorithm rows of products 
are produced then added together to form the final result.  In the baseline algorithm the columns are added together 
after each iteration to get the result instantaneously.  

In the Comba algorithm the columns of the result are produced entirely independently of each other.  That is at 
the $O(n^2)$ level a simple multiplication and addition step is performed.  The carries of the columns are propagated 
after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute 
the product vector $\vec x$ as follows. 

\begin{equation}
\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace
\end{equation}

Where $\vec x_n$ is the $n'th$ column of the output vector.  Consider the following example which computes the vector $\vec x$ for the multiplication
of $576$ and $241$.  

\newpage\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
  \hline &          & 5 & 7 & 6 & First Input\\
  \hline $\times$ & & 2 & 4 & 1 & Second Input\\
\hline            &                        & $1 \cdot 5 = 5$   & $1 \cdot 7 = 7$   & $1 \cdot 6 = 6$ & First pass \\
                  &  $4 \cdot 5 = 20$      & $4 \cdot 7+5=33$  & $4 \cdot 6+7=31$  & 6               & Second pass \\
   $2 \cdot 5 = 10$ &  $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31                & 6             & Third pass \\
\hline 10 & 34 & 45 & 31 & 6 & Final Result \\   
\hline   
\end{tabular}
\end{center}
\end{small}
\caption{Comba Multiplication Diagram}
\end{figure}

At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler.  
Now the columns must be fixed by propagating the carry upwards.  The resultant vector will have one extra dimension over the input vector which is
congruent to adding a leading zero digit.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}







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\end{center}
\end{small}
\caption{Algorithm s\_mp\_mul\_digs}
\end{figure}

\textbf{Algorithm s\_mp\_mul\_digs.}
This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits.  While it may seem
a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent
algorithm.  The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}.
Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the
inputs.

The first thing this algorithm checks for is whether a Comba multiplier can be used instead.   If the minimum digit count of either
input is less than $\delta$, then the Comba method may be used instead.    After the Comba method is ruled out, the baseline algorithm begins.  A
temporary mp\_int variable $t$ is used to hold the intermediate result of the product.  This allows the algorithm to be used to
compute products when either $a = c$ or $b = c$ without overwriting the inputs.

All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output.  The $pb$ variable
is given the count of digits to read from $b$ inside the nested loop.  If $pb \le 1$ then no more output digits can be produced and the algorithm
will exit the loop.  The best way to think of the loops are as a series of $pb \times 1$ multiplications.    That is, in each pass of the
innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$.

For example, consider multiplying $576$ by $241$.  That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best
visualized in the following table.

\begin{figure}[here]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|l|}
\hline   &&          & 5 & 7 & 6 & \\
\hline   $\times$&&  & 2 & 4 & 1 & \\
\hline &&&&&&\\
  &&          & 5 & 7 & 6 & $10^0(1)(576)$ \\
  &2 &   3    & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\
  1 & 3 & 8 & 8 & 1 & 6 &   $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\
\hline
\end{tabular}
\end{center}
\caption{Long-Hand Multiplication Diagram}
\end{figure}

Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate
count.  That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult.

Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable.  The multiplication on that step
is assumed to be a double wide output single precision multiplication.  That is, two single precision variables are multiplied to produce a
double precision result.  The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step
5.4.1 is propagated through the nested loop.  If the carry was not propagated immediately it would overflow the single precision digit
$t_{ix+iy}$ and the result would be lost.

At step 5.5 the nested loop is finished and any carry that was left over should be forwarded.  The carry does not have to be added to the $ix+pb$'th
digit since that digit is assumed to be zero at this point.  However, if $ix + pb \ge digs$ the carry is not set as it would make the result
exceed the precision requested.

EXAM,bn_s_mp_mul_digs.c

First we determine (line @30,if@) if the Comba method can be used first since it's faster.  The conditions for
sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than
\textbf{MP\_WARRAY}.  This new constant is used to control the stack usage in the Comba routines.  By default it is
set to $\delta$ but can be reduced when memory is at a premium.

If we cannot use the Comba method we proceed to setup the baseline routine.  We allocate the the destination mp\_int
$t$ (line @36,init@) to the exact size of the output to avoid further re--allocations.  At this point we now
begin the $O(n^2)$ loop.

This implementation of multiplication has the caveat that it can be trimmed to only produce a variable number of
digits as output.  In each iteration of the outer loop the $pb$ variable is set (line @48,MIN@) to the maximum
number of inner loop iterations.

Inside the inner loop we calculate $\hat r$ as the mp\_word product of the two mp\_digits and the addition of the
carry from the previous iteration.  A particularly important observation is that most modern optimizing
C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that
is required for the product.  In x86 terms for example, this means using the MUL instruction.

Each digit of the product is stored in turn (line @68,tmpt@) and the carry propagated (line @71,>>@) to the
next iteration.

\subsection{Faster Multiplication by the ``Comba'' Method}
MARK,COMBA

One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be
computed and propagated upwards.  This makes the nested loop very sequential and hard to unroll and implement
in parallel.  The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G.
Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations.  As an
interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written
five years before.

At the heart of the Comba technique is once again the long-hand algorithm.  Except in this case a slight
twist is placed on how the columns of the result are produced.  In the standard long-hand algorithm rows of products
are produced then added together to form the final result.  In the baseline algorithm the columns are added together
after each iteration to get the result instantaneously.

In the Comba algorithm the columns of the result are produced entirely independently of each other.  That is at
the $O(n^2)$ level a simple multiplication and addition step is performed.  The carries of the columns are propagated
after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute
the product vector $\vec x$ as follows.

\begin{equation}
\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace
\end{equation}

Where $\vec x_n$ is the $n'th$ column of the output vector.  Consider the following example which computes the vector $\vec x$ for the multiplication
of $576$ and $241$.

\newpage\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
  \hline &          & 5 & 7 & 6 & First Input\\
  \hline $\times$ & & 2 & 4 & 1 & Second Input\\
\hline            &                        & $1 \cdot 5 = 5$   & $1 \cdot 7 = 7$   & $1 \cdot 6 = 6$ & First pass \\
                  &  $4 \cdot 5 = 20$      & $4 \cdot 7+5=33$  & $4 \cdot 6+7=31$  & 6               & Second pass \\
   $2 \cdot 5 = 10$ &  $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31                & 6             & Third pass \\
\hline 10 & 34 & 45 & 31 & 6 & Final Result \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Comba Multiplication Diagram}
\end{figure}

At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler.
Now the columns must be fixed by propagating the carry upwards.  The resultant vector will have one extra dimension over the input vector which is
congruent to adding a leading zero digit.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
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2553
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\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Comba Fixup}
\end{figure}

With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$.  In this case 
$241 \cdot 576$ is in fact $138816$ and the procedure succeeded.  If the algorithm is correct and as will be demonstrated shortly more
efficient than the baseline algorithm why not simply always use this algorithm?

\subsubsection{Column Weight.}
At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output 
independently.  A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix
the carries.  For example, in the multiplication of two three-digit numbers the third column of output will be the sum of
three single precision multiplications.  If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then
an overflow can occur and the carry information will be lost.  For any $m$ and $n$ digit inputs the maximum weight of any column is 
min$(m, n)$ which is fairly obvious.

The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used.  Recall
from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision.  Given these
two quantities we must not violate the following

\begin{equation}
k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha}
\end{equation}

Which reduces to 

\begin{equation}
k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha}
\end{equation}

Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit.  By further re-arrangement of the equation the final solution is
found.

\begin{equation}
k  < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}}
\end{equation}

The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$.  In this configuration 
the smaller input may not have more than $256$ digits if the Comba method is to be used.  This is quite satisfactory for most applications since 
$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\
\textbf{Input}.   mp\_int $a$, mp\_int $b$ and an integer $digs$ \\







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\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Comba Fixup}
\end{figure}

With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$.  In this case
$241 \cdot 576$ is in fact $138816$ and the procedure succeeded.  If the algorithm is correct and as will be demonstrated shortly more
efficient than the baseline algorithm why not simply always use this algorithm?

\subsubsection{Column Weight.}
At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output
independently.  A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix
the carries.  For example, in the multiplication of two three-digit numbers the third column of output will be the sum of
three single precision multiplications.  If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then
an overflow can occur and the carry information will be lost.  For any $m$ and $n$ digit inputs the maximum weight of any column is
min$(m, n)$ which is fairly obvious.

The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used.  Recall
from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision.  Given these
two quantities we must not violate the following

\begin{equation}
k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha}
\end{equation}

Which reduces to

\begin{equation}
k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha}
\end{equation}

Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit.  By further re-arrangement of the equation the final solution is
found.

\begin{equation}
k  < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}}
\end{equation}

The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$.  In this configuration
the smaller input may not have more than $256$ digits if the Comba method is to be used.  This is quite satisfactory for most applications since
$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\
\textbf{Input}.   mp\_int $a$, mp\_int $b$ and an integer $digs$ \\
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\caption{Algorithm fast\_s\_mp\_mul\_digs}
\label{fig:COMBAMULT}
\end{figure}

\textbf{Algorithm fast\_s\_mp\_mul\_digs.}
This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision.

The outer loop of this algorithm is more complicated than that of the baseline multiplier.  This is because on the inside of the 
loop we want to produce one column per pass.  This allows the accumulator $\_ \hat W$ to be placed in CPU registers and
reduce the memory bandwidth to two \textbf{mp\_digit} reads per iteration.

The $ty$ variable is set to the minimum count of $ix$ or the number of digits in $b$.  That way if $a$ has more digits than
$b$ this will be limited to $b.used - 1$.  The $tx$ variable is set to the to the distance past $b.used$ the variable
$ix$ is.  This is used for the immediately subsequent statement where we find $iy$.  

The variable $iy$ is the minimum digits we can read from either $a$ or $b$ before running out.  Computing one column at a time
means we have to scan one integer upwards and the other downwards.  $a$ starts at $tx$ and $b$ starts at $ty$.  In each
pass we are producing the $ix$'th output column and we note that $tx + ty = ix$.  As we move $tx$ upwards we have to 
move $ty$ downards so the equality remains valid.  The $iy$ variable is the number of iterations until 
$tx \ge a.used$ or $ty < 0$ occurs.

After every inner pass we store the lower half of the accumulator into $W_{ix}$ and then propagate the carry of the accumulator
into the next round by dividing $\_ \hat W$ by $\beta$.

To measure the benefits of the Comba method over the baseline method consider the number of operations that are required.  If the 
cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require 
$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers.  The Comba method requires only $O(pn^2 + qn)$ time, however in practice, 
the speed increase is actually much more.  With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply
and addition operations in the nested loop in parallel.  

EXAM,bn_fast_s_mp_mul_digs.c

As per the pseudo--code we first calculate $pa$ (line @47,MIN@) as the number of digits to output.  Next we begin the outer loop
to produce the individual columns of the product.  We use the two aliases $tmpx$ and $tmpy$ (lines @61,tmpx@, @62,tmpy@) to point
inside the two multiplicands quickly.  

The inner loop (lines @70,for@ to @72,}@) of this implementation is where the tradeoff come into play.  Originally this comba 
implementation was ``row--major'' which means it adds to each of the columns in each pass.  After the outer loop it would then fix 
the carries.  This was very fast except it had an annoying drawback.  You had to read a mp\_word and two mp\_digits and write 
one mp\_word per iteration.  On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth 
is very high and it can keep the ALU fed with data.  It did, however, matter on older and embedded cpus where cache is often 
slower and also often doesn't exist.  This new algorithm only performs two reads per iteration under the assumption that the 
compiler has aliased $\_ \hat W$ to a CPU register.

After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines @75,W[ix]@, @78,>>@) to forward it as 
a carry for the next pass.  After the outer loop we use the final carry (line @82,W[ix]@) as the last digit of the product.  

\subsection{Polynomial Basis Multiplication}
To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication.  In the following algorithms
the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and  
$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required.  In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree.
 
The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$.  The coefficients $w_i$ will
directly yield the desired product when $\beta$ is substituted for $x$.  The direct solution to solve for the $2n + 1$ coefficients
requires $O(n^2)$ time and would in practice be slower than the Comba technique.

However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown 
coefficients.   This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with 
Gaussian elimination.  This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in 
effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$.  

The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible.  However, since 
$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place.  The benefit of this technique stems from the 
fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively.  As a result finding the $2n + 1$ relations required 
by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs.

When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$.  The $\zeta_0$ term
is simply the product $W(0) = w_0 = a_0 \cdot b_0$.  The $\zeta_1$ term is the product 
$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$.  The third point $\zeta_{\infty}$ is less obvious but rather
simple to explain.  The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication.  
The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$.  Note that the 
points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly.

If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points} 
$\left (2^q \right )^{2n}  \cdot \zeta_{2^{-q}}$ for small values of $q$.  The term ``mirror point'' stems from the fact that 
$\left (2^q \right )^{2n}  \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$.  For
example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror.

\begin{eqnarray}
\zeta_{2}                  = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\
16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0)
\end{eqnarray}

Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts.  For example, when $n = 2$ the
polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$.  This technique of polynomial representation is known as Horner's method.  

As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications.  Each multiplication is of 
multiplicands that have $n$ times fewer digits than the inputs.  The asymptotic running time of this algorithm is 
$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}).  Figure~\ref{fig:exponent}
summarizes the exponents for various values of $n$.

\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline \textbf{Split into $n$ Parts} & \textbf{Exponent}  & \textbf{Notes}\\







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\caption{Algorithm fast\_s\_mp\_mul\_digs}
\label{fig:COMBAMULT}
\end{figure}

\textbf{Algorithm fast\_s\_mp\_mul\_digs.}
This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision.

The outer loop of this algorithm is more complicated than that of the baseline multiplier.  This is because on the inside of the
loop we want to produce one column per pass.  This allows the accumulator $\_ \hat W$ to be placed in CPU registers and
reduce the memory bandwidth to two \textbf{mp\_digit} reads per iteration.

The $ty$ variable is set to the minimum count of $ix$ or the number of digits in $b$.  That way if $a$ has more digits than
$b$ this will be limited to $b.used - 1$.  The $tx$ variable is set to the to the distance past $b.used$ the variable
$ix$ is.  This is used for the immediately subsequent statement where we find $iy$.

The variable $iy$ is the minimum digits we can read from either $a$ or $b$ before running out.  Computing one column at a time
means we have to scan one integer upwards and the other downwards.  $a$ starts at $tx$ and $b$ starts at $ty$.  In each
pass we are producing the $ix$'th output column and we note that $tx + ty = ix$.  As we move $tx$ upwards we have to
move $ty$ downards so the equality remains valid.  The $iy$ variable is the number of iterations until
$tx \ge a.used$ or $ty < 0$ occurs.

After every inner pass we store the lower half of the accumulator into $W_{ix}$ and then propagate the carry of the accumulator
into the next round by dividing $\_ \hat W$ by $\beta$.

To measure the benefits of the Comba method over the baseline method consider the number of operations that are required.  If the
cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require
$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers.  The Comba method requires only $O(pn^2 + qn)$ time, however in practice,
the speed increase is actually much more.  With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply
and addition operations in the nested loop in parallel.

EXAM,bn_fast_s_mp_mul_digs.c

As per the pseudo--code we first calculate $pa$ (line @47,MIN@) as the number of digits to output.  Next we begin the outer loop
to produce the individual columns of the product.  We use the two aliases $tmpx$ and $tmpy$ (lines @61,tmpx@, @62,tmpy@) to point
inside the two multiplicands quickly.

The inner loop (lines @70,for@ to @72,}@) of this implementation is where the tradeoff come into play.  Originally this comba
implementation was ``row--major'' which means it adds to each of the columns in each pass.  After the outer loop it would then fix
the carries.  This was very fast except it had an annoying drawback.  You had to read a mp\_word and two mp\_digits and write
one mp\_word per iteration.  On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth
is very high and it can keep the ALU fed with data.  It did, however, matter on older and embedded cpus where cache is often
slower and also often doesn't exist.  This new algorithm only performs two reads per iteration under the assumption that the
compiler has aliased $\_ \hat W$ to a CPU register.

After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines @75,W[ix]@, @78,>>@) to forward it as
a carry for the next pass.  After the outer loop we use the final carry (line @82,W[ix]@) as the last digit of the product.

\subsection{Polynomial Basis Multiplication}
To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication.  In the following algorithms
the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and
$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required.  In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree.

The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$.  The coefficients $w_i$ will
directly yield the desired product when $\beta$ is substituted for $x$.  The direct solution to solve for the $2n + 1$ coefficients
requires $O(n^2)$ time and would in practice be slower than the Comba technique.

However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown
coefficients.   This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with
Gaussian elimination.  This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in
effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$.

The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible.  However, since
$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place.  The benefit of this technique stems from the
fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively.  As a result finding the $2n + 1$ relations required
by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs.

When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$.  The $\zeta_0$ term
is simply the product $W(0) = w_0 = a_0 \cdot b_0$.  The $\zeta_1$ term is the product
$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$.  The third point $\zeta_{\infty}$ is less obvious but rather
simple to explain.  The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication.
The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$.  Note that the
points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly.

If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points}
$\left (2^q \right )^{2n}  \cdot \zeta_{2^{-q}}$ for small values of $q$.  The term ``mirror point'' stems from the fact that
$\left (2^q \right )^{2n}  \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$.  For
example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror.

\begin{eqnarray}
\zeta_{2}                  = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\
16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0)
\end{eqnarray}

Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts.  For example, when $n = 2$ the
polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$.  This technique of polynomial representation is known as Horner's method.

As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications.  Each multiplication is of
multiplicands that have $n$ times fewer digits than the inputs.  The asymptotic running time of this algorithm is
$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}).  Figure~\ref{fig:exponent}
summarizes the exponents for various values of $n$.

\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline \textbf{Split into $n$ Parts} & \textbf{Exponent}  & \textbf{Notes}\\
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\end{center}
\caption{Asymptotic Running Time of Polynomial Basis Multiplication}
\label{fig:exponent}
\end{figure}

At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$.  However, the overhead
of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large
numbers.  

\subsubsection{Cutoff Point}
The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach.  However, 
the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved.  This makes the
polynomial basis approach more costly to use with small inputs.

Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}).  There exists a 
point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and 
when $m > y$ the Comba methods are slower than the polynomial basis algorithms.  

The exact location of $y$ depends on several key architectural elements of the computer platform in question.

\begin{enumerate}
\item  The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc.  For example
on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$.  The higher the ratio in favour of multiplication the lower
the cutoff point $y$ will be.  

\item  The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is.  Generally speaking as the number of splits
grows the complexity grows substantially.  Ideally solving the system will only involve addition, subtraction and shifting of integers.  This
directly reflects on the ratio previous mentioned.

\item  To a lesser extent memory bandwidth and function call overheads.  Provided the values are in the processor cache this is less of an
influence over the cutoff point.

\end{enumerate}

A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met.  For example, if the point
is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster.  Finding the cutoff points is fairly simple when
a high resolution timer is available.  

\subsection{Karatsuba Multiplication}
Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for
general purpose multiplication.  Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with 
light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent.

\begin{equation}
f(x) \cdot g(x) = acx^2 + ((a + b)(c + d) - (ac + bd))x + bd
\end{equation}

Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product.  Applying
this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique.  It turns 
out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points 
$\zeta_0$, $\zeta_{\infty}$ and $\zeta_{1}$.  Consider the resultant system of equations.

\begin{center}
\begin{tabular}{rcrcrcrc}
$\zeta_{0}$ &      $=$ &  &  &  & & $w_0$ \\
$\zeta_{1}$ &      $=$ & $w_2$ & $+$ & $w_1$ & $+$ & $w_0$ \\
$\zeta_{\infty}$ & $=$ & $w_2$ &  & &  & \\
\end{tabular}
\end{center}

By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for.  The simplicity
of this system of equations has made Karatsuba fairly popular.  In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.}
making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\
\textbf{Input}.   mp\_int $a$ and mp\_int $b$ \\







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\end{center}
\caption{Asymptotic Running Time of Polynomial Basis Multiplication}
\label{fig:exponent}
\end{figure}

At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$.  However, the overhead
of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large
numbers.

\subsubsection{Cutoff Point}
The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach.  However,
the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved.  This makes the
polynomial basis approach more costly to use with small inputs.

Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}).  There exists a
point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and
when $m > y$ the Comba methods are slower than the polynomial basis algorithms.

The exact location of $y$ depends on several key architectural elements of the computer platform in question.

\begin{enumerate}
\item  The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc.  For example
on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$.  The higher the ratio in favour of multiplication the lower
the cutoff point $y$ will be.

\item  The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is.  Generally speaking as the number of splits
grows the complexity grows substantially.  Ideally solving the system will only involve addition, subtraction and shifting of integers.  This
directly reflects on the ratio previous mentioned.

\item  To a lesser extent memory bandwidth and function call overheads.  Provided the values are in the processor cache this is less of an
influence over the cutoff point.

\end{enumerate}

A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met.  For example, if the point
is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster.  Finding the cutoff points is fairly simple when
a high resolution timer is available.

\subsection{Karatsuba Multiplication}
Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for
general purpose multiplication.  Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with
light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent.

\begin{equation}
f(x) \cdot g(x) = acx^2 + ((a + b)(c + d) - (ac + bd))x + bd
\end{equation}

Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product.  Applying
this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique.  It turns
out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points
$\zeta_0$, $\zeta_{\infty}$ and $\zeta_{1}$.  Consider the resultant system of equations.

\begin{center}
\begin{tabular}{rcrcrcrc}
$\zeta_{0}$ &      $=$ &  &  &  & & $w_0$ \\
$\zeta_{1}$ &      $=$ & $w_2$ & $+$ & $w_1$ & $+$ & $w_0$ \\
$\zeta_{\infty}$ & $=$ & $w_2$ &  & &  & \\
\end{tabular}
\end{center}

By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for.  The simplicity
of this system of equations has made Karatsuba fairly popular.  In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.}
making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\
\textbf{Input}.   mp\_int $a$ and mp\_int $b$ \\
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Calculate the final product. \\
15.  $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\
16.  $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\
17.  $t1 \leftarrow x0y0 + t1$ \\
18.  $c \leftarrow t1 + x1y1$ \\
19.  Clear all of the temporary variables. \\
20.  Return(\textit{MP\_OKAY}).\\
\hline 
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_karatsuba\_mul}
\end{figure}

\textbf{Algorithm mp\_karatsuba\_mul.}
This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm.  It is loosely based on the description
from Knuth \cite[pp. 294-295]{TAOCPV2}.  

\index{radix point}
In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen.  The radix point chosen must
be used for both of the inputs meaning that it must be smaller than the smallest input.  Step 3 chooses the radix point $B$ as half of the 
smallest input \textbf{used} count.  After the radix point is chosen the inputs are split into lower and upper halves.  Step 4 and 5 
compute the lower halves.  Step 6 and 7 computer the upper halves.  

After the halves have been computed the three intermediate half-size products must be computed.  Step 8 and 9 compute the trivial products
$x0 \cdot y0$ and $x1 \cdot y1$.  The mp\_int $x0$ is used as a temporary variable after $x1 + x0$ has been computed.  By using $x0$ instead
of an additional temporary variable, the algorithm can avoid an addition memory allocation operation.

The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations.

EXAM,bn_mp_karatsuba_mul.c

The new coding element in this routine, not  seen in previous routines, is the usage of goto statements.  The conventional
wisdom is that goto statements should be avoided.  This is generally true, however when every single function call can fail, it makes sense
to handle error recovery with a single piece of code.  Lines @61,if@ to @75,if@ handle initializing all of the temporary variables 
required.  Note how each of the if statements goes to a different label in case of failure.  This allows the routine to correctly free only
the temporaries that have been successfully allocated so far.

The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large.  This saves the 
additional reallocation that would have been necessary.  Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective
number of digits for the next section of code.

The first algebraic portion of the algorithm is to split the two inputs into their halves.  However, instead of using mp\_mod\_2d and mp\_rshd
to extract the halves, the respective code has been placed inline within the body of the function.  To initialize the halves, the \textbf{used} and 
\textbf{sign} members are copied first.  The first for loop on line @98,for@ copies the lower halves.  Since they are both the same magnitude it 
is simpler to calculate both lower halves in a single loop.  The for loop on lines @104,for@ and @109,for@ calculate the upper halves $x1$ and 
$y1$ respectively.

By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs.

When line @152,err@ is reached, the algorithm has completed succesfully.  The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that
the same code that handles errors can be used to clear the temporary variables and return.  

\subsection{Toom-Cook $3$-Way Multiplication}
Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points  are 
chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce.  Here, the points $\zeta_{0}$, 
$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients 
of the $W(x)$.

With the five relations that Toom-Cook specifies, the following system of equations is formed.

\begin{center}
\begin{tabular}{rcrcrcrcrcr}
$\zeta_0$                    & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$  \\
$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$  \\
$\zeta_1$                    & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$  \\
$\zeta_2$                    & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$  \\
$\zeta_{\infty}$             & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$  \\
\end{tabular}
\end{center}

A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power
of two, two divisions by three and one multiplication by three.  All of these $19$ sub-operations require less than quadratic time, meaning that
the algorithm can be faster than a baseline multiplication.  However, the greater complexity of this algorithm places the cutoff point
(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_toom\_mul}. \\
\textbf{Input}.   mp\_int $a$ and mp\_int $b$ \\







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Calculate the final product. \\
15.  $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\
16.  $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\
17.  $t1 \leftarrow x0y0 + t1$ \\
18.  $c \leftarrow t1 + x1y1$ \\
19.  Clear all of the temporary variables. \\
20.  Return(\textit{MP\_OKAY}).\\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_karatsuba\_mul}
\end{figure}

\textbf{Algorithm mp\_karatsuba\_mul.}
This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm.  It is loosely based on the description
from Knuth \cite[pp. 294-295]{TAOCPV2}.

\index{radix point}
In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen.  The radix point chosen must
be used for both of the inputs meaning that it must be smaller than the smallest input.  Step 3 chooses the radix point $B$ as half of the
smallest input \textbf{used} count.  After the radix point is chosen the inputs are split into lower and upper halves.  Step 4 and 5
compute the lower halves.  Step 6 and 7 computer the upper halves.

After the halves have been computed the three intermediate half-size products must be computed.  Step 8 and 9 compute the trivial products
$x0 \cdot y0$ and $x1 \cdot y1$.  The mp\_int $x0$ is used as a temporary variable after $x1 + x0$ has been computed.  By using $x0$ instead
of an additional temporary variable, the algorithm can avoid an addition memory allocation operation.

The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations.

EXAM,bn_mp_karatsuba_mul.c

The new coding element in this routine, not  seen in previous routines, is the usage of goto statements.  The conventional
wisdom is that goto statements should be avoided.  This is generally true, however when every single function call can fail, it makes sense
to handle error recovery with a single piece of code.  Lines @61,if@ to @75,if@ handle initializing all of the temporary variables
required.  Note how each of the if statements goes to a different label in case of failure.  This allows the routine to correctly free only
the temporaries that have been successfully allocated so far.

The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large.  This saves the
additional reallocation that would have been necessary.  Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective
number of digits for the next section of code.

The first algebraic portion of the algorithm is to split the two inputs into their halves.  However, instead of using mp\_mod\_2d and mp\_rshd
to extract the halves, the respective code has been placed inline within the body of the function.  To initialize the halves, the \textbf{used} and
\textbf{sign} members are copied first.  The first for loop on line @98,for@ copies the lower halves.  Since they are both the same magnitude it
is simpler to calculate both lower halves in a single loop.  The for loop on lines @104,for@ and @109,for@ calculate the upper halves $x1$ and
$y1$ respectively.

By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs.

When line @152,err@ is reached, the algorithm has completed succesfully.  The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that
the same code that handles errors can be used to clear the temporary variables and return.

\subsection{Toom-Cook $3$-Way Multiplication}
Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points  are
chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce.  Here, the points $\zeta_{0}$,
$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients
of the $W(x)$.

With the five relations that Toom-Cook specifies, the following system of equations is formed.

\begin{center}
\begin{tabular}{rcrcrcrcrcr}
$\zeta_0$                    & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$  \\
$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$  \\
$\zeta_1$                    & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$  \\
$\zeta_2$                    & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$  \\
$\zeta_{\infty}$             & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$  \\
\end{tabular}
\end{center}

A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power
of two, two divisions by three and one multiplication by three.  All of these $19$ sub-operations require less than quadratic time, meaning that
the algorithm can be faster than a baseline multiplication.  However, the greater complexity of this algorithm places the cutoff point
(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_toom\_mul}. \\
\textbf{Input}.   mp\_int $a$ and mp\_int $b$ \\
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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_toom\_mul (continued)}
\end{figure}

\textbf{Algorithm mp\_toom\_mul.}
This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach.  Compared to the Karatsuba multiplication, this 
algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead.  In this
description, several statements have been compounded to save space.  The intention is that the statements are executed from left to right across
any given step.

The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively.  From these smaller
integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required.

The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively.  The relation $w_1, w_2$ and $w_3$ correspond
to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively.  These are found using logical shifts to independently find
$f(y)$ and $g(y)$ which significantly speeds up the algorithm.

After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients 
$w_1, w_2$ and $w_3$ to be isolated.  The steps 18 through 25 perform the system reduction required as previously described.  Each step of
the reduction represents the comparable matrix operation that would be performed had this been performed by pencil.  For example, step 18 indicates
that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$.  

Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known.  By substituting $\beta^{k}$ for $x$, the integer 
result $a \cdot b$ is produced.

EXAM,bn_mp_toom_mul.c

The first obvious thing to note is that this algorithm is complicated.  The complexity is worth it if you are multiplying very 
large numbers.  For example, a 10,000 digit multiplication takes approximaly 99,282,205 fewer single precision multiplications with
Toom--Cook than a Comba or baseline approach (this is a savings of more than 99$\%$).  For most ``crypto'' sized numbers this
algorithm is not practical as Karatsuba has a much lower cutoff point.

First we split $a$ and $b$ into three roughly equal portions.  This has been accomplished (lines @40,mod@ to @69,rshd@) with 
combinations of mp\_rshd() and mp\_mod\_2d() function calls.  At this point $a = a2 \cdot \beta^2 + a1 \cdot \beta + a0$ and similiarly
for $b$.  

Next we compute the five points $w0, w1, w2, w3$ and $w4$.  Recall that $w0$ and $w4$ can be computed directly from the portions so
we get those out of the way first (lines @72,mul@ and @77,mul@).  Next we compute $w1, w2$ and $w3$ using Horners method.

After this point we solve for the actual values of $w1, w2$ and $w3$ by reducing the $5 \times 5$ system which is relatively
straight forward.  

\subsection{Signed Multiplication}
Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required.  So far all
of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mul}. \\
\textbf{Input}.   mp\_int $a$ and mp\_int $b$ \\







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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_toom\_mul (continued)}
\end{figure}

\textbf{Algorithm mp\_toom\_mul.}
This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach.  Compared to the Karatsuba multiplication, this
algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead.  In this
description, several statements have been compounded to save space.  The intention is that the statements are executed from left to right across
any given step.

The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively.  From these smaller
integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required.

The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively.  The relation $w_1, w_2$ and $w_3$ correspond
to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively.  These are found using logical shifts to independently find
$f(y)$ and $g(y)$ which significantly speeds up the algorithm.

After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients
$w_1, w_2$ and $w_3$ to be isolated.  The steps 18 through 25 perform the system reduction required as previously described.  Each step of
the reduction represents the comparable matrix operation that would be performed had this been performed by pencil.  For example, step 18 indicates
that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$.

Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known.  By substituting $\beta^{k}$ for $x$, the integer
result $a \cdot b$ is produced.

EXAM,bn_mp_toom_mul.c

The first obvious thing to note is that this algorithm is complicated.  The complexity is worth it if you are multiplying very
large numbers.  For example, a 10,000 digit multiplication takes approximaly 99,282,205 fewer single precision multiplications with
Toom--Cook than a Comba or baseline approach (this is a savings of more than 99$\%$).  For most ``crypto'' sized numbers this
algorithm is not practical as Karatsuba has a much lower cutoff point.

First we split $a$ and $b$ into three roughly equal portions.  This has been accomplished (lines @40,mod@ to @69,rshd@) with
combinations of mp\_rshd() and mp\_mod\_2d() function calls.  At this point $a = a2 \cdot \beta^2 + a1 \cdot \beta + a0$ and similiarly
for $b$.

Next we compute the five points $w0, w1, w2, w3$ and $w4$.  Recall that $w0$ and $w4$ can be computed directly from the portions so
we get those out of the way first (lines @72,mul@ and @77,mul@).  Next we compute $w1, w2$ and $w3$ using Horners method.

After this point we solve for the actual values of $w1, w2$ and $w3$ by reducing the $5 \times 5$ system which is relatively
straight forward.

\subsection{Signed Multiplication}
Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required.  So far all
of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mul}. \\
\textbf{Input}.   mp\_int $a$ and mp\_int $b$ \\
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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul}
\end{figure}

\textbf{Algorithm mp\_mul.}
This algorithm performs the signed multiplication of two inputs.  It will make use of any of the three unsigned multiplication algorithms 
available when the input is of appropriate size.  The \textbf{sign} of the result is not set until the end of the algorithm since algorithm
s\_mp\_mul\_digs will clear it.  

EXAM,bn_mp_mul.c

The implementation is rather simplistic and is not particularly noteworthy.  Line @22,?@ computes the sign of the result using the ``?'' 
operator from the C programming language.  Line @37,<<@ computes $\delta$ using the fact that $1 << k$ is equal to $2^k$.  

\section{Squaring}
\label{sec:basesquare}

Squaring is a special case of multiplication where both multiplicands are equal.  At first it may seem like there is no significant optimization
available but in fact there is.  Consider the multiplication of $576$ against $241$.  In total there will be nine single precision multiplications
performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot  6$, $2 \cdot 7$ and $2 \cdot 5$.  Now consider 
the multiplication of $123$ against $123$.  The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, 
$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$.  On closer inspection some of the products are equivalent.  For example, $3 \cdot 2 = 2 \cdot 3$ 
and $3 \cdot 1 = 1 \cdot 3$. 

For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$
required for multiplication.  The following diagram gives an example of the operations required.

\begin{figure}[here]
\begin{center}
\begin{tabular}{ccccc|c}
&&1&2&3&\\
$\times$ &&1&2&3&\\
\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\
       & $2 \cdot 1$  & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\
         $1 \cdot 1$  & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\
\end{tabular}
\end{center}
\caption{Squaring Optimization Diagram}
\end{figure}

MARK,SQUARE
Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious.  For the purposes of this discussion let $x$
represent the number being squared.  The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it.  

The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product.  Every non-square term of a column will
appear twice hence the name ``double product''.  Every odd column is made up entirely of double products.  In fact every column is made up of double 
products and at most one square (\textit{see the exercise section}).  

The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row, 
occurs at column $2k + 1$.  For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. 
Column two of row one is a square and column three is the first unique column.

\subsection{The Baseline Squaring Algorithm}
The baseline squaring algorithm is meant to be a catch-all squaring algorithm.  It will handle any of the input sizes that the faster routines
will not handle.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_sqr}. \\
\textbf{Input}.   mp\_int $a$ \\







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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul}
\end{figure}

\textbf{Algorithm mp\_mul.}
This algorithm performs the signed multiplication of two inputs.  It will make use of any of the three unsigned multiplication algorithms
available when the input is of appropriate size.  The \textbf{sign} of the result is not set until the end of the algorithm since algorithm
s\_mp\_mul\_digs will clear it.

EXAM,bn_mp_mul.c

The implementation is rather simplistic and is not particularly noteworthy.  Line @22,?@ computes the sign of the result using the ``?''
operator from the C programming language.  Line @37,<<@ computes $\delta$ using the fact that $1 << k$ is equal to $2^k$.

\section{Squaring}
\label{sec:basesquare}

Squaring is a special case of multiplication where both multiplicands are equal.  At first it may seem like there is no significant optimization
available but in fact there is.  Consider the multiplication of $576$ against $241$.  In total there will be nine single precision multiplications
performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot  6$, $2 \cdot 7$ and $2 \cdot 5$.  Now consider
the multiplication of $123$ against $123$.  The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$,
$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$.  On closer inspection some of the products are equivalent.  For example, $3 \cdot 2 = 2 \cdot 3$
and $3 \cdot 1 = 1 \cdot 3$.

For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$
required for multiplication.  The following diagram gives an example of the operations required.

\begin{figure}[here]
\begin{center}
\begin{tabular}{ccccc|c}
&&1&2&3&\\
$\times$ &&1&2&3&\\
\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\
       & $2 \cdot 1$  & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\
         $1 \cdot 1$  & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\
\end{tabular}
\end{center}
\caption{Squaring Optimization Diagram}
\end{figure}

MARK,SQUARE
Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious.  For the purposes of this discussion let $x$
represent the number being squared.  The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it.

The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product.  Every non-square term of a column will
appear twice hence the name ``double product''.  Every odd column is made up entirely of double products.  In fact every column is made up of double
products and at most one square (\textit{see the exercise section}).

The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row,
occurs at column $2k + 1$.  For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero.
Column two of row one is a square and column three is the first unique column.

\subsection{The Baseline Squaring Algorithm}
The baseline squaring algorithm is meant to be a catch-all squaring algorithm.  It will handle any of the input sizes that the faster routines
will not handle.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_sqr}. \\
\textbf{Input}.   mp\_int $a$ \\
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\end{center}
\end{small}
\caption{Algorithm s\_mp\_sqr}
\end{figure}

\textbf{Algorithm s\_mp\_sqr.}
This algorithm computes the square of an input using the three observations on squaring.  It is based fairly faithfully on  algorithm 14.16 of HAC
\cite[pp.596-597]{HAC}.  Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring.  This allows the 
destination mp\_int to be the same as the source mp\_int.

The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while
the inner loop computes the columns of the partial result.  Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate
the carry and compute the double products.  

The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this
very algorithm.  The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that
when it is multiplied by two, it can be properly represented by a mp\_word.

Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial 
results calculated so far.  This involves expensive carry propagation which will be eliminated in the next algorithm.  

EXAM,bn_s_mp_sqr.c

Inside the outer loop (line @32,for@) the square term is calculated on line @35,r =@.  The carry (line @42,>>@) has been
extracted from the mp\_word accumulator using a right shift.  Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized 
(lines @45,tmpx@ and @48,tmpt@) to simplify the inner loop.  The doubling is performed using two
additions (line @57,r + r@) since it is usually faster than shifting, if not at least as fast.  

The important observation is that the inner loop does not begin at $iy = 0$ like for multiplication.  As such the inner loops
get progressively shorter as the algorithm proceeds.  This is what leads to the savings compared to using a multiplication to
square a number. 

\subsection{Faster Squaring by the ``Comba'' Method}
A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop.  Squaring has an additional
drawback that it must double the product inside the inner loop as well.  As for multiplication, the Comba technique can be used to eliminate these
performance hazards.

The first obvious solution is to make an array of mp\_words which will hold all of the columns.  This will indeed eliminate all of the carry
propagation operations from the inner loop.  However, the inner product must still be doubled $O(n^2)$ times.  The solution stems from the simple fact
that $2a + 2b + 2c = 2(a + b + c)$.  That is the sum of all of the double products is equal to double the sum of all the products.  For example,
$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$.  

However, we cannot simply double all of the columns, since the squares appear only once per row.  The most practical solution is to have two 
mp\_word arrays.  One array will hold the squares and the other array will hold the double products.  With both arrays the doubling and 
carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level.  In this case, we have an even simpler solution in mind.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\







|




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\end{center}
\end{small}
\caption{Algorithm s\_mp\_sqr}
\end{figure}

\textbf{Algorithm s\_mp\_sqr.}
This algorithm computes the square of an input using the three observations on squaring.  It is based fairly faithfully on  algorithm 14.16 of HAC
\cite[pp.596-597]{HAC}.  Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring.  This allows the
destination mp\_int to be the same as the source mp\_int.

The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while
the inner loop computes the columns of the partial result.  Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate
the carry and compute the double products.

The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this
very algorithm.  The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that
when it is multiplied by two, it can be properly represented by a mp\_word.

Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial
results calculated so far.  This involves expensive carry propagation which will be eliminated in the next algorithm.

EXAM,bn_s_mp_sqr.c

Inside the outer loop (line @32,for@) the square term is calculated on line @35,r =@.  The carry (line @42,>>@) has been
extracted from the mp\_word accumulator using a right shift.  Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized
(lines @45,tmpx@ and @48,tmpt@) to simplify the inner loop.  The doubling is performed using two
additions (line @57,r + r@) since it is usually faster than shifting, if not at least as fast.

The important observation is that the inner loop does not begin at $iy = 0$ like for multiplication.  As such the inner loops
get progressively shorter as the algorithm proceeds.  This is what leads to the savings compared to using a multiplication to
square a number.

\subsection{Faster Squaring by the ``Comba'' Method}
A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop.  Squaring has an additional
drawback that it must double the product inside the inner loop as well.  As for multiplication, the Comba technique can be used to eliminate these
performance hazards.

The first obvious solution is to make an array of mp\_words which will hold all of the columns.  This will indeed eliminate all of the carry
propagation operations from the inner loop.  However, the inner product must still be doubled $O(n^2)$ times.  The solution stems from the simple fact
that $2a + 2b + 2c = 2(a + b + c)$.  That is the sum of all of the double products is equal to double the sum of all the products.  For example,
$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$.

However, we cannot simply double all of the columns, since the squares appear only once per row.  The most practical solution is to have two
mp\_word arrays.  One array will hold the squares and the other array will hold the double products.  With both arrays the doubling and
carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level.  In this case, we have an even simpler solution in mind.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\
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6.  $oldused \leftarrow b.used$ \\
7.  $b.used \leftarrow 2 \cdot a.used$ \\
8.  for $ix$ from $0$ to $pa - 1$ do \\
\hspace{3mm}8.1  $b_{ix} \leftarrow W_{ix}$ \\
9.  for $ix$ from $pa$ to $oldused - 1$ do \\
\hspace{3mm}9.1  $b_{ix} \leftarrow 0$ \\
10.  Clamp excess digits from $b$.  (\textit{mp\_clamp}) \\
11.  Return(\textit{MP\_OKAY}). \\ 
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm fast\_s\_mp\_sqr}
\end{figure}

\textbf{Algorithm fast\_s\_mp\_sqr.}
This algorithm computes the square of an input using the Comba technique.  It is designed to be a replacement for algorithm 
s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$.  
This algorithm is very similar to the Comba multiplier except with a few key differences we shall make note of.

First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively.  This is because the inner loop
products are to be doubled.  If we had added the previous carry in we would be doubling too much.  Next we perform an
addition MIN condition on $iy$ (step 5.5) to prevent overlapping digits.  For example, $a_3 \cdot a_5$ is equal
$a_5 \cdot a_3$.  Whereas in the multiplication case we would have $5 < a.used$ and $3 \ge 0$ is maintained since we double the sum
of the products just outside the inner loop we have to avoid doing this.  This is also a good thing since we perform
fewer multiplications and the routine ends up being faster.

Finally the last difference is the addition of the ``square'' term outside the inner loop (step 5.8).  We add in the square
only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rfloor$ position.

EXAM,bn_fast_s_mp_sqr.c

This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for 
the special case of squaring.  

\subsection{Polynomial Basis Squaring}
The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring.  The minor exception
is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$.  Instead of performing $2n + 1$
multiplications to find the $\zeta$ relations, squaring operations are performed instead.  

\subsection{Karatsuba Squaring}
Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square.  
Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial.  The Karatsuba equation can be modified to square a 
number with the following equation.

\begin{equation}
h(x) = a^2x^2 + \left ((a + b)^2 - (a^2 + b^2) \right )x + b^2
\end{equation}

Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a + b)^2$.  As in 
Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of 
$O \left ( n^{lg(3)} \right )$.

If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm 
instead?  The answer to this arises from the cutoff point for squaring.  As in multiplication there exists a cutoff point, at which the 
time required for a Comba based squaring and a Karatsuba based squaring meet.  Due to the overhead inherent in the Karatsuba method, the cutoff 
point is fairly high.  For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits.  

Consider squaring a 200 digit number with this technique.  It will be split into two 100 digit halves which are subsequently squared.  
The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm.  If Karatsuba multiplication
were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\
\textbf{Input}.   mp\_int $a$ \\







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6.  $oldused \leftarrow b.used$ \\
7.  $b.used \leftarrow 2 \cdot a.used$ \\
8.  for $ix$ from $0$ to $pa - 1$ do \\
\hspace{3mm}8.1  $b_{ix} \leftarrow W_{ix}$ \\
9.  for $ix$ from $pa$ to $oldused - 1$ do \\
\hspace{3mm}9.1  $b_{ix} \leftarrow 0$ \\
10.  Clamp excess digits from $b$.  (\textit{mp\_clamp}) \\
11.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm fast\_s\_mp\_sqr}
\end{figure}

\textbf{Algorithm fast\_s\_mp\_sqr.}
This algorithm computes the square of an input using the Comba technique.  It is designed to be a replacement for algorithm
s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$.
This algorithm is very similar to the Comba multiplier except with a few key differences we shall make note of.

First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively.  This is because the inner loop
products are to be doubled.  If we had added the previous carry in we would be doubling too much.  Next we perform an
addition MIN condition on $iy$ (step 5.5) to prevent overlapping digits.  For example, $a_3 \cdot a_5$ is equal
$a_5 \cdot a_3$.  Whereas in the multiplication case we would have $5 < a.used$ and $3 \ge 0$ is maintained since we double the sum
of the products just outside the inner loop we have to avoid doing this.  This is also a good thing since we perform
fewer multiplications and the routine ends up being faster.

Finally the last difference is the addition of the ``square'' term outside the inner loop (step 5.8).  We add in the square
only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rfloor$ position.

EXAM,bn_fast_s_mp_sqr.c

This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for
the special case of squaring.

\subsection{Polynomial Basis Squaring}
The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring.  The minor exception
is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$.  Instead of performing $2n + 1$
multiplications to find the $\zeta$ relations, squaring operations are performed instead.

\subsection{Karatsuba Squaring}
Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square.
Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial.  The Karatsuba equation can be modified to square a
number with the following equation.

\begin{equation}
h(x) = a^2x^2 + \left ((a + b)^2 - (a^2 + b^2) \right )x + b^2
\end{equation}

Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a + b)^2$.  As in
Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of
$O \left ( n^{lg(3)} \right )$.

If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm
instead?  The answer to this arises from the cutoff point for squaring.  As in multiplication there exists a cutoff point, at which the
time required for a Comba based squaring and a Karatsuba based squaring meet.  Due to the overhead inherent in the Karatsuba method, the cutoff
point is fairly high.  For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits.

Consider squaring a 200 digit number with this technique.  It will be split into two 100 digit halves which are subsequently squared.
The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm.  If Karatsuba multiplication
were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\
\textbf{Input}.   mp\_int $a$ \\
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as the radix point.  The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form.

By expanding $\left (x1 + x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $(x0 - x1)^2 - (x1^2 + x0^2)  = 2 \cdot x0 \cdot x1$.
Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then
this method is faster.  Assuming no further recursions occur, the difference can be estimated with the following inequality.

Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or
machine clock cycles.}. 

\begin{equation}
5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2
\end{equation}

For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$.  This implies that the following inequality should hold.
\begin{center}
\begin{tabular}{rcl}
${5n \over 3} + 3n^2 + 3n$     & $<$ & ${n \over 3} + 6n^2$ \\
${5 \over 3} + 3n + 3$     & $<$ & ${1 \over 3} + 6n$ \\
${13 \over 9}$     & $<$ & $n$ \\
\end{tabular}
\end{center}

This results in a cutoff point around $n = 2$.  As a consequence it is actually faster to compute the middle term the ``long way'' on processors
where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication.  On
the Intel P4 processor this ratio is 1:29 making this method even more beneficial.  The only common exception is the ARMv4 processor which has a
ratio of 1:7.  } than simpler operations such as addition.  

EXAM,bn_mp_karatsuba_sqr.c

This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul.  It uses the same inline style to copy and 
shift the input into the two halves.  The loop from line @54,{@ to line @70,}@ has been modified since only one input exists.  The \textbf{used}
count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin.  At this point $x1$ and $x0$ are valid equivalents
to the respective halves as if mp\_rshd and mp\_mod\_2d had been used.  

By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered.  On the Athlon the cutoff point
is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}).  On slower processors such as the Intel P4
it is actually below the Comba limit (\textit{at 110 digits}).

This routine uses the same error trap coding style as mp\_karatsuba\_sqr.  As the temporary variables are initialized errors are 
redirected to the error trap higher up.  If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and 
mp\_clears are executed normally.

\subsection{Toom-Cook Squaring}
The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used
instead of multiplication to find the five relations.  The reader is encouraged to read the description of the latter algorithm and try to 
derive their own Toom-Cook squaring algorithm.  

\subsection{High Level Squaring}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_sqr}. \\







|

















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as the radix point.  The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form.

By expanding $\left (x1 + x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $(x0 - x1)^2 - (x1^2 + x0^2)  = 2 \cdot x0 \cdot x1$.
Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then
this method is faster.  Assuming no further recursions occur, the difference can be estimated with the following inequality.

Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or
machine clock cycles.}.

\begin{equation}
5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2
\end{equation}

For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$.  This implies that the following inequality should hold.
\begin{center}
\begin{tabular}{rcl}
${5n \over 3} + 3n^2 + 3n$     & $<$ & ${n \over 3} + 6n^2$ \\
${5 \over 3} + 3n + 3$     & $<$ & ${1 \over 3} + 6n$ \\
${13 \over 9}$     & $<$ & $n$ \\
\end{tabular}
\end{center}

This results in a cutoff point around $n = 2$.  As a consequence it is actually faster to compute the middle term the ``long way'' on processors
where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication.  On
the Intel P4 processor this ratio is 1:29 making this method even more beneficial.  The only common exception is the ARMv4 processor which has a
ratio of 1:7.  } than simpler operations such as addition.

EXAM,bn_mp_karatsuba_sqr.c

This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul.  It uses the same inline style to copy and
shift the input into the two halves.  The loop from line @54,{@ to line @70,}@ has been modified since only one input exists.  The \textbf{used}
count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin.  At this point $x1$ and $x0$ are valid equivalents
to the respective halves as if mp\_rshd and mp\_mod\_2d had been used.

By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered.  On the Athlon the cutoff point
is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}).  On slower processors such as the Intel P4
it is actually below the Comba limit (\textit{at 110 digits}).

This routine uses the same error trap coding style as mp\_karatsuba\_sqr.  As the temporary variables are initialized errors are
redirected to the error trap higher up.  If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and
mp\_clears are executed normally.

\subsection{Toom-Cook Squaring}
The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used
instead of multiplication to find the five relations.  The reader is encouraged to read the description of the latter algorithm and try to
derive their own Toom-Cook squaring algorithm.

\subsection{High Level Squaring}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_sqr}. \\
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\end{small}
\caption{Algorithm mp\_sqr}
\end{figure}

\textbf{Algorithm mp\_sqr.}
This algorithm computes the square of the input using one of four different algorithms.  If the input is very large and has at least
\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used.  If
neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used.  

EXAM,bn_mp_sqr.c

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\
                      & that have different number of digits in Karatsuba multiplication. \\
                      & \\
$\left [ 2 \right ] $ & In ~SQUARE~ the fact that every column of a squaring is made up \\
                      & of double products and at most one square is stated.  Prove this statement. \\
                      & \\                      
$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\
                      & \\
$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\
                      & \\ 
$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\
                      & required for equation $6.7$ to be true.  \\
                      & \\
$\left [ 3 \right ] $ & Implement a threaded version of Comba multiplication (and squaring) where you \\
                      & compute subsets of the columns in each thread.  Determine a cutoff point where \\
                      & it is effective and add the logic to mp\_mul() and mp\_sqr(). \\
                      &\\
$\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and Toom-Cook.  You must \\
                      & increase the throughput of mp\_exptmod() for random odd moduli in the range \\
                      & $512 \ldots 4096$ bits significantly ($> 2x$) to complete this challenge. \\
                      & \\
\end{tabular}

\chapter{Modular Reduction}
MARK,REDUCTION
\section{Basics of Modular Reduction}
\index{modular residue}
Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms, 
such as factoring.  Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set.  A number $a$ is said to be \textit{reduced}
modulo another number $b$ by finding the remainder of the division $a/b$.  Full integer division with remainder is a topic to be covered 
in~\ref{sec:division}.

Modular reduction is equivalent to solving for $r$ in the following equation.  $a = bq + r$ where $q = \lfloor a/b \rfloor$.  The result 
$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$.  In other vernacular $r$ is known as the 
``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and
other forms of residues.  

Modular reductions are normally used to create either finite groups, rings or fields.  The most common usage for performance driven modular reductions 
is in modular exponentiation algorithms.  That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible.  This operation is used in the 
RSA and Diffie-Hellman public key algorithms, for example.  Modular multiplication and squaring also appears as a fundamental operation in 
elliptic curve cryptographic algorithms.  As will be discussed in the subsequent chapter there exist fast algorithms for computing modular 
exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications.  These algorithms will produce partial results in the 
range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms.   They have also been used to create redundancy check 
algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems.  

\section{The Barrett Reduction}
The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate
division.  Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to 

\begin{equation}
c = a - b \cdot \lfloor a/b \rfloor
\end{equation}

Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper 
targeted the DSP56K processor.}  intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal.  However, 
DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types.  
It would take another common optimization to optimize the algorithm.

\subsection{Fixed Point Arithmetic}
The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers.  Fixed
point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were 
fairly slow if not unavailable.   The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit 
integer and a $q$-bit fraction part (\textit{where $p+q = k$}).  

In this system a $k$-bit integer $n$ would actually represent $n/2^q$.  For example, with $q = 4$ the integer $n = 37$ would actually represent the
value $2.3125$.  To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by 
moving the implied decimal point back to where it should be.  For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted 
to fixed point first by multiplying by $2^q$.  Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the 
fixed point representation of $5$.  The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$.  

This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication
of two fixed point numbers.  Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal.  If $2^q$ is 
equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic.  Using this fact dividing an integer 
$a$ by another integer $b$ can be achieved with the following expression.

\begin{equation}
\lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor
\end{equation}

The precision of the division is proportional to the value of $q$.  If the divisor $b$ is used frequently as is the case with 
modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift.  Both operations
are considerably faster than division on most processors.  

Consider dividing $19$ by $5$.  The correct result is $\lfloor 19/5 \rfloor = 3$.  With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which
leads to a product of $19$ which when divided by $2^q$ produces $2$.  However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and
the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct.  The value of $2^q$ must be close to or ideally
larger than the dividend.  In effect if $a$ is the dividend then $q$ should allow $0 \le \lfloor a/2^q \rfloor \le 1$ in order for this approach
to work correctly.  Plugging this form of divison into the original equation the following modular residue equation arises.

\begin{equation}
c = a - b \cdot \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor
\end{equation}

Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol.  Using the $\mu$
variable also helps re-inforce the idea that it is meant to be computed once and re-used.

\begin{equation}
c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor
\end{equation}

Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one.  In the context of Barrett
reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough
precision.  

Let $n$ represent the number of digits in $b$.  This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and 
another $n^2$ single precision multiplications to find the residue.  In total $3n^2$ single precision multiplications are required to 
reduce the number.  

For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$.  Consider reducing
$a = 180388626447$ modulo $b$ using the above reduction equation.  The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$.
By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found.

\subsection{Choosing a Radix Point}
Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications.  If that were the best
that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$.  
See~\ref{sec:division} for further details.} might as well be used in its place.  The key to optimizing the reduction is to reduce the precision of
the initial multiplication that finds the quotient.  

Let $a$ represent the number of which the residue is sought.  Let $b$ represent the modulus used to find the residue.  Let $m$ represent
the number of digits in $b$.  For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if 
two $m$-digit numbers have been multiplied.  Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer.  Digits below the 
$m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$.  Another way to
express this is by re-writing $a$ as two parts.  If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then 
${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$.  Since $a'$ is bound to be less than $b$ the quotient
is bound by $0 \le {a' \over b} < 1$.

Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero.  However, if the digits 
``might as well be zero'' they might as well not be there in the first place.  Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input
with the irrelevant digits trimmed.  Now the modular reduction is trimmed to the almost equivalent equation

\begin{equation}
c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor
\end{equation}

Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the 
exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$.  If the optimization had not been performed the divisor 
would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient 
$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two.  The original fixed point quotient can be off
by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient
can be off by an additional value of one for a total of at most two.  This implies that 
$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$.  By first subtracting $b$ times the quotient and then conditionally subtracting 
$b$ once or twice the residue is found.

The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single
precision multiplications, ignoring the subtractions required.  In total $2m^2 + m$ single precision multiplications are required to find the residue.  
This is considerably faster than the original attempt.

For example, let $\beta = 10$ represent the radix of the digits.  Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ 
represent the value of which the residue is desired.  In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$.  
With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$.  The quotient is then 
$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$.  Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$ 
is found.  

\subsection{Trimming the Quotient}
So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications.  As 
it stands now the algorithm is already fairly fast compared to a full integer division algorithm.  However, there is still room for
optimization.  

After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower
half of the product.  It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision 
multiplications.  If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly.  
In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed.  

The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number.  Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision
multiplications would be required.  Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number
of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications.  

\subsection{Trimming the Residue}
After the quotient has been calculated it is used to reduce the input.  As previously noted the algorithm is not exact and it can be off by a small
multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$.  If $b$ is $m$ digits than the 
result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are
implicitly zero.  

The next optimization arises from this very fact.  Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full
$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed.  Similarly the value of $a$ can
be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well.  A multiplication that produces 
only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications.  

With both optimizations in place the algorithm is the algorithm Barrett proposed.  It requires $m^2 + 2m - 1$ single precision multiplications which
is considerably faster than the straightforward $3m^2$ method.  

\subsection{The Barrett Algorithm}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce}. \\







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\end{small}
\caption{Algorithm mp\_sqr}
\end{figure}

\textbf{Algorithm mp\_sqr.}
This algorithm computes the square of the input using one of four different algorithms.  If the input is very large and has at least
\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used.  If
neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used.

EXAM,bn_mp_sqr.c

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\
                      & that have different number of digits in Karatsuba multiplication. \\
                      & \\
$\left [ 2 \right ] $ & In ~SQUARE~ the fact that every column of a squaring is made up \\
                      & of double products and at most one square is stated.  Prove this statement. \\
                      & \\
$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\
                      & \\
$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\
                      & \\
$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\
                      & required for equation $6.7$ to be true.  \\
                      & \\
$\left [ 3 \right ] $ & Implement a threaded version of Comba multiplication (and squaring) where you \\
                      & compute subsets of the columns in each thread.  Determine a cutoff point where \\
                      & it is effective and add the logic to mp\_mul() and mp\_sqr(). \\
                      &\\
$\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and Toom-Cook.  You must \\
                      & increase the throughput of mp\_exptmod() for random odd moduli in the range \\
                      & $512 \ldots 4096$ bits significantly ($> 2x$) to complete this challenge. \\
                      & \\
\end{tabular}

\chapter{Modular Reduction}
MARK,REDUCTION
\section{Basics of Modular Reduction}
\index{modular residue}
Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms,
such as factoring.  Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set.  A number $a$ is said to be \textit{reduced}
modulo another number $b$ by finding the remainder of the division $a/b$.  Full integer division with remainder is a topic to be covered
in~\ref{sec:division}.

Modular reduction is equivalent to solving for $r$ in the following equation.  $a = bq + r$ where $q = \lfloor a/b \rfloor$.  The result
$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$.  In other vernacular $r$ is known as the
``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and
other forms of residues.

Modular reductions are normally used to create either finite groups, rings or fields.  The most common usage for performance driven modular reductions
is in modular exponentiation algorithms.  That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible.  This operation is used in the
RSA and Diffie-Hellman public key algorithms, for example.  Modular multiplication and squaring also appears as a fundamental operation in
elliptic curve cryptographic algorithms.  As will be discussed in the subsequent chapter there exist fast algorithms for computing modular
exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications.  These algorithms will produce partial results in the
range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms.   They have also been used to create redundancy check
algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems.

\section{The Barrett Reduction}
The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate
division.  Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to

\begin{equation}
c = a - b \cdot \lfloor a/b \rfloor
\end{equation}

Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper
targeted the DSP56K processor.}  intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal.  However,
DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types.
It would take another common optimization to optimize the algorithm.

\subsection{Fixed Point Arithmetic}
The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers.  Fixed
point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were
fairly slow if not unavailable.   The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit
integer and a $q$-bit fraction part (\textit{where $p+q = k$}).

In this system a $k$-bit integer $n$ would actually represent $n/2^q$.  For example, with $q = 4$ the integer $n = 37$ would actually represent the
value $2.3125$.  To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by
moving the implied decimal point back to where it should be.  For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted
to fixed point first by multiplying by $2^q$.  Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the
fixed point representation of $5$.  The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$.

This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication
of two fixed point numbers.  Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal.  If $2^q$ is
equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic.  Using this fact dividing an integer
$a$ by another integer $b$ can be achieved with the following expression.

\begin{equation}
\lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor
\end{equation}

The precision of the division is proportional to the value of $q$.  If the divisor $b$ is used frequently as is the case with
modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift.  Both operations
are considerably faster than division on most processors.

Consider dividing $19$ by $5$.  The correct result is $\lfloor 19/5 \rfloor = 3$.  With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which
leads to a product of $19$ which when divided by $2^q$ produces $2$.  However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and
the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct.  The value of $2^q$ must be close to or ideally
larger than the dividend.  In effect if $a$ is the dividend then $q$ should allow $0 \le \lfloor a/2^q \rfloor \le 1$ in order for this approach
to work correctly.  Plugging this form of divison into the original equation the following modular residue equation arises.

\begin{equation}
c = a - b \cdot \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor
\end{equation}

Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol.  Using the $\mu$
variable also helps re-inforce the idea that it is meant to be computed once and re-used.

\begin{equation}
c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor
\end{equation}

Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one.  In the context of Barrett
reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough
precision.

Let $n$ represent the number of digits in $b$.  This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and
another $n^2$ single precision multiplications to find the residue.  In total $3n^2$ single precision multiplications are required to
reduce the number.

For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$.  Consider reducing
$a = 180388626447$ modulo $b$ using the above reduction equation.  The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$.
By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found.

\subsection{Choosing a Radix Point}
Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications.  If that were the best
that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$.
See~\ref{sec:division} for further details.} might as well be used in its place.  The key to optimizing the reduction is to reduce the precision of
the initial multiplication that finds the quotient.

Let $a$ represent the number of which the residue is sought.  Let $b$ represent the modulus used to find the residue.  Let $m$ represent
the number of digits in $b$.  For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if
two $m$-digit numbers have been multiplied.  Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer.  Digits below the
$m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$.  Another way to
express this is by re-writing $a$ as two parts.  If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then
${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$.  Since $a'$ is bound to be less than $b$ the quotient
is bound by $0 \le {a' \over b} < 1$.

Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero.  However, if the digits
``might as well be zero'' they might as well not be there in the first place.  Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input
with the irrelevant digits trimmed.  Now the modular reduction is trimmed to the almost equivalent equation

\begin{equation}
c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor
\end{equation}

Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the
exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$.  If the optimization had not been performed the divisor
would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient
$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two.  The original fixed point quotient can be off
by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient
can be off by an additional value of one for a total of at most two.  This implies that
$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$.  By first subtracting $b$ times the quotient and then conditionally subtracting
$b$ once or twice the residue is found.

The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single
precision multiplications, ignoring the subtractions required.  In total $2m^2 + m$ single precision multiplications are required to find the residue.
This is considerably faster than the original attempt.

For example, let $\beta = 10$ represent the radix of the digits.  Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$
represent the value of which the residue is desired.  In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$.
With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$.  The quotient is then
$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$.  Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$
is found.

\subsection{Trimming the Quotient}
So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications.  As
it stands now the algorithm is already fairly fast compared to a full integer division algorithm.  However, there is still room for
optimization.

After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower
half of the product.  It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision
multiplications.  If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly.
In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed.

The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number.  Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision
multiplications would be required.  Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number
of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications.

\subsection{Trimming the Residue}
After the quotient has been calculated it is used to reduce the input.  As previously noted the algorithm is not exact and it can be off by a small
multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$.  If $b$ is $m$ digits than the
result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are
implicitly zero.

The next optimization arises from this very fact.  Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full
$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed.  Similarly the value of $a$ can
be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well.  A multiplication that produces
only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications.

With both optimizations in place the algorithm is the algorithm Barrett proposed.  It requires $m^2 + 2m - 1$ single precision multiplications which
is considerably faster than the straightforward $3m^2$ method.

\subsection{The Barrett Algorithm}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce}. \\
3608
3609
3610
3611
3612
3613
3614
3615
3616
3617
3618
3619
3620
3621
3622
3623
3624
3625
3626
3627
3628
3629
3630
3631
3632
3633
3634
3635
3636
3637
3638
3639
3640
3641
3642
3643
3644
3645
3646
3647
3648
3649
3650
3651
3652
3653
\end{center}
\end{small}
\caption{Algorithm mp\_reduce}
\end{figure}

\textbf{Algorithm mp\_reduce.}
This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm.  It is loosely based on algorithm 14.42 of HAC
\cite[pp.  602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}.  The algorithm has several restrictions and assumptions which must 
be adhered to for the algorithm to work.

First the modulus $b$ is assumed to be positive and greater than one.  If the modulus were less than or equal to one than subtracting
a multiple of it would either accomplish nothing or actually enlarge the input.  The input $a$ must be in the range $0 \le a < b^2$ in order
for the quotient to have enough precision.  If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem.
Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish.  The value of $\mu$ is passed as an argument to this 
algorithm and is assumed to be calculated and stored before the algorithm is used.  

Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position.  An algorithm called 
$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task.  The algorithm is based on $s\_mp\_mul\_digs$ except that
instead of stopping at a given level of precision it starts at a given level of precision.  This optimal algorithm can only be used if the number
of digits in $b$ is very much smaller than $\beta$.  

While it is known that 
$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied 
``borrow'' from the higher digits might leave a negative result.  After the multiple of the modulus has been subtracted from $a$ the residue must be 
fixed up in case it is negative.  The invariant $\beta^{m+1}$ must be added to the residue to make it positive again.  

The while loop at step 9 will subtract $b$ until the residue is less than $b$.  If the algorithm is performed correctly this step is 
performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should.

EXAM,bn_mp_reduce.c

The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up.  This essentially halves
the number of single precision multiplications required.  However, the optimization is only safe if $\beta$ is much larger than the number of digits
in the modulus.  In the source code this is evaluated on lines @36,if@ to @44,}@ where algorithm s\_mp\_mul\_high\_digs is used when it is
safe to do so.  

\subsection{The Barrett Setup Algorithm}
In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance.  Ideally this value should be computed once and stored for
future use so that the Barrett algorithm can be used without delay.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce\_setup}. \\
\textbf{Input}.   mp\_int $a$ ($a > 1$)  \\







|





|
|

|


|

|
|
|
|

|







|



|







3597
3598
3599
3600
3601
3602
3603
3604
3605
3606
3607
3608
3609
3610
3611
3612
3613
3614
3615
3616
3617
3618
3619
3620
3621
3622
3623
3624
3625
3626
3627
3628
3629
3630
3631
3632
3633
3634
3635
3636
3637
3638
3639
3640
3641
3642
\end{center}
\end{small}
\caption{Algorithm mp\_reduce}
\end{figure}

\textbf{Algorithm mp\_reduce.}
This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm.  It is loosely based on algorithm 14.42 of HAC
\cite[pp.  602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}.  The algorithm has several restrictions and assumptions which must
be adhered to for the algorithm to work.

First the modulus $b$ is assumed to be positive and greater than one.  If the modulus were less than or equal to one than subtracting
a multiple of it would either accomplish nothing or actually enlarge the input.  The input $a$ must be in the range $0 \le a < b^2$ in order
for the quotient to have enough precision.  If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem.
Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish.  The value of $\mu$ is passed as an argument to this
algorithm and is assumed to be calculated and stored before the algorithm is used.

Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position.  An algorithm called
$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task.  The algorithm is based on $s\_mp\_mul\_digs$ except that
instead of stopping at a given level of precision it starts at a given level of precision.  This optimal algorithm can only be used if the number
of digits in $b$ is very much smaller than $\beta$.

While it is known that
$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied
``borrow'' from the higher digits might leave a negative result.  After the multiple of the modulus has been subtracted from $a$ the residue must be
fixed up in case it is negative.  The invariant $\beta^{m+1}$ must be added to the residue to make it positive again.

The while loop at step 9 will subtract $b$ until the residue is less than $b$.  If the algorithm is performed correctly this step is
performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should.

EXAM,bn_mp_reduce.c

The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up.  This essentially halves
the number of single precision multiplications required.  However, the optimization is only safe if $\beta$ is much larger than the number of digits
in the modulus.  In the source code this is evaluated on lines @36,if@ to @44,}@ where algorithm s\_mp\_mul\_high\_digs is used when it is
safe to do so.

\subsection{The Barrett Setup Algorithm}
In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance.  Ideally this value should be computed once and stored for
future use so that the Barrett algorithm can be used without delay.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce\_setup}. \\
\textbf{Input}.   mp\_int $a$ ($a > 1$)  \\
3666
3667
3668
3669
3670
3671
3672
3673
3674
3675
3676
3677
3678
3679
3680
3681
3682
3683
3684
3685
3686
3687
3688
3689
3690
3691
3692
3693
3694
3695
3696
3697
\textbf{Algorithm mp\_reduce\_setup.}
This algorithm computes the reciprocal $\mu$ required for Barrett reduction.  First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot  m}$ which
is equivalent and much faster.  The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$.

EXAM,bn_mp_reduce_setup.c

This simple routine calculates the reciprocal $\mu$ required by Barrett reduction.  Note the extended usage of algorithm mp\_div where the variable
which would received the remainder is passed as NULL.  As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the 
remainder to be passed as NULL meaning to ignore the value.  

\section{The Montgomery Reduction}
Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting 
form of reduction in common use.  It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a 
residue times a constant.  However, as perplexing as this may sound the algorithm is relatively simple and very efficient.  

Throughout this entire section the variable $n$ will represent the modulus used to form the residue.  As will be discussed shortly the value of
$n$ must be odd.  The variable $x$ will represent the quantity of which the residue is sought.  Similar to the Barrett algorithm the input
is restricted to $0 \le x < n^2$.  To begin the description some simple number theory facts must be established.

\textbf{Fact 1.}  Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$.  Another way
to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$.  Adding zero will not change the value of the residue.  

\textbf{Fact 2.}  If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$.  Actually
this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to 
multiplication by $k^{-1}$ modulo $n$.  

From these two simple facts the following simple algorithm can be derived.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}







|
|


|
|
|






|


|
|







3655
3656
3657
3658
3659
3660
3661
3662
3663
3664
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3667
3668
3669
3670
3671
3672
3673
3674
3675
3676
3677
3678
3679
3680
3681
3682
3683
3684
3685
3686
\textbf{Algorithm mp\_reduce\_setup.}
This algorithm computes the reciprocal $\mu$ required for Barrett reduction.  First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot  m}$ which
is equivalent and much faster.  The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$.

EXAM,bn_mp_reduce_setup.c

This simple routine calculates the reciprocal $\mu$ required by Barrett reduction.  Note the extended usage of algorithm mp\_div where the variable
which would received the remainder is passed as NULL.  As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the
remainder to be passed as NULL meaning to ignore the value.

\section{The Montgomery Reduction}
Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting
form of reduction in common use.  It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a
residue times a constant.  However, as perplexing as this may sound the algorithm is relatively simple and very efficient.

Throughout this entire section the variable $n$ will represent the modulus used to form the residue.  As will be discussed shortly the value of
$n$ must be odd.  The variable $x$ will represent the quantity of which the residue is sought.  Similar to the Barrett algorithm the input
is restricted to $0 \le x < n^2$.  To begin the description some simple number theory facts must be established.

\textbf{Fact 1.}  Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$.  Another way
to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$.  Adding zero will not change the value of the residue.

\textbf{Fact 2.}  If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$.  Actually
this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to
multiplication by $k^{-1}$ modulo $n$.

From these two simple facts the following simple algorithm can be derived.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
3709
3710
3711
3712
3713
3714
3715
3716
3717
3718
3719
3720
3721
3722
3723
3724
\end{center}
\end{small}
\caption{Algorithm Montgomery Reduction}
\end{figure}

The algorithm reduces the input one bit at a time using the two congruencies stated previously.  Inside the loop $n$, which is odd, is
added to $x$ if $x$ is odd.  This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two.  Since
$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$.  Let $r$ represent the 
final result of the Montgomery algorithm.  If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to 
$0 \le r < \lfloor x/2^k \rfloor + n$.  As a result at most a single subtraction is required to get the residue desired.

\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|l|}
\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\







|
|







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3700
3701
3702
3703
3704
3705
3706
3707
3708
3709
3710
3711
3712
3713
\end{center}
\end{small}
\caption{Algorithm Montgomery Reduction}
\end{figure}

The algorithm reduces the input one bit at a time using the two congruencies stated previously.  Inside the loop $n$, which is odd, is
added to $x$ if $x$ is odd.  This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two.  Since
$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$.  Let $r$ represent the
final result of the Montgomery algorithm.  If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to
$0 \le r < \lfloor x/2^k \rfloor + n$.  As a result at most a single subtraction is required to get the residue desired.

\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|l|}
\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\
3735
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3740
3741
3742
3743
3744
3745
3746
3747
3748
3749
3750
3751
3752
3753
3754
\end{tabular}
\end{center}
\end{small}
\caption{Example of Montgomery Reduction (I)}
\label{fig:MONT1}
\end{figure}

Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 9$ (note $\beta^k = 512$ which is larger than $n$).  The result of 
the algorithm $r = 178$ is congruent to the value of $2^{-9} \cdot 5555 \mbox{ (mod }257\mbox{)}$.  When $r$ is multiplied by $2^9$ modulo $257$ the correct residue 
$r \equiv 158$ is produced.  

Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$.  The current algorithm requires $2k^2$ single precision shifts
and $k^2$ single precision additions.  At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful.  
Fortunately there exists an alternative representation of the algorithm.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Montgomery Reduction} (modified I). \\







|
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|


|







3724
3725
3726
3727
3728
3729
3730
3731
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3734
3735
3736
3737
3738
3739
3740
3741
3742
3743
\end{tabular}
\end{center}
\end{small}
\caption{Example of Montgomery Reduction (I)}
\label{fig:MONT1}
\end{figure}

Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 9$ (note $\beta^k = 512$ which is larger than $n$).  The result of
the algorithm $r = 178$ is congruent to the value of $2^{-9} \cdot 5555 \mbox{ (mod }257\mbox{)}$.  When $r$ is multiplied by $2^9$ modulo $257$ the correct residue
$r \equiv 158$ is produced.

Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$.  The current algorithm requires $2k^2$ single precision shifts
and $k^2$ single precision additions.  At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful.
Fortunately there exists an alternative representation of the algorithm.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Montgomery Reduction} (modified I). \\
3789
3790
3791
3792
3793
3794
3795
3796
3797
3798
3799
3800
3801
3802
3803
3804
3805
3806
\end{tabular}
\end{center}
\end{small}
\caption{Example of Montgomery Reduction (II)}
\label{fig:MONT2}
\end{figure}

Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 9$. 
With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the 
loop.  Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed.  In those iterations the $t$'th bit of $x$ is 
zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero.  

\subsection{Digit Based Montgomery Reduction}
Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis.  Consider the
previous algorithm re-written to compute the Montgomery reduction in this new fashion.

\begin{figure}[!here]
\begin{small}







|
|
|
|







3778
3779
3780
3781
3782
3783
3784
3785
3786
3787
3788
3789
3790
3791
3792
3793
3794
3795
\end{tabular}
\end{center}
\end{small}
\caption{Example of Montgomery Reduction (II)}
\label{fig:MONT2}
\end{figure}

Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 9$.
With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the
loop.  Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed.  In those iterations the $t$'th bit of $x$ is
zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero.

\subsection{Digit Based Montgomery Reduction}
Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis.  Consider the
previous algorithm re-written to compute the Montgomery reduction in this new fashion.

\begin{figure}[!here]
\begin{small}
3816
3817
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3819
3820
3821
3822
3823
3824
3825
3826
3827
3828
3829
3830
3831
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3836
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3864
3865
3866
3867
3868
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Montgomery Reduction (modified II)}
\end{figure}

The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue.  If the first digit of 
the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit.  This
problem breaks down to solving the following congruency.  

\begin{center}
\begin{tabular}{rcl}
$x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\
$\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\
$\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
\end{tabular}
\end{center}

In each iteration of the loop on step 1 a new value of $\mu$ must be calculated.  The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used 
extensively in this algorithm and should be precomputed.  Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$.  

For example, let $\beta = 10$ represent the radix.  Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$.  Let $x = 33$ 
represent the value to reduce.

\newpage\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\
\hline --                 & $33$ & --\\
\hline $0$                 & $33 + \mu n = 50$ & $1$ \\
\hline $1$                 & $50 + \mu n \beta = 900$ & $5$ \\
\hline
\end{tabular}
\end{center}
\caption{Example of Montgomery Reduction}
\end{figure}

The final result $900$ is then divided by $\beta^k$ to produce the final result $9$.  The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ 
which implies the result is not the modular residue of $x$ modulo $n$.  However, recall that the residue is actually multiplied by $\beta^{-k}$ in
the algorithm.  To get the true residue the value must be multiplied by $\beta^k$.  In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and
the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$.  

\subsection{Baseline Montgomery Reduction}
The baseline Montgomery reduction algorithm will produce the residue for any size input.  It is designed to be a catch-all algororithm for 
Montgomery reductions.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_montgomery\_reduce}. \\
\textbf{Input}.   mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\







|

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3805
3806
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3808
3809
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\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Montgomery Reduction (modified II)}
\end{figure}

The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue.  If the first digit of
the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit.  This
problem breaks down to solving the following congruency.

\begin{center}
\begin{tabular}{rcl}
$x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\
$\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\
$\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
\end{tabular}
\end{center}

In each iteration of the loop on step 1 a new value of $\mu$ must be calculated.  The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used
extensively in this algorithm and should be precomputed.  Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$.

For example, let $\beta = 10$ represent the radix.  Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$.  Let $x = 33$
represent the value to reduce.

\newpage\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\
\hline --                 & $33$ & --\\
\hline $0$                 & $33 + \mu n = 50$ & $1$ \\
\hline $1$                 & $50 + \mu n \beta = 900$ & $5$ \\
\hline
\end{tabular}
\end{center}
\caption{Example of Montgomery Reduction}
\end{figure}

The final result $900$ is then divided by $\beta^k$ to produce the final result $9$.  The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$
which implies the result is not the modular residue of $x$ modulo $n$.  However, recall that the residue is actually multiplied by $\beta^{-k}$ in
the algorithm.  To get the true residue the value must be multiplied by $\beta^k$.  In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and
the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$.

\subsection{Baseline Montgomery Reduction}
The baseline Montgomery reduction algorithm will produce the residue for any size input.  It is designed to be a catch-all algororithm for
Montgomery reductions.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_montgomery\_reduce}. \\
\textbf{Input}.   mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\
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3903
3904
3905
3906
3907
3908
3909
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3922
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3943
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3945
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3947
3948
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3951
3952
\end{small}
\caption{Algorithm mp\_montgomery\_reduce}
\end{figure}

\textbf{Algorithm mp\_montgomery\_reduce.}
This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm.  The algorithm is loosely based
on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop.  The
restrictions on this algorithm are fairly easy to adapt to.  First $0 \le x < n^2$ bounds the input to numbers in the same range as 
for the Barrett algorithm.  Additionally if $n > 1$ and $n$ is odd there will exist a modular inverse $\rho$.  $\rho$ must be calculated in
advance of this algorithm.  Finally the variable $k$ is fixed and a pseudonym for $n.used$.  

Step 2 decides whether a faster Montgomery algorithm can be used.  It is based on the Comba technique meaning that there are limits on
the size of the input.  This algorithm is discussed in ~COMBARED~.

Step 5 is the main reduction loop of the algorithm.  The value of $\mu$ is calculated once per iteration in the outer loop.  The inner loop
calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits.  Both the addition and
multiplication are performed in the same loop to save time and memory.  Step 5.4 will handle any additional carries that escape the inner loop.

Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications 
in the inner loop.  In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision
multiplications.  

EXAM,bn_mp_montgomery_reduce.c

This is the baseline implementation of the Montgomery reduction algorithm.  Lines @30,digs@ to @35,}@ determine if the Comba based
routine can be used instead.  Line @47,mu@ computes the value of $\mu$ for that particular iteration of the outer loop.  

The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop.  The alias $tmpx$ refers to the $ix$'th digit of $x$ and
the alias $tmpn$ refers to the modulus $n$.  

\subsection{Faster ``Comba'' Montgomery Reduction}
MARK,COMBARED

The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial
nature of the inner loop.  The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba
technique.  The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates
a $k \times 1$ product $k$ times. 

The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$.  This means the 
carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit.  The solution as it turns out is very simple.  
Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry.  

With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases
the speed of the algorithm.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\
\textbf{Input}.   mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\







|

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|

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|







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\end{small}
\caption{Algorithm mp\_montgomery\_reduce}
\end{figure}

\textbf{Algorithm mp\_montgomery\_reduce.}
This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm.  The algorithm is loosely based
on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop.  The
restrictions on this algorithm are fairly easy to adapt to.  First $0 \le x < n^2$ bounds the input to numbers in the same range as
for the Barrett algorithm.  Additionally if $n > 1$ and $n$ is odd there will exist a modular inverse $\rho$.  $\rho$ must be calculated in
advance of this algorithm.  Finally the variable $k$ is fixed and a pseudonym for $n.used$.

Step 2 decides whether a faster Montgomery algorithm can be used.  It is based on the Comba technique meaning that there are limits on
the size of the input.  This algorithm is discussed in ~COMBARED~.

Step 5 is the main reduction loop of the algorithm.  The value of $\mu$ is calculated once per iteration in the outer loop.  The inner loop
calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits.  Both the addition and
multiplication are performed in the same loop to save time and memory.  Step 5.4 will handle any additional carries that escape the inner loop.

Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications
in the inner loop.  In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision
multiplications.

EXAM,bn_mp_montgomery_reduce.c

This is the baseline implementation of the Montgomery reduction algorithm.  Lines @30,digs@ to @35,}@ determine if the Comba based
routine can be used instead.  Line @47,mu@ computes the value of $\mu$ for that particular iteration of the outer loop.

The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop.  The alias $tmpx$ refers to the $ix$'th digit of $x$ and
the alias $tmpn$ refers to the modulus $n$.

\subsection{Faster ``Comba'' Montgomery Reduction}
MARK,COMBARED

The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial
nature of the inner loop.  The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba
technique.  The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates
a $k \times 1$ product $k$ times.

The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$.  This means the
carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit.  The solution as it turns out is very simple.
Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry.

With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases
the speed of the algorithm.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\
\textbf{Input}.   mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\
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4001
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4085
4086
4087
4088
\end{small}
\caption{Algorithm fast\_mp\_montgomery\_reduce}
\end{figure}

\textbf{Algorithm fast\_mp\_montgomery\_reduce.}
This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique.  It is on most computer platforms significantly
faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}).  The algorithm has the same restrictions
on the input as the baseline reduction algorithm.  An additional two restrictions are imposed on this algorithm.  The number of digits $k$ in the 
the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$.   When $\beta = 2^{28}$ this algorithm can be used to reduce modulo
a modulus of at most $3,556$ bits in length.  

As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product.  It is initially filled with the
contents of $x$ with the excess digits zeroed.  The reduction loop is very similar the to the baseline loop at heart.  The multiplication on step
4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$.  Some multipliers such
as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce.  By performing
a single precision multiplication instead half the amount of time is spent.

Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work.  That is what step
4.3 will do.  In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards.  Note
how the upper bits of those same words are not reduced modulo $\beta$.  This is because those values will be discarded shortly and there is no
point.

Step 5 will propagate the remainder of the carries upwards.  On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are
stored in the destination $x$.  

EXAM,bn_fast_mp_montgomery_reduce.c

The $\hat W$ array is first filled with digits of $x$ on line @49,for@ then the rest of the digits are zeroed on line @54,for@.  Both loops share
the same alias variables to make the code easier to read.  

The value of $\mu$ is calculated in an interesting fashion.  First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit.  This
forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision.   Line @101,>>@ fixes the carry 
for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$.

The for loop on line @113,for@ propagates the rest of the carries upwards through the columns.  The for loop on line @126,for@ reduces the columns
modulo $\beta$ and shifts them $k$ places at the same time.  The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th
digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$.  

\subsection{Montgomery Setup}
To calculate the variable $\rho$ a relatively simple algorithm will be required.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_montgomery\_setup}. \\
\textbf{Input}.   mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\
\textbf{Output}.  $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
\hline \\
1.  $b \leftarrow n_0$ \\
2.  If $b$ is even return(\textit{MP\_VAL}) \\
3.  $x \leftarrow (((b + 2) \mbox{ AND } 4) << 1) + b$ \\
4.  for $k$ from 0 to $\lceil lg(lg(\beta)) \rceil - 2$ do \\
\hspace{3mm}4.1  $x \leftarrow x \cdot (2 - bx)$ \\
5.  $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\
6.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_montgomery\_setup} 
\end{figure}

\textbf{Algorithm mp\_montgomery\_setup.}
This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms.  It uses a very interesting trick 
to calculate $1/n_0$ when $\beta$ is a power of two.  

EXAM,bn_mp_montgomery_setup.c

This source code computes the value of $\rho$ required to perform Montgomery reduction.  It has been modified to avoid performing excess
multiplications when $\beta$ is not the default 28-bits.  

\section{The Diminished Radix Algorithm}
The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett
or Montgomery methods for certain forms of moduli.  The technique is based on the following simple congruence.

\begin{equation}
(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)}
\end{equation}

This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive.  It used the fact that if $n = 2^{31}$ and $k=1$ that 
then a x86 multiplier could produce the 62-bit product and use  the ``shrd'' instruction to perform a double-precision right shift.  The proof
of the above equation is very simple.  First write $x$ in the product form.

\begin{equation}
x = qn + r
\end{equation}

Now reduce both sides modulo $(n - k)$.

\begin{equation}
x \equiv qk + r  \mbox{ (mod }(n-k)\mbox{)}
\end{equation}

The variable $n$ reduces modulo $n - k$ to $k$.  By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ 
into the equation the original congruence is reproduced, thus concluding the proof.  The following algorithm is based on this observation.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Diminished Radix Reduction}. \\







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4077
\end{small}
\caption{Algorithm fast\_mp\_montgomery\_reduce}
\end{figure}

\textbf{Algorithm fast\_mp\_montgomery\_reduce.}
This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique.  It is on most computer platforms significantly
faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}).  The algorithm has the same restrictions
on the input as the baseline reduction algorithm.  An additional two restrictions are imposed on this algorithm.  The number of digits $k$ in the
the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$.   When $\beta = 2^{28}$ this algorithm can be used to reduce modulo
a modulus of at most $3,556$ bits in length.

As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product.  It is initially filled with the
contents of $x$ with the excess digits zeroed.  The reduction loop is very similar the to the baseline loop at heart.  The multiplication on step
4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$.  Some multipliers such
as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce.  By performing
a single precision multiplication instead half the amount of time is spent.

Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work.  That is what step
4.3 will do.  In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards.  Note
how the upper bits of those same words are not reduced modulo $\beta$.  This is because those values will be discarded shortly and there is no
point.

Step 5 will propagate the remainder of the carries upwards.  On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are
stored in the destination $x$.

EXAM,bn_fast_mp_montgomery_reduce.c

The $\hat W$ array is first filled with digits of $x$ on line @49,for@ then the rest of the digits are zeroed on line @54,for@.  Both loops share
the same alias variables to make the code easier to read.

The value of $\mu$ is calculated in an interesting fashion.  First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit.  This
forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision.   Line @101,>>@ fixes the carry
for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$.

The for loop on line @113,for@ propagates the rest of the carries upwards through the columns.  The for loop on line @126,for@ reduces the columns
modulo $\beta$ and shifts them $k$ places at the same time.  The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th
digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$.

\subsection{Montgomery Setup}
To calculate the variable $\rho$ a relatively simple algorithm will be required.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_montgomery\_setup}. \\
\textbf{Input}.   mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\
\textbf{Output}.  $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
\hline \\
1.  $b \leftarrow n_0$ \\
2.  If $b$ is even return(\textit{MP\_VAL}) \\
3.  $x \leftarrow (((b + 2) \mbox{ AND } 4) << 1) + b$ \\
4.  for $k$ from 0 to $\lceil lg(lg(\beta)) \rceil - 2$ do \\
\hspace{3mm}4.1  $x \leftarrow x \cdot (2 - bx)$ \\
5.  $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\
6.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_montgomery\_setup}
\end{figure}

\textbf{Algorithm mp\_montgomery\_setup.}
This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms.  It uses a very interesting trick
to calculate $1/n_0$ when $\beta$ is a power of two.

EXAM,bn_mp_montgomery_setup.c

This source code computes the value of $\rho$ required to perform Montgomery reduction.  It has been modified to avoid performing excess
multiplications when $\beta$ is not the default 28-bits.

\section{The Diminished Radix Algorithm}
The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett
or Montgomery methods for certain forms of moduli.  The technique is based on the following simple congruence.

\begin{equation}
(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)}
\end{equation}

This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive.  It used the fact that if $n = 2^{31}$ and $k=1$ that
then a x86 multiplier could produce the 62-bit product and use  the ``shrd'' instruction to perform a double-precision right shift.  The proof
of the above equation is very simple.  First write $x$ in the product form.

\begin{equation}
x = qn + r
\end{equation}

Now reduce both sides modulo $(n - k)$.

\begin{equation}
x \equiv qk + r  \mbox{ (mod }(n-k)\mbox{)}
\end{equation}

The variable $n$ reduces modulo $n - k$ to $k$.  By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$
into the equation the original congruence is reproduced, thus concluding the proof.  The following algorithm is based on this observation.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Diminished Radix Reduction}. \\
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4123
4124
4125
4126
4127
4128
4129
4130
4131
4132
4133
4134
4135
4136
4137
4138
4139
4140
4141
4142
4143
4144
4145
4146
4147
4148
4149
4150
4151
4152
4153
4154
4155
4156
\caption{Algorithm Diminished Radix Reduction}
\label{fig:DR}
\end{figure}

This algorithm will reduce $x$ modulo $n - k$ and return the residue.  If $0 \le x < (n - k)^2$ then the algorithm will loop almost always
once or twice and occasionally three times.  For simplicity sake the value of $x$ is bounded by the following simple polynomial.

\begin{equation} 
0 \le x < n^2 + k^2 - 2nk
\end{equation}

The true bound is  $0 \le x < (n - k - 1)^2$ but this has quite a few more terms.  The value of $q$ after step 1 is bounded by the following.

\begin{equation}
q < n - 2k - k^2/n
\end{equation}

Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero.  The value of $x$ after step 3 is bounded trivially as
$0 \le x < n$.  By step four the sum $x + q$ is bounded by 

\begin{equation}
0 \le q + x < (k + 1)n - 2k^2 - 1
\end{equation}

With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3.  After the second pass it is highly unlike that the
sum in step 4 will exceed $n - k$.  In practice fewer than three passes of the algorithm are required to reduce virtually every input in the 
range $0 \le x < (n - k - 1)^2$.  

\begin{figure}
\begin{small}
\begin{center}
\begin{tabular}{|l|}
\hline
$x = 123456789, n = 256, k = 3$ \\
\hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\
$q \leftarrow q*k = 1446759$ \\
$x \leftarrow x \mbox{ mod } n = 21$ \\
$x \leftarrow x + q = 1446780$ \\
$x \leftarrow x - (n - k) = 1446527$ \\
\hline 
$q \leftarrow \lfloor x/n \rfloor = 5650$ \\
$q \leftarrow q*k = 16950$ \\
$x \leftarrow x \mbox{ mod } n = 127$ \\
$x \leftarrow x + q = 17077$ \\
$x \leftarrow x - (n - k) = 16824$ \\
\hline 
$q \leftarrow \lfloor x/n \rfloor = 65$ \\
$q \leftarrow q*k = 195$ \\
$x \leftarrow x \mbox{ mod } n = 184$ \\
$x \leftarrow x + q = 379$ \\
$x \leftarrow x - (n - k) = 126$ \\
\hline
\end{tabular}







|










|






|
|












|





|







4093
4094
4095
4096
4097
4098
4099
4100
4101
4102
4103
4104
4105
4106
4107
4108
4109
4110
4111
4112
4113
4114
4115
4116
4117
4118
4119
4120
4121
4122
4123
4124
4125
4126
4127
4128
4129
4130
4131
4132
4133
4134
4135
4136
4137
4138
4139
4140
4141
4142
4143
4144
4145
\caption{Algorithm Diminished Radix Reduction}
\label{fig:DR}
\end{figure}

This algorithm will reduce $x$ modulo $n - k$ and return the residue.  If $0 \le x < (n - k)^2$ then the algorithm will loop almost always
once or twice and occasionally three times.  For simplicity sake the value of $x$ is bounded by the following simple polynomial.

\begin{equation}
0 \le x < n^2 + k^2 - 2nk
\end{equation}

The true bound is  $0 \le x < (n - k - 1)^2$ but this has quite a few more terms.  The value of $q$ after step 1 is bounded by the following.

\begin{equation}
q < n - 2k - k^2/n
\end{equation}

Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero.  The value of $x$ after step 3 is bounded trivially as
$0 \le x < n$.  By step four the sum $x + q$ is bounded by

\begin{equation}
0 \le q + x < (k + 1)n - 2k^2 - 1
\end{equation}

With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3.  After the second pass it is highly unlike that the
sum in step 4 will exceed $n - k$.  In practice fewer than three passes of the algorithm are required to reduce virtually every input in the
range $0 \le x < (n - k - 1)^2$.

\begin{figure}
\begin{small}
\begin{center}
\begin{tabular}{|l|}
\hline
$x = 123456789, n = 256, k = 3$ \\
\hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\
$q \leftarrow q*k = 1446759$ \\
$x \leftarrow x \mbox{ mod } n = 21$ \\
$x \leftarrow x + q = 1446780$ \\
$x \leftarrow x - (n - k) = 1446527$ \\
\hline
$q \leftarrow \lfloor x/n \rfloor = 5650$ \\
$q \leftarrow q*k = 16950$ \\
$x \leftarrow x \mbox{ mod } n = 127$ \\
$x \leftarrow x + q = 17077$ \\
$x \leftarrow x - (n - k) = 16824$ \\
\hline
$q \leftarrow \lfloor x/n \rfloor = 65$ \\
$q \leftarrow q*k = 195$ \\
$x \leftarrow x \mbox{ mod } n = 184$ \\
$x \leftarrow x + q = 379$ \\
$x \leftarrow x - (n - k) = 126$ \\
\hline
\end{tabular}
4165
4166
4167
4168
4169
4170
4171
4172
4173
4174
4175
4176
4177
4178
4179
4180
4181
4182
4183
4184
4185
4186
4187
4188
4189
4190
4191
4192
4193
4194
4195
4196
4197
4198
4199
4200
4201
three passes were required to find the residue $x \equiv 126$.


\subsection{Choice of Moduli}
On the surface this algorithm looks like a very expensive algorithm.  It requires a couple of subtractions followed by multiplication and other
modular reductions.  The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen.

Division in general is a very expensive operation to perform.  The one exception is when the division is by a power of the radix of representation used.  
Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right.  Similarly division 
by two (\textit{or powers of two}) is very simple for binary computers to perform.  It would therefore seem logical to choose $n$ of the form $2^p$ 
which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits.  

However, there is one operation related to division of power of twos that is even faster than this.  If $n = \beta^p$ then the division may be 
performed by moving whole digits to the right $p$ places.  In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$.  
Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$.  

Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted
modulus'' will refer to a modulus of the form $2^p - k$.  The word ``restricted'' in this case refers to the fact that it is based on the 
$2^p$ logic except $p$ must be a multiple of $lg(\beta)$.  

\subsection{Choice of $k$}
Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$
in step 2 is the most expensive operation.  Fortunately the choice of $k$ is not terribly limited.  For all intents and purposes it might
as well be a single digit.  The smaller the value of $k$ is the faster the algorithm will be.  

\subsection{Restricted Diminished Radix Reduction}
The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$.  This algorithm can reduce 
an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}.  The implementation
of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition 
of $x$ and $q$.  The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular 
exponentiations are performed.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_dr\_reduce}. \\







|
|
|
|

|
|
|


|
|




|


|

|
|







4154
4155
4156
4157
4158
4159
4160
4161
4162
4163
4164
4165
4166
4167
4168
4169
4170
4171
4172
4173
4174
4175
4176
4177
4178
4179
4180
4181
4182
4183
4184
4185
4186
4187
4188
4189
4190
three passes were required to find the residue $x \equiv 126$.


\subsection{Choice of Moduli}
On the surface this algorithm looks like a very expensive algorithm.  It requires a couple of subtractions followed by multiplication and other
modular reductions.  The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen.

Division in general is a very expensive operation to perform.  The one exception is when the division is by a power of the radix of representation used.
Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right.  Similarly division
by two (\textit{or powers of two}) is very simple for binary computers to perform.  It would therefore seem logical to choose $n$ of the form $2^p$
which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits.

However, there is one operation related to division of power of twos that is even faster than this.  If $n = \beta^p$ then the division may be
performed by moving whole digits to the right $p$ places.  In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$.
Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$.

Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted
modulus'' will refer to a modulus of the form $2^p - k$.  The word ``restricted'' in this case refers to the fact that it is based on the
$2^p$ logic except $p$ must be a multiple of $lg(\beta)$.

\subsection{Choice of $k$}
Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$
in step 2 is the most expensive operation.  Fortunately the choice of $k$ is not terribly limited.  For all intents and purposes it might
as well be a single digit.  The smaller the value of $k$ is the faster the algorithm will be.

\subsection{Restricted Diminished Radix Reduction}
The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$.  This algorithm can reduce
an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}.  The implementation
of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition
of $x$ and $q$.  The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular
exponentiations are performed.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_dr\_reduce}. \\
4223
4224
4225
4226
4227
4228
4229
4230
4231
4232
4233
4234
4235
4236
4237
4238
4239
4240
4241
4242
4243
4244
4245
4246
4247
4248
4249
4250
4251
4252
4253
4254
4255
4256
4257
4258
4259
4260
4261
\end{center}
\end{small}
\caption{Algorithm mp\_dr\_reduce}
\end{figure}

\textbf{Algorithm mp\_dr\_reduce.}
This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$.  It has similar restrictions to that of the Barrett reduction
with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$.  

This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization.  The division by $\beta^m$, multiplication by $k$
and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4.  The division by $\beta^m$ is emulated by accessing
the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position.  After the loop the $m$'th
digit is set to the carry and the upper digits are zeroed.  Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to 
$x$ before the addition of the multiple of the upper half.  

At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required.  First $n$ is subtracted from $x$ and then the algorithm resumes
at step 3.  

EXAM,bn_mp_dr_reduce.c

The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$.  The label on line @49,top:@ is where
the algorithm will resume if further reduction passes are required.  In theory it could be placed at the top of the function however, the size of
the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time.  

The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits.  By reading digits from $x$ offset by $m$ digits
a division by $\beta^m$ can be simulated virtually for free.  The loop on line @61,for@ performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11})
in this algorithm.

By line @68,mu@ the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed.  Similarly by line @71,for@ the 
same pointer will point to the $m+1$'th digit where the zeroes will be placed.  

Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required.  
With the same logic at line @82,sub@ the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used
as well.  Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code
does not need to be checked.

\subsubsection{Setup}
To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required.  This algorithm is not really complicated but provided for
completeness.







|




|
|


|





|





|
|

|







4212
4213
4214
4215
4216
4217
4218
4219
4220
4221
4222
4223
4224
4225
4226
4227
4228
4229
4230
4231
4232
4233
4234
4235
4236
4237
4238
4239
4240
4241
4242
4243
4244
4245
4246
4247
4248
4249
4250
\end{center}
\end{small}
\caption{Algorithm mp\_dr\_reduce}
\end{figure}

\textbf{Algorithm mp\_dr\_reduce.}
This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$.  It has similar restrictions to that of the Barrett reduction
with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$.

This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization.  The division by $\beta^m$, multiplication by $k$
and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4.  The division by $\beta^m$ is emulated by accessing
the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position.  After the loop the $m$'th
digit is set to the carry and the upper digits are zeroed.  Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to
$x$ before the addition of the multiple of the upper half.

At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required.  First $n$ is subtracted from $x$ and then the algorithm resumes
at step 3.

EXAM,bn_mp_dr_reduce.c

The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$.  The label on line @49,top:@ is where
the algorithm will resume if further reduction passes are required.  In theory it could be placed at the top of the function however, the size of
the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time.

The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits.  By reading digits from $x$ offset by $m$ digits
a division by $\beta^m$ can be simulated virtually for free.  The loop on line @61,for@ performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11})
in this algorithm.

By line @68,mu@ the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed.  Similarly by line @71,for@ the
same pointer will point to the $m+1$'th digit where the zeroes will be placed.

Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required.
With the same logic at line @82,sub@ the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used
as well.  Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code
does not need to be checked.

\subsubsection{Setup}
To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required.  This algorithm is not really complicated but provided for
completeness.
4338
4339
4340
4341
4342
4343
4344
4345
4346
4347
4348
4349
4350
4351
4352
4353
4354
4355
4356
4357
4358
4359
4360
4361
4362
4363
4364
4365
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_2k}
\end{figure}

\textbf{Algorithm mp\_reduce\_2k.}
This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$.  Division by $2^p$ is emulated with a right
shift which makes the algorithm fairly inexpensive to use.  

EXAM,bn_mp_reduce_2k.c

The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$.  The call to mp\_div\_2d
on line @31,mp_div_2d@ calculates both the quotient $q$ and the remainder $a$ required.  By doing both in a single function call the code size
is kept fairly small.  The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without
any multiplications.  

The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are 
positive.  By using the unsigned versions the overhead is kept to a minimum.  

\subsubsection{Unrestricted Setup}
To setup this reduction algorithm the value of $k = 2^p - n$ is required.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\
\textbf{Input}.   mp\_int $n$   \\







|






|

|
|


|







4327
4328
4329
4330
4331
4332
4333
4334
4335
4336
4337
4338
4339
4340
4341
4342
4343
4344
4345
4346
4347
4348
4349
4350
4351
4352
4353
4354
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_2k}
\end{figure}

\textbf{Algorithm mp\_reduce\_2k.}
This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$.  Division by $2^p$ is emulated with a right
shift which makes the algorithm fairly inexpensive to use.

EXAM,bn_mp_reduce_2k.c

The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$.  The call to mp\_div\_2d
on line @31,mp_div_2d@ calculates both the quotient $q$ and the remainder $a$ required.  By doing both in a single function call the code size
is kept fairly small.  The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without
any multiplications.

The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are
positive.  By using the unsigned versions the overhead is kept to a minimum.

\subsubsection{Unrestricted Setup}
To setup this reduction algorithm the value of $k = 2^p - n$ is required.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\
\textbf{Input}.   mp\_int $n$   \\
4375
4376
4377
4378
4379
4380
4381
4382
4383
4384
4385
4386
4387
4388
4389
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_2k\_setup}
\end{figure}

\textbf{Algorithm mp\_reduce\_2k\_setup.}
This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k.  By making a temporary variable $x$ equal to $2^p$ a subtraction
is sufficient to solve for $k$.  Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$.  

EXAM,bn_mp_reduce_2k_setup.c

\subsubsection{Unrestricted Detection}
An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true.

\begin{enumerate}







|







4364
4365
4366
4367
4368
4369
4370
4371
4372
4373
4374
4375
4376
4377
4378
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_2k\_setup}
\end{figure}

\textbf{Algorithm mp\_reduce\_2k\_setup.}
This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k.  By making a temporary variable $x$ equal to $2^p$ a subtraction
is sufficient to solve for $k$.  Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$.

EXAM,bn_mp_reduce_2k_setup.c

\subsubsection{Unrestricted Detection}
An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true.

\begin{enumerate}
4414
4415
4416
4417
4418
4419
4420
4421
4422
4423
4424
4425
4426
4427
4428
4429
4430
4431
4432
4433
4434
4435
4436
4437
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_is\_2k}
\end{figure}

\textbf{Algorithm mp\_reduce\_is\_2k.}
This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly.  

EXAM,bn_mp_reduce_is_2k.c



\section{Algorithm Comparison}
So far three very different algorithms for modular reduction have been discussed.  Each of the algorithms have their own strengths and weaknesses
that makes having such a selection very useful.  The following table sumarizes the three algorithms along with comparisons of work factors.  Since
all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table.  

\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\
\hline Barrett    & $m^2 + 2m - 1$ & None              & $79$ & $1087$ & $4223$ \\
\hline Montgomery & $m^2 + m$      & $n$ must be odd   & $72$ & $1056$ & $4160$ \\







|








|







4403
4404
4405
4406
4407
4408
4409
4410
4411
4412
4413
4414
4415
4416
4417
4418
4419
4420
4421
4422
4423
4424
4425
4426
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_is\_2k}
\end{figure}

\textbf{Algorithm mp\_reduce\_is\_2k.}
This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly.

EXAM,bn_mp_reduce_is_2k.c



\section{Algorithm Comparison}
So far three very different algorithms for modular reduction have been discussed.  Each of the algorithms have their own strengths and weaknesses
that makes having such a selection very useful.  The following table sumarizes the three algorithms along with comparisons of work factors.  Since
all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table.

\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\
\hline Barrett    & $m^2 + 2m - 1$ & None              & $79$ & $1087$ & $4223$ \\
\hline Montgomery & $m^2 + m$      & $n$ must be odd   & $72$ & $1056$ & $4160$ \\
4459
4460
4461
4462
4463
4464
4465
4466
4467
4468
4469
4470
4471
4472
4473
4474
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4477
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4479
4480
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4482
4483
4484
4485
4486
4487
4488
4489
4490
4491
4492
4493
4494
4495
4496
4497
4498
4499
4500
4501
4502
4503
4504
4505
                     & \\
$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly.  \\
                     & \\
$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\
                     & (\textit{figure~\ref{fig:DR}}) terminates.  Also prove the probability that it will \\
                     & terminate within $1 \le k \le 10$ iterations. \\
                     & \\
\end{tabular}                     


\chapter{Exponentiation}
Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$.  A variant of exponentiation, computed
in a finite field or ring, is called modular exponentiation.  This latter style of operation is typically used in public key 
cryptosystems such as RSA and Diffie-Hellman.  The ability to quickly compute modular exponentiations is of great benefit to any
such cryptosystem and many methods have been sought to speed it up.

\section{Exponentiation Basics}
A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired.  However, as $b$ grows in size
the number of multiplications becomes prohibitive.  Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature
with a $1024$-bit key.  Such a calculation could never be completed as it would take simply far too long.

Fortunately there is a very simple algorithm based on the laws of exponents.  Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which
are two trivial relationships between the base and the exponent.  Let $b_i$ represent the $i$'th bit of $b$ starting from the least 
significant bit.  If $b$ is a $k$-bit integer than the following equation is true.

\begin{equation}
a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i}
\end{equation}

By taking the base $a$ logarithm of both sides of the equation the following equation is the result.

\begin{equation}
b = \sum_{i=0}^{k-1}2^i \cdot b_i
\end{equation}

The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to
$a^{2^{i+1}}$.  This observation forms the basis of essentially all fast exponentiation algorithms.  It requires $k$ squarings and on average
$k \over 2$ multiplications to compute the result.  This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times.

While this current method is a considerable speed up there are further improvements to be made.  For example, the $a^{2^i}$ term does not need to 
be computed in an auxilary variable.  Consider the following equivalent algorithm.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Left to Right Exponentiation}. \\







|




|









|
















|







4448
4449
4450
4451
4452
4453
4454
4455
4456
4457
4458
4459
4460
4461
4462
4463
4464
4465
4466
4467
4468
4469
4470
4471
4472
4473
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4475
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4480
4481
4482
4483
4484
4485
4486
4487
4488
4489
4490
4491
4492
4493
4494
                     & \\
$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly.  \\
                     & \\
$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\
                     & (\textit{figure~\ref{fig:DR}}) terminates.  Also prove the probability that it will \\
                     & terminate within $1 \le k \le 10$ iterations. \\
                     & \\
\end{tabular}


\chapter{Exponentiation}
Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$.  A variant of exponentiation, computed
in a finite field or ring, is called modular exponentiation.  This latter style of operation is typically used in public key
cryptosystems such as RSA and Diffie-Hellman.  The ability to quickly compute modular exponentiations is of great benefit to any
such cryptosystem and many methods have been sought to speed it up.

\section{Exponentiation Basics}
A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired.  However, as $b$ grows in size
the number of multiplications becomes prohibitive.  Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature
with a $1024$-bit key.  Such a calculation could never be completed as it would take simply far too long.

Fortunately there is a very simple algorithm based on the laws of exponents.  Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which
are two trivial relationships between the base and the exponent.  Let $b_i$ represent the $i$'th bit of $b$ starting from the least
significant bit.  If $b$ is a $k$-bit integer than the following equation is true.

\begin{equation}
a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i}
\end{equation}

By taking the base $a$ logarithm of both sides of the equation the following equation is the result.

\begin{equation}
b = \sum_{i=0}^{k-1}2^i \cdot b_i
\end{equation}

The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to
$a^{2^{i+1}}$.  This observation forms the basis of essentially all fast exponentiation algorithms.  It requires $k$ squarings and on average
$k \over 2$ multiplications to compute the result.  This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times.

While this current method is a considerable speed up there are further improvements to be made.  For example, the $a^{2^i}$ term does not need to
be computed in an auxilary variable.  Consider the following equivalent algorithm.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Left to Right Exponentiation}. \\
4517
4518
4519
4520
4521
4522
4523
4524
4525
4526
4527
4528
4529
4530
4531
4532
4533
4534
4535
4536
4537
4538
4539
4540
4541
4542
4543
4544
4545
4546
4547
4548
4549
4550
4551
4552
4553
4554
4555
4556
4557
4558
\end{small}
\caption{Left to Right Exponentiation}
\label{fig:LTOR}
\end{figure}

This algorithm starts from the most significant bit and works towards the least significant bit.  When the $i$'th bit of $b$ is set $a$ is
multiplied against the current product.  In each iteration the product is squared which doubles the exponent of the individual terms of the
product.  

For example, let $b = 101100_2 \equiv 44_{10}$.  The following chart demonstrates the actions of the algorithm.

\newpage\begin{figure}
\begin{center}
\begin{tabular}{|c|c|}
\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\
\hline - & $1$ \\
\hline $5$ & $a$ \\
\hline $4$ & $a^2$ \\
\hline $3$ & $a^4 \cdot a$ \\
\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\
\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\
\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\
\hline
\end{tabular}
\end{center}
\caption{Example of Left to Right Exponentiation}
\end{figure}

When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation.  This particular algorithm is 
called ``Left to Right'' because it reads the exponent in that order.  All of the exponentiation algorithms that will be presented are of this nature.  

\subsection{Single Digit Exponentiation}
The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit.  It is intended 
to be used when a small power of an input is required (\textit{e.g. $a^5$}).  It is faster than simply multiplying $b - 1$ times for all values of 
$b$ that are greater than three.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_expt\_d}. \\
\textbf{Input}.   mp\_int $a$ and mp\_digit $b$ \\







|




















|
|


|
|
|







4506
4507
4508
4509
4510
4511
4512
4513
4514
4515
4516
4517
4518
4519
4520
4521
4522
4523
4524
4525
4526
4527
4528
4529
4530
4531
4532
4533
4534
4535
4536
4537
4538
4539
4540
4541
4542
4543
4544
4545
4546
4547
\end{small}
\caption{Left to Right Exponentiation}
\label{fig:LTOR}
\end{figure}

This algorithm starts from the most significant bit and works towards the least significant bit.  When the $i$'th bit of $b$ is set $a$ is
multiplied against the current product.  In each iteration the product is squared which doubles the exponent of the individual terms of the
product.

For example, let $b = 101100_2 \equiv 44_{10}$.  The following chart demonstrates the actions of the algorithm.

\newpage\begin{figure}
\begin{center}
\begin{tabular}{|c|c|}
\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\
\hline - & $1$ \\
\hline $5$ & $a$ \\
\hline $4$ & $a^2$ \\
\hline $3$ & $a^4 \cdot a$ \\
\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\
\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\
\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\
\hline
\end{tabular}
\end{center}
\caption{Example of Left to Right Exponentiation}
\end{figure}

When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation.  This particular algorithm is
called ``Left to Right'' because it reads the exponent in that order.  All of the exponentiation algorithms that will be presented are of this nature.

\subsection{Single Digit Exponentiation}
The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit.  It is intended
to be used when a small power of an input is required (\textit{e.g. $a^5$}).  It is faster than simply multiplying $b - 1$ times for all values of
$b$ that are greater than three.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_expt\_d}. \\
\textbf{Input}.   mp\_int $a$ and mp\_digit $b$ \\
4572
4573
4574
4575
4576
4577
4578
4579
4580
4581
4582
4583
4584
4585
4586
4587
4588
4589
4590
4591
4592
4593
4594
4595
4596
4597
4598
4599
4600
4601
4602
\end{center}
\end{small}
\caption{Algorithm mp\_expt\_d}
\end{figure}

\textbf{Algorithm mp\_expt\_d.}
This algorithm computes the value of $a$ raised to the power of a single digit $b$.  It uses the left to right exponentiation algorithm to
quickly compute the exponentiation.  It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the 
exponent is a fixed width.  

A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$.  The result is set to the initial value of 
$1$ in the subsequent step.

Inside the loop the exponent is read from the most significant bit first down to the least significant bit.  First $c$ is invariably squared
on step 3.1.  In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$.  The value
of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit.  In effect each
iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location.

EXAM,bn_mp_expt_d.c

Line @29,mp_set@ sets the initial value of the result to $1$.  Next the loop on line @31,for@ steps through each bit of the exponent starting from
the most significant down towards the least significant. The invariant squaring operation placed on line @333,mp_sqr@ is performed first.  After 
the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set.  The shift on line
@47,<<@ moves all of the bits of the exponent upwards towards the most significant location.  

\section{$k$-ary Exponentiation}
When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor
slower than squaring.  Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$.  Suppose instead it referred to
the $i$'th $k$-bit digit of the exponent of $b$.  For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY}
computes the same exponentiation.  A group of $k$ bits from the exponent is called a \textit{window}.  That is it is a small window on only a
portion of the entire exponent.  Consider the following modification to the basic left to right exponentiation algorithm.







|
|

|







|

|
|

|







4561
4562
4563
4564
4565
4566
4567
4568
4569
4570
4571
4572
4573
4574
4575
4576
4577
4578
4579
4580
4581
4582
4583
4584
4585
4586
4587
4588
4589
4590
4591
\end{center}
\end{small}
\caption{Algorithm mp\_expt\_d}
\end{figure}

\textbf{Algorithm mp\_expt\_d.}
This algorithm computes the value of $a$ raised to the power of a single digit $b$.  It uses the left to right exponentiation algorithm to
quickly compute the exponentiation.  It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the
exponent is a fixed width.

A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$.  The result is set to the initial value of
$1$ in the subsequent step.

Inside the loop the exponent is read from the most significant bit first down to the least significant bit.  First $c$ is invariably squared
on step 3.1.  In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$.  The value
of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit.  In effect each
iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location.

EXAM,bn_mp_expt_d_ex.c

This describes only the algorithm that is used when the parameter $fast$ is $0$.  Line @31,mp_set@ sets the initial value of the result to $1$.  Next the loop on line @54,for@ steps through each bit of the exponent starting from
the most significant down towards the least significant. The invariant squaring operation placed on line @333,mp_sqr@ is performed first.  After
the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set.  The shift on line
@69,<<@ moves all of the bits of the exponent upwards towards the most significant location.

\section{$k$-ary Exponentiation}
When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor
slower than squaring.  Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$.  Suppose instead it referred to
the $i$'th $k$-bit digit of the exponent of $b$.  For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY}
computes the same exponentiation.  A group of $k$ bits from the exponent is called a \textit{window}.  That is it is a small window on only a
portion of the entire exponent.  Consider the following modification to the basic left to right exponentiation algorithm.
4621
4622
4623
4624
4625
4626
4627
4628
4629
4630
4631
4632
4633
4634
4635
4636
4637
4638
4639
4640
4641
4642
4643
4644
4645
\end{small}
\caption{$k$-ary Exponentiation}
\label{fig:KARY}
\end{figure}

The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times.  If the values of $a^g$ for $0 < g < 2^k$ have been
precomputed this algorithm requires only $t$ multiplications and $tk$ squarings.  The table can be generated with $2^{k - 1} - 1$ squarings and
$2^{k - 1} + 1$ multiplications.  This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$.  
However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}.

Suppose $k = 4$ and $t = 100$.  This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation.  The
original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value.  The total number of squarings
has increased slightly but the number of multiplications has nearly halved.

\subsection{Optimal Values of $k$}
An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$.  The simplest
approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result.  Table~\ref{fig:OPTK} lists optimal values of $k$
for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}.  

\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\
\hline $16$ & $2$ & $27$ & $24$ \\







|









|







4610
4611
4612
4613
4614
4615
4616
4617
4618
4619
4620
4621
4622
4623
4624
4625
4626
4627
4628
4629
4630
4631
4632
4633
4634
\end{small}
\caption{$k$-ary Exponentiation}
\label{fig:KARY}
\end{figure}

The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times.  If the values of $a^g$ for $0 < g < 2^k$ have been
precomputed this algorithm requires only $t$ multiplications and $tk$ squarings.  The table can be generated with $2^{k - 1} - 1$ squarings and
$2^{k - 1} + 1$ multiplications.  This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$.
However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}.

Suppose $k = 4$ and $t = 100$.  This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation.  The
original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value.  The total number of squarings
has increased slightly but the number of multiplications has nearly halved.

\subsection{Optimal Values of $k$}
An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$.  The simplest
approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result.  Table~\ref{fig:OPTK} lists optimal values of $k$
for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}.

\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\
\hline $16$ & $2$ & $27$ & $24$ \\
4657
4658
4659
4660
4661
4662
4663
4664
4665
4666
4667
4668
4669
4670
4671
4672
4673
4674
\end{center}
\caption{Optimal Values of $k$ for $k$-ary Exponentiation}
\label{fig:OPTK}
\end{figure}

\subsection{Sliding-Window Exponentiation}
A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$.  Essentially
this is a table for all values of $g$ where the most significant bit of $g$ is a one.  However, in order for this to be allowed in the 
algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided.  

Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}.  

\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\
\hline $16$ & $3$ & $24$ & $27$ \\







|
|

|







4646
4647
4648
4649
4650
4651
4652
4653
4654
4655
4656
4657
4658
4659
4660
4661
4662
4663
\end{center}
\caption{Optimal Values of $k$ for $k$-ary Exponentiation}
\label{fig:OPTK}
\end{figure}

\subsection{Sliding-Window Exponentiation}
A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$.  Essentially
this is a table for all values of $g$ where the most significant bit of $g$ is a one.  However, in order for this to be allowed in the
algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided.

Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm {\ref{fig:KARY}}.

\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\
\hline $16$ & $3$ & $24$ & $27$ \\
4711
4712
4713
4714
4715
4716
4717
4718
4719
4720
4721
4722
4723
4724
4725
4726
4727
4728
4729
4730
4731
4732
4733
4734
4735
4736
4737
4738
4739
4740
4741
4742
4743
4744
4745
4746
4747
\end{center}
\end{small}
\caption{Sliding Window $k$-ary Exponentiation}
\end{figure}

Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent.  While this
algorithm requires the same number of squarings it can potentially have fewer multiplications.  The pre-computed table $a^g$ is also half
the size as the previous table.  

Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms.  The first algorithm will divide the exponent up as 
the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$.  The second algorithm will break the 
exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$.  The single digit $0$ in the second representation are where
a single squaring took place instead of a squaring and multiplication.  In total the first method requires $10$ multiplications and $18$ 
squarings.  The second method requires $8$ multiplications and $18$ squarings.  

In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster.  

\section{Modular Exponentiation}

Modular exponentiation is essentially computing the power of a base within a finite field or ring.  For example, computing 
$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation.  Instead of first computing $a^b$ and then reducing it 
modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation.  

This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using
one of the algorithms presented in ~REDUCTION~.  

Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first.  This algorithm
will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The
value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}).  If no inverse exists the algorithm
terminates with an error.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_exptmod}. \\
\textbf{Input}.   mp\_int $a$, $b$ and $c$ \\







|

|
|

|
|

|



|
|
|


|




|







4700
4701
4702
4703
4704
4705
4706
4707
4708
4709
4710
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4713
4714
4715
4716
4717
4718
4719
4720
4721
4722
4723
4724
4725
4726
4727
4728
4729
4730
4731
4732
4733
4734
4735
4736
\end{center}
\end{small}
\caption{Sliding Window $k$-ary Exponentiation}
\end{figure}

Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent.  While this
algorithm requires the same number of squarings it can potentially have fewer multiplications.  The pre-computed table $a^g$ is also half
the size as the previous table.

Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms.  The first algorithm will divide the exponent up as
the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$.  The second algorithm will break the
exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$.  The single digit $0$ in the second representation are where
a single squaring took place instead of a squaring and multiplication.  In total the first method requires $10$ multiplications and $18$
squarings.  The second method requires $8$ multiplications and $18$ squarings.

In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster.

\section{Modular Exponentiation}

Modular exponentiation is essentially computing the power of a base within a finite field or ring.  For example, computing
$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation.  Instead of first computing $a^b$ and then reducing it
modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation.

This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using
one of the algorithms presented in ~REDUCTION~.

Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first.  This algorithm
will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The
value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}).  If no inverse exists the algorithm
terminates with an error.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_exptmod}. \\
\textbf{Input}.   mp\_int $a$, $b$ and $c$ \\
4760
4761
4762
4763
4764
4765
4766
4767
4768
4769
4770
4771
4772
4773
4774
4775
4776
4777
4778
4779
4780
4781
4782
4783
4784
4785
4786
4787
4788
4789
4790
4791
4792
4793
4794
4795
4796
4797
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_exptmod}
\end{figure}

\textbf{Algorithm mp\_exptmod.}
The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod.  It is a sliding window $k$-ary algorithm 
which uses Barrett reduction to reduce the product modulo $p$.  The second algorithm mp\_exptmod\_fast performs the same operation 
except it uses either Montgomery or Diminished Radix reduction.  The two latter reduction algorithms are clumped in the same exponentiation
algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}).  

EXAM,bn_mp_exptmod.c

In order to keep the algorithms in a known state the first step on line @29,if@ is to reject any negative modulus as input.  If the exponent is
negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$.  The temporary variable $tmpG$ is assigned
the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$.  The algorithm will recuse with these new values with a positive
exponent.

If the exponent is positive the algorithm resumes the exponentiation.  Line @63,dr_@ determines if the modulus is of the restricted Diminished Radix 
form.  If it is not line @65,reduce@ attempts to determine if it is of a unrestricted Diminished Radix form.  The integer $dr$ will take on one
of three values.

\begin{enumerate}
\item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form.
\item $dr = 1$ means that the modulus is of restricted Diminished Radix form.
\item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form.
\end{enumerate}

Line @69,if@ determines if the fast modular exponentiation algorithm can be used.  It is allowed if $dr \ne 0$ or if the modulus is odd.  Otherwise,
the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction.  

\subsection{Barrett Modular Exponentiation}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}







|
|

|








|










|







4749
4750
4751
4752
4753
4754
4755
4756
4757
4758
4759
4760
4761
4762
4763
4764
4765
4766
4767
4768
4769
4770
4771
4772
4773
4774
4775
4776
4777
4778
4779
4780
4781
4782
4783
4784
4785
4786
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_exptmod}
\end{figure}

\textbf{Algorithm mp\_exptmod.}
The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod.  It is a sliding window $k$-ary algorithm
which uses Barrett reduction to reduce the product modulo $p$.  The second algorithm mp\_exptmod\_fast performs the same operation
except it uses either Montgomery or Diminished Radix reduction.  The two latter reduction algorithms are clumped in the same exponentiation
algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}).

EXAM,bn_mp_exptmod.c

In order to keep the algorithms in a known state the first step on line @29,if@ is to reject any negative modulus as input.  If the exponent is
negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$.  The temporary variable $tmpG$ is assigned
the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$.  The algorithm will recuse with these new values with a positive
exponent.

If the exponent is positive the algorithm resumes the exponentiation.  Line @63,dr_@ determines if the modulus is of the restricted Diminished Radix
form.  If it is not line @65,reduce@ attempts to determine if it is of a unrestricted Diminished Radix form.  The integer $dr$ will take on one
of three values.

\begin{enumerate}
\item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form.
\item $dr = 1$ means that the modulus is of restricted Diminished Radix form.
\item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form.
\end{enumerate}

Line @69,if@ determines if the fast modular exponentiation algorithm can be used.  It is allowed if $dr \ne 0$ or if the modulus is odd.  Otherwise,
the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction.

\subsection{Barrett Modular Exponentiation}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
4888
4889
4890
4891
4892
4893
4894
4895
4896
4897
4898
4899
4900
4901
4902
4903
4904
4905
4906
4907
4908
4909
4910
4911
4912
4913
4914
4915
4916
4917
4918
4919
4920
4921
4922
4923
4924
4925
4926
4927
4928
4929
4930
4931
4932
4933
4934
4935
4936
4937
4938
4939
4940
4941
4942
4943
4944
4945
4946
4947
4948
4949
4950
4951
4952
4953
4954
\caption{Algorithm s\_mp\_exptmod (continued)}
\end{figure}

\textbf{Algorithm s\_mp\_exptmod.}
This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$.  It takes advantage of the Barrett reduction
algorithm to keep the product small throughout the algorithm.

The first two steps determine the optimal window size based on the number of bits in the exponent.  The larger the exponent the 
larger the window size becomes.  After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated.  This
table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$.  

After the table is allocated the first power of $g$ is found.  Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make
the rest of the algorithm more efficient.  The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$
times.  The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$.

Now that the table is available the sliding window may begin.  The following list describes the functions of all the variables in the window.
\begin{enumerate}
\item The variable $mode$ dictates how the bits of the exponent are interpreted.  
\begin{enumerate}
   \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet.  For example, if the exponent were simply 
         $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit.  In this case bits are ignored until a non-zero bit is found.  
   \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet.  In this mode leading $0$ bits 
         are read and a single squaring is performed.  If a non-zero bit is read a new window is created.  
   \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit
         downwards.
\end{enumerate}
\item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read.  When it reaches zero a new digit
      is fetched from the exponent.
\item The variable $buf$ holds the currently read digit of the exponent. 
\item The variable $digidx$ is an index into the exponents digits.  It starts at the leading digit $x.used - 1$ and moves towards the trailing digit.
\item The variable $bitcpy$ indicates how many bits are in the currently formed window.  When it reaches $winsize$ the window is flushed and
      the appropriate operations performed.
\item The variable $bitbuf$ holds the current bits of the window being formed.  
\end{enumerate}

All of step 12 is the window processing loop.  It will iterate while there are digits available form the exponent to read.  The first step
inside this loop is to extract a new digit if no more bits are available in the current digit.  If there are no bits left a new digit is
read and if there are no digits left than the loop terminates.  

After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit
upwards.  In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to 
trailing edges the entire exponent is read from most significant bit to least significant bit.

At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read.  This prevents the 
algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read.  Step 12.6 and 12.7-10 handle
the two cases of $mode = 1$ and $mode = 2$ respectively.  

FIGU,expt_state,Sliding Window State Diagram

By step 13 there are no more digits left in the exponent.  However, there may be partial bits in the window left.  If $mode = 2$ then 
a Left-to-Right algorithm is used to process the remaining few bits.  

EXAM,bn_s_mp_exptmod.c

Lines @31,if@ through @45,}@ determine the optimal window size based on the length of the exponent in bits.  The window divisions are sorted
from smallest to greatest so that in each \textbf{if} statement only one condition must be tested.  For example, by the \textbf{if} statement 
on line @37,if@ the value of $x$ is already known to be greater than $140$.  

The conditional piece of code beginning on line @42,ifdef@ allows the window size to be restricted to five bits.  This logic is used to ensure
the table of precomputed powers of $G$ remains relatively small.  

The for loop on line @60,for@ initializes the $M$ array while lines @71,mp_init@ and @75,mp_reduce@ through @85,}@ initialize the reduction
function that will be used for this modulus.

-- More later.

\section{Quick Power of Two}







|

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|





|



|




|


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|



|
|




|
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|







4877
4878
4879
4880
4881
4882
4883
4884
4885
4886
4887
4888
4889
4890
4891
4892
4893
4894
4895
4896
4897
4898
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4900
4901
4902
4903
4904
4905
4906
4907
4908
4909
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4933
4934
4935
4936
4937
4938
4939
4940
4941
4942
4943
\caption{Algorithm s\_mp\_exptmod (continued)}
\end{figure}

\textbf{Algorithm s\_mp\_exptmod.}
This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$.  It takes advantage of the Barrett reduction
algorithm to keep the product small throughout the algorithm.

The first two steps determine the optimal window size based on the number of bits in the exponent.  The larger the exponent the
larger the window size becomes.  After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated.  This
table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$.

After the table is allocated the first power of $g$ is found.  Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make
the rest of the algorithm more efficient.  The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$
times.  The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$.

Now that the table is available the sliding window may begin.  The following list describes the functions of all the variables in the window.
\begin{enumerate}
\item The variable $mode$ dictates how the bits of the exponent are interpreted.
\begin{enumerate}
   \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet.  For example, if the exponent were simply
         $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit.  In this case bits are ignored until a non-zero bit is found.
   \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet.  In this mode leading $0$ bits
         are read and a single squaring is performed.  If a non-zero bit is read a new window is created.
   \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit
         downwards.
\end{enumerate}
\item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read.  When it reaches zero a new digit
      is fetched from the exponent.
\item The variable $buf$ holds the currently read digit of the exponent.
\item The variable $digidx$ is an index into the exponents digits.  It starts at the leading digit $x.used - 1$ and moves towards the trailing digit.
\item The variable $bitcpy$ indicates how many bits are in the currently formed window.  When it reaches $winsize$ the window is flushed and
      the appropriate operations performed.
\item The variable $bitbuf$ holds the current bits of the window being formed.
\end{enumerate}

All of step 12 is the window processing loop.  It will iterate while there are digits available form the exponent to read.  The first step
inside this loop is to extract a new digit if no more bits are available in the current digit.  If there are no bits left a new digit is
read and if there are no digits left than the loop terminates.

After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit
upwards.  In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to
trailing edges the entire exponent is read from most significant bit to least significant bit.

At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read.  This prevents the
algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read.  Step 12.6 and 12.7-10 handle
the two cases of $mode = 1$ and $mode = 2$ respectively.

FIGU,expt_state,Sliding Window State Diagram

By step 13 there are no more digits left in the exponent.  However, there may be partial bits in the window left.  If $mode = 2$ then
a Left-to-Right algorithm is used to process the remaining few bits.

EXAM,bn_s_mp_exptmod.c

Lines @31,if@ through @45,}@ determine the optimal window size based on the length of the exponent in bits.  The window divisions are sorted
from smallest to greatest so that in each \textbf{if} statement only one condition must be tested.  For example, by the \textbf{if} statement
on line @37,if@ the value of $x$ is already known to be greater than $140$.

The conditional piece of code beginning on line @42,ifdef@ allows the window size to be restricted to five bits.  This logic is used to ensure
the table of precomputed powers of $G$ remains relatively small.

The for loop on line @60,for@ initializes the $M$ array while lines @71,mp_init@ and @75,mp_reduce@ through @85,}@ initialize the reduction
function that will be used for this modulus.

-- More later.

\section{Quick Power of Two}
4978
4979
4980
4981
4982
4983
4984
4985
4986
4987
4988
4989
4990
4991
4992
4993
4994
4995
4996
4997
4998
4999
5000
5001
5002
5003
5004
\textbf{Algorithm mp\_2expt.}

EXAM,bn_mp_2expt.c

\chapter{Higher Level Algorithms}

This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package.  These
routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important.  

The first section describes a method of integer division with remainder that is universally well known.  It provides the signed division logic
for the package.  The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations.  
These algorithms serve mostly to simplify other algorithms where small constants are required.  The last two sections discuss how to manipulate 
various representations of integers.  For example, converting from an mp\_int to a string of character.

\section{Integer Division with Remainder}
\label{sec:division}

Integer division aside from modular exponentiation is the most intensive algorithm to compute.  Like addition, subtraction and multiplication
the basis of this algorithm is the long-hand division algorithm taught to school children.  Throughout this discussion several common variables
will be used.  Let $x$ represent the divisor and $y$ represent the dividend.  Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and 
let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$.  The following simple algorithm will be used to start the discussion.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Radix-$\beta$ Integer Division}. \\







|


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|







4967
4968
4969
4970
4971
4972
4973
4974
4975
4976
4977
4978
4979
4980
4981
4982
4983
4984
4985
4986
4987
4988
4989
4990
4991
4992
4993
\textbf{Algorithm mp\_2expt.}

EXAM,bn_mp_2expt.c

\chapter{Higher Level Algorithms}

This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package.  These
routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important.

The first section describes a method of integer division with remainder that is universally well known.  It provides the signed division logic
for the package.  The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations.
These algorithms serve mostly to simplify other algorithms where small constants are required.  The last two sections discuss how to manipulate
various representations of integers.  For example, converting from an mp\_int to a string of character.

\section{Integer Division with Remainder}
\label{sec:division}

Integer division aside from modular exponentiation is the most intensive algorithm to compute.  Like addition, subtraction and multiplication
the basis of this algorithm is the long-hand division algorithm taught to school children.  Throughout this discussion several common variables
will be used.  Let $x$ represent the divisor and $y$ represent the dividend.  Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and
let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$.  The following simple algorithm will be used to start the discussion.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Radix-$\beta$ Integer Division}. \\
5020
5021
5022
5023
5024
5025
5026
5027
5028
5029
5030
5031
5032
5033
5034
5035
5036
5037
5038
5039
5040
5041
5042
5043
5044
5045
5046
5047
5048
5049
5050
5051
5052
5053
5054
5055
5056
5057
5058
5059
5060
5061
5062
5063
5064
5065
5066
5067
5068
5069
\caption{Algorithm Radix-$\beta$ Integer Division}
\label{fig:raddiv}
\end{figure}

As children we are taught this very simple algorithm for the case of $\beta = 10$.  Almost instinctively several optimizations are taught for which
their reason of existing are never explained.  For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor.

To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and 
simultaneously $(k + 1)x\beta^t$ is greater than $y$.  Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have.  The habitual method
used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient.  By only using leading
digits a much simpler division may be used to form an educated guess at what the value must be.  In this case $k = \lfloor 54/23\rfloor = 2$ quickly 
arises as a possible  solution.  Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$.  
As a  result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$.

Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder 
$y = 841 - 3x\beta = 181$.  Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the
remainder $y = 181 - 7x = 20$.  The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since 
$237 \cdot 23 + 20 = 5471$ is true.  

\subsection{Quotient Estimation}
\label{sec:divest}
As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend.  When $p$ leading
digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows.  Technically
speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the
dividend and divisor are zero.  

The value of the estimation may off by a few values in either direction and in general is fairly correct.  A simplification \cite[pp. 271]{TAOCPV2}
of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$.  The estimate 
using this technique is never too small.  For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$ 
represent the most significant digits of the dividend and divisor respectively.

\textbf{Proof.}\textit{  The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to 
$k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. }
The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger.  For all other 
cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$.  The latter portion of the inequalility
$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values.  Next a series of 
inequalities will prove the hypothesis.

\begin{equation}
y - \hat k x \le y - \hat k x_s\beta^s
\end{equation}

This is trivially true since $x \ge x_s\beta^s$.  Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$.  

\begin{equation}
y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s)
\end{equation}

By simplifying the previous inequality the following inequality is formed.








|


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|







5009
5010
5011
5012
5013
5014
5015
5016
5017
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5019
5020
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5043
5044
5045
5046
5047
5048
5049
5050
5051
5052
5053
5054
5055
5056
5057
5058
\caption{Algorithm Radix-$\beta$ Integer Division}
\label{fig:raddiv}
\end{figure}

As children we are taught this very simple algorithm for the case of $\beta = 10$.  Almost instinctively several optimizations are taught for which
their reason of existing are never explained.  For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor.

To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and
simultaneously $(k + 1)x\beta^t$ is greater than $y$.  Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have.  The habitual method
used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient.  By only using leading
digits a much simpler division may be used to form an educated guess at what the value must be.  In this case $k = \lfloor 54/23\rfloor = 2$ quickly
arises as a possible  solution.  Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$.
As a  result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$.

Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder
$y = 841 - 3x\beta = 181$.  Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the
remainder $y = 181 - 7x = 20$.  The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since
$237 \cdot 23 + 20 = 5471$ is true.

\subsection{Quotient Estimation}
\label{sec:divest}
As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend.  When $p$ leading
digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows.  Technically
speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the
dividend and divisor are zero.

The value of the estimation may off by a few values in either direction and in general is fairly correct.  A simplification \cite[pp. 271]{TAOCPV2}
of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$.  The estimate
using this technique is never too small.  For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$
represent the most significant digits of the dividend and divisor respectively.

\textbf{Proof.}\textit{  The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to
$k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. }
The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger.  For all other
cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$.  The latter portion of the inequalility
$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values.  Next a series of
inequalities will prove the hypothesis.

\begin{equation}
y - \hat k x \le y - \hat k x_s\beta^s
\end{equation}

This is trivially true since $x \ge x_s\beta^s$.  Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$.

\begin{equation}
y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s)
\end{equation}

By simplifying the previous inequality the following inequality is formed.

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5086
5087
5088
5089
5090
5091
5092
5093
5094
5095
5096
5097
5098
5099
5100
Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which concludes the proof.  \textbf{QED}


\subsection{Normalized Integers}
For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$.  By multiplying both
$x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original
remainder.  The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will
lie in the domain of a single digit.  Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$.  

\begin{equation} 
{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta} 
\end{equation}

At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small.  

\subsection{Radix-$\beta$ Division with Remainder}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_div}. \\







|

|
|


|







5069
5070
5071
5072
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5075
5076
5077
5078
5079
5080
5081
5082
5083
5084
5085
5086
5087
5088
5089
Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which concludes the proof.  \textbf{QED}


\subsection{Normalized Integers}
For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$.  By multiplying both
$x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original
remainder.  The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will
lie in the domain of a single digit.  Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$.

\begin{equation}
{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta}
\end{equation}

At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small.

\subsection{Radix-$\beta$ Division with Remainder}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_div}. \\
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5186
5187
5188
5189
5190
5191
5192
5193
5194
5195
5196
5197
5198
5199
5200
5201
5202
5203
5204
5205
5206
5207
5208
5209
5210
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5213
5214
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5217
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5253
5254
5255
5256
5257
5258
5259
5260
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5262
5263
5264
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5266
\end{small}
\caption{Algorithm mp\_div (continued)}
\end{figure}
\textbf{Algorithm mp\_div.}
This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor.  The algorithm is a signed
division and will produce a fully qualified quotient and remainder.

First the divisor $b$ must be non-zero which is enforced in step one.  If the divisor is larger than the dividend than the quotient is implicitly 
zero and the remainder is the dividend.  

After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient.  Two unsigned copies of the
divisor $y$ and dividend $x$ are made as well.  The core of the division algorithm is an unsigned division and will only work if the values are
positive.  Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$.  
This is performed by shifting both to the left by enough bits to get the desired normalization.  

At this point the division algorithm can begin producing digits of the quotient.  Recall that maximum value of the estimation used is 
$2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means.  In this case $y$ is shifted
to the left (\textit{step ten}) so that it has the same number of digits as $x$.  The loop on step eleven will subtract multiples of the 
shifted copy of $y$ until $x$ is smaller.  Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two
times to produce the desired leading digit of the quotient.  

Now the remainder of the digits can be produced.  The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly
accurately approximate the true quotient digit.  The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by
induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$.  

Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high.  The next step of the estimation process is
to refine the estimation.  The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher
order approximation to adjust the quotient digit.

After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced
by optimizing Barrett reduction.}.  Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of
algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large.  

Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the 
remainder.  An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC}
is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie 
outside their respective boundaries.  For example, if $t = 0$ or $i \le 1$ then the digits would be undefined.  In those cases the digits should
respectively be replaced with a zero.  

EXAM,bn_mp_div.c

The implementation of this algorithm differs slightly from the pseudo code presented previously.  In this algorithm either of the quotient $c$ or
remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired.  For example, the C code to call the division
algorithm with only the quotient is 

\begin{verbatim}
mp_div(&a, &b, &c, NULL);  /* c = [a/b] */
\end{verbatim}

Lines @108,if@ and @113,if@ handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor 
respectively.  After the two trivial cases all of the temporary variables are initialized.  Line @147,neg@ determines the sign of 
the quotient and line @148,sign@ ensures that both $x$ and $y$ are positive.  

The number of bits in the leading digit is calculated on line @151,norm@.  Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits
of precision which when reduced modulo $lg(\beta)$ produces the value of $k$.  In this case $k$ is the number of bits in the leading digit which is
exactly what is required.  For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting
them to the left by $lg(\beta) - 1 - k$ bits.

Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively.  These are first used to produce the 
leading digit of the quotient.  The loop beginning on line @184,for@ will produce the remainder of the quotient digits.

The conditional ``continue'' on line @186,continue@ is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the
algorithm eliminates multiple non-zero digits in a single iteration.  This ensures that $x_i$ is always non-zero since by definition the digits
above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}.  

Lines @214,t1@, @216,t1@ and @222,t2@ through @225,t2@ manually construct the high accuracy estimations by setting the digits of the two mp\_int 
variables directly.  

\section{Single Digit Helpers}

This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants.  All of 
the helper functions assume the single digit input is positive and will treat them as such.

\subsection{Single Digit Addition and Subtraction}

Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction 
algorithms.   As a result these algorithms are subtantially simpler with a slight cost in performance.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_add\_d}. \\







|
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|

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|



|







|

|

|

|





|





|
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|




|

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|




|







5173
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5176
5177
5178
5179
5180
5181
5182
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5186
5187
5188
5189
5190
5191
5192
5193
5194
5195
5196
5197
5198
5199
5200
5201
5202
5203
5204
5205
5206
5207
5208
5209
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5212
5213
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5222
5223
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5227
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5232
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5234
5235
5236
5237
5238
5239
5240
5241
5242
5243
5244
5245
5246
5247
5248
5249
5250
5251
5252
5253
5254
5255
\end{small}
\caption{Algorithm mp\_div (continued)}
\end{figure}
\textbf{Algorithm mp\_div.}
This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor.  The algorithm is a signed
division and will produce a fully qualified quotient and remainder.

First the divisor $b$ must be non-zero which is enforced in step one.  If the divisor is larger than the dividend than the quotient is implicitly
zero and the remainder is the dividend.

After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient.  Two unsigned copies of the
divisor $y$ and dividend $x$ are made as well.  The core of the division algorithm is an unsigned division and will only work if the values are
positive.  Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$.
This is performed by shifting both to the left by enough bits to get the desired normalization.

At this point the division algorithm can begin producing digits of the quotient.  Recall that maximum value of the estimation used is
$2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means.  In this case $y$ is shifted
to the left (\textit{step ten}) so that it has the same number of digits as $x$.  The loop on step eleven will subtract multiples of the
shifted copy of $y$ until $x$ is smaller.  Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two
times to produce the desired leading digit of the quotient.

Now the remainder of the digits can be produced.  The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly
accurately approximate the true quotient digit.  The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by
induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$.

Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high.  The next step of the estimation process is
to refine the estimation.  The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher
order approximation to adjust the quotient digit.

After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced
by optimizing Barrett reduction.}.  Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of
algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large.

Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the
remainder.  An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC}
is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie
outside their respective boundaries.  For example, if $t = 0$ or $i \le 1$ then the digits would be undefined.  In those cases the digits should
respectively be replaced with a zero.

EXAM,bn_mp_div.c

The implementation of this algorithm differs slightly from the pseudo code presented previously.  In this algorithm either of the quotient $c$ or
remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired.  For example, the C code to call the division
algorithm with only the quotient is

\begin{verbatim}
mp_div(&a, &b, &c, NULL);  /* c = [a/b] */
\end{verbatim}

Lines @108,if@ and @113,if@ handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor
respectively.  After the two trivial cases all of the temporary variables are initialized.  Line @147,neg@ determines the sign of
the quotient and line @148,sign@ ensures that both $x$ and $y$ are positive.

The number of bits in the leading digit is calculated on line @151,norm@.  Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits
of precision which when reduced modulo $lg(\beta)$ produces the value of $k$.  In this case $k$ is the number of bits in the leading digit which is
exactly what is required.  For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting
them to the left by $lg(\beta) - 1 - k$ bits.

Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively.  These are first used to produce the
leading digit of the quotient.  The loop beginning on line @184,for@ will produce the remainder of the quotient digits.

The conditional ``continue'' on line @186,continue@ is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the
algorithm eliminates multiple non-zero digits in a single iteration.  This ensures that $x_i$ is always non-zero since by definition the digits
above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}.

Lines @214,t1@, @216,t1@ and @222,t2@ through @225,t2@ manually construct the high accuracy estimations by setting the digits of the two mp\_int
variables directly.

\section{Single Digit Helpers}

This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants.  All of
the helper functions assume the single digit input is positive and will treat them as such.

\subsection{Single Digit Addition and Subtraction}

Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction
algorithms.   As a result these algorithms are subtantially simpler with a slight cost in performance.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_add\_d}. \\
5318
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5320
5321
5322
5323
5324
5325
5326
5327
5328
5329
5330
5331
5332
5333
5334
5335
5336
5337
5338
5339
5340
5341
5342
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul\_d}
\end{figure}
\textbf{Algorithm mp\_mul\_d.}
This algorithm quickly multiplies an mp\_int by a small single digit value.  It is specially tailored to the job and has a minimal of overhead.  
Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations.  

EXAM,bn_mp_mul_d.c

In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is 
read from the source.  This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively.  

\subsection{Single Digit Division}
Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion.  Since the
divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_div\_d}. \\
\textbf{Input}.   mp\_int $a$ and a mp\_digit $b$ \\







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|







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5309
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5313
5314
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5320
5321
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5323
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5325
5326
5327
5328
5329
5330
5331
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul\_d}
\end{figure}
\textbf{Algorithm mp\_mul\_d.}
This algorithm quickly multiplies an mp\_int by a small single digit value.  It is specially tailored to the job and has a minimal of overhead.
Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations.

EXAM,bn_mp_mul_d.c

In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is
read from the source.  This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively.

\subsection{Single Digit Division}
Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion.  Since the
divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_div\_d}. \\
\textbf{Input}.   mp\_int $a$ and a mp\_digit $b$ \\
5365
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5367
5368
5369
5370
5371
5372
5373
5374
5375
5376
5377
5378
5379
5380
5381
5382
5383
5384
5385
5386
5387
5388
5389
5390
5391
5392
5393
5394
5395
5396
5397
5398
5399
5400
5401
5402
5403
5404
5405
5406
5407
\end{center}
\end{small}
\caption{Algorithm mp\_div\_d}
\end{figure}
\textbf{Algorithm mp\_div\_d.}
This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach.  Essentially in every iteration of the
algorithm another digit of the dividend is reduced and another digit of quotient produced.  Provided $b < \beta$ the value of $\hat w$
after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$.  

If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3.  It replaces the division by three with
a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup.  In essence it is much like the Barrett reduction
from chapter seven.  

EXAM,bn_mp_div_d.c

Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to
indicate the respective value is not required.  This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created.

The division and remainder on lines @44,/@ and @45,%@ can be replaced often by a single division on most processors.  For example, the 32-bit x86 based 
processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously.  Unfortunately the GCC 
compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively.  

\subsection{Single Digit Root Extraction}

Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned.  Algorithms such as the Newton-Raphson approximation 
(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$.  

\begin{equation}
x_{i+1} = x_i - {f(x_i) \over f'(x_i)}
\label{eqn:newton}
\end{equation}

In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired.  The derivative of $f(x)$ is 
simply $f'(x) = nx^{n - 1}$.  Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain
such as the real numbers.  As a result the root found can be above the true root by few and must be manually adjusted.  Ideally at the end of the 
algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_n\_root}. \\
\textbf{Input}.   mp\_int $a$ and a mp\_digit $b$ \\







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5354
5355
5356
5357
5358
5359
5360
5361
5362
5363
5364
5365
5366
5367
5368
5369
5370
5371
5372
5373
5374
5375
5376
5377
5378
5379
5380
5381
5382
5383
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5385
5386
5387
5388
5389
5390
5391
5392
5393
5394
5395
5396
\end{center}
\end{small}
\caption{Algorithm mp\_div\_d}
\end{figure}
\textbf{Algorithm mp\_div\_d.}
This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach.  Essentially in every iteration of the
algorithm another digit of the dividend is reduced and another digit of quotient produced.  Provided $b < \beta$ the value of $\hat w$
after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$.

If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3.  It replaces the division by three with
a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup.  In essence it is much like the Barrett reduction
from chapter seven.

EXAM,bn_mp_div_d.c

Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to
indicate the respective value is not required.  This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created.

The division and remainder on lines @90,/@ and @91,-@ can be replaced often by a single division on most processors.  For example, the 32-bit x86 based
processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously.  Unfortunately the GCC
compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively.

\subsection{Single Digit Root Extraction}

Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned.  Algorithms such as the Newton-Raphson approximation
(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$.

\begin{equation}
x_{i+1} = x_i - {f(x_i) \over f'(x_i)}
\label{eqn:newton}
\end{equation}

In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired.  The derivative of $f(x)$ is
simply $f'(x) = nx^{n - 1}$.  Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain
such as the real numbers.  As a result the root found can be above the true root by few and must be manually adjusted.  Ideally at the end of the
algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_n\_root}. \\
\textbf{Input}.   mp\_int $a$ and a mp\_digit $b$ \\
5434
5435
5436
5437
5438
5439
5440
5441
5442
5443
5444
5445
5446
5447
5448
5449
5450
5451
5452
5453
5454
5455
5456
5457
5458
5459
5460
\end{center}
\end{small}
\caption{Algorithm mp\_n\_root}
\end{figure}
\textbf{Algorithm mp\_n\_root.}
This algorithm finds the integer $n$'th root of an input using the Newton-Raphson approach.  It is partially optimized based on the observation
that the numerator of ${f(x) \over f'(x)}$ can be derived from a partial denominator.  That is at first the denominator is calculated by finding
$x^{b - 1}$.  This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator.  This saves a total of $b - 1$ 
multiplications by t$1$ inside the loop.  

The initial value of the approximation is t$2 = 2$ which allows the algorithm to start with very small values and quickly converge on the
root.  Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$.  

EXAM,bn_mp_n_root.c

\section{Random Number Generation}

Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms.  Pollard-Rho 
factoring for example, can make use of random values as starting points to find factors of a composite integer.  In this case the algorithm presented
is solely for simulations and not intended for cryptographic use.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_rand}. \\
\textbf{Input}.   An integer $b$ \\







|
|


|





|

|







5423
5424
5425
5426
5427
5428
5429
5430
5431
5432
5433
5434
5435
5436
5437
5438
5439
5440
5441
5442
5443
5444
5445
5446
5447
5448
5449
\end{center}
\end{small}
\caption{Algorithm mp\_n\_root}
\end{figure}
\textbf{Algorithm mp\_n\_root.}
This algorithm finds the integer $n$'th root of an input using the Newton-Raphson approach.  It is partially optimized based on the observation
that the numerator of ${f(x) \over f'(x)}$ can be derived from a partial denominator.  That is at first the denominator is calculated by finding
$x^{b - 1}$.  This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator.  This saves a total of $b - 1$
multiplications by t$1$ inside the loop.

The initial value of the approximation is t$2 = 2$ which allows the algorithm to start with very small values and quickly converge on the
root.  Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$.

EXAM,bn_mp_n_root.c

\section{Random Number Generation}

Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms.  Pollard-Rho
factoring for example, can make use of random values as starting points to find factors of a composite integer.  In this case the algorithm presented
is solely for simulations and not intended for cryptographic use.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_rand}. \\
\textbf{Input}.   An integer $b$ \\
5474
5475
5476
5477
5478
5479
5480
5481
5482
5483
5484
5485
5486
5487
5488
5489
5490
5491
5492
5493
5494
5495
5496
5497
5498
5499
5500
5501
5502
5503
5504
5505
5506
5507
5508
\end{center}
\end{small}
\caption{Algorithm mp\_rand}
\end{figure}
\textbf{Algorithm mp\_rand.}
This algorithm produces a pseudo-random integer of $b$ digits.  By ensuring that the first digit is non-zero the algorithm also guarantees that the
final result has at least $b$ digits.  It relies heavily on a third-part random number generator which should ideally generate uniformly all of
the integers from $0$ to $\beta - 1$.  

EXAM,bn_mp_rand.c

\section{Formatted Representations}
The ability to emit a radix-$n$ textual representation of an integer is useful for interacting with human parties.  For example, the ability to
be given a string of characters such as ``114585'' and turn it into the radix-$\beta$ equivalent would make it easier to enter numbers
into a program.

\subsection{Reading Radix-n Input}
For the purposes of this text we will assume that a simple lower ASCII map (\ref{fig:ASC}) is used for the values of from $0$ to $63$ to 
printable characters.  For example, when the character ``N'' is read it represents the integer $23$.  The first $16$ characters of the
map are for the common representations up to hexadecimal.  After that they match the ``base64'' encoding scheme which are suitable chosen
such that they are printable.  While outputting as base64 may not be too helpful for human operators it does allow communication via non binary
mediums.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{cc|cc|cc|cc}
\hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} &  \textbf{Value} & \textbf{Char} \\
\hline 
0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 \\
4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\
8 & 8 & 9 & 9 & 10 & A & 11 & B \\
12 & C & 13 & D & 14 & E & 15 & F \\
16 & G & 17 & H & 18 & I & 19 & J \\
20 & K & 21 & L & 22 & M & 23 & N \\
24 & O & 25 & P & 26 & Q & 27 & R \\







|









|









|







5463
5464
5465
5466
5467
5468
5469
5470
5471
5472
5473
5474
5475
5476
5477
5478
5479
5480
5481
5482
5483
5484
5485
5486
5487
5488
5489
5490
5491
5492
5493
5494
5495
5496
5497
\end{center}
\end{small}
\caption{Algorithm mp\_rand}
\end{figure}
\textbf{Algorithm mp\_rand.}
This algorithm produces a pseudo-random integer of $b$ digits.  By ensuring that the first digit is non-zero the algorithm also guarantees that the
final result has at least $b$ digits.  It relies heavily on a third-part random number generator which should ideally generate uniformly all of
the integers from $0$ to $\beta - 1$.

EXAM,bn_mp_rand.c

\section{Formatted Representations}
The ability to emit a radix-$n$ textual representation of an integer is useful for interacting with human parties.  For example, the ability to
be given a string of characters such as ``114585'' and turn it into the radix-$\beta$ equivalent would make it easier to enter numbers
into a program.

\subsection{Reading Radix-n Input}
For the purposes of this text we will assume that a simple lower ASCII map (\ref{fig:ASC}) is used for the values of from $0$ to $63$ to
printable characters.  For example, when the character ``N'' is read it represents the integer $23$.  The first $16$ characters of the
map are for the common representations up to hexadecimal.  After that they match the ``base64'' encoding scheme which are suitable chosen
such that they are printable.  While outputting as base64 may not be too helpful for human operators it does allow communication via non binary
mediums.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{cc|cc|cc|cc}
\hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} &  \textbf{Value} & \textbf{Char} \\
\hline
0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 \\
4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\
8 & 8 & 9 & 9 & 10 & A & 11 & B \\
12 & C & 13 & D & 14 & E & 15 & F \\
16 & G & 17 & H & 18 & I & 19 & J \\
20 & K & 21 & L & 22 & M & 23 & N \\
24 & O & 25 & P & 26 & Q & 27 & R \\
5548
5549
5550
5551
5552
5553
5554
5555
5556
5557
5558
5559
5560
5561
5562
5563
5564
5565
5566
5567
5568
5569
5570
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_read\_radix}
\end{figure}
\textbf{Algorithm mp\_read\_radix.}
This algorithm will read an ASCII string and produce the radix-$\beta$ mp\_int representation of the same integer.  A minus symbol ``-'' may precede the 
string  to indicate the value is negative, otherwise it is assumed to be positive.  The algorithm will read up to $sn$ characters from the input
and will stop when it reads a character it cannot map the algorithm stops reading characters from the string.  This allows numbers to be embedded
as part of larger input without any significant problem.

EXAM,bn_mp_read_radix.c

\subsection{Generating Radix-$n$ Output}
Generating radix-$n$ output is fairly trivial with a division and remainder algorithm.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_toradix}. \\
\textbf{Input}.   A mp\_int $a$ and an integer $r$\\







|







|







5537
5538
5539
5540
5541
5542
5543
5544
5545
5546
5547
5548
5549
5550
5551
5552
5553
5554
5555
5556
5557
5558
5559
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_read\_radix}
\end{figure}
\textbf{Algorithm mp\_read\_radix.}
This algorithm will read an ASCII string and produce the radix-$\beta$ mp\_int representation of the same integer.  A minus symbol ``-'' may precede the
string  to indicate the value is negative, otherwise it is assumed to be positive.  The algorithm will read up to $sn$ characters from the input
and will stop when it reads a character it cannot map the algorithm stops reading characters from the string.  This allows numbers to be embedded
as part of larger input without any significant problem.

EXAM,bn_mp_read_radix.c

\subsection{Generating Radix-$n$ Output}
Generating radix-$n$ output is fairly trivial with a division and remainder algorithm.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_toradix}. \\
\textbf{Input}.   A mp\_int $a$ and an integer $r$\\
5590
5591
5592
5593
5594
5595
5596
5597
5598
5599
5600
5601
5602
5603
5604
5605
5606
5607
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_toradix}
\end{figure}
\textbf{Algorithm mp\_toradix.}
This algorithm computes the radix-$r$ representation of an mp\_int $a$.  The ``digits'' of the representation are extracted by reducing 
successive powers of $\lfloor a / r^k \rfloor$ the input modulo $r$ until $r^k > a$.  Note that instead of actually dividing by $r^k$ in
each iteration the quotient $\lfloor a / r \rfloor$ is saved for the next iteration.  As a result a series of trivial $n \times 1$ divisions
are required instead of a series of $n \times k$ divisions.  One design flaw of this approach is that the digits are produced in the reverse order 
(see~\ref{fig:mpradix}).  To remedy this flaw the digits must be swapped or simply ``reversed''.

\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline \textbf{Value of $a$} & \textbf{Value of $d$} & \textbf{Value of $str$} \\
\hline $1234$ & -- & -- \\







|


|







5579
5580
5581
5582
5583
5584
5585
5586
5587
5588
5589
5590
5591
5592
5593
5594
5595
5596
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_toradix}
\end{figure}
\textbf{Algorithm mp\_toradix.}
This algorithm computes the radix-$r$ representation of an mp\_int $a$.  The ``digits'' of the representation are extracted by reducing
successive powers of $\lfloor a / r^k \rfloor$ the input modulo $r$ until $r^k > a$.  Note that instead of actually dividing by $r^k$ in
each iteration the quotient $\lfloor a / r \rfloor$ is saved for the next iteration.  As a result a series of trivial $n \times 1$ divisions
are required instead of a series of $n \times k$ divisions.  One design flaw of this approach is that the digits are produced in the reverse order
(see~\ref{fig:mpradix}).  To remedy this flaw the digits must be swapped or simply ``reversed''.

\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline \textbf{Value of $a$} & \textbf{Value of $d$} & \textbf{Value of $str$} \\
\hline $1234$ & -- & -- \\
5615
5616
5617
5618
5619
5620
5621
5622
5623
5624
5625
5626
5627
5628
5629
5630
5631
5632
5633
5634
5635
5636
5637
5638
5639
\caption{Example of Algorithm mp\_toradix.}
\label{fig:mpradix}
\end{figure}

EXAM,bn_mp_toradix.c

\chapter{Number Theoretic Algorithms}
This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi 
symbol computation.  These algorithms arise as essential components in several key cryptographic algorithms such as the RSA public key algorithm and
various Sieve based factoring algorithms.

\section{Greatest Common Divisor}
The greatest common divisor of two integers $a$ and $b$, often denoted as $(a, b)$ is the largest integer $k$ that is a proper divisor of
both $a$ and $b$.  That is, $k$ is the largest integer such that $0 \equiv a \mbox{ (mod }k\mbox{)}$ and $0 \equiv b \mbox{ (mod }k\mbox{)}$ occur
simultaneously.

The most common approach (cite) is to reduce one input modulo another.  That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then
$r$ is also divisible by $k$.  The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Greatest Common Divisor (I)}. \\
\textbf{Input}.   Two positive integers $a$ and $b$ greater than zero. \\







|









|







5604
5605
5606
5607
5608
5609
5610
5611
5612
5613
5614
5615
5616
5617
5618
5619
5620
5621
5622
5623
5624
5625
5626
5627
5628
\caption{Example of Algorithm mp\_toradix.}
\label{fig:mpradix}
\end{figure}

EXAM,bn_mp_toradix.c

\chapter{Number Theoretic Algorithms}
This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi
symbol computation.  These algorithms arise as essential components in several key cryptographic algorithms such as the RSA public key algorithm and
various Sieve based factoring algorithms.

\section{Greatest Common Divisor}
The greatest common divisor of two integers $a$ and $b$, often denoted as $(a, b)$ is the largest integer $k$ that is a proper divisor of
both $a$ and $b$.  That is, $k$ is the largest integer such that $0 \equiv a \mbox{ (mod }k\mbox{)}$ and $0 \equiv b \mbox{ (mod }k\mbox{)}$ occur
simultaneously.

The most common approach (cite) is to reduce one input modulo another.  That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then
$r$ is also divisible by $k$.  The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Greatest Common Divisor (I)}. \\
\textbf{Input}.   Two positive integers $a$ and $b$ greater than zero. \\
5649
5650
5651
5652
5653
5654
5655
5656
5657
5658
5659
5660
5661
5662
5663
5664
5665
\end{center}
\end{small}
\caption{Algorithm Greatest Common Divisor (I)}
\label{fig:gcd1}
\end{figure}

This algorithm will quickly converge on the greatest common divisor since the residue $r$ tends diminish rapidly.  However, divisions are
relatively expensive operations to perform and should ideally be avoided.  There is another approach based on a similar relationship of 
greatest common divisors.  The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$.  
In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Greatest Common Divisor (II)}. \\
\textbf{Input}.   Two positive integers $a$ and $b$ greater than zero. \\







|
|
|







5638
5639
5640
5641
5642
5643
5644
5645
5646
5647
5648
5649
5650
5651
5652
5653
5654
\end{center}
\end{small}
\caption{Algorithm Greatest Common Divisor (I)}
\label{fig:gcd1}
\end{figure}

This algorithm will quickly converge on the greatest common divisor since the residue $r$ tends diminish rapidly.  However, divisions are
relatively expensive operations to perform and should ideally be avoided.  There is another approach based on a similar relationship of
greatest common divisors.  The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$.
In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Greatest Common Divisor (II)}. \\
\textbf{Input}.   Two positive integers $a$ and $b$ greater than zero. \\
5675
5676
5677
5678
5679
5680
5681
5682
5683
5684
5685
5686
5687
5688
5689
5690
5691
5692
5693
5694
5695
5696
5697
5698
5699
\end{small}
\caption{Algorithm Greatest Common Divisor (II)}
\label{fig:gcd2}
\end{figure}

\textbf{Proof} \textit{Algorithm~\ref{fig:gcd2} will return the greatest common divisor of $a$ and $b$.}
The algorithm in figure~\ref{fig:gcd2} will eventually terminate since $b \ge a$ the subtraction in step 1.2 will be a value less than $b$.  In other
words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$.  Since both $a$ and $b$ are always 
divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the 
second to last iteration of the algorithm $b = a$ and clearly $(a, a) = a$ which concludes the proof.  \textbf{QED}.

As a matter of practicality algorithm \ref{fig:gcd1} decreases far too slowly to be useful.  Specially if $b$ is much larger than $a$ such that 
$b - a$ is still very much larger than $a$.  A simple addition to the algorithm is to divide $b - a$ by a power of some integer $p$ which does
not divide the greatest common divisor but will divide $b - a$.  In this case ${b - a} \over p$ is also an integer and still divisible by
the greatest common divisor.

However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first.  
Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Greatest Common Divisor (III)}. \\
\textbf{Input}.   Two positive integers $a$ and $b$ greater than zero. \\







|
|


|




|
|







5664
5665
5666
5667
5668
5669
5670
5671
5672
5673
5674
5675
5676
5677
5678
5679
5680
5681
5682
5683
5684
5685
5686
5687
5688
\end{small}
\caption{Algorithm Greatest Common Divisor (II)}
\label{fig:gcd2}
\end{figure}

\textbf{Proof} \textit{Algorithm~\ref{fig:gcd2} will return the greatest common divisor of $a$ and $b$.}
The algorithm in figure~\ref{fig:gcd2} will eventually terminate since $b \ge a$ the subtraction in step 1.2 will be a value less than $b$.  In other
words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$.  Since both $a$ and $b$ are always
divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the
second to last iteration of the algorithm $b = a$ and clearly $(a, a) = a$ which concludes the proof.  \textbf{QED}.

As a matter of practicality algorithm \ref{fig:gcd1} decreases far too slowly to be useful.  Specially if $b$ is much larger than $a$ such that
$b - a$ is still very much larger than $a$.  A simple addition to the algorithm is to divide $b - a$ by a power of some integer $p$ which does
not divide the greatest common divisor but will divide $b - a$.  In this case ${b - a} \over p$ is also an integer and still divisible by
the greatest common divisor.

However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first.
Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Greatest Common Divisor (III)}. \\
\textbf{Input}.   Two positive integers $a$ and $b$ greater than zero. \\
5718
5719
5720
5721
5722
5723
5724
5725
5726
5727
5728
5729
5730
5731
5732
5733
5734
5735
5736
5737
5738
5739
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Greatest Common Divisor (III)}
\label{fig:gcd3}
\end{figure}

This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$ 
decreases more rapidly.  The first loop on step two removes powers of $p$ that are in common.  A count, $k$, is kept which will present a common
divisor of $p^k$.  After step two the remaining common divisor of $a$ and $b$ cannot be divisible by $p$.  This means that $p$ can be safely 
divided out of the difference $b - a$ so long as the division leaves no remainder.  

In particular the value of $p$ should be chosen such that the division on step 5.3.1 occur often.  It also helps that division by $p$ be easy
to compute.  The ideal choice of $p$ is two since division by two amounts to a right logical shift.  Another important observation is that by
step five both $a$ and $b$ are odd.  Therefore, the diffrence $b - a$ must be even which means that each iteration removes one bit from the 
largest of the pair.

\subsection{Complete Greatest Common Divisor}
The algorithms presented so far cannot handle inputs which are zero or negative.  The following algorithm can handle all input cases properly
and will produce the greatest common divisor.

\newpage\begin{figure}[!here]







|

|
|



|







5707
5708
5709
5710
5711
5712
5713
5714
5715
5716
5717
5718
5719
5720
5721
5722
5723
5724
5725
5726
5727
5728
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Greatest Common Divisor (III)}
\label{fig:gcd3}
\end{figure}

This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$
decreases more rapidly.  The first loop on step two removes powers of $p$ that are in common.  A count, $k$, is kept which will present a common
divisor of $p^k$.  After step two the remaining common divisor of $a$ and $b$ cannot be divisible by $p$.  This means that $p$ can be safely
divided out of the difference $b - a$ so long as the division leaves no remainder.

In particular the value of $p$ should be chosen such that the division on step 5.3.1 occur often.  It also helps that division by $p$ be easy
to compute.  The ideal choice of $p$ is two since division by two amounts to a right logical shift.  Another important observation is that by
step five both $a$ and $b$ are odd.  Therefore, the diffrence $b - a$ must be even which means that each iteration removes one bit from the
largest of the pair.

\subsection{Complete Greatest Common Divisor}
The algorithms presented so far cannot handle inputs which are zero or negative.  The following algorithm can handle all input cases properly
and will produce the greatest common divisor.

\newpage\begin{figure}[!here]
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\end{center}
\end{small}
\caption{Algorithm mp\_gcd}
\end{figure}
\textbf{Algorithm mp\_gcd.}
This algorithm will produce the greatest common divisor of two mp\_ints $a$ and $b$.  The algorithm was originally based on Algorithm B of
Knuth \cite[pp. 338]{TAOCPV2} but has been modified to be simpler to explain.  In theory it achieves the same asymptotic working time as
Algorithm B and in practice this appears to be true.  

The first two steps handle the cases where either one of or both inputs are zero.  If either input is zero the greatest common divisor is the 
largest input or zero if they are both zero.  If the inputs are not trivial than $u$ and $v$ are assigned the absolute values of 
$a$ and $b$ respectively and the algorithm will proceed to reduce the pair.

Step five will divide out any common factors of two and keep track of the count in the variable $k$.  After this step, two is no longer a
factor of the remaining greatest common divisor between $u$ and $v$ and can be safely evenly divided out of either whenever they are even.  Step 
six and seven ensure that the $u$ and $v$ respectively have no more factors of two.  At most only one of the while--loops will iterate since 
they cannot both be even.

By step eight both of $u$ and $v$ are odd which is required for the inner logic.  First the pair are swapped such that $v$ is equal to
or greater than $u$.  This ensures that the subtraction on step 8.2 will always produce a positive and even result.  Step 8.3 removes any
factors of two from the difference $u$ to ensure that in the next iteration of the loop both are once again odd.

After $v = 0$ occurs the variable $u$ has the greatest common divisor of the pair $\left < u, v \right >$ just after step six.  The result
must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier.  

EXAM,bn_mp_gcd.c

This function makes use of the macros mp\_iszero and mp\_iseven.  The former evaluates to $1$ if the input mp\_int is equivalent to the 
integer zero otherwise it evaluates to $0$.  The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise
it evaluates to $0$.  Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero.  The three 
trivial cases of inputs are handled on lines @23,zero@ through @29,}@.  After those lines the inputs are assumed to be non-zero.

Lines @32,if@ and @36,if@ make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively.  At this point the common factors of two 
must be divided out of the two inputs.  The block starting at line @43,common@ removes common factors of two by first counting the number of trailing
zero bits in both.  The local integer $k$ is used to keep track of how many factors of $2$ are pulled out of both values.  It is assumed that 
the number of factors will not exceed the maximum value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than 
entries than are accessible by an ``int'' so this is not a limitation.}.  

At this point there are no more common factors of two in the two values.  The divisions by a power of two on lines @60,div_2d@ and @67,div_2d@ remove 
any independent factors of two such that both $u$ and $v$ are guaranteed to be an odd integer before hitting the main body of the algorithm.  The while loop
on line @72, while@ performs the reduction of the pair until $v$ is equal to zero.  The unsigned comparison and subtraction algorithms are used in
place of the full signed routines since both values are guaranteed to be positive and the result of the subtraction is guaranteed to be non-negative.

\section{Least Common Multiple}
The least common multiple of a pair of integers is their product divided by their greatest common divisor.  For two integers $a$ and $b$ the
least common multiple is normally denoted as $[ a, b ]$ and numerically equivalent to ${ab} \over {(a, b)}$.  For example, if $a = 2 \cdot 2 \cdot 3 = 12$
and $b = 2 \cdot 3 \cdot 3 \cdot 7 = 126$ the least common multiple is ${126 \over {(12, 126)}} = {126 \over 6} = 21$.

The least common multiple arises often in coding theory as well as number theory.  If two functions have periods of $a$ and $b$ respectively they will
collide, that is be in synchronous states, after only $[ a, b ]$ iterations.  This is why, for example, random number generators based on 
Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}).  
Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_lcm}. \\







|

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|



|
|







|



|

|


|

|
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|

|










|
|







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\end{center}
\end{small}
\caption{Algorithm mp\_gcd}
\end{figure}
\textbf{Algorithm mp\_gcd.}
This algorithm will produce the greatest common divisor of two mp\_ints $a$ and $b$.  The algorithm was originally based on Algorithm B of
Knuth \cite[pp. 338]{TAOCPV2} but has been modified to be simpler to explain.  In theory it achieves the same asymptotic working time as
Algorithm B and in practice this appears to be true.

The first two steps handle the cases where either one of or both inputs are zero.  If either input is zero the greatest common divisor is the
largest input or zero if they are both zero.  If the inputs are not trivial than $u$ and $v$ are assigned the absolute values of
$a$ and $b$ respectively and the algorithm will proceed to reduce the pair.

Step five will divide out any common factors of two and keep track of the count in the variable $k$.  After this step, two is no longer a
factor of the remaining greatest common divisor between $u$ and $v$ and can be safely evenly divided out of either whenever they are even.  Step
six and seven ensure that the $u$ and $v$ respectively have no more factors of two.  At most only one of the while--loops will iterate since
they cannot both be even.

By step eight both of $u$ and $v$ are odd which is required for the inner logic.  First the pair are swapped such that $v$ is equal to
or greater than $u$.  This ensures that the subtraction on step 8.2 will always produce a positive and even result.  Step 8.3 removes any
factors of two from the difference $u$ to ensure that in the next iteration of the loop both are once again odd.

After $v = 0$ occurs the variable $u$ has the greatest common divisor of the pair $\left < u, v \right >$ just after step six.  The result
must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier.

EXAM,bn_mp_gcd.c

This function makes use of the macros mp\_iszero and mp\_iseven.  The former evaluates to $1$ if the input mp\_int is equivalent to the
integer zero otherwise it evaluates to $0$.  The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise
it evaluates to $0$.  Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero.  The three
trivial cases of inputs are handled on lines @23,zero@ through @29,}@.  After those lines the inputs are assumed to be non-zero.

Lines @32,if@ and @36,if@ make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively.  At this point the common factors of two
must be divided out of the two inputs.  The block starting at line @43,common@ removes common factors of two by first counting the number of trailing
zero bits in both.  The local integer $k$ is used to keep track of how many factors of $2$ are pulled out of both values.  It is assumed that
the number of factors will not exceed the maximum value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than
entries than are accessible by an ``int'' so this is not a limitation.}.

At this point there are no more common factors of two in the two values.  The divisions by a power of two on lines @60,div_2d@ and @67,div_2d@ remove
any independent factors of two such that both $u$ and $v$ are guaranteed to be an odd integer before hitting the main body of the algorithm.  The while loop
on line @72, while@ performs the reduction of the pair until $v$ is equal to zero.  The unsigned comparison and subtraction algorithms are used in
place of the full signed routines since both values are guaranteed to be positive and the result of the subtraction is guaranteed to be non-negative.

\section{Least Common Multiple}
The least common multiple of a pair of integers is their product divided by their greatest common divisor.  For two integers $a$ and $b$ the
least common multiple is normally denoted as $[ a, b ]$ and numerically equivalent to ${ab} \over {(a, b)}$.  For example, if $a = 2 \cdot 2 \cdot 3 = 12$
and $b = 2 \cdot 3 \cdot 3 \cdot 7 = 126$ the least common multiple is ${126 \over {(12, 126)}} = {126 \over 6} = 21$.

The least common multiple arises often in coding theory as well as number theory.  If two functions have periods of $a$ and $b$ respectively they will
collide, that is be in synchronous states, after only $[ a, b ]$ iterations.  This is why, for example, random number generators based on
Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}).
Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_lcm}. \\
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\textbf{Algorithm mp\_lcm.}
This algorithm computes the least common multiple of two mp\_int inputs $a$ and $b$.  It computes the least common multiple directly by
dividing the product of the two inputs by their greatest common divisor.

EXAM,bn_mp_lcm.c

\section{Jacobi Symbol Computation}
To explain the Jacobi Symbol we shall first discuss the Legendre function\footnote{Arrg.  What is the name of this?} off which the Jacobi symbol is 
defined.  The Legendre function computes whether or not an integer $a$ is a quadratic residue modulo an odd prime $p$.  Numerically it is
equivalent to equation \ref{eqn:legendre}.

\textit{-- Tom, don't be an ass, cite your source here...!}

\begin{equation}
a^{(p-1)/2} \equiv \begin{array}{rl}
                              -1 &  \mbox{if }a\mbox{ is a quadratic non-residue.} \\
                              0  &  \mbox{if }a\mbox{ divides }p\mbox{.} \\
                              1  &  \mbox{if }a\mbox{ is a quadratic residue}. 
                              \end{array} \mbox{ (mod }p\mbox{)}
\label{eqn:legendre}                              
\end{equation}

\textbf{Proof.} \textit{Equation \ref{eqn:legendre} correctly identifies the residue status of an integer $a$ modulo a prime $p$.}
An integer $a$ is a quadratic residue if the following equation has a solution.

\begin{equation}
x^2 \equiv a \mbox{ (mod }p\mbox{)}







|









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\textbf{Algorithm mp\_lcm.}
This algorithm computes the least common multiple of two mp\_int inputs $a$ and $b$.  It computes the least common multiple directly by
dividing the product of the two inputs by their greatest common divisor.

EXAM,bn_mp_lcm.c

\section{Jacobi Symbol Computation}
To explain the Jacobi Symbol we shall first discuss the Legendre function\footnote{Arrg.  What is the name of this?} off which the Jacobi symbol is
defined.  The Legendre function computes whether or not an integer $a$ is a quadratic residue modulo an odd prime $p$.  Numerically it is
equivalent to equation \ref{eqn:legendre}.

\textit{-- Tom, don't be an ass, cite your source here...!}

\begin{equation}
a^{(p-1)/2} \equiv \begin{array}{rl}
                              -1 &  \mbox{if }a\mbox{ is a quadratic non-residue.} \\
                              0  &  \mbox{if }a\mbox{ divides }p\mbox{.} \\
                              1  &  \mbox{if }a\mbox{ is a quadratic residue}.
                              \end{array} \mbox{ (mod }p\mbox{)}
\label{eqn:legendre}
\end{equation}

\textbf{Proof.} \textit{Equation \ref{eqn:legendre} correctly identifies the residue status of an integer $a$ modulo a prime $p$.}
An integer $a$ is a quadratic residue if the following equation has a solution.

\begin{equation}
x^2 \equiv a \mbox{ (mod }p\mbox{)}
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Whether equation \ref{eqn:root} has a solution or not equation \ref{eqn:rooti} is always true.  If $a^{(p-1)/2} - 1 \equiv 0 \mbox{ (mod }p\mbox{)}$
then the quantity in the braces must be zero.  By reduction,

\begin{eqnarray}
\left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \equiv 0  \nonumber \\
\left (x^2 \right )^{(p-1)/2} \equiv a^{(p-1)/2} \nonumber \\
x^2 \equiv a \mbox{ (mod }p\mbox{)} 
\end{eqnarray}

As a result there must be a solution to the quadratic equation and in turn $a$ must be a quadratic residue.  If $a$ does not divide $p$ and $a$
is not a quadratic residue then the only other value $a^{(p-1)/2}$ may be congruent to is $-1$ since
\begin{equation}
0 \equiv a^{p - 1} - 1 \equiv (a^{(p-1)/2} + 1)(a^{(p-1)/2} - 1) \mbox{ (mod }p\mbox{)}
\end{equation}
One of the terms on the right hand side must be zero.  \textbf{QED}

\subsection{Jacobi Symbol}
The Jacobi symbol is a generalization of the Legendre function for any odd non prime moduli $p$ greater than 2.  If $p = \prod_{i=0}^n p_i$ then
the Jacobi symbol $\left ( { a \over p } \right )$ is equal to the following equation.

\begin{equation}
\left ( { a \over p } \right ) = \left ( { a \over p_0} \right ) \left ( { a \over p_1} \right ) \ldots \left ( { a \over p_n} \right )
\end{equation}

By inspection if $p$ is prime the Jacobi symbol is equivalent to the Legendre function.  The following facts\footnote{See HAC \cite[pp. 72-74]{HAC} for
further details.} will be used to derive an efficient Jacobi symbol algorithm.  Where $p$ is an odd integer greater than two and $a, b \in \Z$ the
following are true.  

\begin{enumerate}
\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$. 
\item $\left ( { ab \over p} \right ) = \left ( { a \over p} \right )\left ( { b \over p} \right )$.
\item If $a \equiv b$ then $\left ( { a \over p} \right ) = \left ( { b \over p} \right )$.
\item $\left ( { 2 \over p} \right )$ equals $1$ if $p \equiv 1$ or $7 \mbox{ (mod }8\mbox{)}$.  Otherwise, it equals $-1$.
\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$.  More specifically 
$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$.  
\end{enumerate}

Using these facts if $a = 2^k \cdot a'$ then

\begin{eqnarray}
\left ( { a \over p } \right ) = \left ( {{2^k} \over p } \right ) \left ( {a' \over p} \right ) \nonumber \\
                               = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right ) 
\label{eqn:jacobi}
\end{eqnarray}

By fact five, 

\begin{equation}
\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} 
\end{equation}

Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then 

\begin{equation}
\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} 
\end{equation}

By putting both observations into equation \ref{eqn:jacobi} the following simplified equation is formed.

\begin{equation}
\left ( { a \over p } \right ) = \left ( {2 \over p } \right )^k \left ( {{p\mbox{ mod }a'} \over a'} \right )  \cdot (-1)^{(p-1)(a'-1)/4} 
\end{equation}

The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively.  The value of 
$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$.  Using this approach the 
factors of $p$ do not have to be known.  Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the 
Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_jacobi}. \\
\textbf{Input}.   mp\_int $a$ and $p$, $a \ge 0$, $p \ge 3$, $p \equiv 1 \mbox{ (mod }2\mbox{)}$ \\







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Whether equation \ref{eqn:root} has a solution or not equation \ref{eqn:rooti} is always true.  If $a^{(p-1)/2} - 1 \equiv 0 \mbox{ (mod }p\mbox{)}$
then the quantity in the braces must be zero.  By reduction,

\begin{eqnarray}
\left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \equiv 0  \nonumber \\
\left (x^2 \right )^{(p-1)/2} \equiv a^{(p-1)/2} \nonumber \\
x^2 \equiv a \mbox{ (mod }p\mbox{)}
\end{eqnarray}

As a result there must be a solution to the quadratic equation and in turn $a$ must be a quadratic residue.  If $a$ does not divide $p$ and $a$
is not a quadratic residue then the only other value $a^{(p-1)/2}$ may be congruent to is $-1$ since
\begin{equation}
0 \equiv a^{p - 1} - 1 \equiv (a^{(p-1)/2} + 1)(a^{(p-1)/2} - 1) \mbox{ (mod }p\mbox{)}
\end{equation}
One of the terms on the right hand side must be zero.  \textbf{QED}

\subsection{Jacobi Symbol}
The Jacobi symbol is a generalization of the Legendre function for any odd non prime moduli $p$ greater than 2.  If $p = \prod_{i=0}^n p_i$ then
the Jacobi symbol $\left ( { a \over p } \right )$ is equal to the following equation.

\begin{equation}
\left ( { a \over p } \right ) = \left ( { a \over p_0} \right ) \left ( { a \over p_1} \right ) \ldots \left ( { a \over p_n} \right )
\end{equation}

By inspection if $p$ is prime the Jacobi symbol is equivalent to the Legendre function.  The following facts\footnote{See HAC \cite[pp. 72-74]{HAC} for
further details.} will be used to derive an efficient Jacobi symbol algorithm.  Where $p$ is an odd integer greater than two and $a, b \in \Z$ the
following are true.

\begin{enumerate}
\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$.
\item $\left ( { ab \over p} \right ) = \left ( { a \over p} \right )\left ( { b \over p} \right )$.
\item If $a \equiv b$ then $\left ( { a \over p} \right ) = \left ( { b \over p} \right )$.
\item $\left ( { 2 \over p} \right )$ equals $1$ if $p \equiv 1$ or $7 \mbox{ (mod }8\mbox{)}$.  Otherwise, it equals $-1$.
\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$.  More specifically
$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$.
\end{enumerate}

Using these facts if $a = 2^k \cdot a'$ then

\begin{eqnarray}
\left ( { a \over p } \right ) = \left ( {{2^k} \over p } \right ) \left ( {a' \over p} \right ) \nonumber \\
                               = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right )
\label{eqn:jacobi}
\end{eqnarray}

By fact five,

\begin{equation}
\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4}
\end{equation}

Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then

\begin{equation}
\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4}
\end{equation}

By putting both observations into equation \ref{eqn:jacobi} the following simplified equation is formed.

\begin{equation}
\left ( { a \over p } \right ) = \left ( {2 \over p } \right )^k \left ( {{p\mbox{ mod }a'} \over a'} \right )  \cdot (-1)^{(p-1)(a'-1)/4}
\end{equation}

The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively.  The value of
$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$.  Using this approach the
factors of $p$ do not have to be known.  Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the
Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_jacobi}. \\
\textbf{Input}.   mp\_int $a$ and $p$, $a \ge 0$, $p \ge 3$, $p \equiv 1 \mbox{ (mod }2\mbox{)}$ \\
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6045
6046
6047
6048
6049
6050
6051
6052
6053
6054
6055
6056
6057
6058
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_jacobi}
\end{figure}
\textbf{Algorithm mp\_jacobi.}
This algorithm computes the Jacobi symbol for an arbitrary positive integer $a$ with respect to an odd integer $p$ greater than three.  The algorithm
is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}.  

Step numbers one and two handle the trivial cases of $a = 0$ and $a = 1$ respectively.  Step five determines the number of two factors in the
input $a$.  If $k$ is even than the term $\left ( { 2 \over p } \right )^k$ must always evaluate to one.  If $k$ is odd than the term evaluates to one 
if $p_0$ is congruent to one or seven modulo eight, otherwise it evaluates to $-1$. After the the $\left ( { 2 \over p } \right )^k$ term is handled 
the $(-1)^{(p-1)(a'-1)/4}$ is computed and multiplied against the current product $s$.  The latter term evaluates to one if both $p$ and $a'$ 
are congruent to one modulo four, otherwise it evaluates to negative one.

By step nine if $a'$ does not equal one a recursion is required.  Step 9.1 computes $p' \equiv p \mbox{ (mod }a'\mbox{)}$ and will recurse to compute
$\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi product.

EXAM,bn_mp_jacobi.c

As a matter of practicality the variable $a'$ as per the pseudo-code is reprensented by the variable $a1$ since the $'$ symbol is not valid for a C 
variable name character. 

The two simple cases of $a = 0$ and $a = 1$ are handled at the very beginning to simplify the algorithm.  If the input is non-trivial the algorithm
has to proceed compute the Jacobi.  The variable $s$ is used to hold the current Jacobi product.  Note that $s$ is merely a C ``int'' data type since
the values it may obtain are merely $-1$, $0$ and $1$.  

After a local copy of $a$ is made all of the factors of two are divided out and the total stored in $k$.  Technically only the least significant
bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same 
processor requirements and neither is faster than the other.

Line @59, if@ through @70, }@ determines the value of $\left ( { 2 \over p } \right )^k$.  If the least significant bit of $k$ is zero than
$k$ is even and the value is one.  Otherwise, the value of $s$ depends on which residue class $p$ belongs to modulo eight.  The value of
$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines @73, if@ through @75, }@.  

Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$.  

\textit{-- Comment about default $s$ and such...}

\section{Modular Inverse}
\label{sec:modinv}
The modular inverse of a number actually refers to the modular multiplicative inverse.  Essentially for any integer $a$ such that $(a, p) = 1$ there
exist another integer $b$ such that $ab \equiv 1 \mbox{ (mod }p\mbox{)}$.  The integer $b$ is called the multiplicative inverse of $a$ which is
denoted as $b = a^{-1}$.  Technically speaking modular inversion is a well defined operation for any finite ring or field not just for rings and 
fields of integers.  However, the former will be the matter of discussion.

The simplest approach is to compute the algebraic inverse of the input.  That is to compute $b \equiv a^{\Phi(p) - 1}$.  If $\Phi(p)$ is the 
order of the multiplicative subgroup modulo $p$ then $b$ must be the multiplicative inverse of $a$.  The proof of which is trivial.

\begin{equation}
ab \equiv a \left (a^{\Phi(p) - 1} \right ) \equiv a^{\Phi(p)} \equiv a^0 \equiv 1 \mbox{ (mod }p\mbox{)}
\end{equation}

However, as simple as this approach may be it has two serious flaws.  It requires that the value of $\Phi(p)$ be known which if $p$ is composite 
requires all of the prime factors.  This approach also is very slow as the size of $p$ grows.  

A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear 
Diophantine\footnote{See LeVeque \cite[pp. 40-43]{LeVeque} for more information.} equation.

\begin{equation}
ab + pq = 1
\end{equation}

Where $a$, $b$, $p$ and $q$ are all integers.  If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of 
$a$ modulo $p$.  The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$.  
However, instead of using that algorithm directly a variant known as the binary Extended Euclidean algorithm will be used in its place.  The
binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine 
equation.  

\subsection{General Case}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_invmod}. \\







|


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|



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|




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5973
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5990
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5992
5993
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6000
6001
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6039
6040
6041
6042
6043
6044
6045
6046
6047
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_jacobi}
\end{figure}
\textbf{Algorithm mp\_jacobi.}
This algorithm computes the Jacobi symbol for an arbitrary positive integer $a$ with respect to an odd integer $p$ greater than three.  The algorithm
is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}.

Step numbers one and two handle the trivial cases of $a = 0$ and $a = 1$ respectively.  Step five determines the number of two factors in the
input $a$.  If $k$ is even than the term $\left ( { 2 \over p } \right )^k$ must always evaluate to one.  If $k$ is odd than the term evaluates to one
if $p_0$ is congruent to one or seven modulo eight, otherwise it evaluates to $-1$. After the the $\left ( { 2 \over p } \right )^k$ term is handled
the $(-1)^{(p-1)(a'-1)/4}$ is computed and multiplied against the current product $s$.  The latter term evaluates to one if both $p$ and $a'$
are congruent to one modulo four, otherwise it evaluates to negative one.

By step nine if $a'$ does not equal one a recursion is required.  Step 9.1 computes $p' \equiv p \mbox{ (mod }a'\mbox{)}$ and will recurse to compute
$\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi product.

EXAM,bn_mp_jacobi.c

As a matter of practicality the variable $a'$ as per the pseudo-code is reprensented by the variable $a1$ since the $'$ symbol is not valid for a C
variable name character.

The two simple cases of $a = 0$ and $a = 1$ are handled at the very beginning to simplify the algorithm.  If the input is non-trivial the algorithm
has to proceed compute the Jacobi.  The variable $s$ is used to hold the current Jacobi product.  Note that $s$ is merely a C ``int'' data type since
the values it may obtain are merely $-1$, $0$ and $1$.

After a local copy of $a$ is made all of the factors of two are divided out and the total stored in $k$.  Technically only the least significant
bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same
processor requirements and neither is faster than the other.

Line @59, if@ through @70, }@ determines the value of $\left ( { 2 \over p } \right )^k$.  If the least significant bit of $k$ is zero than
$k$ is even and the value is one.  Otherwise, the value of $s$ depends on which residue class $p$ belongs to modulo eight.  The value of
$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines @73, if@ through @75, }@.

Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$.

\textit{-- Comment about default $s$ and such...}

\section{Modular Inverse}
\label{sec:modinv}
The modular inverse of a number actually refers to the modular multiplicative inverse.  Essentially for any integer $a$ such that $(a, p) = 1$ there
exist another integer $b$ such that $ab \equiv 1 \mbox{ (mod }p\mbox{)}$.  The integer $b$ is called the multiplicative inverse of $a$ which is
denoted as $b = a^{-1}$.  Technically speaking modular inversion is a well defined operation for any finite ring or field not just for rings and
fields of integers.  However, the former will be the matter of discussion.

The simplest approach is to compute the algebraic inverse of the input.  That is to compute $b \equiv a^{\Phi(p) - 1}$.  If $\Phi(p)$ is the
order of the multiplicative subgroup modulo $p$ then $b$ must be the multiplicative inverse of $a$.  The proof of which is trivial.

\begin{equation}
ab \equiv a \left (a^{\Phi(p) - 1} \right ) \equiv a^{\Phi(p)} \equiv a^0 \equiv 1 \mbox{ (mod }p\mbox{)}
\end{equation}

However, as simple as this approach may be it has two serious flaws.  It requires that the value of $\Phi(p)$ be known which if $p$ is composite
requires all of the prime factors.  This approach also is very slow as the size of $p$ grows.

A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear
Diophantine\footnote{See LeVeque \cite[pp. 40-43]{LeVeque} for more information.} equation.

\begin{equation}
ab + pq = 1
\end{equation}

Where $a$, $b$, $p$ and $q$ are all integers.  If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of
$a$ modulo $p$.  The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$.
However, instead of using that algorithm directly a variant known as the binary Extended Euclidean algorithm will be used in its place.  The
binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine
equation.

\subsection{General Case}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_invmod}. \\
6096
6097
6098
6099
6100
6101
6102
6103
6104
6105
6106
6107
6108
6109
6110
6111
6112
6113
6114
6115
6116
6117
6118
6119
6120
6121
6122
6123
6124
6125
6126
6127
6128
6129
6130
6131
6132
6133
6134
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6136
6137
6138
6139
6140
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6143
6144
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6149
6150
6151
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6155
6156
6157
6158
6159
6160
6161
6162
6163
6164
6165
6166
6167
6168
6169
15.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\end{figure}
\textbf{Algorithm mp\_invmod.}
This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$.  This algorithm is a variation of the 
extended binary Euclidean algorithm from HAC \cite[pp. 608]{HAC}.  It has been modified to only compute the modular inverse and not a complete
Diophantine solution.  

If $b \le 0$ than the modulus is invalid and MP\_VAL is returned.  Similarly if both $a$ and $b$ are even then there cannot be a multiplicative
inverse for $a$ and the error is reported.  

The astute reader will observe that steps seven through nine are very similar to the binary greatest common divisor algorithm mp\_gcd.  In this case
the other variables to the Diophantine equation are solved.  The algorithm terminates when $u = 0$ in which case the solution is

\begin{equation}
Ca + Db = v
\end{equation}

If $v$, the greatest common divisor of $a$ and $b$ is not equal to one then the algorithm will report an error as no inverse exists.  Otherwise, $C$
is the modular inverse of $a$.  The actual value of $C$ is congruent to, but not necessarily equal to, the ideal modular inverse which should lie 
within $1 \le a^{-1} < b$.  Step numbers twelve and thirteen adjust the inverse until it is in range.  If the original input $a$ is within $0 < a < p$ 
then only a couple of additions or subtractions will be required to adjust the inverse.

EXAM,bn_mp_invmod.c

\subsubsection{Odd Moduli}

When the modulus $b$ is odd the variables $A$ and $C$ are fixed and are not required to compute the inverse.  In particular by attempting to solve
the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$.  

The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed.  This 
optimization will halve the time required to compute the modular inverse.

\section{Primality Tests}

A non-zero integer $a$ is said to be prime if it is not divisible by any other integer excluding one and itself.  For example, $a = 7$ is prime 
since the integers $2 \ldots 6$ do not evenly divide $a$.  By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$. 

Prime numbers arise in cryptography considerably as they allow finite fields to be formed.  The ability to determine whether an integer is prime or
not quickly has been a viable subject in cryptography and number theory for considerable time.  The algorithms that will be presented are all
probablistic algorithms in that when they report an integer is composite it must be composite.  However, when the algorithms report an integer is
prime the algorithm may be incorrect.  

As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as 
well be zero.  For the purposes of these discussions let $n$ represent the candidate integer of which the primality is in question.

\subsection{Trial Division}

Trial division means to attempt to evenly divide a candidate integer by small prime integers.  If the candidate can be evenly divided it obviously
cannot be prime.  By dividing by all primes $1 < p \le \sqrt{n}$ this test can actually prove whether an integer is prime.  However, such a test
would require a prohibitive amount of time as $n$ grows.

Instead of dividing by every prime, a smaller, more mangeable set of primes may be used instead.  By performing trial division with only a subset
of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate is prime.  However, often it can prove a candidate is not prime.

The benefit of this test is that trial division by small values is fairly efficient.  Specially compared to the other algorithms that will be
discussed shortly.  The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by
$1 - {1.12 \over ln(q)}$.  The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range 
$3 \le q \le 100$.  

At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly.  At $q = 90$ further testing is generally not going to 
be of any practical use.  In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate 
approximately $80\%$ of all candidate integers.  The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base.  The 
array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_prime\_is\_divisible}. \\
\textbf{Input}.   mp\_int $a$ \\







|

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6085
6086
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6101
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6158
15.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\end{figure}
\textbf{Algorithm mp\_invmod.}
This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$.  This algorithm is a variation of the
extended binary Euclidean algorithm from HAC \cite[pp. 608]{HAC}.  It has been modified to only compute the modular inverse and not a complete
Diophantine solution.

If $b \le 0$ than the modulus is invalid and MP\_VAL is returned.  Similarly if both $a$ and $b$ are even then there cannot be a multiplicative
inverse for $a$ and the error is reported.

The astute reader will observe that steps seven through nine are very similar to the binary greatest common divisor algorithm mp\_gcd.  In this case
the other variables to the Diophantine equation are solved.  The algorithm terminates when $u = 0$ in which case the solution is

\begin{equation}
Ca + Db = v
\end{equation}

If $v$, the greatest common divisor of $a$ and $b$ is not equal to one then the algorithm will report an error as no inverse exists.  Otherwise, $C$
is the modular inverse of $a$.  The actual value of $C$ is congruent to, but not necessarily equal to, the ideal modular inverse which should lie
within $1 \le a^{-1} < b$.  Step numbers twelve and thirteen adjust the inverse until it is in range.  If the original input $a$ is within $0 < a < p$
then only a couple of additions or subtractions will be required to adjust the inverse.

EXAM,bn_mp_invmod.c

\subsubsection{Odd Moduli}

When the modulus $b$ is odd the variables $A$ and $C$ are fixed and are not required to compute the inverse.  In particular by attempting to solve
the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$.

The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed.  This
optimization will halve the time required to compute the modular inverse.

\section{Primality Tests}

A non-zero integer $a$ is said to be prime if it is not divisible by any other integer excluding one and itself.  For example, $a = 7$ is prime
since the integers $2 \ldots 6$ do not evenly divide $a$.  By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$.

Prime numbers arise in cryptography considerably as they allow finite fields to be formed.  The ability to determine whether an integer is prime or
not quickly has been a viable subject in cryptography and number theory for considerable time.  The algorithms that will be presented are all
probablistic algorithms in that when they report an integer is composite it must be composite.  However, when the algorithms report an integer is
prime the algorithm may be incorrect.

As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as
well be zero.  For the purposes of these discussions let $n$ represent the candidate integer of which the primality is in question.

\subsection{Trial Division}

Trial division means to attempt to evenly divide a candidate integer by small prime integers.  If the candidate can be evenly divided it obviously
cannot be prime.  By dividing by all primes $1 < p \le \sqrt{n}$ this test can actually prove whether an integer is prime.  However, such a test
would require a prohibitive amount of time as $n$ grows.

Instead of dividing by every prime, a smaller, more mangeable set of primes may be used instead.  By performing trial division with only a subset
of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate is prime.  However, often it can prove a candidate is not prime.

The benefit of this test is that trial division by small values is fairly efficient.  Specially compared to the other algorithms that will be
discussed shortly.  The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by
$1 - {1.12 \over ln(q)}$.  The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range
$3 \le q \le 100$.

At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly.  At $q = 90$ further testing is generally not going to
be of any practical use.  In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate
approximately $80\%$ of all candidate integers.  The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base.  The
array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_prime\_is\_divisible}. \\
\textbf{Input}.   mp\_int $a$ \\
6179
6180
6181
6182
6183
6184
6185
6186
6187
6188
6189
6190
6191
6192
6193
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6205
6206
6207
6208
6209
6210
6211
6212
6213
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_prime\_is\_divisible}
\end{figure}
\textbf{Algorithm mp\_prime\_is\_divisible.}
This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions.  

EXAM,bn_mp_prime_is_divisible.c

The algorithm defaults to a return of $0$ in case an error occurs.  The values in the prime table are all specified to be in the range of a 
mp\_digit.  The table \_\_prime\_tab is defined in the following file.

EXAM,bn_prime_tab.c

Note that there are two possible tables.  When an mp\_digit is 7-bits long only the primes upto $127$ may be included, otherwise the primes
upto $1619$ are used.  Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit. 

\subsection{The Fermat Test}
The Fermat test is probably one the oldest tests to have a non-trivial probability of success.  It is based on the fact that if $n$ is in 
fact prime then $a^{n} \equiv a \mbox{ (mod }n\mbox{)}$ for all $0 < a < n$.  The reason being that if $n$ is prime than the order of
the multiplicative sub group is $n - 1$.  Any base $a$ must have an order which divides $n - 1$ and as such $a^n$ is equivalent to 
$a^1 = a$.  

If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$.  In which case 
it is possible that $a^n \nequiv a \mbox{ (mod }n\mbox{)}$.  However, this test is not absolute as it is possible that the order
of a base will divide $n - 1$ which would then be reported as prime.  Such a base yields what is known as a Fermat pseudo-prime.  Several 
integers known as Carmichael numbers will be a pseudo-prime to all valid bases.  Fortunately such numbers are extremely rare as $n$ grows
in size.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}







|



|





|


|

|
|

|

|







6168
6169
6170
6171
6172
6173
6174
6175
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6177
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6182
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6186
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6195
6196
6197
6198
6199
6200
6201
6202
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_prime\_is\_divisible}
\end{figure}
\textbf{Algorithm mp\_prime\_is\_divisible.}
This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions.

EXAM,bn_mp_prime_is_divisible.c

The algorithm defaults to a return of $0$ in case an error occurs.  The values in the prime table are all specified to be in the range of a
mp\_digit.  The table \_\_prime\_tab is defined in the following file.

EXAM,bn_prime_tab.c

Note that there are two possible tables.  When an mp\_digit is 7-bits long only the primes upto $127$ may be included, otherwise the primes
upto $1619$ are used.  Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit.

\subsection{The Fermat Test}
The Fermat test is probably one the oldest tests to have a non-trivial probability of success.  It is based on the fact that if $n$ is in
fact prime then $a^{n} \equiv a \mbox{ (mod }n\mbox{)}$ for all $0 < a < n$.  The reason being that if $n$ is prime than the order of
the multiplicative sub group is $n - 1$.  Any base $a$ must have an order which divides $n - 1$ and as such $a^n$ is equivalent to
$a^1 = a$.

If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$.  In which case
it is possible that $a^n \nequiv a \mbox{ (mod }n\mbox{)}$.  However, this test is not absolute as it is possible that the order
of a base will divide $n - 1$ which would then be reported as prime.  Such a base yields what is known as a Fermat pseudo-prime.  Several
integers known as Carmichael numbers will be a pseudo-prime to all valid bases.  Fortunately such numbers are extremely rare as $n$ grows
in size.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
6225
6226
6227
6228
6229
6230
6231
6232
6233
6234
6235
6236
6237
6238
6239
6240
6241
6242
6243
6244
6245
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_prime\_fermat}
\end{figure}
\textbf{Algorithm mp\_prime\_fermat.}
This algorithm determines whether an mp\_int $a$ is a Fermat prime to the base $b$ or not.  It uses a single modular exponentiation to
determine the result.  

EXAM,bn_mp_prime_fermat.c

\subsection{The Miller-Rabin Test}
The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen 
candidate  integers.  The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the 
value must be equal to $-1$.  The squarings are stopped as soon as $-1$ is observed.  If the value of $1$ is observed first it means that
some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}







|




|
|







6214
6215
6216
6217
6218
6219
6220
6221
6222
6223
6224
6225
6226
6227
6228
6229
6230
6231
6232
6233
6234
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_prime\_fermat}
\end{figure}
\textbf{Algorithm mp\_prime\_fermat.}
This algorithm determines whether an mp\_int $a$ is a Fermat prime to the base $b$ or not.  It uses a single modular exponentiation to
determine the result.

EXAM,bn_mp_prime_fermat.c

\subsection{The Miller-Rabin Test}
The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen
candidate  integers.  The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the
value must be equal to $-1$.  The squarings are stopped as soon as $-1$ is observed.  If the value of $1$ is observed first it means that
some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
6267
6268
6269
6270
6271
6272
6273
6274
6275
6276
6277
6278
6279
6280
6281
6282
6283
6284
6285
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_prime\_miller\_rabin}
\end{figure}
\textbf{Algorithm mp\_prime\_miller\_rabin.}
This algorithm performs one trial round of the Miller-Rabin algorithm to the base $b$.  It will set $c = 1$ if the algorithm cannot determine
if $b$ is composite or $c = 0$ if $b$ is provably composite.  The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$.  

If the value $y \equiv b^r$ is congruent to $\pm 1$ then the algorithm cannot prove if $a$ is composite or not.  Otherwise, the algorithm will
square $y$ upto $s - 1$ times stopping only when $y \equiv -1$.  If $y^2 \equiv 1$ and $y \nequiv \pm 1$ then the algorithm can report that $a$
is provably composite.  If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite.  If $a$ is not provably 
composite then it is \textit{probably} prime.

EXAM,bn_mp_prime_miller_rabin.c











|



|







6256
6257
6258
6259
6260
6261
6262
6263
6264
6265
6266
6267
6268
6269
6270
6271
6272
6273
6274
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_prime\_miller\_rabin}
\end{figure}
\textbf{Algorithm mp\_prime\_miller\_rabin.}
This algorithm performs one trial round of the Miller-Rabin algorithm to the base $b$.  It will set $c = 1$ if the algorithm cannot determine
if $b$ is composite or $c = 0$ if $b$ is provably composite.  The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$.

If the value $y \equiv b^r$ is congruent to $\pm 1$ then the algorithm cannot prove if $a$ is composite or not.  Otherwise, the algorithm will
square $y$ upto $s - 1$ times stopping only when $y \equiv -1$.  If $y^2 \equiv 1$ and $y \nequiv \pm 1$ then the algorithm can report that $a$
is provably composite.  If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite.  If $a$ is not provably
composite then it is \textit{probably} prime.

EXAM,bn_mp_prime_miller_rabin.c




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Greg Rose \\
QUALCOMM Australia \\
\end{tabular}
%\end{small}
}
}
\maketitle
This text has been placed in the public domain.  This text corresponds to the v0.39 release of the 
LibTomMath project.

\begin{alltt}
Tom St Denis
111 Banning Rd
Ottawa, Ontario
K2L 1C3
Canada

Phone: 1-613-836-3160
Email: [email protected]
\end{alltt}

This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{} 
{\em book} macro package and the Perl {\em booker} package.

\tableofcontents
\listoffigures
\chapter*{Prefaces}
When I tell people about my LibTom projects and that I release them as public domain they are often puzzled.  
They ask why I did it and especially why I continue to work on them for free.  The best I can explain it is ``Because I can.''  
Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which 
perhaps explains it better.  I am the first to admit there is not anything that special with what I have done.  Perhaps
others can see that too and then we would have a society to be proud of.  My LibTom projects are what I am doing to give 
back to society in the form of tools and knowledge that can help others in their endeavours.

I started writing this book because it was the most logical task to further my goal of open academia.  The LibTomMath source
code itself was written to be easy to follow and learn from.  There are times, however, where pure C source code does not
explain the algorithms properly.  Hence this book.  The book literally starts with the foundation of the library and works
itself outwards to the more complicated algorithms.  The use of both pseudo--code and verbatim source code provides a duality
of ``theory'' and ``practice'' that the computer science students of the world shall appreciate.  I never deviate too far
from relatively straightforward algebra and I hope that this book can be a valuable learning asset.

This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora
of kind people donating their time, resources and kind words to help support my work.  Writing a text of significant
length (along with the source code) is a tiresome and lengthy process.  Currently the LibTom project is four years old,
comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material.  People like Mads and Greg 
were there at the beginning to encourage me to work well.  It is amazing how timely validation from others can boost morale to 
continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003.  

To my many friends whom I have met through the years I thank you for the good times and the words of encouragement.  I hope I
honour your kind gestures with this project.

Open Source.  Open Academia.  Open Minds.

\begin{flushright} Tom St Denis \end{flushright}

\newpage
I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also 
contribute to educate others facing the problem of having to handle big number mathematical calculations.

This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of 
how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about 
the layout and language used.

I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the 
practical aspects of cryptography. 

Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a 
great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up 
multiple precision calculations is often very important since we deal with outdated machine architecture where modular 
reductions, for example, become painfully slow.

This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks 
themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?''

\begin{flushright}
Mads Rasmussen

S\~{a}o Paulo - SP

Brazil
\end{flushright}

\newpage
It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about 
Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not 
really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once.

At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the 
sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real
contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity. 
Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake.

When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully, 
and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close 
friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort, 
and I'm pleased to be involved with it.

\begin{flushright}
Greg Rose, Sydney, Australia, June 2003. 
\end{flushright}

\mainmatter
\pagestyle{headings}
\chapter{Introduction}
\section{Multiple Precision Arithmetic}

\subsection{What is Multiple Precision Arithmetic?}
When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively
raise or lower the precision of the numbers we are dealing with.  For example, in decimal we almost immediate can 
reason that $7$ times $6$ is $42$.  However, $42$ has two digits of precision as opposed to one digit we started with.  
Further multiplications of say $3$ result in a larger precision result $126$.  In these few examples we have multiple 
precisions for the numbers we are working with.  Despite the various levels of precision a single subset\footnote{With the occasional optimization.}
 of algorithms can be designed to accomodate them.  

By way of comparison a fixed or single precision operation would lose precision on various operations.  For example, in
the decimal system with fixed precision $6 \cdot 7 = 2$.

Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in
schools to manually add, subtract, multiply and divide.  

\subsection{The Need for Multiple Precision Arithmetic}
The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation
of public-key cryptography algorithms.   Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require 
integers of significant magnitude to resist known cryptanalytic attacks.  For example, at the time of this writing a 
typical RSA modulus would be at least greater than $10^{309}$.  However, modern programming languages such as ISO C \cite{ISOC} and 
Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision.

\begin{figure}[!here]
\begin{center}
\begin{tabular}{|r|c|}
\hline \textbf{Data Type} & \textbf{Range} \\
\hline char  & $-128 \ldots 127$ \\
\hline short & $-32768 \ldots 32767$ \\
\hline long  & $-2147483648 \ldots 2147483647$ \\
\hline long long & $-9223372036854775808 \ldots 9223372036854775807$ \\
\hline
\end{tabular}
\end{center}
\caption{Typical Data Types for the C Programming Language}
\label{fig:ISOC}
\end{figure}

The largest data type guaranteed to be provided by the ISO C programming 
language\footnote{As per the ISO C standard.  However, each compiler vendor is allowed to augment the precision as they 
see fit.}  can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is 
insufficient to accomodate the magnitude required for the problem at hand.  An RSA modulus of magnitude $10^{19}$ could be 
trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, 
rendering any protocol based on the algorithm insecure.  Multiple precision algorithms solve this very problem by 
extending the range of representable integers while using single precision data types.

Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic 
primitives.  Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in 
various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient.  In fact, several 
major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and 
deployment of efficient algorithms.

However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines.  
Another auxiliary use of multiple precision integers is high precision floating point data types.  
The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$.  
Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE.  Since IEEE 
floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small 
(\textit{23, 48 and 64 bits}).  The mantissa is merely an integer and a multiple precision integer could be used to create
a mantissa of much larger precision than hardware alone can efficiently support.  This approach could be useful where 
scientific applications must minimize the total output error over long calculations.

Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$).
In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}.

\subsection{Benefits of Multiple Precision Arithmetic}
\index{precision}
The benefit of multiple precision representations over single or fixed precision representations is that 
no precision is lost while representing the result of an operation which requires excess precision.  For example, 
the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully.  A multiple 
precision algorithm would augment the precision of the destination to accomodate the result while a single precision system 
would truncate excess bits to maintain a fixed level of precision.

It is possible to implement algorithms which require large integers with fixed precision algorithms.  For example, elliptic
curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum 
size the system will ever need.  Such an approach can lead to vastly simpler algorithms which can accomodate the 
integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard 
processor has an 8 bit accumulator.}.  However, as efficient as such an approach may be, the resulting source code is not
normally very flexible.  It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated.

Multiple precision algorithms have the most overhead of any style of arithmetic.  For the the most part the 
overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved
platforms.  However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the 
inputs.  That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input 
without the designer's explicit forethought.  This leads to lower cost of ownership for the code as it only has to 
be written and tested once.

\section{Purpose of This Text}
The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms.  
That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' 
elements that are neglected by authors of other texts on the subject.  Several well reknowned texts \cite{TAOCPV2,HAC} 
give considerably detailed explanations of the theoretical aspects of algorithms and often very little information 
regarding the practical implementation aspects.  

In most cases how an algorithm is explained and how it is actually implemented are two very different concepts.  For 
example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple 
algorithm for performing multiple precision integer addition.  However, the description lacks any discussion concerning 
the fact that the two integer inputs may be of differing magnitudes.  As a result the implementation is not as simple
as the text would lead people to believe.  Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not 
discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}).

Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers 
and fast modular inversion, which we consider practical oversights.  These optimal algorithms are vital to achieve 
any form of useful performance in non-trivial applications.  

To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer
package.  As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.com}} package is used 
to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field 
tested and work very well.  The LibTomMath library is freely available on the Internet for all uses and this text 
discusses a very large portion of the inner workings of the library.

The algorithms that are presented will always include at least one ``pseudo-code'' description followed 
by the actual C source code that implements the algorithm.  The pseudo-code can be used to implement the same 
algorithm in other programming languages as the reader sees fit.  

This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch.  Showing
the reader how the algorithms fit together as well as where to start on various taskings.  

\section{Discussion and Notation}
\subsection{Notation}
A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent
the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$.  The elements of the array $x$ are said to be the radix $\beta$ digits 
of the integer.  For example, $x = (1,2,3)_{10}$ would represent the integer 
$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$.  

\index{mp\_int}
The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well 
as auxilary data required to manipulate the data.  These additional members are discussed further in section 
\ref{sec:MPINT}.  For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be 
synonymous.  When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members 
are present as well.  An expression of the type \textit{variablename.item} implies that it should evaluate to the 
member named ``item'' of the variable.  For example, a string of characters may have a member ``length'' which would 
evaluate to the number of characters in the string.  If the string $a$ equals ``hello'' then it follows that 
$a.length = 5$.  

For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used
to solve a given problem.  When an algorithm is described as accepting an integer input it is assumed the input is 
a plain integer with no additional multiple-precision members.  That is, algorithms that use integers as opposed to 
mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management.  These 
algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple
precision algorithm to solve the same problem.  

\subsection{Precision Notation}
The variable $\beta$ represents the radix of a single digit of a multiple precision integer and 
must be of the form $q^p$ for $q, p \in \Z^+$.  A single precision variable must be able to represent integers in 
the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range 
$0 \le x < q \beta^2$.  The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the 
carry.  Since all modern computers are binary, it is assumed that $q$ is two.

\index{mp\_digit} \index{mp\_word}
Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent 
a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type.  In 
several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words.  
For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to 
the $j$'th digit of a double precision array.  Whenever an expression is to be assigned to a double precision
variable it is assumed that all single precision variables are promoted to double precision during the evaluation.  
Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single
precision data type.

For example, if $\beta = 10^2$ a single precision data type may represent a value in the 
range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$.  Let
$a = 23$ and $b = 49$ represent two single precision variables.  The single precision product shall be written
as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$.
In this particular case, $\hat c = 1127$ and $c = 127$.  The most significant digit of the product would not fit 
in a single precision data type and as a result $c \ne \hat c$.  

\subsection{Algorithm Inputs and Outputs}
Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision
as indicated.  The only exception to this rule is when variables have been indicated to be of type mp\_int.  This 
distinction is important as scalars are often used as array indicies and various other counters.  

\subsection{Mathematical Expressions}
The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression 
itself.  For example, $\lfloor 5.7 \rfloor = 5$.  Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression
rounded to an integer not less than the expression itself.  For example, $\lceil 5.1 \rceil = 6$.  Typically when 
the $/$ division symbol is used the intention is to perform an integer division with truncation.  For example, 
$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity.  When an expression is written as a 
fraction a real value division is implied, for example ${5 \over 2} = 2.5$.  

The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation
of the integer.  For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$.  

\subsection{Work Effort}
\index{big-Oh}
To measure the efficiency of the specified algorithms, a modified big-Oh notation is used.  In this system all 
single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}.  
That is a single precision addition, multiplication and division are assumed to take the same time to 
complete.  While this is generally not true in practice, it will simplify the discussions considerably.

Some algorithms have slight advantages over others which is why some constants will not be removed in 
the notation.  For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a 
baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work.  In standard big-Oh notation these 
would both be said to be equivalent to $O(n^2)$.  However, 
in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small.  As a 
result small constant factors in the work effort will make an observable difference in algorithm efficiency.

All of the algorithms presented in this text have a polynomial time work level.  That is, of the form 
$O(n^k)$ for $n, k \in \Z^{+}$.  This will help make useful comparisons in terms of the speed of the algorithms and how 
various optimizations will help pay off in the long run.

\section{Exercises}
Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to
the discussion at hand.  These exercises are not designed to be prize winning problems, but instead to be thought 
provoking.  Wherever possible the problems are forward minded, stating problems that will be answered in subsequent 
chapters.  The reader is encouraged to finish the exercises as they appear to get a better understanding of the 
subject material.  

That being said, the problems are designed to affirm knowledge of a particular subject matter.  Students in particular
are encouraged to verify they can answer the problems correctly before moving on.

Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of
the problem.  However, unlike \cite{TAOCPV2} the problems do not get nearly as hard.  The scoring of these 
exercises ranges from one (the easiest) to five (the hardest).  The following table sumarizes the 
scoring system used.

\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|c|l|}
\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\







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Greg Rose \\
QUALCOMM Australia \\
\end{tabular}
%\end{small}
}
}
\maketitle
This text has been placed in the public domain.  This text corresponds to the v0.39 release of the
LibTomMath project.












This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{}
{\em book} macro package and the Perl {\em booker} package.

\tableofcontents
\listoffigures
\chapter*{Prefaces}
When I tell people about my LibTom projects and that I release them as public domain they are often puzzled.
They ask why I did it and especially why I continue to work on them for free.  The best I can explain it is ``Because I can.''
Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which
perhaps explains it better.  I am the first to admit there is not anything that special with what I have done.  Perhaps
others can see that too and then we would have a society to be proud of.  My LibTom projects are what I am doing to give
back to society in the form of tools and knowledge that can help others in their endeavours.

I started writing this book because it was the most logical task to further my goal of open academia.  The LibTomMath source
code itself was written to be easy to follow and learn from.  There are times, however, where pure C source code does not
explain the algorithms properly.  Hence this book.  The book literally starts with the foundation of the library and works
itself outwards to the more complicated algorithms.  The use of both pseudo--code and verbatim source code provides a duality
of ``theory'' and ``practice'' that the computer science students of the world shall appreciate.  I never deviate too far
from relatively straightforward algebra and I hope that this book can be a valuable learning asset.

This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora
of kind people donating their time, resources and kind words to help support my work.  Writing a text of significant
length (along with the source code) is a tiresome and lengthy process.  Currently the LibTom project is four years old,
comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material.  People like Mads and Greg
were there at the beginning to encourage me to work well.  It is amazing how timely validation from others can boost morale to
continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003.

To my many friends whom I have met through the years I thank you for the good times and the words of encouragement.  I hope I
honour your kind gestures with this project.

Open Source.  Open Academia.  Open Minds.

\begin{flushright} Tom St Denis \end{flushright}

\newpage
I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also
contribute to educate others facing the problem of having to handle big number mathematical calculations.

This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of
how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about
the layout and language used.

I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the
practical aspects of cryptography.

Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a
great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up
multiple precision calculations is often very important since we deal with outdated machine architecture where modular
reductions, for example, become painfully slow.

This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks
themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?''

\begin{flushright}
Mads Rasmussen

S\~{a}o Paulo - SP

Brazil
\end{flushright}

\newpage
It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about
Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not
really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once.

At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the
sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real
contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity.
Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake.

When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully,
and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close
friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort,
and I'm pleased to be involved with it.

\begin{flushright}
Greg Rose, Sydney, Australia, June 2003.
\end{flushright}

\mainmatter
\pagestyle{headings}
\chapter{Introduction}
\section{Multiple Precision Arithmetic}

\subsection{What is Multiple Precision Arithmetic?}
When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively
raise or lower the precision of the numbers we are dealing with.  For example, in decimal we almost immediate can
reason that $7$ times $6$ is $42$.  However, $42$ has two digits of precision as opposed to one digit we started with.
Further multiplications of say $3$ result in a larger precision result $126$.  In these few examples we have multiple
precisions for the numbers we are working with.  Despite the various levels of precision a single subset\footnote{With the occasional optimization.}
 of algorithms can be designed to accomodate them.

By way of comparison a fixed or single precision operation would lose precision on various operations.  For example, in
the decimal system with fixed precision $6 \cdot 7 = 2$.

Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in
schools to manually add, subtract, multiply and divide.

\subsection{The Need for Multiple Precision Arithmetic}
The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation
of public-key cryptography algorithms.   Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require
integers of significant magnitude to resist known cryptanalytic attacks.  For example, at the time of this writing a
typical RSA modulus would be at least greater than $10^{309}$.  However, modern programming languages such as ISO C \cite{ISOC} and
Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision.

\begin{figure}[!here]
\begin{center}
\begin{tabular}{|r|c|}
\hline \textbf{Data Type} & \textbf{Range} \\
\hline char  & $-128 \ldots 127$ \\
\hline short & $-32768 \ldots 32767$ \\
\hline long  & $-2147483648 \ldots 2147483647$ \\
\hline long long & $-9223372036854775808 \ldots 9223372036854775807$ \\
\hline
\end{tabular}
\end{center}
\caption{Typical Data Types for the C Programming Language}
\label{fig:ISOC}
\end{figure}

The largest data type guaranteed to be provided by the ISO C programming
language\footnote{As per the ISO C standard.  However, each compiler vendor is allowed to augment the precision as they
see fit.}  can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is
insufficient to accomodate the magnitude required for the problem at hand.  An RSA modulus of magnitude $10^{19}$ could be
trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer,
rendering any protocol based on the algorithm insecure.  Multiple precision algorithms solve this very problem by
extending the range of representable integers while using single precision data types.

Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic
primitives.  Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in
various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient.  In fact, several
major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and
deployment of efficient algorithms.

However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines.
Another auxiliary use of multiple precision integers is high precision floating point data types.
The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$.
Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE.  Since IEEE
floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small
(\textit{23, 48 and 64 bits}).  The mantissa is merely an integer and a multiple precision integer could be used to create
a mantissa of much larger precision than hardware alone can efficiently support.  This approach could be useful where
scientific applications must minimize the total output error over long calculations.

Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$).
In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}.

\subsection{Benefits of Multiple Precision Arithmetic}
\index{precision}
The benefit of multiple precision representations over single or fixed precision representations is that
no precision is lost while representing the result of an operation which requires excess precision.  For example,
the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully.  A multiple
precision algorithm would augment the precision of the destination to accomodate the result while a single precision system
would truncate excess bits to maintain a fixed level of precision.

It is possible to implement algorithms which require large integers with fixed precision algorithms.  For example, elliptic
curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum
size the system will ever need.  Such an approach can lead to vastly simpler algorithms which can accomodate the
integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard
processor has an 8 bit accumulator.}.  However, as efficient as such an approach may be, the resulting source code is not
normally very flexible.  It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated.

Multiple precision algorithms have the most overhead of any style of arithmetic.  For the the most part the
overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved
platforms.  However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the
inputs.  That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input
without the designer's explicit forethought.  This leads to lower cost of ownership for the code as it only has to
be written and tested once.

\section{Purpose of This Text}
The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms.
That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping''
elements that are neglected by authors of other texts on the subject.  Several well reknowned texts \cite{TAOCPV2,HAC}
give considerably detailed explanations of the theoretical aspects of algorithms and often very little information
regarding the practical implementation aspects.

In most cases how an algorithm is explained and how it is actually implemented are two very different concepts.  For
example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple
algorithm for performing multiple precision integer addition.  However, the description lacks any discussion concerning
the fact that the two integer inputs may be of differing magnitudes.  As a result the implementation is not as simple
as the text would lead people to believe.  Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not
discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}).

Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers
and fast modular inversion, which we consider practical oversights.  These optimal algorithms are vital to achieve
any form of useful performance in non-trivial applications.

To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer
package.  As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.com}} package is used
to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field
tested and work very well.  The LibTomMath library is freely available on the Internet for all uses and this text
discusses a very large portion of the inner workings of the library.

The algorithms that are presented will always include at least one ``pseudo-code'' description followed
by the actual C source code that implements the algorithm.  The pseudo-code can be used to implement the same
algorithm in other programming languages as the reader sees fit.

This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch.  Showing
the reader how the algorithms fit together as well as where to start on various taskings.

\section{Discussion and Notation}
\subsection{Notation}
A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent
the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$.  The elements of the array $x$ are said to be the radix $\beta$ digits
of the integer.  For example, $x = (1,2,3)_{10}$ would represent the integer
$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$.

\index{mp\_int}
The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well
as auxilary data required to manipulate the data.  These additional members are discussed further in section
\ref{sec:MPINT}.  For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be
synonymous.  When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members
are present as well.  An expression of the type \textit{variablename.item} implies that it should evaluate to the
member named ``item'' of the variable.  For example, a string of characters may have a member ``length'' which would
evaluate to the number of characters in the string.  If the string $a$ equals ``hello'' then it follows that
$a.length = 5$.

For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used
to solve a given problem.  When an algorithm is described as accepting an integer input it is assumed the input is
a plain integer with no additional multiple-precision members.  That is, algorithms that use integers as opposed to
mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management.  These
algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple
precision algorithm to solve the same problem.

\subsection{Precision Notation}
The variable $\beta$ represents the radix of a single digit of a multiple precision integer and
must be of the form $q^p$ for $q, p \in \Z^+$.  A single precision variable must be able to represent integers in
the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range
$0 \le x < q \beta^2$.  The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the
carry.  Since all modern computers are binary, it is assumed that $q$ is two.

\index{mp\_digit} \index{mp\_word}
Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent
a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type.  In
several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words.
For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to
the $j$'th digit of a double precision array.  Whenever an expression is to be assigned to a double precision
variable it is assumed that all single precision variables are promoted to double precision during the evaluation.
Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single
precision data type.

For example, if $\beta = 10^2$ a single precision data type may represent a value in the
range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$.  Let
$a = 23$ and $b = 49$ represent two single precision variables.  The single precision product shall be written
as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$.
In this particular case, $\hat c = 1127$ and $c = 127$.  The most significant digit of the product would not fit
in a single precision data type and as a result $c \ne \hat c$.

\subsection{Algorithm Inputs and Outputs}
Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision
as indicated.  The only exception to this rule is when variables have been indicated to be of type mp\_int.  This
distinction is important as scalars are often used as array indicies and various other counters.

\subsection{Mathematical Expressions}
The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression
itself.  For example, $\lfloor 5.7 \rfloor = 5$.  Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression
rounded to an integer not less than the expression itself.  For example, $\lceil 5.1 \rceil = 6$.  Typically when
the $/$ division symbol is used the intention is to perform an integer division with truncation.  For example,
$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity.  When an expression is written as a
fraction a real value division is implied, for example ${5 \over 2} = 2.5$.

The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation
of the integer.  For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$.

\subsection{Work Effort}
\index{big-Oh}
To measure the efficiency of the specified algorithms, a modified big-Oh notation is used.  In this system all
single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}.
That is a single precision addition, multiplication and division are assumed to take the same time to
complete.  While this is generally not true in practice, it will simplify the discussions considerably.

Some algorithms have slight advantages over others which is why some constants will not be removed in
the notation.  For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a
baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work.  In standard big-Oh notation these
would both be said to be equivalent to $O(n^2)$.  However,
in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small.  As a
result small constant factors in the work effort will make an observable difference in algorithm efficiency.

All of the algorithms presented in this text have a polynomial time work level.  That is, of the form
$O(n^k)$ for $n, k \in \Z^{+}$.  This will help make useful comparisons in terms of the speed of the algorithms and how
various optimizations will help pay off in the long run.

\section{Exercises}
Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to
the discussion at hand.  These exercises are not designed to be prize winning problems, but instead to be thought
provoking.  Wherever possible the problems are forward minded, stating problems that will be answered in subsequent
chapters.  The reader is encouraged to finish the exercises as they appear to get a better understanding of the
subject material.

That being said, the problems are designed to affirm knowledge of a particular subject matter.  Students in particular
are encouraged to verify they can answer the problems correctly before moving on.

Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of
the problem.  However, unlike \cite{TAOCPV2} the problems do not get nearly as hard.  The scoring of these
exercises ranges from one (the easiest) to five (the hardest).  The following table sumarizes the
scoring system used.

\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|c|l|}
\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\
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\end{tabular}
\end{small}
\end{center}
\caption{Exercise Scoring System}
\end{figure}

Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or
devising new theory.  These problems are quick tests to see if the material is understood.  Problems at the second level 
are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer.  These
two levels are essentially entry level questions.  

Problems at the third level are meant to be a bit more difficult than the first two levels.  The answer is often 
fairly obvious but arriving at an exacting solution requires some thought and skill.  These problems will almost always 
involve devising a new algorithm or implementing a variation of another algorithm previously presented.  Readers who can
answer these questions will feel comfortable with the concepts behind the topic at hand.

Problems at the fourth level are meant to be similar to those of the level three questions except they will require 
additional research to be completed.  The reader will most likely not know the answer right away, nor will the text provide 
the exact details of the answer until a subsequent chapter.  

Problems at the fifth level are meant to be the hardest 
problems relative to all the other problems in the chapter.  People who can correctly answer fifth level problems have a 
mastery of the subject matter at hand.

Often problems will be tied together.  The purpose of this is to start a chain of thought that will be discussed in future chapters.  The reader
is encouraged to answer the follow-up problems and try to draw the relevance of problems.

\section{Introduction to LibTomMath}

\subsection{What is LibTomMath?}
LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C.  By portable it 
is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on 
any given platform.  

The library has been successfully tested under numerous operating systems including Unix\footnote{All of these
trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such 
as the Gameboy Advance.  The library is designed to contain enough functionality to be able to develop applications such 
as public key cryptosystems and still maintain a relatively small footprint.

\subsection{Goals of LibTomMath}

Libraries which obtain the most efficiency are rarely written in a high level programming language such as C.  However, 
even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the 
library.  Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM 
processors.  Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window 
exponentiation and Montgomery reduction have been provided to make the library more efficient.  

Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface 
(\textit{API}) has been kept as simple as possible.  Often generic place holder routines will make use of specialized 
algorithms automatically without the developer's specific attention.  One such example is the generic multiplication 
algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication 
based on the magnitude of the inputs and the configuration of the library.  

Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project.  Ideally the library should 
be source compatible with another popular library which makes it more attractive for developers to use.  In this case the
MPI library was used as a API template for all the basic functions.  MPI was chosen because it is another library that fits 
in the same niche as LibTomMath.  Even though LibTomMath uses MPI as the template for the function names and argument 
passing conventions, it has been written from scratch by Tom St Denis.

The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' 
library exists which can be used to teach computer science students how to perform fast and reliable multiple precision 
integer arithmetic.  To this end the source code has been given quite a few comments and algorithm discussion points.  

\section{Choice of LibTomMath}
LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but
for more worthy reasons.  Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL 
\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for 
reasons that will be explained in the following sub-sections.

\subsection{Code Base}
The LibTomMath code base is all portable ISO C source code.  This means that there are no platform dependent conditional
segments of code littered throughout the source.  This clean and uncluttered approach to the library means that a
developer can more readily discern the true intent of a given section of source code without trying to keep track of
what conditional code will be used.

The code base of LibTomMath is well organized.  Each function is in its own separate source code file 
which allows the reader to find a given function very quickly.  On average there are $76$ lines of code per source
file which makes the source very easily to follow.  By comparison MPI and LIP are single file projects making code tracing
very hard.  GMP has many conditional code segments which also hinder tracing.  

When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.}
 which is fairly small compared to GMP (over $250$KiB).  LibTomMath is slightly larger than MPI (which compiles to about 
$50$KiB) but LibTomMath is also much faster and more complete than MPI.

\subsection{API Simplicity}
LibTomMath is designed after the MPI library and shares the API design.  Quite often programs that use MPI will build 
with LibTomMath without change. The function names correlate directly to the action they perform.  Almost all of the 
functions share the same parameter passing convention.  The learning curve is fairly shallow with the API provided 
which is an extremely valuable benefit for the student and developer alike.  

The LIP library is an example of a library with an API that is awkward to work with.  LIP uses function names that are often ``compressed'' to 
illegible short hand.  LibTomMath does not share this characteristic.  

The GMP library also does not return error codes.  Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors
are signaled to the host application.  This happens to be the fastest approach but definitely not the most versatile.  In
effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely 
undersireable in many situations.

\subsection{Optimizations}
While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does
feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring.  GMP 
and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations.  GMP lacks a few
of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP
only had Barrett and Montgomery modular reduction algorithms.}.  

LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular
exponentiation.  In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually  
slower than the best libraries such as GMP and OpenSSL by only a small factor.

\subsection{Portability and Stability}
LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler 
(\textit{GCC}).  This means that without changes the library will build without configuration or setting up any 
variables.  LIP and MPI will build ``out of the box'' as well but have numerous known bugs.  Most notably the author of 
MPI has recently stopped working on his library and LIP has long since been discontinued.  

GMP requires a configuration script to run and will not build out of the box.   GMP and LibTomMath are still in active
development and are very stable across a variety of platforms.

\subsection{Choice}
LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for
the case study of this text.  Various source files from the LibTomMath project will be included within the text.  However, 
the reader is encouraged to download their own copy of the library to actually be able to work with the library.  

\chapter{Getting Started}
\section{Library Basics}
The trick to writing any useful library of source code is to build a solid foundation and work outwards from it.  First, 
a problem along with allowable solution parameters should be identified and analyzed.  In this particular case the 
inability to accomodate multiple precision integers is the problem.  Futhermore, the solution must be written
as portable source code that is reasonably efficient across several different computer platforms.

After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion.  
That is, to implement the lowest level dependencies first and work towards the most abstract functions last.  For example, 
before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm.
By building outwards from a base foundation instead of using a parallel design methodology the resulting project is 
highly modular.  Being highly modular is a desirable property of any project as it often means the resulting product
has a small footprint and updates are easy to perform.  

Usually when I start a project I will begin with the header files.  I define the data types I think I will need and 
prototype the initial functions that are not dependent on other functions (within the library).  After I 
implement these base functions I prototype more dependent functions and implement them.   The process repeats until
I implement all of the functions I require.  For example, in the case of LibTomMath I implemented functions such as 
mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod().  As an example as to 
why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the 
dependent function mp\_exptmod() was written.  Adding the new multiplication algorithms did not require changes to the 
mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development 
for new algorithms.  This methodology allows new algorithms to be tested in a complete framework with relative ease.

\begin{center}
\begin{figure}[here]
\includegraphics{pics/design_process.ps}
\caption{Design Flow of the First Few Original LibTomMath Functions.}
\label{pic:design_process}
\end{figure}
\end{center}

Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing
the source code.  For example, one day I may audit the multipliers and the next day the polynomial basis functions.  

It only makes sense to begin the text with the preliminary data types and support algorithms required as well.  
This chapter discusses the core algorithms of the library which are the dependents for every other algorithm.

\section{What is a Multiple Precision Integer?}
Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot 
be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is 
to use fixed precision data types to create and manipulate multiple precision integers which may represent values 
that are very large.  

As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits.  In the decimal system
the largest single digit value is $9$.  However, by concatenating digits together larger numbers may be represented.  Newly prepended digits 
(\textit{to the left}) are said to be in a different power of ten column.  That is, the number $123$ can be described as having a $1$ in the hundreds 
column, $2$ in the tens column and $3$ in the ones column.  Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$.  Computer based 
multiple precision arithmetic is essentially the same concept.  Larger integers are represented by adjoining fixed 
precision computer words with the exception that a different radix is used.

What most people probably do not think about explicitly are the various other attributes that describe a multiple precision 
integer.  For example, the integer $154_{10}$ has two immediately obvious properties.  First, the integer is positive, 
that is the sign of this particular integer is positive as opposed to negative.  Second, the integer has three digits in 
its representation.  There is an additional property that the integer posesses that does not concern pencil-and-paper 
arithmetic.  The third property is how many digits placeholders are available to hold the integer.  

The human analogy of this third property is ensuring there is enough space on the paper to write the integer.  For example,
if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left.  
Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer
will not exceed the allowed boundaries.  These three properties make up what is known as a multiple precision 
integer or mp\_int for short.  

\subsection{The mp\_int Structure}
\label{sec:MPINT}
The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer.  The ISO C standard does not provide for 
any such data type but it does provide for making composite data types known as structures.  The following is the structure definition 
used within LibTomMath.

\index{mp\_int}
\begin{figure}[here]
\begin{center}
\begin{small}
%\begin{verbatim}







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\end{tabular}
\end{small}
\end{center}
\caption{Exercise Scoring System}
\end{figure}

Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or
devising new theory.  These problems are quick tests to see if the material is understood.  Problems at the second level
are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer.  These
two levels are essentially entry level questions.

Problems at the third level are meant to be a bit more difficult than the first two levels.  The answer is often
fairly obvious but arriving at an exacting solution requires some thought and skill.  These problems will almost always
involve devising a new algorithm or implementing a variation of another algorithm previously presented.  Readers who can
answer these questions will feel comfortable with the concepts behind the topic at hand.

Problems at the fourth level are meant to be similar to those of the level three questions except they will require
additional research to be completed.  The reader will most likely not know the answer right away, nor will the text provide
the exact details of the answer until a subsequent chapter.

Problems at the fifth level are meant to be the hardest
problems relative to all the other problems in the chapter.  People who can correctly answer fifth level problems have a
mastery of the subject matter at hand.

Often problems will be tied together.  The purpose of this is to start a chain of thought that will be discussed in future chapters.  The reader
is encouraged to answer the follow-up problems and try to draw the relevance of problems.

\section{Introduction to LibTomMath}

\subsection{What is LibTomMath?}
LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C.  By portable it
is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on
any given platform.

The library has been successfully tested under numerous operating systems including Unix\footnote{All of these
trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such
as the Gameboy Advance.  The library is designed to contain enough functionality to be able to develop applications such
as public key cryptosystems and still maintain a relatively small footprint.

\subsection{Goals of LibTomMath}

Libraries which obtain the most efficiency are rarely written in a high level programming language such as C.  However,
even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the
library.  Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM
processors.  Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window
exponentiation and Montgomery reduction have been provided to make the library more efficient.

Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface
(\textit{API}) has been kept as simple as possible.  Often generic place holder routines will make use of specialized
algorithms automatically without the developer's specific attention.  One such example is the generic multiplication
algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication
based on the magnitude of the inputs and the configuration of the library.

Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project.  Ideally the library should
be source compatible with another popular library which makes it more attractive for developers to use.  In this case the
MPI library was used as a API template for all the basic functions.  MPI was chosen because it is another library that fits
in the same niche as LibTomMath.  Even though LibTomMath uses MPI as the template for the function names and argument
passing conventions, it has been written from scratch by Tom St Denis.

The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum''
library exists which can be used to teach computer science students how to perform fast and reliable multiple precision
integer arithmetic.  To this end the source code has been given quite a few comments and algorithm discussion points.

\section{Choice of LibTomMath}
LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but
for more worthy reasons.  Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL
\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for
reasons that will be explained in the following sub-sections.

\subsection{Code Base}
The LibTomMath code base is all portable ISO C source code.  This means that there are no platform dependent conditional
segments of code littered throughout the source.  This clean and uncluttered approach to the library means that a
developer can more readily discern the true intent of a given section of source code without trying to keep track of
what conditional code will be used.

The code base of LibTomMath is well organized.  Each function is in its own separate source code file
which allows the reader to find a given function very quickly.  On average there are $76$ lines of code per source
file which makes the source very easily to follow.  By comparison MPI and LIP are single file projects making code tracing
very hard.  GMP has many conditional code segments which also hinder tracing.

When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.}
 which is fairly small compared to GMP (over $250$KiB).  LibTomMath is slightly larger than MPI (which compiles to about
$50$KiB) but LibTomMath is also much faster and more complete than MPI.

\subsection{API Simplicity}
LibTomMath is designed after the MPI library and shares the API design.  Quite often programs that use MPI will build
with LibTomMath without change. The function names correlate directly to the action they perform.  Almost all of the
functions share the same parameter passing convention.  The learning curve is fairly shallow with the API provided
which is an extremely valuable benefit for the student and developer alike.

The LIP library is an example of a library with an API that is awkward to work with.  LIP uses function names that are often ``compressed'' to
illegible short hand.  LibTomMath does not share this characteristic.

The GMP library also does not return error codes.  Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors
are signaled to the host application.  This happens to be the fastest approach but definitely not the most versatile.  In
effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely
undersireable in many situations.

\subsection{Optimizations}
While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does
feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring.  GMP
and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations.  GMP lacks a few
of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP
only had Barrett and Montgomery modular reduction algorithms.}.

LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular
exponentiation.  In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually
slower than the best libraries such as GMP and OpenSSL by only a small factor.

\subsection{Portability and Stability}
LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler
(\textit{GCC}).  This means that without changes the library will build without configuration or setting up any
variables.  LIP and MPI will build ``out of the box'' as well but have numerous known bugs.  Most notably the author of
MPI has recently stopped working on his library and LIP has long since been discontinued.

GMP requires a configuration script to run and will not build out of the box.   GMP and LibTomMath are still in active
development and are very stable across a variety of platforms.

\subsection{Choice}
LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for
the case study of this text.  Various source files from the LibTomMath project will be included within the text.  However,
the reader is encouraged to download their own copy of the library to actually be able to work with the library.

\chapter{Getting Started}
\section{Library Basics}
The trick to writing any useful library of source code is to build a solid foundation and work outwards from it.  First,
a problem along with allowable solution parameters should be identified and analyzed.  In this particular case the
inability to accomodate multiple precision integers is the problem.  Futhermore, the solution must be written
as portable source code that is reasonably efficient across several different computer platforms.

After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion.
That is, to implement the lowest level dependencies first and work towards the most abstract functions last.  For example,
before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm.
By building outwards from a base foundation instead of using a parallel design methodology the resulting project is
highly modular.  Being highly modular is a desirable property of any project as it often means the resulting product
has a small footprint and updates are easy to perform.

Usually when I start a project I will begin with the header files.  I define the data types I think I will need and
prototype the initial functions that are not dependent on other functions (within the library).  After I
implement these base functions I prototype more dependent functions and implement them.   The process repeats until
I implement all of the functions I require.  For example, in the case of LibTomMath I implemented functions such as
mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod().  As an example as to
why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the
dependent function mp\_exptmod() was written.  Adding the new multiplication algorithms did not require changes to the
mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development
for new algorithms.  This methodology allows new algorithms to be tested in a complete framework with relative ease.

\begin{center}
\begin{figure}[here]
\includegraphics{pics/design_process.ps}
\caption{Design Flow of the First Few Original LibTomMath Functions.}
\label{pic:design_process}
\end{figure}
\end{center}

Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing
the source code.  For example, one day I may audit the multipliers and the next day the polynomial basis functions.

It only makes sense to begin the text with the preliminary data types and support algorithms required as well.
This chapter discusses the core algorithms of the library which are the dependents for every other algorithm.

\section{What is a Multiple Precision Integer?}
Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot
be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is
to use fixed precision data types to create and manipulate multiple precision integers which may represent values
that are very large.

As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits.  In the decimal system
the largest single digit value is $9$.  However, by concatenating digits together larger numbers may be represented.  Newly prepended digits
(\textit{to the left}) are said to be in a different power of ten column.  That is, the number $123$ can be described as having a $1$ in the hundreds
column, $2$ in the tens column and $3$ in the ones column.  Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$.  Computer based
multiple precision arithmetic is essentially the same concept.  Larger integers are represented by adjoining fixed
precision computer words with the exception that a different radix is used.

What most people probably do not think about explicitly are the various other attributes that describe a multiple precision
integer.  For example, the integer $154_{10}$ has two immediately obvious properties.  First, the integer is positive,
that is the sign of this particular integer is positive as opposed to negative.  Second, the integer has three digits in
its representation.  There is an additional property that the integer posesses that does not concern pencil-and-paper
arithmetic.  The third property is how many digits placeholders are available to hold the integer.

The human analogy of this third property is ensuring there is enough space on the paper to write the integer.  For example,
if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left.
Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer
will not exceed the allowed boundaries.  These three properties make up what is known as a multiple precision
integer or mp\_int for short.

\subsection{The mp\_int Structure}
\label{sec:MPINT}
The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer.  The ISO C standard does not provide for
any such data type but it does provide for making composite data types known as structures.  The following is the structure definition
used within LibTomMath.

\index{mp\_int}
\begin{figure}[here]
\begin{center}
\begin{small}
%\begin{verbatim}
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\end{center}
\end{figure}

The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows.

\begin{enumerate}
\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent
a given integer.  The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count.  

\item The \textbf{alloc} parameter denotes how 
many digits are available in the array to use by functions before it has to increase in size.  When the \textbf{used} count 
of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the 
array to accommodate the precision of the result.  

\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple 
precision integer.  It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits.  The array is maintained in a least 
significant digit order.  As a pencil and paper analogy the array is organized such that the right most digits are stored
first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array.  For example, 
if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then 
it would represent the integer $a + b\beta + c\beta^2 + \ldots$  

\index{MP\_ZPOS} \index{MP\_NEG}
\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}).  
\end{enumerate}

\subsubsection{Valid mp\_int Structures}
Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency.  
The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy().

\begin{enumerate}
\item The value of \textbf{alloc} may not be less than one.  That is \textbf{dp} always points to a previously allocated
array of digits.
\item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero.
\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero.  That is, 
leading zero digits in the most significant positions must be trimmed.
   \begin{enumerate}
   \item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero.
   \end{enumerate}
\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero; 
this represents the mp\_int value of zero.
\end{enumerate}

\section{Argument Passing}
A convention of argument passing must be adopted early on in the development of any library.  Making the function 
prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity.  
In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int 
structures.  That means that the source (input) operands are placed on the left and the destination (output) on the right.   
Consider the following examples.

\begin{verbatim}
   mp_mul(&a, &b, &c);   /* c = a * b */
   mp_add(&a, &b, &a);   /* a = a + b */
   mp_sqr(&a, &b);       /* b = a * a */
\end{verbatim}

The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the
functions and make sense of them.  For example, the first function would read ``multiply a and b and store in c''.

Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order
of assignment expressions.  That is, the destination (output) is on the left and arguments (inputs) are on the right.  In 
truth, it is entirely a matter of preference.  In the case of LibTomMath the convention from the MPI library has been 
adopted.  

Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a 
destination.  For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$.  This is an important 
feature to implement since it allows the calling functions to cut down on the number of variables it must maintain.  
However, to implement this feature specific care has to be given to ensure the destination is not modified before the 
source is fully read.

\section{Return Values}
A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them 
to the caller.  By catching runtime errors a library can be guaranteed to prevent undefined behaviour.  However, the end 
developer can still manage to cause a library to crash.  For example, by passing an invalid pointer an application may
fault by dereferencing memory not owned by the application.

In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for 
instance) and memory allocation errors.  It will not check that the mp\_int passed to any function is valid nor 
will it check pointers for validity.  Any function that can cause a runtime error will return an error code as an 
\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}).

\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM}
\begin{figure}[here]
\begin{center}
\begin{tabular}{|l|l|}
\hline \textbf{Value} & \textbf{Meaning} \\
\hline \textbf{MP\_OKAY} & The function was successful \\
\hline \textbf{MP\_VAL}  & One of the input value(s) was invalid \\
\hline \textbf{MP\_MEM}  & The function ran out of heap memory \\
\hline
\end{tabular}
\end{center}
\caption{LibTomMath Error Codes}
\label{fig:errcodes}
\end{figure}

When an error is detected within a function it should free any memory it allocated, often during the initialization of
temporary mp\_ints, and return as soon as possible.  The goal is to leave the system in the same state it was when the 
function was called.  Error checking with this style of API is fairly simple.

\begin{verbatim}
   int err;
   if ((err = mp_add(&a, &b, &c)) != MP_OKAY) {
      printf("Error: %s\n", mp_error_to_string(err));
      exit(EXIT_FAILURE);
   }
\end{verbatim}

The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use.  Not all errors are fatal 
and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases.

\section{Initialization and Clearing}
The logical starting point when actually writing multiple precision integer functions is the initialization and 
clearing of the mp\_int structures.  These two algorithms will be used by the majority of the higher level algorithms.

Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of
the integer.  Often it is optimal to allocate a sufficiently large pre-set number of digits even though
the initial integer will represent zero.  If only a single digit were allocated quite a few subsequent re-allocations
would occur when operations are performed on the integers.  There is a tradeoff between how many default digits to allocate
and how many re-allocations are tolerable.  Obviously allocating an excessive amount of digits initially will waste 
memory and become unmanageable.  

If the memory for the digits has been successfully allocated then the rest of the members of the structure must
be initialized.  Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set
to zero.  The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}.

\subsection{Initializing an mp\_int}
An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the







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\end{center}
\end{figure}

The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows.

\begin{enumerate}
\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent
a given integer.  The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count.

\item The \textbf{alloc} parameter denotes how
many digits are available in the array to use by functions before it has to increase in size.  When the \textbf{used} count
of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the
array to accommodate the precision of the result.

\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple
precision integer.  It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits.  The array is maintained in a least
significant digit order.  As a pencil and paper analogy the array is organized such that the right most digits are stored
first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array.  For example,
if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then
it would represent the integer $a + b\beta + c\beta^2 + \ldots$

\index{MP\_ZPOS} \index{MP\_NEG}
\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}).
\end{enumerate}

\subsubsection{Valid mp\_int Structures}
Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency.
The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy().

\begin{enumerate}
\item The value of \textbf{alloc} may not be less than one.  That is \textbf{dp} always points to a previously allocated
array of digits.
\item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero.
\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero.  That is,
leading zero digits in the most significant positions must be trimmed.
   \begin{enumerate}
   \item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero.
   \end{enumerate}
\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero;
this represents the mp\_int value of zero.
\end{enumerate}

\section{Argument Passing}
A convention of argument passing must be adopted early on in the development of any library.  Making the function
prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity.
In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int
structures.  That means that the source (input) operands are placed on the left and the destination (output) on the right.
Consider the following examples.

\begin{verbatim}
   mp_mul(&a, &b, &c);   /* c = a * b */
   mp_add(&a, &b, &a);   /* a = a + b */
   mp_sqr(&a, &b);       /* b = a * a */
\end{verbatim}

The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the
functions and make sense of them.  For example, the first function would read ``multiply a and b and store in c''.

Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order
of assignment expressions.  That is, the destination (output) is on the left and arguments (inputs) are on the right.  In
truth, it is entirely a matter of preference.  In the case of LibTomMath the convention from the MPI library has been
adopted.

Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a
destination.  For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$.  This is an important
feature to implement since it allows the calling functions to cut down on the number of variables it must maintain.
However, to implement this feature specific care has to be given to ensure the destination is not modified before the
source is fully read.

\section{Return Values}
A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them
to the caller.  By catching runtime errors a library can be guaranteed to prevent undefined behaviour.  However, the end
developer can still manage to cause a library to crash.  For example, by passing an invalid pointer an application may
fault by dereferencing memory not owned by the application.

In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for
instance) and memory allocation errors.  It will not check that the mp\_int passed to any function is valid nor
will it check pointers for validity.  Any function that can cause a runtime error will return an error code as an
\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}).

\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM}
\begin{figure}[here]
\begin{center}
\begin{tabular}{|l|l|}
\hline \textbf{Value} & \textbf{Meaning} \\
\hline \textbf{MP\_OKAY} & The function was successful \\
\hline \textbf{MP\_VAL}  & One of the input value(s) was invalid \\
\hline \textbf{MP\_MEM}  & The function ran out of heap memory \\
\hline
\end{tabular}
\end{center}
\caption{LibTomMath Error Codes}
\label{fig:errcodes}
\end{figure}

When an error is detected within a function it should free any memory it allocated, often during the initialization of
temporary mp\_ints, and return as soon as possible.  The goal is to leave the system in the same state it was when the
function was called.  Error checking with this style of API is fairly simple.

\begin{verbatim}
   int err;
   if ((err = mp_add(&a, &b, &c)) != MP_OKAY) {
      printf("Error: %s\n", mp_error_to_string(err));
      exit(EXIT_FAILURE);
   }
\end{verbatim}

The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use.  Not all errors are fatal
and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases.

\section{Initialization and Clearing}
The logical starting point when actually writing multiple precision integer functions is the initialization and
clearing of the mp\_int structures.  These two algorithms will be used by the majority of the higher level algorithms.

Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of
the integer.  Often it is optimal to allocate a sufficiently large pre-set number of digits even though
the initial integer will represent zero.  If only a single digit were allocated quite a few subsequent re-allocations
would occur when operations are performed on the integers.  There is a tradeoff between how many default digits to allocate
and how many re-allocations are tolerable.  Obviously allocating an excessive amount of digits initially will waste
memory and become unmanageable.

If the memory for the digits has been successfully allocated then the rest of the members of the structure must
be initialized.  Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set
to zero.  The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}.

\subsection{Initializing an mp\_int}
An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the
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\end{center}
\caption{Algorithm mp\_init}
\end{figure}

\textbf{Algorithm mp\_init.}
The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly
manipulte it.  It is assumed that the input may not have had any of its members previously initialized which is certainly
a valid assumption if the input resides on the stack.  

Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for
the digits is allocated.  If this fails the function returns before setting any of the other members.  The \textbf{MP\_PREC} 
name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} 
used to dictate the minimum precision of newly initialized mp\_int integers.  Ideally, it is at least equal to the smallest
precision number you'll be working with.

Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow
heap operations later functions will have to perform in the future.  If \textbf{MP\_PREC} is set correctly the slack 
memory and the number of heap operations will be trivial.

Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and
\textbf{alloc} members initialized.  This ensures that the mp\_int will always represent the default state of zero regardless
of the original condition of the input.

\textbf{Remark.}
This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally
when the ``to'' keyword is placed between two expressions.  For example, ``for $a$ from $b$ to $c$ do'' means that
a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$.  In each
iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$.  If $b > c$ occured
the loop would not iterate.  By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate 
decrementally.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_init.c
\vspace{-3mm}
\begin{alltt}



























\end{alltt}
\end{small}

One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure.  It 
is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack.  The 
call to mp\_init() is used only to initialize the members of the structure to a known default state.  

Here we see (line 24) the memory allocation is performed first.  This allows us to exit cleanly and quickly
if there is an error.  If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there
was a memory error.  The function XMALLOC is what actually allocates the memory.  Technically XMALLOC is not a function
but a macro defined in ``tommath.h``.  By default, XMALLOC will evaluate to malloc() which is the C library's built--in
memory allocation routine.

In order to assure the mp\_int is in a known state the digits must be set to zero.  On most platforms this could have been
accomplished by using calloc() instead of malloc().  However,  to correctly initialize a integer type to a given value in a 
portable fashion you have to actually assign the value.  The for loop (line 30) performs this required
operation.

After the memory has been successfully initialized the remainder of the members are initialized 
(lines 34 through 35) to their respective default states.  At this point the algorithm has succeeded and
a success code is returned to the calling function.  If this function returns \textbf{MP\_OKAY} it is safe to assume the 
mp\_int structure has been properly initialized and is safe to use with other functions within the library.  

\subsection{Clearing an mp\_int}
When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be 
returned to the application's memory pool with the mp\_clear algorithm.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clear}. \\
\textbf{Input}.   An mp\_int $a$ \\







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\end{center}
\caption{Algorithm mp\_init}
\end{figure}

\textbf{Algorithm mp\_init.}
The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly
manipulte it.  It is assumed that the input may not have had any of its members previously initialized which is certainly
a valid assumption if the input resides on the stack.

Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for
the digits is allocated.  If this fails the function returns before setting any of the other members.  The \textbf{MP\_PREC}
name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.}
used to dictate the minimum precision of newly initialized mp\_int integers.  Ideally, it is at least equal to the smallest
precision number you'll be working with.

Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow
heap operations later functions will have to perform in the future.  If \textbf{MP\_PREC} is set correctly the slack
memory and the number of heap operations will be trivial.

Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and
\textbf{alloc} members initialized.  This ensures that the mp\_int will always represent the default state of zero regardless
of the original condition of the input.

\textbf{Remark.}
This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally
when the ``to'' keyword is placed between two expressions.  For example, ``for $a$ from $b$ to $c$ do'' means that
a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$.  In each
iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$.  If $b > c$ occured
the loop would not iterate.  By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate
decrementally.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_init.c
\vspace{-3mm}
\begin{alltt}
016
017   /* init a new mp_int */
018   int mp_init (mp_int * a)
019   \{
020     int i;
021
022     /* allocate memory required and clear it */
023     a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * MP_PREC);
024     if (a->dp == NULL) \{
025       return MP_MEM;
026     \}
027
028     /* set the digits to zero */
029     for (i = 0; i < MP_PREC; i++) \{
030         a->dp[i] = 0;
031     \}
032
033     /* set the used to zero, allocated digits to the default precision
034      * and sign to positive */
035     a->used  = 0;
036     a->alloc = MP_PREC;
037     a->sign  = MP_ZPOS;
038
039     return MP_OKAY;
040   \}
041   #endif
042
\end{alltt}
\end{small}

One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure.  It
is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack.  The
call to mp\_init() is used only to initialize the members of the structure to a known default state.

Here we see (line 23) the memory allocation is performed first.  This allows us to exit cleanly and quickly
if there is an error.  If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there
was a memory error.  The function XMALLOC is what actually allocates the memory.  Technically XMALLOC is not a function
but a macro defined in ``tommath.h``.  By default, XMALLOC will evaluate to malloc() which is the C library's built--in
memory allocation routine.

In order to assure the mp\_int is in a known state the digits must be set to zero.  On most platforms this could have been
accomplished by using calloc() instead of malloc().  However,  to correctly initialize a integer type to a given value in a
portable fashion you have to actually assign the value.  The for loop (line 29) performs this required
operation.

After the memory has been successfully initialized the remainder of the members are initialized
(lines 33 through 34) to their respective default states.  At this point the algorithm has succeeded and
a success code is returned to the calling function.  If this function returns \textbf{MP\_OKAY} it is safe to assume the
mp\_int structure has been properly initialized and is safe to use with other functions within the library.

\subsection{Clearing an mp\_int}
When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be
returned to the application's memory pool with the mp\_clear algorithm.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clear}. \\
\textbf{Input}.   An mp\_int $a$ \\
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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_clear}
\end{figure}

\textbf{Algorithm mp\_clear.}
This algorithm accomplishes two goals.  First, it clears the digits and the other mp\_int members.  This ensures that 
if a developer accidentally re-uses a cleared structure it is less likely to cause problems.  The second goal
is to free the allocated memory.

The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this
algorithm will not try to free the memory multiple times.  Cleared mp\_ints are detectable by having a pre-defined invalid 
digit pointer \textbf{dp} setting.

Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm
with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_clear.c
\vspace{-3mm}
\begin{alltt}

























\end{alltt}
\end{small}

The algorithm only operates on the mp\_int if it hasn't been previously cleared.  The if statement (line 25)
checks to see if the \textbf{dp} member is not \textbf{NULL}.  If the mp\_int is a valid mp\_int then \textbf{dp} cannot be
\textbf{NULL} in which case the if statement will evaluate to true.

The digits of the mp\_int are cleared by the for loop (line 27) which assigns a zero to every digit.  Similar to mp\_init()
the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable.  

The digits are deallocated off the heap via the XFREE macro.  Similar to XMALLOC the XFREE macro actually evaluates to
a standard C library function.  In this case the free() function.  Since free() only deallocates the memory the pointer
still has to be reset to \textbf{NULL} manually (line 35).  

Now that the digits have been cleared and deallocated the other members are set to their final values (lines 36 and 37).

\section{Maintenance Algorithms}

The previous sections describes how to initialize and clear an mp\_int structure.  To further support operations
that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be
able to augment the precision of an mp\_int and 
initialize mp\_ints with differing initial conditions.  

These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level
algorithms such as addition, multiplication and modular exponentiation.

\subsection{Augmenting an mp\_int's Precision}
When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire 
result of an operation without loss of precision.  Quite often the size of the array given by the \textbf{alloc} member 
is large enough to simply increase the \textbf{used} digit count.  However, when the size of the array is too small it 
must be re-sized appropriately to accomodate the result.  The mp\_grow algorithm will provide this functionality.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_grow}. \\
\textbf{Input}.   An mp\_int $a$ and an integer $b$. \\







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867
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923
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928
929
930
931
932
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_clear}
\end{figure}

\textbf{Algorithm mp\_clear.}
This algorithm accomplishes two goals.  First, it clears the digits and the other mp\_int members.  This ensures that
if a developer accidentally re-uses a cleared structure it is less likely to cause problems.  The second goal
is to free the allocated memory.

The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this
algorithm will not try to free the memory multiple times.  Cleared mp\_ints are detectable by having a pre-defined invalid
digit pointer \textbf{dp} setting.

Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm
with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_clear.c
\vspace{-3mm}
\begin{alltt}
016
017   /* clear one (frees)  */
018   void
019   mp_clear (mp_int * a)
020   \{
021     int i;
022
023     /* only do anything if a hasn't been freed previously */
024     if (a->dp != NULL) \{
025       /* first zero the digits */
026       for (i = 0; i < a->used; i++) \{
027           a->dp[i] = 0;
028       \}
029
030       /* free ram */
031       XFREE(a->dp);
032
033       /* reset members to make debugging easier */
034       a->dp    = NULL;
035       a->alloc = a->used = 0;
036       a->sign  = MP_ZPOS;
037     \}
038   \}
039   #endif
040
\end{alltt}
\end{small}

The algorithm only operates on the mp\_int if it hasn't been previously cleared.  The if statement (line 24)
checks to see if the \textbf{dp} member is not \textbf{NULL}.  If the mp\_int is a valid mp\_int then \textbf{dp} cannot be
\textbf{NULL} in which case the if statement will evaluate to true.

The digits of the mp\_int are cleared by the for loop (line 26) which assigns a zero to every digit.  Similar to mp\_init()
the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable.

The digits are deallocated off the heap via the XFREE macro.  Similar to XMALLOC the XFREE macro actually evaluates to
a standard C library function.  In this case the free() function.  Since free() only deallocates the memory the pointer
still has to be reset to \textbf{NULL} manually (line 34).

Now that the digits have been cleared and deallocated the other members are set to their final values (lines 35 and 36).

\section{Maintenance Algorithms}

The previous sections describes how to initialize and clear an mp\_int structure.  To further support operations
that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be
able to augment the precision of an mp\_int and
initialize mp\_ints with differing initial conditions.

These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level
algorithms such as addition, multiplication and modular exponentiation.

\subsection{Augmenting an mp\_int's Precision}
When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire
result of an operation without loss of precision.  Quite often the size of the array given by the \textbf{alloc} member
is large enough to simply increase the \textbf{used} digit count.  However, when the size of the array is too small it
must be re-sized appropriately to accomodate the result.  The mp\_grow algorithm will provide this functionality.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_grow}. \\
\textbf{Input}.   An mp\_int $a$ and an integer $b$. \\
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
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921
922
923






































924
925
926
927
928
929
930
931
932
933
934
935
936
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938
939
940
941
942
943
944
945
946
947
948
949
950
951
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_grow}
\end{figure}

\textbf{Algorithm mp\_grow.}
It is ideal to prevent re-allocations from being performed if they are not required (step one).  This is useful to 
prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow.  

The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three).  
This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values.  

It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact.  This is much 
akin to how the \textit{realloc} function from the standard C library works.  Since the newly allocated digits are 
assumed to contain undefined values they are initially set to zero.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_grow.c
\vspace{-3mm}
\begin{alltt}






































\end{alltt}
\end{small}

A quick optimization is to first determine if a memory re-allocation is required at all.  The if statement (line 24) checks
if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count.  If the count is not larger than \textbf{alloc}
the function skips the re-allocation part thus saving time.

When a re-allocation is performed it is turned into an optimal request to save time in the future.  The requested digit count is
padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line 25).  The XREALLOC function is used
to re-allocate the memory.  As per the other functions XREALLOC is actually a macro which evaluates to realloc by default.  The realloc
function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before
the re-allocation.  All	that is left is to clear the newly allocated digits and return.

Note that the re-allocation result is actually stored in a temporary pointer $tmp$.  This is to allow this function to return
an error with a valid pointer.  Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$.  That would
result in a memory leak if XREALLOC ever failed.  

\subsection{Initializing Variable Precision mp\_ints}
Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size 
of input mp\_ints to a given algorithm.  The purpose of algorithm mp\_init\_size is similar to mp\_init except that it 
will allocate \textit{at least} a specified number of digits.  

\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_size}. \\
\textbf{Input}.   An mp\_int $a$ and the requested number of digits $b$. \\







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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_grow}
\end{figure}

\textbf{Algorithm mp\_grow.}
It is ideal to prevent re-allocations from being performed if they are not required (step one).  This is useful to
prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow.

The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three).
This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values.

It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact.  This is much
akin to how the \textit{realloc} function from the standard C library works.  Since the newly allocated digits are
assumed to contain undefined values they are initially set to zero.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_grow.c
\vspace{-3mm}
\begin{alltt}
016
017   /* grow as required */
018   int mp_grow (mp_int * a, int size)
019   \{
020     int     i;
021     mp_digit *tmp;
022
023     /* if the alloc size is smaller alloc more ram */
024     if (a->alloc < size) \{
025       /* ensure there are always at least MP_PREC digits extra on top */
026       size += (MP_PREC * 2) - (size % MP_PREC);
027
028       /* reallocate the array a->dp
029        *
030        * We store the return in a temporary variable
031        * in case the operation failed we don't want
032        * to overwrite the dp member of a.
033        */
034       tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * size);
035       if (tmp == NULL) \{
036         /* reallocation failed but "a" is still valid [can be freed] */
037         return MP_MEM;
038       \}
039
040       /* reallocation succeeded so set a->dp */
041       a->dp = tmp;
042
043       /* zero excess digits */
044       i        = a->alloc;
045       a->alloc = size;
046       for (; i < a->alloc; i++) \{
047         a->dp[i] = 0;
048       \}
049     \}
050     return MP_OKAY;
051   \}
052   #endif
053
\end{alltt}
\end{small}

A quick optimization is to first determine if a memory re-allocation is required at all.  The if statement (line 24) checks
if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count.  If the count is not larger than \textbf{alloc}
the function skips the re-allocation part thus saving time.

When a re-allocation is performed it is turned into an optimal request to save time in the future.  The requested digit count is
padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line 26).  The XREALLOC function is used
to re-allocate the memory.  As per the other functions XREALLOC is actually a macro which evaluates to realloc by default.  The realloc
function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before
the re-allocation.  All	that is left is to clear the newly allocated digits and return.

Note that the re-allocation result is actually stored in a temporary pointer $tmp$.  This is to allow this function to return
an error with a valid pointer.  Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$.  That would
result in a memory leak if XREALLOC ever failed.

\subsection{Initializing Variable Precision mp\_ints}
Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size
of input mp\_ints to a given algorithm.  The purpose of algorithm mp\_init\_size is similar to mp\_init except that it
will allocate \textit{at least} a specified number of digits.

\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_size}. \\
\textbf{Input}.   An mp\_int $a$ and the requested number of digits $b$. \\
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983





























984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_init\_size}
\end{figure}

\textbf{Algorithm mp\_init\_size.}
This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of 
digits allocated can be controlled by the second input argument $b$.  The input size is padded upwards so it is a 
multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits.  This padding is used to prevent trivial 
allocations from becoming a bottleneck in the rest of the algorithms.

Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero.  This 
particular algorithm is useful if it is known ahead of time the approximate size of the input.  If the approximation is
correct no further memory re-allocations are required to work with the mp\_int.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_size.c
\vspace{-3mm}
\begin{alltt}





























\end{alltt}
\end{small}

The number of digits $b$ requested is padded (line 24) by first augmenting it to the next multiple of 
\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result.  If the memory can be successfully allocated the 
mp\_int is placed in a default state representing the integer zero.  Otherwise, the error code \textbf{MP\_MEM} will be 
returned (line 29).  

The digits are allocated and set to zero at the same time with the calloc() function (line @25,XCALLOC@).  The 
\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set 
to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines 33, 34 and 35).  If the function 
returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the 
functions to work with.

\subsection{Multiple Integer Initializations and Clearings}
Occasionally a function will require a series of mp\_int data types to be made available simultaneously.  
The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single
statement.  It is essentially a shortcut to multiple initializations.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_multi}. \\







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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_init\_size}
\end{figure}

\textbf{Algorithm mp\_init\_size.}
This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of
digits allocated can be controlled by the second input argument $b$.  The input size is padded upwards so it is a
multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits.  This padding is used to prevent trivial
allocations from becoming a bottleneck in the rest of the algorithms.

Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero.  This
particular algorithm is useful if it is known ahead of time the approximate size of the input.  If the approximation is
correct no further memory re-allocations are required to work with the mp\_int.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_size.c
\vspace{-3mm}
\begin{alltt}
016
017   /* init an mp_init for a given size */
018   int mp_init_size (mp_int * a, int size)
019   \{
020     int x;
021
022     /* pad size so there are always extra digits */
023     size += (MP_PREC * 2) - (size % MP_PREC);
024
025     /* alloc mem */
026     a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * size);
027     if (a->dp == NULL) \{
028       return MP_MEM;
029     \}
030
031     /* set the members */
032     a->used  = 0;
033     a->alloc = size;
034     a->sign  = MP_ZPOS;
035
036     /* zero the digits */
037     for (x = 0; x < size; x++) \{
038         a->dp[x] = 0;
039     \}
040
041     return MP_OKAY;
042   \}
043   #endif
044
\end{alltt}
\end{small}

The number of digits $b$ requested is padded (line 23) by first augmenting it to the next multiple of
\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result.  If the memory can be successfully allocated the
mp\_int is placed in a default state representing the integer zero.  Otherwise, the error code \textbf{MP\_MEM} will be
returned (line 28).

The digits are allocated with the malloc() function (line 26) and set to zero afterwards (line 37).  The
\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set
to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines 32, 33 and 34).  If the function
returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the
functions to work with.

\subsection{Multiple Integer Initializations and Clearings}
Occasionally a function will require a series of mp\_int data types to be made available simultaneously.
The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single
statement.  It is essentially a shortcut to multiple initializations.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_multi}. \\
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031








































1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
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1060
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1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init\_multi}
\end{figure}

\textbf{Algorithm mp\_init\_multi.}
The algorithm will initialize the array of mp\_int variables one at a time.  If a runtime error has been detected 
(\textit{step 1.2}) all of the previously initialized variables are cleared.  The goal is an ``all or nothing'' 
initialization which allows for quick recovery from runtime errors.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_multi.c
\vspace{-3mm}
\begin{alltt}








































\end{alltt}
\end{small}

This function intializes a variable length list of mp\_int structure pointers.  However, instead of having the mp\_int
structures in an actual C array they are simply passed as arguments to the function.  This function makes use of the 
``...'' argument syntax of the C programming language.  The list is terminated with a final \textbf{NULL} argument 
appended on the right.  

The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function.  A count
$n$ of succesfully initialized mp\_int structures is maintained (line 48) such that if a failure does occur,
the algorithm can backtrack and free the previously initialized structures (lines 28 to 47).  


\subsection{Clamping Excess Digits}
When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of 
the function instead of checking during the computation.  For example, a multiplication of a $i$ digit number by a 
$j$ digit produces a result of at most $i + j$ digits.  It is entirely possible that the result is $i + j - 1$ 
though, with no final carry into the last position.  However, suppose the destination had to be first expanded 
(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry.  
That would be a considerable waste of time since heap operations are relatively slow.

The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function
terminates.  This way a single heap operation (\textit{at most}) is required.  However, if the result was not checked
there would be an excess high order zero digit.  

For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$.  The leading zero digit 
will not contribute to the precision of the result.  In fact, through subsequent operations more leading zero digits would
accumulate to the point the size of the integer would be prohibitive.  As a result even though the precision is very 
low the representation is excessively large.  

The mp\_clamp algorithm is designed to solve this very problem.  It will trim high-order zeros by decrementing the 
\textbf{used} count until a non-zero most significant digit is found.  Also in this system, zero is considered to be a 
positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to 
\textbf{MP\_ZPOS}.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clamp}. \\
\textbf{Input}.   An mp\_int $a$ \\







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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init\_multi}
\end{figure}

\textbf{Algorithm mp\_init\_multi.}
The algorithm will initialize the array of mp\_int variables one at a time.  If a runtime error has been detected
(\textit{step 1.2}) all of the previously initialized variables are cleared.  The goal is an ``all or nothing''
initialization which allows for quick recovery from runtime errors.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_multi.c
\vspace{-3mm}
\begin{alltt}
016   #include <stdarg.h>
017
018   int mp_init_multi(mp_int *mp, ...)
019   \{
020       mp_err res = MP_OKAY;      /* Assume ok until proven otherwise */
021       int n = 0;                 /* Number of ok inits */
022       mp_int* cur_arg = mp;
023       va_list args;
024
025       va_start(args, mp);        /* init args to next argument from caller */
026       while (cur_arg != NULL) \{
027           if (mp_init(cur_arg) != MP_OKAY) \{
028               /* Oops - error! Back-track and mp_clear what we already
029                  succeeded in init-ing, then return error.
030               */
031               va_list clean_args;
032
033               /* end the current list */
034               va_end(args);
035
036               /* now start cleaning up */
037               cur_arg = mp;
038               va_start(clean_args, mp);
039               while (n-- != 0) \{
040                   mp_clear(cur_arg);
041                   cur_arg = va_arg(clean_args, mp_int*);
042               \}
043               va_end(clean_args);
044               res = MP_MEM;
045               break;
046           \}
047           n++;
048           cur_arg = va_arg(args, mp_int*);
049       \}
050       va_end(args);
051       return res;                /* Assumed ok, if error flagged above. */
052   \}
053
054   #endif
055
\end{alltt}
\end{small}

This function intializes a variable length list of mp\_int structure pointers.  However, instead of having the mp\_int
structures in an actual C array they are simply passed as arguments to the function.  This function makes use of the
``...'' argument syntax of the C programming language.  The list is terminated with a final \textbf{NULL} argument
appended on the right.

The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function.  A count
$n$ of succesfully initialized mp\_int structures is maintained (line 47) such that if a failure does occur,
the algorithm can backtrack and free the previously initialized structures (lines 27 to 46).


\subsection{Clamping Excess Digits}
When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of
the function instead of checking during the computation.  For example, a multiplication of a $i$ digit number by a
$j$ digit produces a result of at most $i + j$ digits.  It is entirely possible that the result is $i + j - 1$
though, with no final carry into the last position.  However, suppose the destination had to be first expanded
(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry.
That would be a considerable waste of time since heap operations are relatively slow.

The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function
terminates.  This way a single heap operation (\textit{at most}) is required.  However, if the result was not checked
there would be an excess high order zero digit.

For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$.  The leading zero digit
will not contribute to the precision of the result.  In fact, through subsequent operations more leading zero digits would
accumulate to the point the size of the integer would be prohibitive.  As a result even though the precision is very
low the representation is excessively large.

The mp\_clamp algorithm is designed to solve this very problem.  It will trim high-order zeros by decrementing the
\textbf{used} count until a non-zero most significant digit is found.  Also in this system, zero is considered to be a
positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to
\textbf{MP\_ZPOS}.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clamp}. \\
\textbf{Input}.   An mp\_int $a$ \\
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\end{tabular}
\end{center}
\caption{Algorithm mp\_clamp}
\end{figure}

\textbf{Algorithm mp\_clamp.}
As can be expected this algorithm is very simple.  The loop on step one is expected to iterate only once or twice at
the most.  For example, this will happen in cases where there is not a carry to fill the last position.  Step two fixes the sign for 
when all of the digits are zero to ensure that the mp\_int is valid at all times.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_clamp.c
\vspace{-3mm}
\begin{alltt}

























\end{alltt}
\end{small}

Note on line 28 how to test for the \textbf{used} count is made on the left of the \&\& operator.  In the C programming
language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails.  This is 
important since if the \textbf{used} is zero the test on the right would fetch below the array.  That is obviously 
undesirable.  The parenthesis on line 31 is used to make sure the \textbf{used} count is decremented and not
the pointer ``a''.  

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\
                     & \\
$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations.  \\
                     & \\







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\end{tabular}
\end{center}
\caption{Algorithm mp\_clamp}
\end{figure}

\textbf{Algorithm mp\_clamp.}
As can be expected this algorithm is very simple.  The loop on step one is expected to iterate only once or twice at
the most.  For example, this will happen in cases where there is not a carry to fill the last position.  Step two fixes the sign for
when all of the digits are zero to ensure that the mp\_int is valid at all times.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_clamp.c
\vspace{-3mm}
\begin{alltt}
016
017   /* trim unused digits
018    *
019    * This is used to ensure that leading zero digits are
020    * trimed and the leading "used" digit will be non-zero
021    * Typically very fast.  Also fixes the sign if there
022    * are no more leading digits
023    */
024   void
025   mp_clamp (mp_int * a)
026   \{
027     /* decrease used while the most significant digit is
028      * zero.
029      */
030     while ((a->used > 0) && (a->dp[a->used - 1] == 0)) \{
031       --(a->used);
032     \}
033
034     /* reset the sign flag if used == 0 */
035     if (a->used == 0) \{
036       a->sign = MP_ZPOS;
037     \}
038   \}
039   #endif
040
\end{alltt}
\end{small}

Note on line 27 how to test for the \textbf{used} count is made on the left of the \&\& operator.  In the C programming
language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails.  This is
important since if the \textbf{used} is zero the test on the right would fetch below the array.  That is obviously
undesirable.  The parenthesis on line 30 is used to make sure the \textbf{used} count is decremented and not
the pointer ``a''.

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\
                     & \\
$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations.  \\
                     & \\
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% CHAPTER FOUR
%%%

\chapter{Basic Operations}

\section{Introduction}
In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining
mp\_int structures.  This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low 
level basis of the entire library.  While these algorithm are relatively trivial it is important to understand how they
work before proceeding since these algorithms will be used almost intrinsically in the following chapters.

The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of
mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures
represent.   

\section{Assigning Values to mp\_int Structures}
\subsection{Copying an mp\_int}
Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making
a copy for the purposes of this text.  The copy of the mp\_int will be a separate entity that represents the same
value as the mp\_int it was copied from.  The mp\_copy algorithm provides this functionality. 

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_copy}. \\
\textbf{Input}.  An mp\_int $a$ and $b$. \\
\textbf{Output}.  Store a copy of $a$ in $b$. \\







|





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% CHAPTER FOUR
%%%

\chapter{Basic Operations}

\section{Introduction}
In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining
mp\_int structures.  This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low
level basis of the entire library.  While these algorithm are relatively trivial it is important to understand how they
work before proceeding since these algorithms will be used almost intrinsically in the following chapters.

The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of
mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures
represent.

\section{Assigning Values to mp\_int Structures}
\subsection{Copying an mp\_int}
Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making
a copy for the purposes of this text.  The copy of the mp\_int will be a separate entity that represents the same
value as the mp\_int it was copied from.  The mp\_copy algorithm provides this functionality.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_copy}. \\
\textbf{Input}.  An mp\_int $a$ and $b$. \\
\textbf{Output}.  Store a copy of $a$ in $b$. \\
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\end{tabular}
\end{center}
\caption{Algorithm mp\_copy}
\end{figure}

\textbf{Algorithm mp\_copy.}
This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will
represent the same integer as the mp\_int $a$.  The mp\_int $b$ shall be a complete and distinct copy of the 
mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$.

If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow 
algorithm.  The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two
and three).  The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of
$b$.

\textbf{Remark.}  This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the
text.  The error return codes of other algorithms are not explicitly checked in the pseudo-code presented.  For example, in 
step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded.  Text space is 
limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return
the error code itself.  However, the C code presented will demonstrate all of the error handling logic required to 
implement the pseudo-code.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_copy.c
\vspace{-3mm}
\begin{alltt}

















































\end{alltt}
\end{small}

Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output
mp\_int structures passed to a function are one and the same.  For this case it is optimal to return immediately without 
copying digits (line 25).  

The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$.  If $b.alloc$ is less than
$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines 30 to 33).  In order to
simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits
of the mp\_ints $a$ and $b$ respectively.  These aliases (lines 43 and 46) allow the compiler to access the digits without first dereferencing the
mp\_int pointers and then subsequently the pointer to the digits.  

After the aliases are established the digits from $a$ are copied into $b$ (lines 49 to 51) and then the excess 
digits of $b$ are set to zero (lines 54 to 56).  Both ``for'' loops make use of the pointer aliases and in 
fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits.  This optimization 
allows the alias to stay in a machine register fairly easy between the two loops.

\textbf{Remarks.}  The use of pointer aliases is an implementation methodology first introduced in this function that will
be used considerably in other functions.  Technically, a pointer alias is simply a short hand alias used to lower the 
number of pointer dereferencing operations required to access data.  For example, a for loop may resemble

\begin{alltt}
for (x = 0; x < 100; x++) \{
    a->num[4]->dp[x] = 0;
\}
\end{alltt}

This could be re-written using aliases as 

\begin{alltt}
mp_digit *tmpa;
a = a->num[4]->dp;
for (x = 0; x < 100; x++) \{
    *a++ = 0;
\}
\end{alltt}

In this case an alias is used to access the 
array of digits within an mp\_int structure directly.  It may seem that a pointer alias is strictly not required 
as a compiler may optimize out the redundant pointer operations.  However, there are two dominant reasons to use aliases.

The first reason is that most compilers will not effectively optimize pointer arithmetic.  For example, some optimizations 
may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC).  Also some optimizations may 
work for GCC and not MSVC.  As such it is ideal to find a common ground for as many compilers as possible.  Pointer 
aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code 
stands a better chance of being faster.

The second reason is that pointer aliases often can make an algorithm simpler to read.  Consider the first ``for'' 
loop of the function mp\_copy() re-written to not use pointer aliases.

\begin{alltt}
    /* copy all the digits */
    for (n = 0; n < a->used; n++) \{
      b->dp[n] = a->dp[n];
    \}
\end{alltt}

Whether this code is harder to read depends strongly on the individual.  However, it is quantifiably slightly more 
complicated as there are four variables within the statement instead of just two.

\subsubsection{Nested Statements}
Another commonly used technique in the source routines is that certain sections of code are nested.  This is used in
particular with the pointer aliases to highlight code phases.  For example, a Comba multiplier (discussed in chapter six)
will typically have three different phases.  First the temporaries are initialized, then the columns calculated and 
finally the carries are propagated.  In this example the middle column production phase will typically be nested as it
uses temporary variables and aliases the most.

The nesting also simplies the source code as variables that are nested are only valid for their scope.  As a result
the various temporary variables required do not propagate into other sections of code.


\subsection{Creating a Clone}
Another common operation is to make a local temporary copy of an mp\_int argument.  To initialize an mp\_int 
and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone.  This is 
useful within functions that need to modify an argument but do not wish to actually modify the original copy.  The 
mp\_init\_copy algorithm has been designed to help perform this task.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_copy}. \\
\textbf{Input}.   An mp\_int $a$ and $b$\\
\textbf{Output}.  $a$ is initialized to be a copy of $b$. \\
\hline \\
1.  Init $a$.  (\textit{mp\_init}) \\
2.  Copy $b$ to $a$.  (\textit{mp\_copy}) \\
3.  Return the status of the copy operation. \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init\_copy}
\end{figure}

\textbf{Algorithm mp\_init\_copy.}
This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it.  As 
such this algorithm will perform two operations in one step.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_copy.c
\vspace{-3mm}
\begin{alltt}













\end{alltt}
\end{small}

This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}.  Note that 
\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call
and \textbf{a} will be left intact.  

\section{Zeroing an Integer}
Reseting an mp\_int to the default state is a common step in many algorithms.  The mp\_zero algorithm will be the algorithm used to
perform this task.

\begin{figure}[here]
\begin{center}







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\end{tabular}
\end{center}
\caption{Algorithm mp\_copy}
\end{figure}

\textbf{Algorithm mp\_copy.}
This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will
represent the same integer as the mp\_int $a$.  The mp\_int $b$ shall be a complete and distinct copy of the
mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$.

If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow
algorithm.  The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two
and three).  The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of
$b$.

\textbf{Remark.}  This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the
text.  The error return codes of other algorithms are not explicitly checked in the pseudo-code presented.  For example, in
step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded.  Text space is
limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return
the error code itself.  However, the C code presented will demonstrate all of the error handling logic required to
implement the pseudo-code.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_copy.c
\vspace{-3mm}
\begin{alltt}
016
017   /* copy, b = a */
018   int
019   mp_copy (mp_int * a, mp_int * b)
020   \{
021     int     res, n;
022
023     /* if dst == src do nothing */
024     if (a == b) \{
025       return MP_OKAY;
026     \}
027
028     /* grow dest */
029     if (b->alloc < a->used) \{
030        if ((res = mp_grow (b, a->used)) != MP_OKAY) \{
031           return res;
032        \}
033     \}
034
035     /* zero b and copy the parameters over */
036     \{
037       mp_digit *tmpa, *tmpb;
038
039       /* pointer aliases */
040
041       /* source */
042       tmpa = a->dp;
043
044       /* destination */
045       tmpb = b->dp;
046
047       /* copy all the digits */
048       for (n = 0; n < a->used; n++) \{
049         *tmpb++ = *tmpa++;
050       \}
051
052       /* clear high digits */
053       for (; n < b->used; n++) \{
054         *tmpb++ = 0;
055       \}
056     \}
057
058     /* copy used count and sign */
059     b->used = a->used;
060     b->sign = a->sign;
061     return MP_OKAY;
062   \}
063   #endif
064
\end{alltt}
\end{small}

Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output
mp\_int structures passed to a function are one and the same.  For this case it is optimal to return immediately without
copying digits (line 24).

The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$.  If $b.alloc$ is less than
$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines 29 to 33).  In order to
simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits
of the mp\_ints $a$ and $b$ respectively.  These aliases (lines 42 and 45) allow the compiler to access the digits without first dereferencing the
mp\_int pointers and then subsequently the pointer to the digits.

After the aliases are established the digits from $a$ are copied into $b$ (lines 48 to 50) and then the excess
digits of $b$ are set to zero (lines 53 to 55).  Both ``for'' loops make use of the pointer aliases and in
fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits.  This optimization
allows the alias to stay in a machine register fairly easy between the two loops.

\textbf{Remarks.}  The use of pointer aliases is an implementation methodology first introduced in this function that will
be used considerably in other functions.  Technically, a pointer alias is simply a short hand alias used to lower the
number of pointer dereferencing operations required to access data.  For example, a for loop may resemble

\begin{alltt}
for (x = 0; x < 100; x++) \{
    a->num[4]->dp[x] = 0;
\}
\end{alltt}

This could be re-written using aliases as

\begin{alltt}
mp_digit *tmpa;
a = a->num[4]->dp;
for (x = 0; x < 100; x++) \{
    *a++ = 0;
\}
\end{alltt}

In this case an alias is used to access the
array of digits within an mp\_int structure directly.  It may seem that a pointer alias is strictly not required
as a compiler may optimize out the redundant pointer operations.  However, there are two dominant reasons to use aliases.

The first reason is that most compilers will not effectively optimize pointer arithmetic.  For example, some optimizations
may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC).  Also some optimizations may
work for GCC and not MSVC.  As such it is ideal to find a common ground for as many compilers as possible.  Pointer
aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code
stands a better chance of being faster.

The second reason is that pointer aliases often can make an algorithm simpler to read.  Consider the first ``for''
loop of the function mp\_copy() re-written to not use pointer aliases.

\begin{alltt}
    /* copy all the digits */
    for (n = 0; n < a->used; n++) \{
      b->dp[n] = a->dp[n];
    \}
\end{alltt}

Whether this code is harder to read depends strongly on the individual.  However, it is quantifiably slightly more
complicated as there are four variables within the statement instead of just two.

\subsubsection{Nested Statements}
Another commonly used technique in the source routines is that certain sections of code are nested.  This is used in
particular with the pointer aliases to highlight code phases.  For example, a Comba multiplier (discussed in chapter six)
will typically have three different phases.  First the temporaries are initialized, then the columns calculated and
finally the carries are propagated.  In this example the middle column production phase will typically be nested as it
uses temporary variables and aliases the most.

The nesting also simplies the source code as variables that are nested are only valid for their scope.  As a result
the various temporary variables required do not propagate into other sections of code.


\subsection{Creating a Clone}
Another common operation is to make a local temporary copy of an mp\_int argument.  To initialize an mp\_int
and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone.  This is
useful within functions that need to modify an argument but do not wish to actually modify the original copy.  The
mp\_init\_copy algorithm has been designed to help perform this task.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_copy}. \\
\textbf{Input}.   An mp\_int $a$ and $b$\\
\textbf{Output}.  $a$ is initialized to be a copy of $b$. \\
\hline \\
1.  Init $a$.  (\textit{mp\_init}) \\
2.  Copy $b$ to $a$.  (\textit{mp\_copy}) \\
3.  Return the status of the copy operation. \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init\_copy}
\end{figure}

\textbf{Algorithm mp\_init\_copy.}
This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it.  As
such this algorithm will perform two operations in one step.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_copy.c
\vspace{-3mm}
\begin{alltt}
016
017   /* creates "a" then copies b into it */
018   int mp_init_copy (mp_int * a, mp_int * b)
019   \{
020     int     res;
021
022     if ((res = mp_init_size (a, b->used)) != MP_OKAY) \{
023       return res;
024     \}
025     return mp_copy (b, a);
026   \}
027   #endif
028
\end{alltt}
\end{small}

This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}.  Note that
\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call
and \textbf{a} will be left intact.

\section{Zeroing an Integer}
Reseting an mp\_int to the default state is a common step in many algorithms.  The mp\_zero algorithm will be the algorithm used to
perform this task.

\begin{figure}[here]
\begin{center}
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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_zero}
\end{figure}

\textbf{Algorithm mp\_zero.}
This algorithm simply resets a mp\_int to the default state.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_zero.c
\vspace{-3mm}
\begin{alltt}

















\end{alltt}
\end{small}

After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the 
\textbf{sign} variable is set to \textbf{MP\_ZPOS}.

\section{Sign Manipulation}
\subsection{Absolute Value}
With the mp\_int representation of an integer, calculating the absolute value is trivial.  The mp\_abs algorithm will compute
the absolute value of an mp\_int.








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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_zero}
\end{figure}

\textbf{Algorithm mp\_zero.}
This algorithm simply resets a mp\_int to the default state.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_zero.c
\vspace{-3mm}
\begin{alltt}
016
017   /* set to zero */
018   void mp_zero (mp_int * a)
019   \{
020     int       n;
021     mp_digit *tmp;
022
023     a->sign = MP_ZPOS;
024     a->used = 0;
025
026     tmp = a->dp;
027     for (n = 0; n < a->alloc; n++) \{
028        *tmp++ = 0;
029     \}
030   \}
031   #endif
032
\end{alltt}
\end{small}

After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the
\textbf{sign} variable is set to \textbf{MP\_ZPOS}.

\section{Sign Manipulation}
\subsection{Absolute Value}
With the mp\_int representation of an integer, calculating the absolute value is trivial.  The mp\_abs algorithm will compute
the absolute value of an mp\_int.

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\end{center}
\caption{Algorithm mp\_abs}
\end{figure}

\textbf{Algorithm mp\_abs.}
This algorithm computes the absolute of an mp\_int input.  First it copies $a$ over $b$.  This is an example of an
algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful.  This allows,
for instance, the developer to pass the same mp\_int as the source and destination to this function without addition 
logic to handle it.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_abs.c
\vspace{-3mm}
\begin{alltt}
























\end{alltt}
\end{small}

This fairly trivial algorithm first eliminates non--required duplications (line 28) and then sets the
\textbf{sign} flag to \textbf{MP\_ZPOS}.

\subsection{Integer Negation}
With the mp\_int representation of an integer, calculating the negation is also trivial.  The mp\_neg algorithm will compute
the negative of an mp\_int input.

\begin{figure}[here]







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\end{center}
\caption{Algorithm mp\_abs}
\end{figure}

\textbf{Algorithm mp\_abs.}
This algorithm computes the absolute of an mp\_int input.  First it copies $a$ over $b$.  This is an example of an
algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful.  This allows,
for instance, the developer to pass the same mp\_int as the source and destination to this function without addition
logic to handle it.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_abs.c
\vspace{-3mm}
\begin{alltt}
016
017   /* b = |a|
018    *
019    * Simple function copies the input and fixes the sign to positive
020    */
021   int
022   mp_abs (mp_int * a, mp_int * b)
023   \{
024     int     res;
025
026     /* copy a to b */
027     if (a != b) \{
028        if ((res = mp_copy (a, b)) != MP_OKAY) \{
029          return res;
030        \}
031     \}
032
033     /* force the sign of b to positive */
034     b->sign = MP_ZPOS;
035
036     return MP_OKAY;
037   \}
038   #endif
039
\end{alltt}
\end{small}

This fairly trivial algorithm first eliminates non--required duplications (line 27) and then sets the
\textbf{sign} flag to \textbf{MP\_ZPOS}.

\subsection{Integer Negation}
With the mp\_int representation of an integer, calculating the negation is also trivial.  The mp\_neg algorithm will compute
the negative of an mp\_int input.

\begin{figure}[here]
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\end{tabular}
\end{center}
\caption{Algorithm mp\_neg}
\end{figure}

\textbf{Algorithm mp\_neg.}
This algorithm computes the negation of an input.  First it copies $a$ over $b$.  If $a$ has no used digits then
the algorithm returns immediately.  Otherwise it flips the sign flag and stores the result in $b$.  Note that if 
$a$ had no digits then it must be positive by definition.  Had step three been omitted then the algorithm would return
zero as negative.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_neg.c
\vspace{-3mm}
\begin{alltt}





















\end{alltt}
\end{small}

Like mp\_abs() this function avoids non--required duplications (line 22) and then sets the sign.  We
have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}.  If the mp\_int is zero
than the \textbf{sign} is hard--coded to \textbf{MP\_ZPOS}.

\section{Small Constants}
\subsection{Setting Small Constants}
Often a mp\_int must be set to a relatively small value such as $1$ or $2$.  For these cases the mp\_set algorithm is useful.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_set}. \\
\textbf{Input}.   An mp\_int $a$ and a digit $b$ \\
\textbf{Output}.  Make $a$ equivalent to $b$ \\
\hline \\
1.  Zero $a$ (\textit{mp\_zero}). \\
2.  $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\
3.  $a.used \leftarrow  \left \lbrace \begin{array}{ll}
                              1 &  \mbox{if }a_0 > 0 \\
                              0 &  \mbox{if }a_0 = 0 
                              \end{array} \right .$ \\
\hline                              
\end{tabular}
\end{center}
\caption{Algorithm mp\_set}
\end{figure}

\textbf{Algorithm mp\_set.}
This algorithm sets a mp\_int to a small single digit value.  Step number 1 ensures that the integer is reset to the default state.  The
single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_set.c
\vspace{-3mm}
\begin{alltt}










\end{alltt}
\end{small}

First we zero (line 21) the mp\_int to make sure that the other members are initialized for a 
small positive constant.  mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count
is zero.  Next we set the digit and reduce it modulo $\beta$ (line 22).  After this step we have to 
check if the resulting digit is zero or not.  If it is not then we set the \textbf{used} count to one, otherwise
to zero.

We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with 
$2^k - 1$ will perform the same operation.

One important limitation of this function is that it will only set one digit.  The size of a digit is not fixed, meaning source that uses 
this function should take that into account.  Only trivially small constants can be set using this function.

\subsection{Setting Large Constants}
To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal.  It accepts a ``long''
data type as input and will always treat it as a 32-bit integer.

\begin{figure}[here]







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\end{tabular}
\end{center}
\caption{Algorithm mp\_neg}
\end{figure}

\textbf{Algorithm mp\_neg.}
This algorithm computes the negation of an input.  First it copies $a$ over $b$.  If $a$ has no used digits then
the algorithm returns immediately.  Otherwise it flips the sign flag and stores the result in $b$.  Note that if
$a$ had no digits then it must be positive by definition.  Had step three been omitted then the algorithm would return
zero as negative.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_neg.c
\vspace{-3mm}
\begin{alltt}
016
017   /* b = -a */
018   int mp_neg (mp_int * a, mp_int * b)
019   \{
020     int     res;
021     if (a != b) \{
022        if ((res = mp_copy (a, b)) != MP_OKAY) \{
023           return res;
024        \}
025     \}
026
027     if (mp_iszero(b) != MP_YES) \{
028        b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS;
029     \} else \{
030        b->sign = MP_ZPOS;
031     \}
032
033     return MP_OKAY;
034   \}
035   #endif
036
\end{alltt}
\end{small}

Like mp\_abs() this function avoids non--required duplications (line 21) and then sets the sign.  We
have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}.  If the mp\_int is zero
than the \textbf{sign} is hard--coded to \textbf{MP\_ZPOS}.

\section{Small Constants}
\subsection{Setting Small Constants}
Often a mp\_int must be set to a relatively small value such as $1$ or $2$.  For these cases the mp\_set algorithm is useful.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_set}. \\
\textbf{Input}.   An mp\_int $a$ and a digit $b$ \\
\textbf{Output}.  Make $a$ equivalent to $b$ \\
\hline \\
1.  Zero $a$ (\textit{mp\_zero}). \\
2.  $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\
3.  $a.used \leftarrow  \left \lbrace \begin{array}{ll}
                              1 &  \mbox{if }a_0 > 0 \\
                              0 &  \mbox{if }a_0 = 0
                              \end{array} \right .$ \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_set}
\end{figure}

\textbf{Algorithm mp\_set.}
This algorithm sets a mp\_int to a small single digit value.  Step number 1 ensures that the integer is reset to the default state.  The
single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_set.c
\vspace{-3mm}
\begin{alltt}
016
017   /* set to a digit */
018   void mp_set (mp_int * a, mp_digit b)
019   \{
020     mp_zero (a);
021     a->dp[0] = b & MP_MASK;
022     a->used  = (a->dp[0] != 0) ? 1 : 0;
023   \}
024   #endif
025
\end{alltt}
\end{small}

First we zero (line 20) the mp\_int to make sure that the other members are initialized for a
small positive constant.  mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count
is zero.  Next we set the digit and reduce it modulo $\beta$ (line 21).  After this step we have to
check if the resulting digit is zero or not.  If it is not then we set the \textbf{used} count to one, otherwise
to zero.

We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with
$2^k - 1$ will perform the same operation.

One important limitation of this function is that it will only set one digit.  The size of a digit is not fixed, meaning source that uses
this function should take that into account.  Only trivially small constants can be set using this function.

\subsection{Setting Large Constants}
To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal.  It accepts a ``long''
data type as input and will always treat it as a 32-bit integer.

\begin{figure}[here]
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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_set\_int}
\end{figure}

\textbf{Algorithm mp\_set\_int.}
The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the 
mp\_int.  Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions.  In step 2.2 the
next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is 
incremented to reflect the addition.  The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have
zero digits used and the newly added four bits would be ignored.

Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_set\_int.c
\vspace{-3mm}
\begin{alltt}





























\end{alltt}
\end{small}

This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes.  The weird
addition on line 39 ensures that the newly added in bits are added to the number of digits.  While it may not 
seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line 28 
as well as the  call to mp\_clamp() on line 41.  Both functions will clamp excess leading digits which keeps 
the number of used digits low.

\section{Comparisons}
\subsection{Unsigned Comparisions}
Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers.  For example,
to compare $1,234$ to $1,264$ the digits are extracted by their positions.  That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$
to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude 
positions.  If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater.  

The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two
mp\_int variables alone.  It will ignore the sign of the two inputs.  Such a function is useful when an absolute comparison is required or if the 
signs are known to agree in advance.

To facilitate working with the results of the comparison functions three constants are required.  

\begin{figure}[here]
\begin{center}
\begin{tabular}{|r|l|}
\hline \textbf{Constant} & \textbf{Meaning} \\
\hline \textbf{MP\_GT} & Greater Than \\
\hline \textbf{MP\_EQ} & Equal To \\







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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_set\_int}
\end{figure}

\textbf{Algorithm mp\_set\_int.}
The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the
mp\_int.  Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions.  In step 2.2 the
next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is
incremented to reflect the addition.  The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have
zero digits used and the newly added four bits would be ignored.

Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_set\_int.c
\vspace{-3mm}
\begin{alltt}
016
017   /* set a 32-bit const */
018   int mp_set_int (mp_int * a, unsigned long b)
019   \{
020     int     x, res;
021
022     mp_zero (a);
023
024     /* set four bits at a time */
025     for (x = 0; x < 8; x++) \{
026       /* shift the number up four bits */
027       if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) \{
028         return res;
029       \}
030
031       /* OR in the top four bits of the source */
032       a->dp[0] |= (b >> 28) & 15;
033
034       /* shift the source up to the next four bits */
035       b <<= 4;
036
037       /* ensure that digits are not clamped off */
038       a->used += 1;
039     \}
040     mp_clamp (a);
041     return MP_OKAY;
042   \}
043   #endif
044
\end{alltt}
\end{small}

This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes.  The weird
addition on line 38 ensures that the newly added in bits are added to the number of digits.  While it may not
seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line 27
as well as the  call to mp\_clamp() on line 40.  Both functions will clamp excess leading digits which keeps
the number of used digits low.

\section{Comparisons}
\subsection{Unsigned Comparisions}
Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers.  For example,
to compare $1,234$ to $1,264$ the digits are extracted by their positions.  That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$
to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude
positions.  If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater.

The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two
mp\_int variables alone.  It will ignore the sign of the two inputs.  Such a function is useful when an absolute comparison is required or if the
signs are known to agree in advance.

To facilitate working with the results of the comparison functions three constants are required.

\begin{figure}[here]
\begin{center}
\begin{tabular}{|r|l|}
\hline \textbf{Constant} & \textbf{Meaning} \\
\hline \textbf{MP\_GT} & Greater Than \\
\hline \textbf{MP\_EQ} & Equal To \\
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\end{tabular}
\end{center}
\caption{Algorithm mp\_cmp\_mag}
\end{figure}

\textbf{Algorithm mp\_cmp\_mag.}
By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return
\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$.  The first two steps compare the number of digits used in both $a$ and $b$.  
Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is.  
If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit.  

By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to
the zero'th digit.  If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp\_mag.c
\vspace{-3mm}
\begin{alltt}




































\end{alltt}
\end{small}

The two if statements (lines 25 and 29) compare the number of digits in the two inputs.  These two are 
performed before all of the digits are compared since it is a very cheap test to perform and can potentially save 
considerable time.  The implementation given is also not valid without those two statements.  $b.alloc$ may be 
smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits.



\subsection{Signed Comparisons}
Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}).  Based on an unsigned magnitude 
comparison a trivial signed comparison algorithm can be written.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_cmp}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$ \\







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\end{tabular}
\end{center}
\caption{Algorithm mp\_cmp\_mag}
\end{figure}

\textbf{Algorithm mp\_cmp\_mag.}
By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return
\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$.  The first two steps compare the number of digits used in both $a$ and $b$.
Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is.
If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit.

By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to
the zero'th digit.  If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp\_mag.c
\vspace{-3mm}
\begin{alltt}
016
017   /* compare maginitude of two ints (unsigned) */
018   int mp_cmp_mag (mp_int * a, mp_int * b)
019   \{
020     int     n;
021     mp_digit *tmpa, *tmpb;
022
023     /* compare based on # of non-zero digits */
024     if (a->used > b->used) \{
025       return MP_GT;
026     \}
027
028     if (a->used < b->used) \{
029       return MP_LT;
030     \}
031
032     /* alias for a */
033     tmpa = a->dp + (a->used - 1);
034
035     /* alias for b */
036     tmpb = b->dp + (a->used - 1);
037
038     /* compare based on digits  */
039     for (n = 0; n < a->used; ++n, --tmpa, --tmpb) \{
040       if (*tmpa > *tmpb) \{
041         return MP_GT;
042       \}
043
044       if (*tmpa < *tmpb) \{
045         return MP_LT;
046       \}
047     \}
048     return MP_EQ;
049   \}
050   #endif
051
\end{alltt}
\end{small}

The two if statements (lines 24 and 28) compare the number of digits in the two inputs.  These two are
performed before all of the digits are compared since it is a very cheap test to perform and can potentially save
considerable time.  The implementation given is also not valid without those two statements.  $b.alloc$ may be
smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits.



\subsection{Signed Comparisons}
Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}).  Based on an unsigned magnitude
comparison a trivial signed comparison algorithm can be written.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_cmp}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$ \\
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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_cmp}
\end{figure}

\textbf{Algorithm mp\_cmp.}
The first two steps compare the signs of the two inputs.  If the signs do not agree then it can return right away with the appropriate 
comparison code.  When the signs are equal the digits of the inputs must be compared to determine the correct result.  In step 
three the unsigned comparision flips the order of the arguments since they are both negative.  For instance, if $-a > -b$ then 
$\vert a \vert < \vert b \vert$.  Step number four will compare the two when they are both positive.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp.c
\vspace{-3mm}
\begin{alltt}
























\end{alltt}
\end{small}

The two if statements (lines 23 and 24) perform the initial sign comparison.  If the signs are not the equal then which ever
has the positive sign is larger.   The inputs are compared (line 32) based on magnitudes.  If the signs were both 
negative then the unsigned comparison is performed in the opposite direction (line 34).  Otherwise, the signs are assumed to 
be both positive and a forward direction unsigned comparison is performed.

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\
                     & \\
$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits  \\
                     & of two random digits (of equal magnitude) before a difference is found. \\
                     & \\
$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based  \\
                     & on the observations made in the previous problem. \\
                     &
\end{tabular}

\chapter{Basic Arithmetic}
\section{Introduction}
At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been 
established.  The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms.  These 
algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms.  It is very important 
that these algorithms are highly optimized.  On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms 
which easily places them at $O(n^2)$ or even $O(n^3)$ work levels.  

All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right 
logical shifts respectively.  A logical shift is analogous to sliding the decimal point of radix-10 representations.  For example, the real 
number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}).  
Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two.  
For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$.

One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed
from the number.  For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$.  However, with a logical shift the 
result is $110_2$.  

\section{Addition and Subtraction}
In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus.  For example, with 32-bit integers
$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$  since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$.  
As a result subtraction can be performed with a trivial series of logical operations and an addition.

However, in multiple precision arithmetic negative numbers are not represented in the same way.  Instead a sign flag is used to keep track of the
sign of the integer.  As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or 
subtraction algorithms with the sign fixed up appropriately.

The lower level algorithms will add or subtract integers without regard to the sign flag.  That is they will add or subtract the magnitude of
the integers respectively.

\subsection{Low Level Addition}
An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers.  That is to add the 
trailing digits first and propagate the resulting carry upwards.  Since this is a lower level algorithm the name will have a ``s\_'' prefix.  
Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely.

\newpage
\begin{figure}[!here]
\begin{center}
\begin{small}
\begin{tabular}{l}







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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_cmp}
\end{figure}

\textbf{Algorithm mp\_cmp.}
The first two steps compare the signs of the two inputs.  If the signs do not agree then it can return right away with the appropriate
comparison code.  When the signs are equal the digits of the inputs must be compared to determine the correct result.  In step
three the unsigned comparision flips the order of the arguments since they are both negative.  For instance, if $-a > -b$ then
$\vert a \vert < \vert b \vert$.  Step number four will compare the two when they are both positive.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp.c
\vspace{-3mm}
\begin{alltt}
016
017   /* compare two ints (signed)*/
018   int
019   mp_cmp (mp_int * a, mp_int * b)
020   \{
021     /* compare based on sign */
022     if (a->sign != b->sign) \{
023        if (a->sign == MP_NEG) \{
024           return MP_LT;
025        \} else \{
026           return MP_GT;
027        \}
028     \}
029
030     /* compare digits */
031     if (a->sign == MP_NEG) \{
032        /* if negative compare opposite direction */
033        return mp_cmp_mag(b, a);
034     \} else \{
035        return mp_cmp_mag(a, b);
036     \}
037   \}
038   #endif
039
\end{alltt}
\end{small}

The two if statements (lines 22 and 23) perform the initial sign comparison.  If the signs are not the equal then which ever
has the positive sign is larger.   The inputs are compared (line 31) based on magnitudes.  If the signs were both
negative then the unsigned comparison is performed in the opposite direction (line 33).  Otherwise, the signs are assumed to
be both positive and a forward direction unsigned comparison is performed.

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\
                     & \\
$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits  \\
                     & of two random digits (of equal magnitude) before a difference is found. \\
                     & \\
$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based  \\
                     & on the observations made in the previous problem. \\
                     &
\end{tabular}

\chapter{Basic Arithmetic}
\section{Introduction}
At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been
established.  The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms.  These
algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms.  It is very important
that these algorithms are highly optimized.  On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms
which easily places them at $O(n^2)$ or even $O(n^3)$ work levels.

All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right
logical shifts respectively.  A logical shift is analogous to sliding the decimal point of radix-10 representations.  For example, the real
number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}).
Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two.
For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$.

One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed
from the number.  For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$.  However, with a logical shift the
result is $110_2$.

\section{Addition and Subtraction}
In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus.  For example, with 32-bit integers
$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$  since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$.
As a result subtraction can be performed with a trivial series of logical operations and an addition.

However, in multiple precision arithmetic negative numbers are not represented in the same way.  Instead a sign flag is used to keep track of the
sign of the integer.  As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or
subtraction algorithms with the sign fixed up appropriately.

The lower level algorithms will add or subtract integers without regard to the sign flag.  That is they will add or subtract the magnitude of
the integers respectively.

\subsection{Low Level Addition}
An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers.  That is to add the
trailing digits first and propagate the resulting carry upwards.  Since this is a lower level algorithm the name will have a ``s\_'' prefix.
Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely.

\newpage
\begin{figure}[!here]
\begin{center}
\begin{small}
\begin{tabular}{l}
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\end{tabular}
\end{small}
\end{center}
\caption{Algorithm s\_mp\_add}
\end{figure}

\textbf{Algorithm s\_mp\_add.}
This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes.  
Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}.  Even the 
MIX pseudo  machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes.

The first thing that has to be accomplished is to sort out which of the two inputs is the largest.  The addition logic
will simply add all of the smallest input to the largest input and store that first part of the result in the
destination.  Then it will apply a simpler addition loop to excess digits of the larger input.

The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two 
inputs.  The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the
same number of digits.  After the inputs are sorted the destination $c$ is grown as required to accomodate the sum 
of the two inputs.  The original \textbf{used} count of $c$ is copied and set to the new used count.  

At this point the first addition loop will go through as many digit positions that both inputs have.  The carry
variable $\mu$ is set to zero outside the loop.  Inside the loop an ``addition'' step requires three statements to produce
one digit of the summand.  First
two digits from $a$ and $b$ are added together along with the carry $\mu$.  The carry of this step is extracted and stored
in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$.

Now all of the digit positions that both inputs have in common have been exhausted.  If $min \ne max$ then $x$ is an alias
for one of the inputs that has more digits.  A simplified addition loop is then used to essentially copy the remaining digits
and the carry to the destination.

The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are zeroed which completes the addition.


\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_add.c
\vspace{-3mm}
\begin{alltt}


























































































\end{alltt}
\end{small}

We first sort (lines 28 to 36) the inputs based on magnitude and determine the $min$ and $max$ variables.
Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias.  Next we
grow the destination (38 to 42) ensure that it can accomodate the result of the addition. 

Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style.  The three aliases that are on 
lines 56, 59 and 62 represent the two inputs and destination variables respectively.  These aliases are used to ensure the
compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int.

The initial carry $u$ will be cleared (line 65), note that $u$ is of type mp\_digit which ensures type 
compatibility within the implementation.  The initial addition (line 66 to 75) adds digits from
both inputs until the smallest input runs out of digits.  Similarly the conditional addition loop
(line 81 to 90) adds the remaining digits from the larger of the two inputs.  The addition is finished 
with the final carry being stored in $tmpc$ (line 94).  Note the ``++'' operator within the same expression.
After line 94, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$.  This is useful
for the next loop (line 97 to 99) which set any old upper digits to zero.

\subsection{Low Level Subtraction}
The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm.  The principle difference is that the
unsigned subtraction algorithm requires the result to be positive.  That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must 
be met for this algorithm to function properly.  Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly.  
This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms.


For this algorithm a new variable is required to make the description simpler.  Recall from section 1.3.1 that a mp\_digit must be able to represent
the range $0 \le x < 2\beta$ for the algorithms to work correctly.  However, it is allowable that a mp\_digit represent a larger range of values.  For 
this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a 
mp\_digit (\textit{this implies $2^{\gamma} > \beta$}).  

For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$.  In ISO C an ``unsigned long''
data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$.

\newpage\begin{figure}[!here]
\begin{center}
\begin{small}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_sub}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\
\textbf{Output}.  The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\
\hline \\
1.  $min \leftarrow b.used$ \\
2.  $max \leftarrow a.used$ \\
3.  If $c.alloc < max$ then grow $c$ to hold at least $max$ digits.  (\textit{mp\_grow}) \\
4.  $oldused \leftarrow c.used$ \\ 
5.  $c.used \leftarrow max$ \\
6.  $u \leftarrow 0$ \\
7.  for $n$ from $0$ to $min - 1$ do \\
\hspace{3mm}7.1  $c_n \leftarrow a_n - b_n - u$ \\
\hspace{3mm}7.2  $u   \leftarrow c_n >> (\gamma - 1)$ \\
\hspace{3mm}7.3  $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
8.  if $min < max$ then do \\







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\end{tabular}
\end{small}
\end{center}
\caption{Algorithm s\_mp\_add}
\end{figure}

\textbf{Algorithm s\_mp\_add.}
This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes.
Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}.  Even the
MIX pseudo  machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes.

The first thing that has to be accomplished is to sort out which of the two inputs is the largest.  The addition logic
will simply add all of the smallest input to the largest input and store that first part of the result in the
destination.  Then it will apply a simpler addition loop to excess digits of the larger input.

The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two
inputs.  The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the
same number of digits.  After the inputs are sorted the destination $c$ is grown as required to accomodate the sum
of the two inputs.  The original \textbf{used} count of $c$ is copied and set to the new used count.

At this point the first addition loop will go through as many digit positions that both inputs have.  The carry
variable $\mu$ is set to zero outside the loop.  Inside the loop an ``addition'' step requires three statements to produce
one digit of the summand.  First
two digits from $a$ and $b$ are added together along with the carry $\mu$.  The carry of this step is extracted and stored
in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$.

Now all of the digit positions that both inputs have in common have been exhausted.  If $min \ne max$ then $x$ is an alias
for one of the inputs that has more digits.  A simplified addition loop is then used to essentially copy the remaining digits
and the carry to the destination.

The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are zeroed which completes the addition.


\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_add.c
\vspace{-3mm}
\begin{alltt}
016
017   /* low level addition, based on HAC pp.594, Algorithm 14.7 */
018   int
019   s_mp_add (mp_int * a, mp_int * b, mp_int * c)
020   \{
021     mp_int *x;
022     int     olduse, res, min, max;
023
024     /* find sizes, we let |a| <= |b| which means we have to sort
025      * them.  "x" will point to the input with the most digits
026      */
027     if (a->used > b->used) \{
028       min = b->used;
029       max = a->used;
030       x = a;
031     \} else \{
032       min = a->used;
033       max = b->used;
034       x = b;
035     \}
036
037     /* init result */
038     if (c->alloc < (max + 1)) \{
039       if ((res = mp_grow (c, max + 1)) != MP_OKAY) \{
040         return res;
041       \}
042     \}
043
044     /* get old used digit count and set new one */
045     olduse = c->used;
046     c->used = max + 1;
047
048     \{
049       mp_digit u, *tmpa, *tmpb, *tmpc;
050       int i;
051
052       /* alias for digit pointers */
053
054       /* first input */
055       tmpa = a->dp;
056
057       /* second input */
058       tmpb = b->dp;
059
060       /* destination */
061       tmpc = c->dp;
062
063       /* zero the carry */
064       u = 0;
065       for (i = 0; i < min; i++) \{
066         /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */
067         *tmpc = *tmpa++ + *tmpb++ + u;
068
069         /* U = carry bit of T[i] */
070         u = *tmpc >> ((mp_digit)DIGIT_BIT);
071
072         /* take away carry bit from T[i] */
073         *tmpc++ &= MP_MASK;
074       \}
075
076       /* now copy higher words if any, that is in A+B
077        * if A or B has more digits add those in
078        */
079       if (min != max) \{
080         for (; i < max; i++) \{
081           /* T[i] = X[i] + U */
082           *tmpc = x->dp[i] + u;
083
084           /* U = carry bit of T[i] */
085           u = *tmpc >> ((mp_digit)DIGIT_BIT);
086
087           /* take away carry bit from T[i] */
088           *tmpc++ &= MP_MASK;
089         \}
090       \}
091
092       /* add carry */
093       *tmpc++ = u;
094
095       /* clear digits above oldused */
096       for (i = c->used; i < olduse; i++) \{
097         *tmpc++ = 0;
098       \}
099     \}
100
101     mp_clamp (c);
102     return MP_OKAY;
103   \}
104   #endif
105
\end{alltt}
\end{small}

We first sort (lines 27 to 35) the inputs based on magnitude and determine the $min$ and $max$ variables.
Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias.  Next we
grow the destination (37 to 42) ensure that it can accomodate the result of the addition.

Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style.  The three aliases that are on
lines 55, 58 and 61 represent the two inputs and destination variables respectively.  These aliases are used to ensure the
compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int.

The initial carry $u$ will be cleared (line 64), note that $u$ is of type mp\_digit which ensures type
compatibility within the implementation.  The initial addition (line 65 to 74) adds digits from
both inputs until the smallest input runs out of digits.  Similarly the conditional addition loop
(line 80 to 90) adds the remaining digits from the larger of the two inputs.  The addition is finished
with the final carry being stored in $tmpc$ (line 93).  Note the ``++'' operator within the same expression.
After line 93, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$.  This is useful
for the next loop (line 96 to 99) which set any old upper digits to zero.

\subsection{Low Level Subtraction}
The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm.  The principle difference is that the
unsigned subtraction algorithm requires the result to be positive.  That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must
be met for this algorithm to function properly.  Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly.
This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms.


For this algorithm a new variable is required to make the description simpler.  Recall from section 1.3.1 that a mp\_digit must be able to represent
the range $0 \le x < 2\beta$ for the algorithms to work correctly.  However, it is allowable that a mp\_digit represent a larger range of values.  For
this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a
mp\_digit (\textit{this implies $2^{\gamma} > \beta$}).

For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$.  In ISO C an ``unsigned long''
data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$.

\newpage\begin{figure}[!here]
\begin{center}
\begin{small}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_sub}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\
\textbf{Output}.  The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\
\hline \\
1.  $min \leftarrow b.used$ \\
2.  $max \leftarrow a.used$ \\
3.  If $c.alloc < max$ then grow $c$ to hold at least $max$ digits.  (\textit{mp\_grow}) \\
4.  $oldused \leftarrow c.used$ \\
5.  $c.used \leftarrow max$ \\
6.  $u \leftarrow 0$ \\
7.  for $n$ from $0$ to $min - 1$ do \\
\hspace{3mm}7.1  $c_n \leftarrow a_n - b_n - u$ \\
\hspace{3mm}7.2  $u   \leftarrow c_n >> (\gamma - 1)$ \\
\hspace{3mm}7.3  $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
8.  if $min < max$ then do \\
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\textbf{Algorithm s\_mp\_sub.}
This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive.  That is when
passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly.  This
algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well.  As was the case
of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude.

The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$.  Steps 1 and 2 
set the $min$ and $max$ variables.  Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at 
most $max$ digits in length as opposed to $max + 1$.  Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and 
set to the maximal count for the operation.

The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision 
subtraction is used instead.  Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction 
loops.  Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry.  

For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$.  The least significant bit will force a carry upwards to 
the third bit which will be set to zero after the borrow.  After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain,  When the 
third bit of $0101_2$ is subtracted from the result it will cause another carry.  In this case though the carry will be forced to propagate all the 
way to the most significant bit.  

Recall that $\beta < 2^{\gamma}$.  This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most 
significant bit.  Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that
is needed is a single zero or one bit for the carry.  Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the 
carry.  This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed.  

If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$.  Step
10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sub.c
\vspace{-3mm}
\begin{alltt}







































































\end{alltt}
\end{small}

Like low level addition we ``sort'' the inputs.  Except in this case the sorting is hardcoded 
(lines 25 and 26).  In reality the $min$ and $max$ variables are only aliases and are only 
used to make the source code easier to read.  Again the pointer alias optimization is used 
within this algorithm.  The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized
(lines 42, 43 and 44) for $a$, $b$ and $c$ respectively.

The first subtraction loop (lines 47 through 61) subtract digits from both inputs until the smaller of
the two inputs has been exhausted.  As remarked earlier there is an implementation reason for using the ``awkward'' 
method of extracting the carry (line 57).  The traditional method for extracting the carry would be to shift 
by $lg(\beta)$ positions and logically AND the least significant bit.  The AND operation is required because all of 
the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction.  This carry 
extraction requires two relatively cheap operations to extract the carry.  The other method is to simply shift the 
most significant bit to the least significant bit thus extracting the carry with a single cheap operation.  This 
optimization only works on twos compliment machines which is a safe assumption to make.

If $a$ has a larger magnitude than $b$ an additional loop (lines 64 through 73) is required to propagate 
the carry through $a$ and copy the result to $c$.  

\subsection{High Level Addition}
Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be
established.  This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data 
types.  

Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} 
flag.  A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases.

\begin{figure}[!here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_add}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$  \\







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\textbf{Algorithm s\_mp\_sub.}
This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive.  That is when
passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly.  This
algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well.  As was the case
of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude.

The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$.  Steps 1 and 2
set the $min$ and $max$ variables.  Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at
most $max$ digits in length as opposed to $max + 1$.  Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and
set to the maximal count for the operation.

The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision
subtraction is used instead.  Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction
loops.  Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry.

For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$.  The least significant bit will force a carry upwards to
the third bit which will be set to zero after the borrow.  After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain,  When the
third bit of $0101_2$ is subtracted from the result it will cause another carry.  In this case though the carry will be forced to propagate all the
way to the most significant bit.

Recall that $\beta < 2^{\gamma}$.  This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most
significant bit.  Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that
is needed is a single zero or one bit for the carry.  Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the
carry.  This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed.

If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$.  Step
10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sub.c
\vspace{-3mm}
\begin{alltt}
016
017   /* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */
018   int
019   s_mp_sub (mp_int * a, mp_int * b, mp_int * c)
020   \{
021     int     olduse, res, min, max;
022
023     /* find sizes */
024     min = b->used;
025     max = a->used;
026
027     /* init result */
028     if (c->alloc < max) \{
029       if ((res = mp_grow (c, max)) != MP_OKAY) \{
030         return res;
031       \}
032     \}
033     olduse = c->used;
034     c->used = max;
035
036     \{
037       mp_digit u, *tmpa, *tmpb, *tmpc;
038       int i;
039
040       /* alias for digit pointers */
041       tmpa = a->dp;
042       tmpb = b->dp;
043       tmpc = c->dp;
044
045       /* set carry to zero */
046       u = 0;
047       for (i = 0; i < min; i++) \{
048         /* T[i] = A[i] - B[i] - U */
049         *tmpc = (*tmpa++ - *tmpb++) - u;
050
051         /* U = carry bit of T[i]
052          * Note this saves performing an AND operation since
053          * if a carry does occur it will propagate all the way to the
054          * MSB.  As a result a single shift is enough to get the carry
055          */
056         u = *tmpc >> ((mp_digit)((CHAR_BIT * sizeof(mp_digit)) - 1));
057
058         /* Clear carry from T[i] */
059         *tmpc++ &= MP_MASK;
060       \}
061
062       /* now copy higher words if any, e.g. if A has more digits than B  */
063       for (; i < max; i++) \{
064         /* T[i] = A[i] - U */
065         *tmpc = *tmpa++ - u;
066
067         /* U = carry bit of T[i] */
068         u = *tmpc >> ((mp_digit)((CHAR_BIT * sizeof(mp_digit)) - 1));
069
070         /* Clear carry from T[i] */
071         *tmpc++ &= MP_MASK;
072       \}
073
074       /* clear digits above used (since we may not have grown result above) */

075       for (i = c->used; i < olduse; i++) \{
076         *tmpc++ = 0;
077       \}
078     \}
079
080     mp_clamp (c);
081     return MP_OKAY;
082   \}
083
084   #endif
085
\end{alltt}
\end{small}

Like low level addition we ``sort'' the inputs.  Except in this case the sorting is hardcoded
(lines 24 and 25).  In reality the $min$ and $max$ variables are only aliases and are only
used to make the source code easier to read.  Again the pointer alias optimization is used
within this algorithm.  The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized
(lines 41, 42 and 43) for $a$, $b$ and $c$ respectively.

The first subtraction loop (lines 46 through 60) subtract digits from both inputs until the smaller of
the two inputs has been exhausted.  As remarked earlier there is an implementation reason for using the ``awkward''
method of extracting the carry (line 56).  The traditional method for extracting the carry would be to shift
by $lg(\beta)$ positions and logically AND the least significant bit.  The AND operation is required because all of
the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction.  This carry
extraction requires two relatively cheap operations to extract the carry.  The other method is to simply shift the
most significant bit to the least significant bit thus extracting the carry with a single cheap operation.  This
optimization only works on twos compliment machines which is a safe assumption to make.

If $a$ has a larger magnitude than $b$ an additional loop (lines 63 through 72) is required to propagate
the carry through $a$ and copy the result to $c$.

\subsection{High Level Addition}
Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be
established.  This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data
types.

Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign}
flag.  A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases.

\begin{figure}[!here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_add}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$  \\
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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_add}
\end{figure}

\textbf{Algorithm mp\_add.}
This algorithm performs the signed addition of two mp\_int variables.  There is no reference algorithm to draw upon from 
either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations.  The algorithm is fairly 
straightforward but restricted since subtraction can only produce positive results.

\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\







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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_add}
\end{figure}

\textbf{Algorithm mp\_add.}
This algorithm performs the signed addition of two mp\_int variables.  There is no reference algorithm to draw upon from
either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations.  The algorithm is fairly
straightforward but restricted since subtraction can only produce positive results.

\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\
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\end{tabular}
\end{center}
\end{small}
\caption{Addition Guide Chart}
\label{fig:AddChart}
\end{figure}

Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three 
specific cases need to be handled.  The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are 
forwarded to step three to check for errors.  This simplifies the description of the algorithm considerably and best 
follows how the implementation actually was achieved.

Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed.  Recall from the descriptions of algorithms
s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits.  The mp\_clamp algorithm will set the \textbf{sign}
to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero.

For example, consider performing $-a + a$ with algorithm mp\_add.  By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would
produce a result of $-0$.  However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp 
within algorithm s\_mp\_add will force $-0$ to become $0$.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_add.c
\vspace{-3mm}
\begin{alltt}


































\end{alltt}
\end{small}

The source code follows the algorithm fairly closely.  The most notable new source code addition is the usage of the $res$ integer variable which
is used to pass result of the unsigned operations forward.  Unlike in the algorithm, the variable $res$ is merely returned as is without
explicitly checking it and returning the constant \textbf{MP\_OKAY}.  The observation is this algorithm will succeed or fail only if the lower
level functions do so.  Returning their return code is sufficient.

\subsection{High Level Subtraction}
The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm.  

\newpage\begin{figure}[!here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_sub}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$  \\
\textbf{Output}.  The signed subtraction $c = a - b$. \\







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\end{tabular}
\end{center}
\end{small}
\caption{Addition Guide Chart}
\label{fig:AddChart}
\end{figure}

Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three
specific cases need to be handled.  The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are
forwarded to step three to check for errors.  This simplifies the description of the algorithm considerably and best
follows how the implementation actually was achieved.

Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed.  Recall from the descriptions of algorithms
s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits.  The mp\_clamp algorithm will set the \textbf{sign}
to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero.

For example, consider performing $-a + a$ with algorithm mp\_add.  By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would
produce a result of $-0$.  However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp
within algorithm s\_mp\_add will force $-0$ to become $0$.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_add.c
\vspace{-3mm}
\begin{alltt}
016
017   /* high level addition (handles signs) */
018   int mp_add (mp_int * a, mp_int * b, mp_int * c)
019   \{
020     int     sa, sb, res;
021
022     /* get sign of both inputs */
023     sa = a->sign;
024     sb = b->sign;
025
026     /* handle two cases, not four */
027     if (sa == sb) \{
028       /* both positive or both negative */
029       /* add their magnitudes, copy the sign */
030       c->sign = sa;
031       res = s_mp_add (a, b, c);
032     \} else \{
033       /* one positive, the other negative */
034       /* subtract the one with the greater magnitude from */
035       /* the one of the lesser magnitude.  The result gets */
036       /* the sign of the one with the greater magnitude. */
037       if (mp_cmp_mag (a, b) == MP_LT) \{
038         c->sign = sb;
039         res = s_mp_sub (b, a, c);
040       \} else \{
041         c->sign = sa;
042         res = s_mp_sub (a, b, c);
043       \}
044     \}
045     return res;
046   \}
047
048   #endif
049
\end{alltt}
\end{small}

The source code follows the algorithm fairly closely.  The most notable new source code addition is the usage of the $res$ integer variable which
is used to pass result of the unsigned operations forward.  Unlike in the algorithm, the variable $res$ is merely returned as is without
explicitly checking it and returning the constant \textbf{MP\_OKAY}.  The observation is this algorithm will succeed or fail only if the lower
level functions do so.  Returning their return code is sufficient.

\subsection{High Level Subtraction}
The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm.

\newpage\begin{figure}[!here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_sub}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$  \\
\textbf{Output}.  The signed subtraction $c = a - b$. \\
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1976
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_sub}
\end{figure}

\textbf{Algorithm mp\_sub.}
This algorithm performs the signed subtraction of two inputs.  Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or 
\cite{HAC}.  Also this algorithm is restricted by algorithm s\_mp\_sub.  Chart \ref{fig:SubChart} lists the eight possible inputs and
the operations required.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}







|







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\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_sub}
\end{figure}

\textbf{Algorithm mp\_sub.}
This algorithm performs the signed subtraction of two inputs.  Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or
\cite{HAC}.  Also this algorithm is restricted by algorithm s\_mp\_sub.  Chart \ref{fig:SubChart} lists the eight possible inputs and
the operations required.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
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1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002








































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2006
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2012
2013
2014
2015
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\end{tabular}
\end{center}
\end{small}
\caption{Subtraction Guide Chart}
\label{fig:SubChart}
\end{figure}

Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction.  That is to prevent the 
algorithm from producing $-a - -a = -0$ as a result.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_sub.c
\vspace{-3mm}
\begin{alltt}








































\end{alltt}
\end{small}

Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations
and forward it to the end of the function.  On line 39 the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a 
``greater than or equal to'' comparison.  

\section{Bit and Digit Shifting}
It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$.  
This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring.  

In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established.  That is to shift
the digits left or right as well to shift individual bits of the digits left and right.  It is important to note that not all ``shift'' operations
are on radix-$\beta$ digits.  

\subsection{Multiplication by Two}

In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient 
operation to perform.  A single precision logical shift left is sufficient to multiply a single digit by two.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mul\_2}. \\
\textbf{Input}.   One mp\_int $a$ \\







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\end{tabular}
\end{center}
\end{small}
\caption{Subtraction Guide Chart}
\label{fig:SubChart}
\end{figure}

Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction.  That is to prevent the
algorithm from producing $-a - -a = -0$ as a result.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_sub.c
\vspace{-3mm}
\begin{alltt}
016
017   /* high level subtraction (handles signs) */
018   int
019   mp_sub (mp_int * a, mp_int * b, mp_int * c)
020   \{
021     int     sa, sb, res;
022
023     sa = a->sign;
024     sb = b->sign;
025
026     if (sa != sb) \{
027       /* subtract a negative from a positive, OR */
028       /* subtract a positive from a negative. */
029       /* In either case, ADD their magnitudes, */
030       /* and use the sign of the first number. */
031       c->sign = sa;
032       res = s_mp_add (a, b, c);
033     \} else \{
034       /* subtract a positive from a positive, OR */
035       /* subtract a negative from a negative. */
036       /* First, take the difference between their */
037       /* magnitudes, then... */
038       if (mp_cmp_mag (a, b) != MP_LT) \{
039         /* Copy the sign from the first */
040         c->sign = sa;
041         /* The first has a larger or equal magnitude */
042         res = s_mp_sub (a, b, c);
043       \} else \{
044         /* The result has the *opposite* sign from */
045         /* the first number. */
046         c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS;
047         /* The second has a larger magnitude */
048         res = s_mp_sub (b, a, c);
049       \}
050     \}
051     return res;
052   \}
053
054   #endif
055
\end{alltt}
\end{small}

Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations
and forward it to the end of the function.  On line 38 the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a
``greater than or equal to'' comparison.

\section{Bit and Digit Shifting}
It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$.
This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring.

In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established.  That is to shift
the digits left or right as well to shift individual bits of the digits left and right.  It is important to note that not all ``shift'' operations
are on radix-$\beta$ digits.

\subsection{Multiplication by Two}

In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient
operation to perform.  A single precision logical shift left is sufficient to multiply a single digit by two.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mul\_2}. \\
\textbf{Input}.   One mp\_int $a$ \\
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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul\_2}
\end{figure}

\textbf{Algorithm mp\_mul\_2.}
This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two.  Neither \cite{TAOCPV2} nor \cite{HAC} describe such 
an algorithm despite the fact it arises often in other algorithms.  The algorithm is setup much like the lower level algorithm s\_mp\_add since 
it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$.  

Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result.  The initial \textbf{used} count
is set to $a.used$ at step 4.  Only if there is a final carry will the \textbf{used} count require adjustment.

Step 6 is an optimization implementation of the addition loop for this specific case.  That is since the two values being added together 
are the same there is no need to perform two reads from the digits of $a$.  Step 6.1 performs a single precision shift on the current digit $a_n$ to
obtain what will be the carry for the next iteration.  Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus
the previous carry.  Recall from section 4.1 that $a_n << 1$ is equivalent to $a_n \cdot 2$.  An iteration of the addition loop is finished with 
forwarding the carry to the next iteration.

Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$.  
Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2.c
\vspace{-3mm}
\begin{alltt}































































\end{alltt}
\end{small}

This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input.  The only noteworthy difference
is the use of the logical shift operator on line 52 to perform a single precision doubling.  

\subsection{Division by Two}
A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}







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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul\_2}
\end{figure}

\textbf{Algorithm mp\_mul\_2.}
This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two.  Neither \cite{TAOCPV2} nor \cite{HAC} describe such
an algorithm despite the fact it arises often in other algorithms.  The algorithm is setup much like the lower level algorithm s\_mp\_add since
it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$.

Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result.  The initial \textbf{used} count
is set to $a.used$ at step 4.  Only if there is a final carry will the \textbf{used} count require adjustment.

Step 6 is an optimization implementation of the addition loop for this specific case.  That is since the two values being added together
are the same there is no need to perform two reads from the digits of $a$.  Step 6.1 performs a single precision shift on the current digit $a_n$ to
obtain what will be the carry for the next iteration.  Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus
the previous carry.  Recall from section 4.1 that $a_n << 1$ is equivalent to $a_n \cdot 2$.  An iteration of the addition loop is finished with
forwarding the carry to the next iteration.

Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$.
Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2.c
\vspace{-3mm}
\begin{alltt}
016
017   /* b = a*2 */
018   int mp_mul_2(mp_int * a, mp_int * b)
019   \{
020     int     x, res, oldused;
021
022     /* grow to accomodate result */
023     if (b->alloc < (a->used + 1)) \{
024       if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) \{
025         return res;
026       \}
027     \}
028
029     oldused = b->used;
030     b->used = a->used;
031
032     \{
033       mp_digit r, rr, *tmpa, *tmpb;
034
035       /* alias for source */
036       tmpa = a->dp;
037
038       /* alias for dest */
039       tmpb = b->dp;
040
041       /* carry */
042       r = 0;
043       for (x = 0; x < a->used; x++) \{
044
045         /* get what will be the *next* carry bit from the
046          * MSB of the current digit
047          */
048         rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1));
049
050         /* now shift up this digit, add in the carry [from the previous] */
051         *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK;
052
053         /* copy the carry that would be from the source
054          * digit into the next iteration
055          */
056         r = rr;
057       \}
058
059       /* new leading digit? */
060       if (r != 0) \{
061         /* add a MSB which is always 1 at this point */
062         *tmpb = 1;
063         ++(b->used);
064       \}
065
066       /* now zero any excess digits on the destination
067        * that we didn't write to
068        */
069       tmpb = b->dp + b->used;
070       for (x = b->used; x < oldused; x++) \{
071         *tmpb++ = 0;
072       \}
073     \}
074     b->sign = a->sign;
075     return MP_OKAY;
076   \}
077   #endif
078
\end{alltt}
\end{small}

This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input.  The only noteworthy difference
is the use of the logical shift operator on line 51 to perform a single precision doubling.

\subsection{Division by Two}
A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
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2125
2126

















































2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151

\textbf{Algorithm mp\_div\_2.}
This algorithm will divide an mp\_int by two using logical shifts to the right.  Like mp\_mul\_2 it uses a modified low level addition
core as the basis of the algorithm.  Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit.  The algorithm
could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent
reading past the end of the array of digits.

Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the 
least significant bit not the most significant bit.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2.c
\vspace{-3mm}
\begin{alltt}

















































\end{alltt}
\end{small}

\section{Polynomial Basis Operations}
Recall from section 4.3 that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$.  Such a representation is also known as
the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single 
place.  The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer
division and Karatsuba multiplication.  

Converting from an array of digits to polynomial basis is very simple.  Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that
$y = \sum_{i=0}^{2} a_i \beta^i$.  Simply replace $\beta$ with $x$ and the expression is in polynomial basis.  For example, $f(x) = 8x + 9$ is the
polynomial basis representation for $89$ using radix ten.  That is, $f(10) = 8(10) + 9 = 89$.  

\subsection{Multiplication by $x$}

Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one 
degree.  In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$.  From a scalar basis point of view multiplying by $x$ is equivalent to
multiplying by the integer $\beta$.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_lshd}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\







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2883
2884
2885
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2887
2888
2889
2890
2891
2892
2893
2894

\textbf{Algorithm mp\_div\_2.}
This algorithm will divide an mp\_int by two using logical shifts to the right.  Like mp\_mul\_2 it uses a modified low level addition
core as the basis of the algorithm.  Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit.  The algorithm
could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent
reading past the end of the array of digits.

Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the
least significant bit not the most significant bit.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2.c
\vspace{-3mm}
\begin{alltt}
016
017   /* b = a/2 */
018   int mp_div_2(mp_int * a, mp_int * b)
019   \{
020     int     x, res, oldused;
021
022     /* copy */
023     if (b->alloc < a->used) \{
024       if ((res = mp_grow (b, a->used)) != MP_OKAY) \{
025         return res;
026       \}
027     \}
028
029     oldused = b->used;
030     b->used = a->used;
031     \{
032       mp_digit r, rr, *tmpa, *tmpb;
033
034       /* source alias */
035       tmpa = a->dp + b->used - 1;
036
037       /* dest alias */
038       tmpb = b->dp + b->used - 1;
039
040       /* carry */
041       r = 0;
042       for (x = b->used - 1; x >= 0; x--) \{
043         /* get the carry for the next iteration */
044         rr = *tmpa & 1;
045
046         /* shift the current digit, add in carry and store */
047         *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1));
048
049         /* forward carry to next iteration */
050         r = rr;
051       \}
052
053       /* zero excess digits */
054       tmpb = b->dp + b->used;
055       for (x = b->used; x < oldused; x++) \{
056         *tmpb++ = 0;
057       \}
058     \}
059     b->sign = a->sign;
060     mp_clamp (b);
061     return MP_OKAY;
062   \}
063   #endif
064
\end{alltt}
\end{small}

\section{Polynomial Basis Operations}
Recall from section 4.3 that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$.  Such a representation is also known as
the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single
place.  The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer
division and Karatsuba multiplication.

Converting from an array of digits to polynomial basis is very simple.  Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that
$y = \sum_{i=0}^{2} a_i \beta^i$.  Simply replace $\beta$ with $x$ and the expression is in polynomial basis.  For example, $f(x) = 8x + 9$ is the
polynomial basis representation for $89$ using radix ten.  That is, $f(10) = 8(10) + 9 = 89$.

\subsection{Multiplication by $x$}

Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one
degree.  In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$.  From a scalar basis point of view multiplying by $x$ is equivalent to
multiplying by the integer $\beta$.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_lshd}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
















































2200
2201
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2205
2206
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2209
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2212
2213
2214
2215
2216
2217
2218
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_lshd}
\end{figure}

\textbf{Algorithm mp\_lshd.}
This algorithm multiplies an mp\_int by the $b$'th power of $x$.  This is equivalent to multiplying by $\beta^b$.  The algorithm differs 
from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location.  The
motivation behind this change is due to the way this function is typically used.  Algorithms such as mp\_add store the result in an optionally
different third mp\_int because the original inputs are often still required.  Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is
typically used on values where the original value is no longer required.  The algorithm will return success immediately if 
$b \le 0$ since the rest of algorithm is only valid when $b > 0$.  

First the destination $a$ is grown as required to accomodate the result.  The counters $i$ and $j$ are used to form a \textit{sliding window} over
the digits of $a$ of length $b$.  The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}).  
The loop on step 7 copies the digit from the tail to the head.  In each iteration the window is moved down one digit.   The last loop on 
step 8 sets the lower $b$ digits to zero.

\newpage
\begin{center}
\begin{figure}[here]
\includegraphics{pics/sliding_window.ps}
\caption{Sliding Window Movement}
\label{pic:sliding_window}
\end{figure}
\end{center}

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_lshd.c
\vspace{-3mm}
\begin{alltt}
















































\end{alltt}
\end{small}

The if statement (line 24) ensures that the $b$ variable is greater than zero since we do not interpret negative
shift counts properly.  The \textbf{used} count is incremented by $b$ before the copy loop begins.  This elminates 
the need for an additional variable in the for loop.  The variable $top$ (line 42) is an alias
for the leading digit while $bottom$ (line 45) is an alias for the trailing edge.  The aliases form a 
window of exactly $b$ digits over the input.  

\subsection{Division by $x$}

Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_rshd}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\







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2911
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3001
3002
3003
3004
3005
3006
3007
3008
3009
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_lshd}
\end{figure}

\textbf{Algorithm mp\_lshd.}
This algorithm multiplies an mp\_int by the $b$'th power of $x$.  This is equivalent to multiplying by $\beta^b$.  The algorithm differs
from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location.  The
motivation behind this change is due to the way this function is typically used.  Algorithms such as mp\_add store the result in an optionally
different third mp\_int because the original inputs are often still required.  Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is
typically used on values where the original value is no longer required.  The algorithm will return success immediately if
$b \le 0$ since the rest of algorithm is only valid when $b > 0$.

First the destination $a$ is grown as required to accomodate the result.  The counters $i$ and $j$ are used to form a \textit{sliding window} over
the digits of $a$ of length $b$.  The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}).
The loop on step 7 copies the digit from the tail to the head.  In each iteration the window is moved down one digit.   The last loop on
step 8 sets the lower $b$ digits to zero.

\newpage
\begin{center}
\begin{figure}[here]
\includegraphics{pics/sliding_window.ps}
\caption{Sliding Window Movement}
\label{pic:sliding_window}
\end{figure}
\end{center}

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_lshd.c
\vspace{-3mm}
\begin{alltt}
016
017   /* shift left a certain amount of digits */
018   int mp_lshd (mp_int * a, int b)
019   \{
020     int     x, res;
021
022     /* if its less than zero return */
023     if (b <= 0) \{
024       return MP_OKAY;
025     \}
026
027     /* grow to fit the new digits */
028     if (a->alloc < (a->used + b)) \{
029        if ((res = mp_grow (a, a->used + b)) != MP_OKAY) \{
030          return res;
031        \}
032     \}
033
034     \{
035       mp_digit *top, *bottom;
036
037       /* increment the used by the shift amount then copy upwards */
038       a->used += b;
039
040       /* top */
041       top = a->dp + a->used - 1;
042
043       /* base */
044       bottom = (a->dp + a->used - 1) - b;
045
046       /* much like mp_rshd this is implemented using a sliding window
047        * except the window goes the otherway around.  Copying from
048        * the bottom to the top.  see bn_mp_rshd.c for more info.
049        */
050       for (x = a->used - 1; x >= b; x--) \{
051         *top-- = *bottom--;
052       \}
053
054       /* zero the lower digits */
055       top = a->dp;
056       for (x = 0; x < b; x++) \{
057         *top++ = 0;
058       \}
059     \}
060     return MP_OKAY;
061   \}
062   #endif
063
\end{alltt}
\end{small}

The if statement (line 23) ensures that the $b$ variable is greater than zero since we do not interpret negative
shift counts properly.  The \textbf{used} count is incremented by $b$ before the copy loop begins.  This elminates
the need for an additional variable in the for loop.  The variable $top$ (line 41) is an alias
for the leading digit while $bottom$ (line 44) is an alias for the trailing edge.  The aliases form a
window of exactly $b$ digits over the input.

\subsection{Division by $x$}

Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_rshd}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258





















































2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
\end{center}
\end{small}
\caption{Algorithm mp\_rshd}
\end{figure}

\textbf{Algorithm mp\_rshd.}
This algorithm divides the input in place by the $b$'th power of $x$.  It is analogous to dividing by a $\beta^b$ but much quicker since
it does not require single precision division.  This algorithm does not actually return an error code as it cannot fail.  

If the input $b$ is less than one the algorithm quickly returns without performing any work.  If the \textbf{used} count is less than or equal
to the shift count $b$ then it will simply zero the input and return.

After the trivial cases of inputs have been handled the sliding window is setup.  Much like the case of algorithm mp\_lshd a sliding window that
is $b$ digits wide is used to copy the digits.  Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit.  
Also the digits are copied from the leading to the trailing edge.

Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_rshd.c
\vspace{-3mm}
\begin{alltt}





















































\end{alltt}
\end{small}

The only noteworthy element of this routine is the lack of a return type since it cannot fail.  Like mp\_lshd() we
form a sliding window except we copy in the other direction.  After the window (line 60) we then zero
the upper digits of the input to make sure the result is correct.

\section{Powers of Two}

Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required.  For 
example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful.  Instead of performing single
shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed.  

\subsection{Multiplication by Power of Two}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}







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3118
3119
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3121
\end{center}
\end{small}
\caption{Algorithm mp\_rshd}
\end{figure}

\textbf{Algorithm mp\_rshd.}
This algorithm divides the input in place by the $b$'th power of $x$.  It is analogous to dividing by a $\beta^b$ but much quicker since
it does not require single precision division.  This algorithm does not actually return an error code as it cannot fail.

If the input $b$ is less than one the algorithm quickly returns without performing any work.  If the \textbf{used} count is less than or equal
to the shift count $b$ then it will simply zero the input and return.

After the trivial cases of inputs have been handled the sliding window is setup.  Much like the case of algorithm mp\_lshd a sliding window that
is $b$ digits wide is used to copy the digits.  Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit.
Also the digits are copied from the leading to the trailing edge.

Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_rshd.c
\vspace{-3mm}
\begin{alltt}
016
017   /* shift right a certain amount of digits */
018   void mp_rshd (mp_int * a, int b)
019   \{
020     int     x;
021
022     /* if b <= 0 then ignore it */
023     if (b <= 0) \{
024       return;
025     \}
026
027     /* if b > used then simply zero it and return */
028     if (a->used <= b) \{
029       mp_zero (a);
030       return;
031     \}
032
033     \{
034       mp_digit *bottom, *top;
035
036       /* shift the digits down */
037
038       /* bottom */
039       bottom = a->dp;
040
041       /* top [offset into digits] */
042       top = a->dp + b;
043
044       /* this is implemented as a sliding window where
045        * the window is b-digits long and digits from
046        * the top of the window are copied to the bottom
047        *
048        * e.g.
049
050        b-2 | b-1 | b0 | b1 | b2 | ... | bb |   ---->
051                    /\symbol{92}                   |      ---->
052                     \symbol{92}-------------------/      ---->
053        */
054       for (x = 0; x < (a->used - b); x++) \{
055         *bottom++ = *top++;
056       \}
057
058       /* zero the top digits */
059       for (; x < a->used; x++) \{
060         *bottom++ = 0;
061       \}
062     \}
063
064     /* remove excess digits */
065     a->used -= b;
066   \}
067   #endif
068
\end{alltt}
\end{small}

The only noteworthy element of this routine is the lack of a return type since it cannot fail.  Like mp\_lshd() we
form a sliding window except we copy in the other direction.  After the window (line 59) we then zero
the upper digits of the input to make sure the result is correct.

\section{Powers of Two}

Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required.  For
example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful.  Instead of performing single
shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed.

\subsection{Multiplication by Power of Two}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326


































































2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
\caption{Algorithm mp\_mul\_2d}
\end{figure}

\textbf{Algorithm mp\_mul\_2d.}
This algorithm multiplies $a$ by $2^b$ and stores the result in $c$.  The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to
quickly compute the product.

First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than 
$\beta$.  For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ 
left.

After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform.  Step 5 calculates the number of remaining shifts 
required.  If it is non-zero a modified shift loop is used to calculate the remaining product.  
Essentially the loop is a generic version of algorithm mp\_mul\_2 designed to handle any shift count in the range $1 \le x < lg(\beta)$.  The $mask$
variable is used to extract the upper $d$ bits to form the carry for the next iteration.  

This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to 
complete.  It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2d.c
\vspace{-3mm}
\begin{alltt}


































































\end{alltt}
\end{small}

The shifting is performed in--place which means the first step (line 25) is to copy the input to the 
destination.  We avoid calling mp\_copy() by making sure the mp\_ints are different.  The destination then
has to be grown (line 32) to accomodate the result.

If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples 
of $lg(\beta)$.  Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left.  Inside the actual shift 
loop (lines 46 to 76) we make use of pre--computed values $shift$ and $mask$.   These are used to
extract the carry bit(s) to pass into the next iteration of the loop.  The $r$ and $rr$ variables form a 
chain between consecutive iterations to propagate the carry.  

\subsection{Division by Power of Two}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}







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\caption{Algorithm mp\_mul\_2d}
\end{figure}

\textbf{Algorithm mp\_mul\_2d.}
This algorithm multiplies $a$ by $2^b$ and stores the result in $c$.  The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to
quickly compute the product.

First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than
$\beta$.  For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$
left.

After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform.  Step 5 calculates the number of remaining shifts
required.  If it is non-zero a modified shift loop is used to calculate the remaining product.
Essentially the loop is a generic version of algorithm mp\_mul\_2 designed to handle any shift count in the range $1 \le x < lg(\beta)$.  The $mask$
variable is used to extract the upper $d$ bits to form the carry for the next iteration.

This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to
complete.  It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2d.c
\vspace{-3mm}
\begin{alltt}
016
017   /* shift left by a certain bit count */
018   int mp_mul_2d (mp_int * a, int b, mp_int * c)
019   \{
020     mp_digit d;
021     int      res;
022
023     /* copy */
024     if (a != c) \{
025        if ((res = mp_copy (a, c)) != MP_OKAY) \{
026          return res;
027        \}
028     \}
029
030     if (c->alloc < (int)(c->used + (b / DIGIT_BIT) + 1)) \{
031        if ((res = mp_grow (c, c->used + (b / DIGIT_BIT) + 1)) != MP_OKAY) \{
032          return res;
033        \}
034     \}
035
036     /* shift by as many digits in the bit count */
037     if (b >= (int)DIGIT_BIT) \{
038       if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) \{
039         return res;
040       \}
041     \}
042
043     /* shift any bit count < DIGIT_BIT */
044     d = (mp_digit) (b % DIGIT_BIT);
045     if (d != 0) \{
046       mp_digit *tmpc, shift, mask, r, rr;
047       int x;
048
049       /* bitmask for carries */
050       mask = (((mp_digit)1) << d) - 1;
051
052       /* shift for msbs */
053       shift = DIGIT_BIT - d;
054
055       /* alias */
056       tmpc = c->dp;
057
058       /* carry */
059       r    = 0;
060       for (x = 0; x < c->used; x++) \{
061         /* get the higher bits of the current word */
062         rr = (*tmpc >> shift) & mask;
063
064         /* shift the current word and OR in the carry */
065         *tmpc = ((*tmpc << d) | r) & MP_MASK;
066         ++tmpc;
067
068         /* set the carry to the carry bits of the current word */
069         r = rr;
070       \}
071
072       /* set final carry */
073       if (r != 0) \{
074          c->dp[(c->used)++] = r;
075       \}
076     \}
077     mp_clamp (c);
078     return MP_OKAY;
079   \}
080   #endif
081
\end{alltt}
\end{small}

The shifting is performed in--place which means the first step (line 24) is to copy the input to the
destination.  We avoid calling mp\_copy() by making sure the mp\_ints are different.  The destination then
has to be grown (line 31) to accomodate the result.

If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples
of $lg(\beta)$.  Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left.  Inside the actual shift
loop (lines 45 to 76) we make use of pre--computed values $shift$ and $mask$.   These are used to
extract the carry bit(s) to pass into the next iteration of the loop.  The $r$ and $rr$ variables form a
chain between consecutive iterations to propagate the carry.

\subsection{Division by Power of Two}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
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2375
2376
2377
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2379
2380
2381
2382
2383
















































































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2386
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2389
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2391
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2393
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2395
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2401
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2404
2405
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_div\_2d}
\end{figure}

\textbf{Algorithm mp\_div\_2d.}
This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder.  The algorithm is designed much like algorithm 
mp\_mul\_2d by first using whole digit shifts then single precision shifts.  This algorithm will also produce the remainder of the division
by using algorithm mp\_mod\_2d.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2d.c
\vspace{-3mm}
\begin{alltt}
















































































\end{alltt}
\end{small}

The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies.  The remainder $d$ may be optionally 
ignored by passing \textbf{NULL} as the pointer to the mp\_int variable.    The temporary mp\_int variable $t$ is used to hold the 
result of the remainder operation until the end.  This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before
the quotient is obtained.

The remainder of the source code is essentially the same as the source code for mp\_mul\_2d.  The only significant difference is
the direction of the shifts.

\subsection{Remainder of Division by Power of Two}

The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$.  This
algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mod\_2d}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\







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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_div\_2d}
\end{figure}

\textbf{Algorithm mp\_div\_2d.}
This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder.  The algorithm is designed much like algorithm
mp\_mul\_2d by first using whole digit shifts then single precision shifts.  This algorithm will also produce the remainder of the division
by using algorithm mp\_mod\_2d.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2d.c
\vspace{-3mm}
\begin{alltt}
016
017   /* shift right by a certain bit count (store quotient in c, optional remaind
      er in d) */
018   int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d)
019   \{
020     mp_digit D, r, rr;
021     int     x, res;
022     mp_int  t;
023
024
025     /* if the shift count is <= 0 then we do no work */
026     if (b <= 0) \{
027       res = mp_copy (a, c);
028       if (d != NULL) \{
029         mp_zero (d);
030       \}
031       return res;
032     \}
033
034     if ((res = mp_init (&t)) != MP_OKAY) \{
035       return res;
036     \}
037
038     /* get the remainder */
039     if (d != NULL) \{
040       if ((res = mp_mod_2d (a, b, &t)) != MP_OKAY) \{
041         mp_clear (&t);
042         return res;
043       \}
044     \}
045
046     /* copy */
047     if ((res = mp_copy (a, c)) != MP_OKAY) \{
048       mp_clear (&t);
049       return res;
050     \}
051
052     /* shift by as many digits in the bit count */
053     if (b >= (int)DIGIT_BIT) \{
054       mp_rshd (c, b / DIGIT_BIT);
055     \}
056
057     /* shift any bit count < DIGIT_BIT */
058     D = (mp_digit) (b % DIGIT_BIT);
059     if (D != 0) \{
060       mp_digit *tmpc, mask, shift;
061
062       /* mask */
063       mask = (((mp_digit)1) << D) - 1;
064
065       /* shift for lsb */
066       shift = DIGIT_BIT - D;
067
068       /* alias */
069       tmpc = c->dp + (c->used - 1);
070
071       /* carry */
072       r = 0;
073       for (x = c->used - 1; x >= 0; x--) \{
074         /* get the lower  bits of this word in a temp */
075         rr = *tmpc & mask;
076
077         /* shift the current word and mix in the carry bits from the previous
      word */
078         *tmpc = (*tmpc >> D) | (r << shift);
079         --tmpc;
080
081         /* set the carry to the carry bits of the current word found above */
082         r = rr;
083       \}
084     \}
085     mp_clamp (c);
086     if (d != NULL) \{
087       mp_exch (&t, d);
088     \}
089     mp_clear (&t);
090     return MP_OKAY;
091   \}
092   #endif
093
\end{alltt}
\end{small}

The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies.  The remainder $d$ may be optionally
ignored by passing \textbf{NULL} as the pointer to the mp\_int variable.    The temporary mp\_int variable $t$ is used to hold the
result of the remainder operation until the end.  This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before
the quotient is obtained.

The remainder of the source code is essentially the same as the source code for mp\_mul\_2d.  The only significant difference is
the direction of the shifts.

\subsection{Remainder of Division by Power of Two}

The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$.  This
algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mod\_2d}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\
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2437






































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2454
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mod\_2d}
\end{figure}

\textbf{Algorithm mp\_mod\_2d.}
This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$.  First if $b$ is less than or equal to zero the 
result is set to zero.  If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns.  Otherwise, $a$ 
is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mod\_2d.c
\vspace{-3mm}
\begin{alltt}






































\end{alltt}
\end{small}

We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases.  Next if $2^b$ is larger
than the input we just mp\_copy() the input and return right away.  After this point we know we must actually
perform some work to produce the remainder.

Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce 
the number.  First we zero any digits above the last digit in $2^b$ (line 42).  Next we reduce the 
leading digit of both (line 46) and then mp\_clamp().

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\
                      & in $O(n)$ time. \\
                      &\\
$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming  \\







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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mod\_2d}
\end{figure}

\textbf{Algorithm mp\_mod\_2d.}
This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$.  First if $b$ is less than or equal to zero the
result is set to zero.  If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns.  Otherwise, $a$
is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mod\_2d.c
\vspace{-3mm}
\begin{alltt}
016
017   /* calc a value mod 2**b */
018   int
019   mp_mod_2d (mp_int * a, int b, mp_int * c)
020   \{
021     int     x, res;
022
023     /* if b is <= 0 then zero the int */
024     if (b <= 0) \{
025       mp_zero (c);
026       return MP_OKAY;
027     \}
028
029     /* if the modulus is larger than the value than return */
030     if (b >= (int) (a->used * DIGIT_BIT)) \{
031       res = mp_copy (a, c);
032       return res;
033     \}
034
035     /* copy */
036     if ((res = mp_copy (a, c)) != MP_OKAY) \{
037       return res;
038     \}
039
040     /* zero digits above the last digit of the modulus */
041     for (x = (b / DIGIT_BIT) + (((b % DIGIT_BIT) == 0) ? 0 : 1); x < c->used;
      x++) \{
042       c->dp[x] = 0;
043     \}
044     /* clear the digit that is not completely outside/inside the modulus */
045     c->dp[b / DIGIT_BIT] &=
046       (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digi
      t) 1));
047     mp_clamp (c);
048     return MP_OKAY;
049   \}
050   #endif
051
\end{alltt}
\end{small}

We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases.  Next if $2^b$ is larger
than the input we just mp\_copy() the input and return right away.  After this point we know we must actually
perform some work to produce the remainder.

Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce
the number.  First we zero any digits above the last digit in $2^b$ (line 41).  Next we reduce the
leading digit of both (line 45) and then mp\_clamp().

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\
                      & in $O(n)$ time. \\
                      &\\
$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming  \\
2473
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2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
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2491
2492
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2512
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$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\
                      & calculating the result of a signed comparison. \\
                      &
\end{tabular}

\chapter{Multiplication and Squaring}
\section{The Multipliers}
For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of 
algorithms of any multiple precision integer package.  The set of multiplier algorithms include integer multiplication, squaring and modular reduction 
where in each of the algorithms single precision multiplication is the dominant operation performed.  This chapter will discuss integer multiplication 
and squaring, leaving modular reductions for the subsequent chapter.  

The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular 
exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$.  During a modular
exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions, 
35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision 
multiplications.

For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied 
against every digit of the other multiplicand.  Traditional long-hand multiplication is based on this process;  while the techniques can differ the 
overall algorithm used is essentially the same.  Only ``recently'' have faster algorithms been studied.  First Karatsuba multiplication was discovered in 
1962.  This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach.  
This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions.  

\section{Multiplication}
\subsection{The Baseline Multiplication}
\label{sec:basemult}
\index{baseline multiplication}
Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication
algorithm that school children are taught.  The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision 
multiplications are required.  More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required.  To 
simplify most discussions, it will be assumed that the inputs have comparable number of digits.  

The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be 
used.  This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible.    One important 
facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution.  The importance of this 
modification will become evident during the discussion of Barrett modular reduction.  Recall that for a $n$ and $m$ digit input the product 
will be at most $n + m$ digits.  Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product.  

Recall from sub-section 4.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}.  We shall now extend the variable set to 
include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}.  This implies that $2^{\alpha} > 2 \cdot \beta^2$.  The 
constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see sub-section 5.2.2 for more information}).

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\







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$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\
                      & calculating the result of a signed comparison. \\
                      &
\end{tabular}

\chapter{Multiplication and Squaring}
\section{The Multipliers}
For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of
algorithms of any multiple precision integer package.  The set of multiplier algorithms include integer multiplication, squaring and modular reduction
where in each of the algorithms single precision multiplication is the dominant operation performed.  This chapter will discuss integer multiplication
and squaring, leaving modular reductions for the subsequent chapter.

The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular
exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$.  During a modular
exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions,
35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision
multiplications.

For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied
against every digit of the other multiplicand.  Traditional long-hand multiplication is based on this process;  while the techniques can differ the
overall algorithm used is essentially the same.  Only ``recently'' have faster algorithms been studied.  First Karatsuba multiplication was discovered in
1962.  This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach.
This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions.

\section{Multiplication}
\subsection{The Baseline Multiplication}
\label{sec:basemult}
\index{baseline multiplication}
Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication
algorithm that school children are taught.  The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision
multiplications are required.  More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required.  To
simplify most discussions, it will be assumed that the inputs have comparable number of digits.

The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be
used.  This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible.    One important
facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution.  The importance of this
modification will become evident during the discussion of Barrett modular reduction.  Recall that for a $n$ and $m$ digit input the product
will be at most $n + m$ digits.  Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product.

Recall from sub-section 4.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}.  We shall now extend the variable set to
include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}.  This implies that $2^{\alpha} > 2 \cdot \beta^2$.  The
constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see sub-section 5.2.2 for more information}).

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\
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\end{center}
\end{small}
\caption{Algorithm s\_mp\_mul\_digs}
\end{figure}

\textbf{Algorithm s\_mp\_mul\_digs.}
This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits.  While it may seem
a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent 
algorithm.  The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}.  
Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the 
inputs.

The first thing this algorithm checks for is whether a Comba multiplier can be used instead.   If the minimum digit count of either
input is less than $\delta$, then the Comba method may be used instead.    After the Comba method is ruled out, the baseline algorithm begins.  A 
temporary mp\_int variable $t$ is used to hold the intermediate result of the product.  This allows the algorithm to be used to 
compute products when either $a = c$ or $b = c$ without overwriting the inputs.  

All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output.  The $pb$ variable
is given the count of digits to read from $b$ inside the nested loop.  If $pb \le 1$ then no more output digits can be produced and the algorithm
will exit the loop.  The best way to think of the loops are as a series of $pb \times 1$ multiplications.    That is, in each pass of the 
innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$.  

For example, consider multiplying $576$ by $241$.  That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best
visualized in the following table.

\begin{figure}[here]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|l|}
\hline   &&          & 5 & 7 & 6 & \\
\hline   $\times$&&  & 2 & 4 & 1 & \\
\hline &&&&&&\\
  &&          & 5 & 7 & 6 & $10^0(1)(576)$ \\
  &2 &   3    & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\
  1 & 3 & 8 & 8 & 1 & 6 &   $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\
\hline  
\end{tabular}
\end{center}
\caption{Long-Hand Multiplication Diagram}
\end{figure}

Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate 
count.  That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult.

Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable.  The multiplication on that step
is assumed to be a double wide output single precision multiplication.  That is, two single precision variables are multiplied to produce a
double precision result.  The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step
5.4.1 is propagated through the nested loop.  If the carry was not propagated immediately it would overflow the single precision digit 
$t_{ix+iy}$ and the result would be lost.  

At step 5.5 the nested loop is finished and any carry that was left over should be forwarded.  The carry does not have to be added to the $ix+pb$'th
digit since that digit is assumed to be zero at this point.  However, if $ix + pb \ge digs$ the carry is not set as it would make the result
exceed the precision requested.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_mul\_digs.c
\vspace{-3mm}
\begin{alltt}







































































\end{alltt}
\end{small}

First we determine (line 31) if the Comba method can be used first since it's faster.  The conditions for 
sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than 
\textbf{MP\_WARRAY}.  This new constant is used to control the stack usage in the Comba routines.  By default it is 
set to $\delta$ but can be reduced when memory is at a premium.

If we cannot use the Comba method we proceed to setup the baseline routine.  We allocate the the destination mp\_int
$t$ (line 37) to the exact size of the output to avoid further re--allocations.  At this point we now 
begin the $O(n^2)$ loop.

This implementation of multiplication has the caveat that it can be trimmed to only produce a variable number of
digits as output.  In each iteration of the outer loop the $pb$ variable is set (line 49) to the maximum 
number of inner loop iterations.  

Inside the inner loop we calculate $\hat r$ as the mp\_word product of the two mp\_digits and the addition of the
carry from the previous iteration.  A particularly important observation is that most modern optimizing 
C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that 
is required for the product.  In x86 terms for example, this means using the MUL instruction.

Each digit of the product is stored in turn (line 69) and the carry propagated (line 72) to the 
next iteration.

\subsection{Faster Multiplication by the ``Comba'' Method}

One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be 
computed and propagated upwards.  This makes the nested loop very sequential and hard to unroll and implement 
in parallel.  The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G. 
Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations.  As an 
interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written 
five years before.

At the heart of the Comba technique is once again the long-hand algorithm.  Except in this case a slight 
twist is placed on how the columns of the result are produced.  In the standard long-hand algorithm rows of products 
are produced then added together to form the final result.  In the baseline algorithm the columns are added together 
after each iteration to get the result instantaneously.  

In the Comba algorithm the columns of the result are produced entirely independently of each other.  That is at 
the $O(n^2)$ level a simple multiplication and addition step is performed.  The carries of the columns are propagated 
after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute 
the product vector $\vec x$ as follows. 

\begin{equation}
\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace
\end{equation}

Where $\vec x_n$ is the $n'th$ column of the output vector.  Consider the following example which computes the vector $\vec x$ for the multiplication
of $576$ and $241$.  

\newpage\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
  \hline &          & 5 & 7 & 6 & First Input\\
  \hline $\times$ & & 2 & 4 & 1 & Second Input\\
\hline            &                        & $1 \cdot 5 = 5$   & $1 \cdot 7 = 7$   & $1 \cdot 6 = 6$ & First pass \\
                  &  $4 \cdot 5 = 20$      & $4 \cdot 7+5=33$  & $4 \cdot 6+7=31$  & 6               & Second pass \\
   $2 \cdot 5 = 10$ &  $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31                & 6             & Third pass \\
\hline 10 & 34 & 45 & 31 & 6 & Final Result \\   
\hline   
\end{tabular}
\end{center}
\end{small}
\caption{Comba Multiplication Diagram}
\end{figure}

At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler.  
Now the columns must be fixed by propagating the carry upwards.  The resultant vector will have one extra dimension over the input vector which is
congruent to adding a leading zero digit.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}







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\end{center}
\end{small}
\caption{Algorithm s\_mp\_mul\_digs}
\end{figure}

\textbf{Algorithm s\_mp\_mul\_digs.}
This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits.  While it may seem
a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent
algorithm.  The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}.
Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the
inputs.

The first thing this algorithm checks for is whether a Comba multiplier can be used instead.   If the minimum digit count of either
input is less than $\delta$, then the Comba method may be used instead.    After the Comba method is ruled out, the baseline algorithm begins.  A
temporary mp\_int variable $t$ is used to hold the intermediate result of the product.  This allows the algorithm to be used to
compute products when either $a = c$ or $b = c$ without overwriting the inputs.

All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output.  The $pb$ variable
is given the count of digits to read from $b$ inside the nested loop.  If $pb \le 1$ then no more output digits can be produced and the algorithm
will exit the loop.  The best way to think of the loops are as a series of $pb \times 1$ multiplications.    That is, in each pass of the
innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$.

For example, consider multiplying $576$ by $241$.  That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best
visualized in the following table.

\begin{figure}[here]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|l|}
\hline   &&          & 5 & 7 & 6 & \\
\hline   $\times$&&  & 2 & 4 & 1 & \\
\hline &&&&&&\\
  &&          & 5 & 7 & 6 & $10^0(1)(576)$ \\
  &2 &   3    & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\
  1 & 3 & 8 & 8 & 1 & 6 &   $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\
\hline
\end{tabular}
\end{center}
\caption{Long-Hand Multiplication Diagram}
\end{figure}

Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate
count.  That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult.

Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable.  The multiplication on that step
is assumed to be a double wide output single precision multiplication.  That is, two single precision variables are multiplied to produce a
double precision result.  The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step
5.4.1 is propagated through the nested loop.  If the carry was not propagated immediately it would overflow the single precision digit
$t_{ix+iy}$ and the result would be lost.

At step 5.5 the nested loop is finished and any carry that was left over should be forwarded.  The carry does not have to be added to the $ix+pb$'th
digit since that digit is assumed to be zero at this point.  However, if $ix + pb \ge digs$ the carry is not set as it would make the result
exceed the precision requested.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_mul\_digs.c
\vspace{-3mm}
\begin{alltt}
016
017   /* multiplies |a| * |b| and only computes upto digs digits of result
018    * HAC pp. 595, Algorithm 14.12  Modified so you can control how
019    * many digits of output are created.
020    */
021   int s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
022   \{
023     mp_int  t;
024     int     res, pa, pb, ix, iy;
025     mp_digit u;
026     mp_word r;
027     mp_digit tmpx, *tmpt, *tmpy;
028
029     /* can we use the fast multiplier? */
030     if (((digs) < MP_WARRAY) &&
031         (MIN (a->used, b->used) <
032             (1 << ((CHAR_BIT * sizeof(mp_word)) - (2 * DIGIT_BIT))))) \{
033       return fast_s_mp_mul_digs (a, b, c, digs);
034     \}
035
036     if ((res = mp_init_size (&t, digs)) != MP_OKAY) \{
037       return res;
038     \}
039     t.used = digs;
040
041     /* compute the digits of the product directly */
042     pa = a->used;
043     for (ix = 0; ix < pa; ix++) \{
044       /* set the carry to zero */
045       u = 0;
046
047       /* limit ourselves to making digs digits of output */
048       pb = MIN (b->used, digs - ix);
049
050       /* setup some aliases */
051       /* copy of the digit from a used within the nested loop */
052       tmpx = a->dp[ix];
053
054       /* an alias for the destination shifted ix places */
055       tmpt = t.dp + ix;
056
057       /* an alias for the digits of b */
058       tmpy = b->dp;
059
060       /* compute the columns of the output and propagate the carry */
061       for (iy = 0; iy < pb; iy++) \{
062         /* compute the column as a mp_word */
063         r       = (mp_word)*tmpt +
064                   ((mp_word)tmpx * (mp_word)*tmpy++) +
065                   (mp_word)u;
066
067         /* the new column is the lower part of the result */
068         *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
069
070         /* get the carry word from the result */
071         u       = (mp_digit) (r >> ((mp_word) DIGIT_BIT));
072       \}
073       /* set carry if it is placed below digs */
074       if ((ix + iy) < digs) \{
075         *tmpt = u;
076       \}
077     \}
078
079     mp_clamp (&t);
080     mp_exch (&t, c);
081
082     mp_clear (&t);
083     return MP_OKAY;
084   \}
085   #endif
086
\end{alltt}
\end{small}

First we determine (line 30) if the Comba method can be used first since it's faster.  The conditions for
sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than
\textbf{MP\_WARRAY}.  This new constant is used to control the stack usage in the Comba routines.  By default it is
set to $\delta$ but can be reduced when memory is at a premium.

If we cannot use the Comba method we proceed to setup the baseline routine.  We allocate the the destination mp\_int
$t$ (line 36) to the exact size of the output to avoid further re--allocations.  At this point we now
begin the $O(n^2)$ loop.

This implementation of multiplication has the caveat that it can be trimmed to only produce a variable number of
digits as output.  In each iteration of the outer loop the $pb$ variable is set (line 48) to the maximum
number of inner loop iterations.

Inside the inner loop we calculate $\hat r$ as the mp\_word product of the two mp\_digits and the addition of the
carry from the previous iteration.  A particularly important observation is that most modern optimizing
C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that
is required for the product.  In x86 terms for example, this means using the MUL instruction.

Each digit of the product is stored in turn (line 68) and the carry propagated (line 71) to the
next iteration.

\subsection{Faster Multiplication by the ``Comba'' Method}

One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be
computed and propagated upwards.  This makes the nested loop very sequential and hard to unroll and implement
in parallel.  The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G.
Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations.  As an
interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written
five years before.

At the heart of the Comba technique is once again the long-hand algorithm.  Except in this case a slight
twist is placed on how the columns of the result are produced.  In the standard long-hand algorithm rows of products
are produced then added together to form the final result.  In the baseline algorithm the columns are added together
after each iteration to get the result instantaneously.

In the Comba algorithm the columns of the result are produced entirely independently of each other.  That is at
the $O(n^2)$ level a simple multiplication and addition step is performed.  The carries of the columns are propagated
after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute
the product vector $\vec x$ as follows.

\begin{equation}
\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace
\end{equation}

Where $\vec x_n$ is the $n'th$ column of the output vector.  Consider the following example which computes the vector $\vec x$ for the multiplication
of $576$ and $241$.

\newpage\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
  \hline &          & 5 & 7 & 6 & First Input\\
  \hline $\times$ & & 2 & 4 & 1 & Second Input\\
\hline            &                        & $1 \cdot 5 = 5$   & $1 \cdot 7 = 7$   & $1 \cdot 6 = 6$ & First pass \\
                  &  $4 \cdot 5 = 20$      & $4 \cdot 7+5=33$  & $4 \cdot 6+7=31$  & 6               & Second pass \\
   $2 \cdot 5 = 10$ &  $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31                & 6             & Third pass \\
\hline 10 & 34 & 45 & 31 & 6 & Final Result \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Comba Multiplication Diagram}
\end{figure}

At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler.
Now the columns must be fixed by propagating the carry upwards.  The resultant vector will have one extra dimension over the input vector which is
congruent to adding a leading zero digit.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
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\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Comba Fixup}
\end{figure}

With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$.  In this case 
$241 \cdot 576$ is in fact $138816$ and the procedure succeeded.  If the algorithm is correct and as will be demonstrated shortly more
efficient than the baseline algorithm why not simply always use this algorithm?

\subsubsection{Column Weight.}
At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output 
independently.  A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix
the carries.  For example, in the multiplication of two three-digit numbers the third column of output will be the sum of
three single precision multiplications.  If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then
an overflow can occur and the carry information will be lost.  For any $m$ and $n$ digit inputs the maximum weight of any column is 
min$(m, n)$ which is fairly obvious.

The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used.  Recall
from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision.  Given these
two quantities we must not violate the following

\begin{equation}
k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha}
\end{equation}

Which reduces to 

\begin{equation}
k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha}
\end{equation}

Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit.  By further re-arrangement of the equation the final solution is
found.

\begin{equation}
k  < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}}
\end{equation}

The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$.  In this configuration 
the smaller input may not have more than $256$ digits if the Comba method is to be used.  This is quite satisfactory for most applications since 
$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\
\textbf{Input}.   mp\_int $a$, mp\_int $b$ and an integer $digs$ \\







|




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\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Comba Fixup}
\end{figure}

With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$.  In this case
$241 \cdot 576$ is in fact $138816$ and the procedure succeeded.  If the algorithm is correct and as will be demonstrated shortly more
efficient than the baseline algorithm why not simply always use this algorithm?

\subsubsection{Column Weight.}
At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output
independently.  A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix
the carries.  For example, in the multiplication of two three-digit numbers the third column of output will be the sum of
three single precision multiplications.  If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then
an overflow can occur and the carry information will be lost.  For any $m$ and $n$ digit inputs the maximum weight of any column is
min$(m, n)$ which is fairly obvious.

The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used.  Recall
from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision.  Given these
two quantities we must not violate the following

\begin{equation}
k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha}
\end{equation}

Which reduces to

\begin{equation}
k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha}
\end{equation}

Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit.  By further re-arrangement of the equation the final solution is
found.

\begin{equation}
k  < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}}
\end{equation}

The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$.  In this configuration
the smaller input may not have more than $256$ digits if the Comba method is to be used.  This is quite satisfactory for most applications since
$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\
\textbf{Input}.   mp\_int $a$, mp\_int $b$ and an integer $digs$ \\
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\caption{Algorithm fast\_s\_mp\_mul\_digs}
\label{fig:COMBAMULT}
\end{figure}

\textbf{Algorithm fast\_s\_mp\_mul\_digs.}
This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision.

The outer loop of this algorithm is more complicated than that of the baseline multiplier.  This is because on the inside of the 
loop we want to produce one column per pass.  This allows the accumulator $\_ \hat W$ to be placed in CPU registers and
reduce the memory bandwidth to two \textbf{mp\_digit} reads per iteration.

The $ty$ variable is set to the minimum count of $ix$ or the number of digits in $b$.  That way if $a$ has more digits than
$b$ this will be limited to $b.used - 1$.  The $tx$ variable is set to the to the distance past $b.used$ the variable
$ix$ is.  This is used for the immediately subsequent statement where we find $iy$.  

The variable $iy$ is the minimum digits we can read from either $a$ or $b$ before running out.  Computing one column at a time
means we have to scan one integer upwards and the other downwards.  $a$ starts at $tx$ and $b$ starts at $ty$.  In each
pass we are producing the $ix$'th output column and we note that $tx + ty = ix$.  As we move $tx$ upwards we have to 
move $ty$ downards so the equality remains valid.  The $iy$ variable is the number of iterations until 
$tx \ge a.used$ or $ty < 0$ occurs.

After every inner pass we store the lower half of the accumulator into $W_{ix}$ and then propagate the carry of the accumulator
into the next round by dividing $\_ \hat W$ by $\beta$.

To measure the benefits of the Comba method over the baseline method consider the number of operations that are required.  If the 
cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require 
$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers.  The Comba method requires only $O(pn^2 + qn)$ time, however in practice, 
the speed increase is actually much more.  With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply
and addition operations in the nested loop in parallel.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_mul\_digs.c
\vspace{-3mm}
\begin{alltt}
























































































\end{alltt}
\end{small}

As per the pseudo--code we first calculate $pa$ (line 48) as the number of digits to output.  Next we begin the outer loop
to produce the individual columns of the product.  We use the two aliases $tmpx$ and $tmpy$ (lines 62, 63) to point
inside the two multiplicands quickly.  

The inner loop (lines 71 to 74) of this implementation is where the tradeoff come into play.  Originally this comba 
implementation was ``row--major'' which means it adds to each of the columns in each pass.  After the outer loop it would then fix 
the carries.  This was very fast except it had an annoying drawback.  You had to read a mp\_word and two mp\_digits and write 
one mp\_word per iteration.  On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth 
is very high and it can keep the ALU fed with data.  It did, however, matter on older and embedded cpus where cache is often 
slower and also often doesn't exist.  This new algorithm only performs two reads per iteration under the assumption that the 
compiler has aliased $\_ \hat W$ to a CPU register.

After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines 77, 80) to forward it as 
a carry for the next pass.  After the outer loop we use the final carry (line 77) as the last digit of the product.  

\subsection{Polynomial Basis Multiplication}
To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication.  In the following algorithms
the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and  
$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required.  In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree.
 
The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$.  The coefficients $w_i$ will
directly yield the desired product when $\beta$ is substituted for $x$.  The direct solution to solve for the $2n + 1$ coefficients
requires $O(n^2)$ time and would in practice be slower than the Comba technique.

However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown 
coefficients.   This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with 
Gaussian elimination.  This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in 
effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$.  

The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible.  However, since 
$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place.  The benefit of this technique stems from the 
fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively.  As a result finding the $2n + 1$ relations required 
by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs.

When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$.  The $\zeta_0$ term
is simply the product $W(0) = w_0 = a_0 \cdot b_0$.  The $\zeta_1$ term is the product 
$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$.  The third point $\zeta_{\infty}$ is less obvious but rather
simple to explain.  The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication.  
The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$.  Note that the 
points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly.

If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points} 
$\left (2^q \right )^{2n}  \cdot \zeta_{2^{-q}}$ for small values of $q$.  The term ``mirror point'' stems from the fact that 
$\left (2^q \right )^{2n}  \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$.  For
example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror.

\begin{eqnarray}
\zeta_{2}                  = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\
16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0)
\end{eqnarray}

Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts.  For example, when $n = 2$ the
polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$.  This technique of polynomial representation is known as Horner's method.  

As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications.  Each multiplication is of 
multiplicands that have $n$ times fewer digits than the inputs.  The asymptotic running time of this algorithm is 
$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}).  Figure~\ref{fig:exponent}
summarizes the exponents for various values of $n$.

\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline \textbf{Split into $n$ Parts} & \textbf{Exponent}  & \textbf{Notes}\\







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\caption{Algorithm fast\_s\_mp\_mul\_digs}
\label{fig:COMBAMULT}
\end{figure}

\textbf{Algorithm fast\_s\_mp\_mul\_digs.}
This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision.

The outer loop of this algorithm is more complicated than that of the baseline multiplier.  This is because on the inside of the
loop we want to produce one column per pass.  This allows the accumulator $\_ \hat W$ to be placed in CPU registers and
reduce the memory bandwidth to two \textbf{mp\_digit} reads per iteration.

The $ty$ variable is set to the minimum count of $ix$ or the number of digits in $b$.  That way if $a$ has more digits than
$b$ this will be limited to $b.used - 1$.  The $tx$ variable is set to the to the distance past $b.used$ the variable
$ix$ is.  This is used for the immediately subsequent statement where we find $iy$.

The variable $iy$ is the minimum digits we can read from either $a$ or $b$ before running out.  Computing one column at a time
means we have to scan one integer upwards and the other downwards.  $a$ starts at $tx$ and $b$ starts at $ty$.  In each
pass we are producing the $ix$'th output column and we note that $tx + ty = ix$.  As we move $tx$ upwards we have to
move $ty$ downards so the equality remains valid.  The $iy$ variable is the number of iterations until
$tx \ge a.used$ or $ty < 0$ occurs.

After every inner pass we store the lower half of the accumulator into $W_{ix}$ and then propagate the carry of the accumulator
into the next round by dividing $\_ \hat W$ by $\beta$.

To measure the benefits of the Comba method over the baseline method consider the number of operations that are required.  If the
cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require
$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers.  The Comba method requires only $O(pn^2 + qn)$ time, however in practice,
the speed increase is actually much more.  With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply
and addition operations in the nested loop in parallel.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_mul\_digs.c
\vspace{-3mm}
\begin{alltt}
016
017   /* Fast (comba) multiplier
018    *
019    * This is the fast column-array [comba] multiplier.  It is
020    * designed to compute the columns of the product first
021    * then handle the carries afterwards.  This has the effect
022    * of making the nested loops that compute the columns very
023    * simple and schedulable on super-scalar processors.
024    *
025    * This has been modified to produce a variable number of
026    * digits of output so if say only a half-product is required
027    * you don't have to compute the upper half (a feature
028    * required for fast Barrett reduction).
029    *
030    * Based on Algorithm 14.12 on pp.595 of HAC.
031    *
032    */
033   int fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
034   \{
035     int     olduse, res, pa, ix, iz;
036     mp_digit W[MP_WARRAY];
037     mp_word  _W;
038
039     /* grow the destination as required */
040     if (c->alloc < digs) \{
041       if ((res = mp_grow (c, digs)) != MP_OKAY) \{
042         return res;
043       \}
044     \}
045
046     /* number of output digits to produce */
047     pa = MIN(digs, a->used + b->used);
048
049     /* clear the carry */
050     _W = 0;
051     for (ix = 0; ix < pa; ix++) \{
052         int      tx, ty;
053         int      iy;
054         mp_digit *tmpx, *tmpy;
055
056         /* get offsets into the two bignums */
057         ty = MIN(b->used-1, ix);
058         tx = ix - ty;
059
060         /* setup temp aliases */
061         tmpx = a->dp + tx;
062         tmpy = b->dp + ty;
063
064         /* this is the number of times the loop will iterrate, essentially
065            while (tx++ < a->used && ty-- >= 0) \{ ... \}
066          */
067         iy = MIN(a->used-tx, ty+1);
068
069         /* execute loop */
070         for (iz = 0; iz < iy; ++iz) \{
071            _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--);
072
073         \}
074
075         /* store term */
076         W[ix] = ((mp_digit)_W) & MP_MASK;
077
078         /* make next carry */
079         _W = _W >> ((mp_word)DIGIT_BIT);
080     \}
081
082     /* setup dest */
083     olduse  = c->used;
084     c->used = pa;
085
086     \{
087       mp_digit *tmpc;
088       tmpc = c->dp;
089       for (ix = 0; ix < (pa + 1); ix++) \{
090         /* now extract the previous digit [below the carry] */
091         *tmpc++ = W[ix];
092       \}
093
094       /* clear unused digits [that existed in the old copy of c] */
095       for (; ix < olduse; ix++) \{
096         *tmpc++ = 0;
097       \}
098     \}
099     mp_clamp (c);
100     return MP_OKAY;
101   \}
102   #endif
103
\end{alltt}
\end{small}

As per the pseudo--code we first calculate $pa$ (line 47) as the number of digits to output.  Next we begin the outer loop
to produce the individual columns of the product.  We use the two aliases $tmpx$ and $tmpy$ (lines 61, 62) to point
inside the two multiplicands quickly.

The inner loop (lines 70 to 73) of this implementation is where the tradeoff come into play.  Originally this comba
implementation was ``row--major'' which means it adds to each of the columns in each pass.  After the outer loop it would then fix
the carries.  This was very fast except it had an annoying drawback.  You had to read a mp\_word and two mp\_digits and write
one mp\_word per iteration.  On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth
is very high and it can keep the ALU fed with data.  It did, however, matter on older and embedded cpus where cache is often
slower and also often doesn't exist.  This new algorithm only performs two reads per iteration under the assumption that the
compiler has aliased $\_ \hat W$ to a CPU register.

After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines 76, 79) to forward it as
a carry for the next pass.  After the outer loop we use the final carry (line 76) as the last digit of the product.

\subsection{Polynomial Basis Multiplication}
To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication.  In the following algorithms
the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and
$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required.  In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree.

The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$.  The coefficients $w_i$ will
directly yield the desired product when $\beta$ is substituted for $x$.  The direct solution to solve for the $2n + 1$ coefficients
requires $O(n^2)$ time and would in practice be slower than the Comba technique.

However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown
coefficients.   This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with
Gaussian elimination.  This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in
effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$.

The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible.  However, since
$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place.  The benefit of this technique stems from the
fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively.  As a result finding the $2n + 1$ relations required
by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs.

When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$.  The $\zeta_0$ term
is simply the product $W(0) = w_0 = a_0 \cdot b_0$.  The $\zeta_1$ term is the product
$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$.  The third point $\zeta_{\infty}$ is less obvious but rather
simple to explain.  The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication.
The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$.  Note that the
points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly.

If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points}
$\left (2^q \right )^{2n}  \cdot \zeta_{2^{-q}}$ for small values of $q$.  The term ``mirror point'' stems from the fact that
$\left (2^q \right )^{2n}  \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$.  For
example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror.

\begin{eqnarray}
\zeta_{2}                  = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\
16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0)
\end{eqnarray}

Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts.  For example, when $n = 2$ the
polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$.  This technique of polynomial representation is known as Horner's method.

As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications.  Each multiplication is of
multiplicands that have $n$ times fewer digits than the inputs.  The asymptotic running time of this algorithm is
$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}).  Figure~\ref{fig:exponent}
summarizes the exponents for various values of $n$.

\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline \textbf{Split into $n$ Parts} & \textbf{Exponent}  & \textbf{Notes}\\
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\end{center}
\caption{Asymptotic Running Time of Polynomial Basis Multiplication}
\label{fig:exponent}
\end{figure}

At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$.  However, the overhead
of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large
numbers.  

\subsubsection{Cutoff Point}
The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach.  However, 
the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved.  This makes the
polynomial basis approach more costly to use with small inputs.

Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}).  There exists a 
point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and 
when $m > y$ the Comba methods are slower than the polynomial basis algorithms.  

The exact location of $y$ depends on several key architectural elements of the computer platform in question.

\begin{enumerate}
\item  The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc.  For example
on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$.  The higher the ratio in favour of multiplication the lower
the cutoff point $y$ will be.  

\item  The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is.  Generally speaking as the number of splits
grows the complexity grows substantially.  Ideally solving the system will only involve addition, subtraction and shifting of integers.  This
directly reflects on the ratio previous mentioned.

\item  To a lesser extent memory bandwidth and function call overheads.  Provided the values are in the processor cache this is less of an
influence over the cutoff point.

\end{enumerate}

A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met.  For example, if the point
is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster.  Finding the cutoff points is fairly simple when
a high resolution timer is available.  

\subsection{Karatsuba Multiplication}
Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for
general purpose multiplication.  Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with 
light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent.

\begin{equation}
f(x) \cdot g(x) = acx^2 + ((a + b)(c + d) - (ac + bd))x + bd
\end{equation}

Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product.  Applying
this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique.  It turns 
out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points 
$\zeta_0$, $\zeta_{\infty}$ and $\zeta_{1}$.  Consider the resultant system of equations.

\begin{center}
\begin{tabular}{rcrcrcrc}
$\zeta_{0}$ &      $=$ &  &  &  & & $w_0$ \\
$\zeta_{1}$ &      $=$ & $w_2$ & $+$ & $w_1$ & $+$ & $w_0$ \\
$\zeta_{\infty}$ & $=$ & $w_2$ &  & &  & \\
\end{tabular}
\end{center}

By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for.  The simplicity
of this system of equations has made Karatsuba fairly popular.  In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.}
making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\
\textbf{Input}.   mp\_int $a$ and mp\_int $b$ \\







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\end{center}
\caption{Asymptotic Running Time of Polynomial Basis Multiplication}
\label{fig:exponent}
\end{figure}

At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$.  However, the overhead
of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large
numbers.

\subsubsection{Cutoff Point}
The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach.  However,
the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved.  This makes the
polynomial basis approach more costly to use with small inputs.

Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}).  There exists a
point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and
when $m > y$ the Comba methods are slower than the polynomial basis algorithms.

The exact location of $y$ depends on several key architectural elements of the computer platform in question.

\begin{enumerate}
\item  The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc.  For example
on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$.  The higher the ratio in favour of multiplication the lower
the cutoff point $y$ will be.

\item  The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is.  Generally speaking as the number of splits
grows the complexity grows substantially.  Ideally solving the system will only involve addition, subtraction and shifting of integers.  This
directly reflects on the ratio previous mentioned.

\item  To a lesser extent memory bandwidth and function call overheads.  Provided the values are in the processor cache this is less of an
influence over the cutoff point.

\end{enumerate}

A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met.  For example, if the point
is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster.  Finding the cutoff points is fairly simple when
a high resolution timer is available.

\subsection{Karatsuba Multiplication}
Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for
general purpose multiplication.  Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with
light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent.

\begin{equation}
f(x) \cdot g(x) = acx^2 + ((a + b)(c + d) - (ac + bd))x + bd
\end{equation}

Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product.  Applying
this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique.  It turns
out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points
$\zeta_0$, $\zeta_{\infty}$ and $\zeta_{1}$.  Consider the resultant system of equations.

\begin{center}
\begin{tabular}{rcrcrcrc}
$\zeta_{0}$ &      $=$ &  &  &  & & $w_0$ \\
$\zeta_{1}$ &      $=$ & $w_2$ & $+$ & $w_1$ & $+$ & $w_0$ \\
$\zeta_{\infty}$ & $=$ & $w_2$ &  & &  & \\
\end{tabular}
\end{center}

By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for.  The simplicity
of this system of equations has made Karatsuba fairly popular.  In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.}
making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\
\textbf{Input}.   mp\_int $a$ and mp\_int $b$ \\
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Calculate the final product. \\
15.  $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\
16.  $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\
17.  $t1 \leftarrow x0y0 + t1$ \\
18.  $c \leftarrow t1 + x1y1$ \\
19.  Clear all of the temporary variables. \\
20.  Return(\textit{MP\_OKAY}).\\
\hline 
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_karatsuba\_mul}
\end{figure}

\textbf{Algorithm mp\_karatsuba\_mul.}
This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm.  It is loosely based on the description
from Knuth \cite[pp. 294-295]{TAOCPV2}.  

\index{radix point}
In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen.  The radix point chosen must
be used for both of the inputs meaning that it must be smaller than the smallest input.  Step 3 chooses the radix point $B$ as half of the 
smallest input \textbf{used} count.  After the radix point is chosen the inputs are split into lower and upper halves.  Step 4 and 5 
compute the lower halves.  Step 6 and 7 computer the upper halves.  

After the halves have been computed the three intermediate half-size products must be computed.  Step 8 and 9 compute the trivial products
$x0 \cdot y0$ and $x1 \cdot y1$.  The mp\_int $x0$ is used as a temporary variable after $x1 + x0$ has been computed.  By using $x0$ instead
of an additional temporary variable, the algorithm can avoid an addition memory allocation operation.

The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_mul.c
\vspace{-3mm}
\begin{alltt}




















































































































































\end{alltt}
\end{small}

The new coding element in this routine, not  seen in previous routines, is the usage of goto statements.  The conventional
wisdom is that goto statements should be avoided.  This is generally true, however when every single function call can fail, it makes sense
to handle error recovery with a single piece of code.  Lines 62 to 76 handle initializing all of the temporary variables 
required.  Note how each of the if statements goes to a different label in case of failure.  This allows the routine to correctly free only
the temporaries that have been successfully allocated so far.

The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large.  This saves the 
additional reallocation that would have been necessary.  Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective
number of digits for the next section of code.

The first algebraic portion of the algorithm is to split the two inputs into their halves.  However, instead of using mp\_mod\_2d and mp\_rshd
to extract the halves, the respective code has been placed inline within the body of the function.  To initialize the halves, the \textbf{used} and 
\textbf{sign} members are copied first.  The first for loop on line 96 copies the lower halves.  Since they are both the same magnitude it 
is simpler to calculate both lower halves in a single loop.  The for loop on lines 102 and 107 calculate the upper halves $x1$ and 
$y1$ respectively.

By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs.

When line 151 is reached, the algorithm has completed succesfully.  The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that
the same code that handles errors can be used to clear the temporary variables and return.  

\subsection{Toom-Cook $3$-Way Multiplication}
Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points  are 
chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce.  Here, the points $\zeta_{0}$, 
$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients 
of the $W(x)$.

With the five relations that Toom-Cook specifies, the following system of equations is formed.

\begin{center}
\begin{tabular}{rcrcrcrcrcr}
$\zeta_0$                    & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$  \\
$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$  \\
$\zeta_1$                    & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$  \\
$\zeta_2$                    & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$  \\
$\zeta_{\infty}$             & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$  \\
\end{tabular}
\end{center}

A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power
of two, two divisions by three and one multiplication by three.  All of these $19$ sub-operations require less than quadratic time, meaning that
the algorithm can be faster than a baseline multiplication.  However, the greater complexity of this algorithm places the cutoff point
(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_toom\_mul}. \\
\textbf{Input}.   mp\_int $a$ and mp\_int $b$ \\







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Calculate the final product. \\
15.  $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\
16.  $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\
17.  $t1 \leftarrow x0y0 + t1$ \\
18.  $c \leftarrow t1 + x1y1$ \\
19.  Clear all of the temporary variables. \\
20.  Return(\textit{MP\_OKAY}).\\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_karatsuba\_mul}
\end{figure}

\textbf{Algorithm mp\_karatsuba\_mul.}
This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm.  It is loosely based on the description
from Knuth \cite[pp. 294-295]{TAOCPV2}.

\index{radix point}
In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen.  The radix point chosen must
be used for both of the inputs meaning that it must be smaller than the smallest input.  Step 3 chooses the radix point $B$ as half of the
smallest input \textbf{used} count.  After the radix point is chosen the inputs are split into lower and upper halves.  Step 4 and 5
compute the lower halves.  Step 6 and 7 computer the upper halves.

After the halves have been computed the three intermediate half-size products must be computed.  Step 8 and 9 compute the trivial products
$x0 \cdot y0$ and $x1 \cdot y1$.  The mp\_int $x0$ is used as a temporary variable after $x1 + x0$ has been computed.  By using $x0$ instead
of an additional temporary variable, the algorithm can avoid an addition memory allocation operation.

The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_mul.c
\vspace{-3mm}
\begin{alltt}
016
017   /* c = |a| * |b| using Karatsuba Multiplication using
018    * three half size multiplications
019    *
020    * Let B represent the radix [e.g. 2**DIGIT_BIT] and
021    * let n represent half of the number of digits in
022    * the min(a,b)
023    *
024    * a = a1 * B**n + a0
025    * b = b1 * B**n + b0
026    *
027    * Then, a * b =>
028      a1b1 * B**2n + ((a1 + a0)(b1 + b0) - (a0b0 + a1b1)) * B + a0b0
029    *
030    * Note that a1b1 and a0b0 are used twice and only need to be
031    * computed once.  So in total three half size (half # of
032    * digit) multiplications are performed, a0b0, a1b1 and
033    * (a1+b1)(a0+b0)
034    *
035    * Note that a multiplication of half the digits requires
036    * 1/4th the number of single precision multiplications so in
037    * total after one call 25% of the single precision multiplications
038    * are saved.  Note also that the call to mp_mul can end up back
039    * in this function if the a0, a1, b0, or b1 are above the threshold.
040    * This is known as divide-and-conquer and leads to the famous
041    * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than
042    * the standard O(N**2) that the baseline/comba methods use.
043    * Generally though the overhead of this method doesn't pay off
044    * until a certain size (N ~ 80) is reached.
045    */
046   int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c)
047   \{
048     mp_int  x0, x1, y0, y1, t1, x0y0, x1y1;
049     int     B, err;
050
051     /* default the return code to an error */
052     err = MP_MEM;
053
054     /* min # of digits */
055     B = MIN (a->used, b->used);
056
057     /* now divide in two */
058     B = B >> 1;
059
060     /* init copy all the temps */
061     if (mp_init_size (&x0, B) != MP_OKAY)
062       goto ERR;
063     if (mp_init_size (&x1, a->used - B) != MP_OKAY)
064       goto X0;
065     if (mp_init_size (&y0, B) != MP_OKAY)
066       goto X1;
067     if (mp_init_size (&y1, b->used - B) != MP_OKAY)
068       goto Y0;
069
070     /* init temps */
071     if (mp_init_size (&t1, B * 2) != MP_OKAY)
072       goto Y1;
073     if (mp_init_size (&x0y0, B * 2) != MP_OKAY)
074       goto T1;
075     if (mp_init_size (&x1y1, B * 2) != MP_OKAY)
076       goto X0Y0;
077
078     /* now shift the digits */
079     x0.used = y0.used = B;
080     x1.used = a->used - B;
081     y1.used = b->used - B;
082
083     \{
084       int x;
085       mp_digit *tmpa, *tmpb, *tmpx, *tmpy;
086
087       /* we copy the digits directly instead of using higher level functions
088        * since we also need to shift the digits
089        */
090       tmpa = a->dp;
091       tmpb = b->dp;
092
093       tmpx = x0.dp;
094       tmpy = y0.dp;
095       for (x = 0; x < B; x++) \{
096         *tmpx++ = *tmpa++;
097         *tmpy++ = *tmpb++;
098       \}
099
100       tmpx = x1.dp;
101       for (x = B; x < a->used; x++) \{
102         *tmpx++ = *tmpa++;
103       \}
104
105       tmpy = y1.dp;
106       for (x = B; x < b->used; x++) \{
107         *tmpy++ = *tmpb++;
108       \}
109     \}
110
111     /* only need to clamp the lower words since by definition the
112      * upper words x1/y1 must have a known number of digits
113      */
114     mp_clamp (&x0);
115     mp_clamp (&y0);
116
117     /* now calc the products x0y0 and x1y1 */
118     /* after this x0 is no longer required, free temp [x0==t2]! */
119     if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY)
120       goto X1Y1;          /* x0y0 = x0*y0 */
121     if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY)
122       goto X1Y1;          /* x1y1 = x1*y1 */
123
124     /* now calc x1+x0 and y1+y0 */
125     if (s_mp_add (&x1, &x0, &t1) != MP_OKAY)
126       goto X1Y1;          /* t1 = x1 - x0 */
127     if (s_mp_add (&y1, &y0, &x0) != MP_OKAY)
128       goto X1Y1;          /* t2 = y1 - y0 */
129     if (mp_mul (&t1, &x0, &t1) != MP_OKAY)
130       goto X1Y1;          /* t1 = (x1 + x0) * (y1 + y0) */
131
132     /* add x0y0 */
133     if (mp_add (&x0y0, &x1y1, &x0) != MP_OKAY)
134       goto X1Y1;          /* t2 = x0y0 + x1y1 */
135     if (s_mp_sub (&t1, &x0, &t1) != MP_OKAY)
136       goto X1Y1;          /* t1 = (x1+x0)*(y1+y0) - (x1y1 + x0y0) */
137
138     /* shift by B */
139     if (mp_lshd (&t1, B) != MP_OKAY)
140       goto X1Y1;          /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */
141     if (mp_lshd (&x1y1, B * 2) != MP_OKAY)
142       goto X1Y1;          /* x1y1 = x1y1 << 2*B */
143
144     if (mp_add (&x0y0, &t1, &t1) != MP_OKAY)
145       goto X1Y1;          /* t1 = x0y0 + t1 */
146     if (mp_add (&t1, &x1y1, c) != MP_OKAY)
147       goto X1Y1;          /* t1 = x0y0 + t1 + x1y1 */
148
149     /* Algorithm succeeded set the return code to MP_OKAY */
150     err = MP_OKAY;
151
152   X1Y1:mp_clear (&x1y1);
153   X0Y0:mp_clear (&x0y0);
154   T1:mp_clear (&t1);
155   Y1:mp_clear (&y1);
156   Y0:mp_clear (&y0);
157   X1:mp_clear (&x1);
158   X0:mp_clear (&x0);
159   ERR:
160     return err;
161   \}
162   #endif
163
\end{alltt}
\end{small}

The new coding element in this routine, not  seen in previous routines, is the usage of goto statements.  The conventional
wisdom is that goto statements should be avoided.  This is generally true, however when every single function call can fail, it makes sense
to handle error recovery with a single piece of code.  Lines 61 to 75 handle initializing all of the temporary variables
required.  Note how each of the if statements goes to a different label in case of failure.  This allows the routine to correctly free only
the temporaries that have been successfully allocated so far.

The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large.  This saves the
additional reallocation that would have been necessary.  Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective
number of digits for the next section of code.

The first algebraic portion of the algorithm is to split the two inputs into their halves.  However, instead of using mp\_mod\_2d and mp\_rshd
to extract the halves, the respective code has been placed inline within the body of the function.  To initialize the halves, the \textbf{used} and
\textbf{sign} members are copied first.  The first for loop on line 101 copies the lower halves.  Since they are both the same magnitude it
is simpler to calculate both lower halves in a single loop.  The for loop on lines 106 and 106 calculate the upper halves $x1$ and
$y1$ respectively.

By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs.

When line 150 is reached, the algorithm has completed succesfully.  The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that
the same code that handles errors can be used to clear the temporary variables and return.

\subsection{Toom-Cook $3$-Way Multiplication}
Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points  are
chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce.  Here, the points $\zeta_{0}$,
$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients
of the $W(x)$.

With the five relations that Toom-Cook specifies, the following system of equations is formed.

\begin{center}
\begin{tabular}{rcrcrcrcrcr}
$\zeta_0$                    & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$  \\
$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$  \\
$\zeta_1$                    & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$  \\
$\zeta_2$                    & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$  \\
$\zeta_{\infty}$             & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$  \\
\end{tabular}
\end{center}

A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power
of two, two divisions by three and one multiplication by three.  All of these $19$ sub-operations require less than quadratic time, meaning that
the algorithm can be faster than a baseline multiplication.  However, the greater complexity of this algorithm places the cutoff point
(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_toom\_mul}. \\
\textbf{Input}.   mp\_int $a$ and mp\_int $b$ \\
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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_toom\_mul (continued)}
\end{figure}

\textbf{Algorithm mp\_toom\_mul.}
This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach.  Compared to the Karatsuba multiplication, this 
algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead.  In this
description, several statements have been compounded to save space.  The intention is that the statements are executed from left to right across
any given step.

The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively.  From these smaller
integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required.

The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively.  The relation $w_1, w_2$ and $w_3$ correspond
to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively.  These are found using logical shifts to independently find
$f(y)$ and $g(y)$ which significantly speeds up the algorithm.

After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients 
$w_1, w_2$ and $w_3$ to be isolated.  The steps 18 through 25 perform the system reduction required as previously described.  Each step of
the reduction represents the comparable matrix operation that would be performed had this been performed by pencil.  For example, step 18 indicates
that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$.  

Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known.  By substituting $\beta^{k}$ for $x$, the integer 
result $a \cdot b$ is produced.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_toom\_mul.c
\vspace{-3mm}
\begin{alltt}











































































































































































































































































\end{alltt}
\end{small}

The first obvious thing to note is that this algorithm is complicated.  The complexity is worth it if you are multiplying very 
large numbers.  For example, a 10,000 digit multiplication takes approximaly 99,282,205 fewer single precision multiplications with
Toom--Cook than a Comba or baseline approach (this is a savings of more than 99$\%$).  For most ``crypto'' sized numbers this
algorithm is not practical as Karatsuba has a much lower cutoff point.

First we split $a$ and $b$ into three roughly equal portions.  This has been accomplished (lines 41 to 70) with 
combinations of mp\_rshd() and mp\_mod\_2d() function calls.  At this point $a = a2 \cdot \beta^2 + a1 \cdot \beta + a0$ and similiarly
for $b$.  

Next we compute the five points $w0, w1, w2, w3$ and $w4$.  Recall that $w0$ and $w4$ can be computed directly from the portions so
we get those out of the way first (lines 73 and 78).  Next we compute $w1, w2$ and $w3$ using Horners method.

After this point we solve for the actual values of $w1, w2$ and $w3$ by reducing the $5 \times 5$ system which is relatively
straight forward.  

\subsection{Signed Multiplication}
Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required.  So far all
of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mul}. \\
\textbf{Input}.   mp\_int $a$ and mp\_int $b$ \\







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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_toom\_mul (continued)}
\end{figure}

\textbf{Algorithm mp\_toom\_mul.}
This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach.  Compared to the Karatsuba multiplication, this
algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead.  In this
description, several statements have been compounded to save space.  The intention is that the statements are executed from left to right across
any given step.

The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively.  From these smaller
integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required.

The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively.  The relation $w_1, w_2$ and $w_3$ correspond
to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively.  These are found using logical shifts to independently find
$f(y)$ and $g(y)$ which significantly speeds up the algorithm.

After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients
$w_1, w_2$ and $w_3$ to be isolated.  The steps 18 through 25 perform the system reduction required as previously described.  Each step of
the reduction represents the comparable matrix operation that would be performed had this been performed by pencil.  For example, step 18 indicates
that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$.

Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known.  By substituting $\beta^{k}$ for $x$, the integer
result $a \cdot b$ is produced.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_toom\_mul.c
\vspace{-3mm}
\begin{alltt}
016
017   /* multiplication using the Toom-Cook 3-way algorithm
018    *
019    * Much more complicated than Karatsuba but has a lower
020    * asymptotic running time of O(N**1.464).  This algorithm is
021    * only particularly useful on VERY large inputs
022    * (we're talking 1000s of digits here...).
023   */
024   int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c)
025   \{
026       mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2;
027       int res, B;
028
029       /* init temps */
030       if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4,
031                                &a0, &a1, &a2, &b0, &b1,
032                                &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) \{
033          return res;
034       \}
035
036       /* B */
037       B = MIN(a->used, b->used) / 3;
038
039       /* a = a2 * B**2 + a1 * B + a0 */
040       if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) \{
041          goto ERR;
042       \}
043
044       if ((res = mp_copy(a, &a1)) != MP_OKAY) \{
045          goto ERR;
046       \}
047       mp_rshd(&a1, B);
048       if ((res = mp_mod_2d(&a1, DIGIT_BIT * B, &a1)) != MP_OKAY) \{
049          goto ERR;
050       \}
051
052       if ((res = mp_copy(a, &a2)) != MP_OKAY) \{
053          goto ERR;
054       \}
055       mp_rshd(&a2, B*2);
056
057       /* b = b2 * B**2 + b1 * B + b0 */
058       if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) \{
059          goto ERR;
060       \}
061
062       if ((res = mp_copy(b, &b1)) != MP_OKAY) \{
063          goto ERR;
064       \}
065       mp_rshd(&b1, B);
066       (void)mp_mod_2d(&b1, DIGIT_BIT * B, &b1);
067
068       if ((res = mp_copy(b, &b2)) != MP_OKAY) \{
069          goto ERR;
070       \}
071       mp_rshd(&b2, B*2);
072
073       /* w0 = a0*b0 */
074       if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) \{
075          goto ERR;
076       \}
077
078       /* w4 = a2 * b2 */
079       if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) \{
080          goto ERR;
081       \}
082
083       /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */
084       if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) \{
085          goto ERR;
086       \}
087       if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{
088          goto ERR;
089       \}
090       if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{
091          goto ERR;
092       \}
093       if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) \{
094          goto ERR;
095       \}
096
097       if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) \{
098          goto ERR;
099       \}
100       if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{
101          goto ERR;
102       \}
103       if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{
104          goto ERR;
105       \}
106       if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) \{
107          goto ERR;
108       \}
109
110       if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) \{
111          goto ERR;
112       \}
113
114       /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */
115       if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) \{
116          goto ERR;
117       \}
118       if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{
119          goto ERR;
120       \}
121       if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{
122          goto ERR;
123       \}
124       if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{
125          goto ERR;
126       \}
127
128       if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) \{
129          goto ERR;
130       \}
131       if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{
132          goto ERR;
133       \}
134       if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{
135          goto ERR;
136       \}
137       if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{
138          goto ERR;
139       \}
140
141       if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) \{
142          goto ERR;
143       \}
144
145
146       /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */
147       if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) \{
148          goto ERR;
149       \}
150       if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{
151          goto ERR;
152       \}
153       if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) \{
154          goto ERR;
155       \}
156       if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{
157          goto ERR;
158       \}
159       if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) \{
160          goto ERR;
161       \}
162
163       /* now solve the matrix
164
165          0  0  0  0  1
166          1  2  4  8  16
167          1  1  1  1  1
168          16 8  4  2  1
169          1  0  0  0  0
170
171          using 12 subtractions, 4 shifts,
172                 2 small divisions and 1 small multiplication
173        */
174
175       /* r1 - r4 */
176       if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) \{
177          goto ERR;
178       \}
179       /* r3 - r0 */
180       if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) \{
181          goto ERR;
182       \}
183       /* r1/2 */
184       if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) \{
185          goto ERR;
186       \}
187       /* r3/2 */
188       if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) \{
189          goto ERR;
190       \}
191       /* r2 - r0 - r4 */
192       if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) \{
193          goto ERR;
194       \}
195       if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) \{
196          goto ERR;
197       \}
198       /* r1 - r2 */
199       if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{
200          goto ERR;
201       \}
202       /* r3 - r2 */
203       if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{
204          goto ERR;
205       \}
206       /* r1 - 8r0 */
207       if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) \{
208          goto ERR;
209       \}
210       if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) \{
211          goto ERR;
212       \}
213       /* r3 - 8r4 */
214       if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) \{
215          goto ERR;
216       \}
217       if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) \{
218          goto ERR;
219       \}
220       /* 3r2 - r1 - r3 */
221       if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) \{
222          goto ERR;
223       \}
224       if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) \{
225          goto ERR;
226       \}
227       if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) \{
228          goto ERR;
229       \}
230       /* r1 - r2 */
231       if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{
232          goto ERR;
233       \}
234       /* r3 - r2 */
235       if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{
236          goto ERR;
237       \}
238       /* r1/3 */
239       if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) \{
240          goto ERR;
241       \}
242       /* r3/3 */
243       if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) \{
244          goto ERR;
245       \}
246
247       /* at this point shift W[n] by B*n */
248       if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) \{
249          goto ERR;
250       \}
251       if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) \{
252          goto ERR;
253       \}
254       if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) \{
255          goto ERR;
256       \}
257       if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) \{
258          goto ERR;
259       \}
260
261       if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) \{
262          goto ERR;
263       \}
264       if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) \{
265          goto ERR;
266       \}
267       if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) \{
268          goto ERR;
269       \}
270       if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) \{
271          goto ERR;
272       \}
273
274   ERR:
275       mp_clear_multi(&w0, &w1, &w2, &w3, &w4,
276                      &a0, &a1, &a2, &b0, &b1,
277                      &b2, &tmp1, &tmp2, NULL);
278       return res;
279   \}
280
281   #endif
282
\end{alltt}
\end{small}

The first obvious thing to note is that this algorithm is complicated.  The complexity is worth it if you are multiplying very
large numbers.  For example, a 10,000 digit multiplication takes approximaly 99,282,205 fewer single precision multiplications with
Toom--Cook than a Comba or baseline approach (this is a savings of more than 99$\%$).  For most ``crypto'' sized numbers this
algorithm is not practical as Karatsuba has a much lower cutoff point.

First we split $a$ and $b$ into three roughly equal portions.  This has been accomplished (lines 40 to 71) with
combinations of mp\_rshd() and mp\_mod\_2d() function calls.  At this point $a = a2 \cdot \beta^2 + a1 \cdot \beta + a0$ and similiarly
for $b$.

Next we compute the five points $w0, w1, w2, w3$ and $w4$.  Recall that $w0$ and $w4$ can be computed directly from the portions so
we get those out of the way first (lines 74 and 79).  Next we compute $w1, w2$ and $w3$ using Horners method.

After this point we solve for the actual values of $w1, w2$ and $w3$ by reducing the $5 \times 5$ system which is relatively
straight forward.

\subsection{Signed Multiplication}
Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required.  So far all
of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mul}. \\
\textbf{Input}.   mp\_int $a$ and mp\_int $b$ \\
3194
3195
3196
3197
3198
3199
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3201
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3204
3205
3206
3207
3208
















































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3261
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul}
\end{figure}

\textbf{Algorithm mp\_mul.}
This algorithm performs the signed multiplication of two inputs.  It will make use of any of the three unsigned multiplication algorithms 
available when the input is of appropriate size.  The \textbf{sign} of the result is not set until the end of the algorithm since algorithm
s\_mp\_mul\_digs will clear it.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul.c
\vspace{-3mm}
\begin{alltt}
















































\end{alltt}
\end{small}

The implementation is rather simplistic and is not particularly noteworthy.  Line 22 computes the sign of the result using the ``?'' 
operator from the C programming language.  Line 48 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$.  

\section{Squaring}
\label{sec:basesquare}

Squaring is a special case of multiplication where both multiplicands are equal.  At first it may seem like there is no significant optimization
available but in fact there is.  Consider the multiplication of $576$ against $241$.  In total there will be nine single precision multiplications
performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot  6$, $2 \cdot 7$ and $2 \cdot 5$.  Now consider 
the multiplication of $123$ against $123$.  The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, 
$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$.  On closer inspection some of the products are equivalent.  For example, $3 \cdot 2 = 2 \cdot 3$ 
and $3 \cdot 1 = 1 \cdot 3$. 

For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$
required for multiplication.  The following diagram gives an example of the operations required.

\begin{figure}[here]
\begin{center}
\begin{tabular}{ccccc|c}
&&1&2&3&\\
$\times$ &&1&2&3&\\
\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\
       & $2 \cdot 1$  & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\
         $1 \cdot 1$  & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\
\end{tabular}
\end{center}
\caption{Squaring Optimization Diagram}
\end{figure}

Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious.  For the purposes of this discussion let $x$
represent the number being squared.  The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it.  

The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product.  Every non-square term of a column will
appear twice hence the name ``double product''.  Every odd column is made up entirely of double products.  In fact every column is made up of double 
products and at most one square (\textit{see the exercise section}).  

The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row, 
occurs at column $2k + 1$.  For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. 
Column two of row one is a square and column three is the first unique column.

\subsection{The Baseline Squaring Algorithm}
The baseline squaring algorithm is meant to be a catch-all squaring algorithm.  It will handle any of the input sizes that the faster routines
will not handle.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_sqr}. \\
\textbf{Input}.   mp\_int $a$ \\







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4902
4903
4904
4905
4906
4907
4908
4909
4910
4911
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul}
\end{figure}

\textbf{Algorithm mp\_mul.}
This algorithm performs the signed multiplication of two inputs.  It will make use of any of the three unsigned multiplication algorithms
available when the input is of appropriate size.  The \textbf{sign} of the result is not set until the end of the algorithm since algorithm
s\_mp\_mul\_digs will clear it.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul.c
\vspace{-3mm}
\begin{alltt}
016
017   /* high level multiplication (handles sign) */
018   int mp_mul (mp_int * a, mp_int * b, mp_int * c)
019   \{
020     int     res, neg;
021     neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;
022
023     /* use Toom-Cook? */
024   #ifdef BN_MP_TOOM_MUL_C
025     if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) \{
026       res = mp_toom_mul(a, b, c);
027     \} else
028   #endif
029   #ifdef BN_MP_KARATSUBA_MUL_C
030     /* use Karatsuba? */
031     if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) \{
032       res = mp_karatsuba_mul (a, b, c);
033     \} else
034   #endif
035     \{
036       /* can we use the fast multiplier?
037        *
038        * The fast multiplier can be used if the output will
039        * have less than MP_WARRAY digits and the number of
040        * digits won't affect carry propagation
041        */
042       int     digs = a->used + b->used + 1;
043
044   #ifdef BN_FAST_S_MP_MUL_DIGS_C
045       if ((digs < MP_WARRAY) &&
046           (MIN(a->used, b->used) <=
047            (1 << ((CHAR_BIT * sizeof(mp_word)) - (2 * DIGIT_BIT))))) \{
048         res = fast_s_mp_mul_digs (a, b, c, digs);
049       \} else
050   #endif
051       \{
052   #ifdef BN_S_MP_MUL_DIGS_C
053         res = s_mp_mul (a, b, c); /* uses s_mp_mul_digs */
054   #else
055         res = MP_VAL;
056   #endif
057       \}
058     \}
059     c->sign = (c->used > 0) ? neg : MP_ZPOS;
060     return res;
061   \}
062   #endif
063
\end{alltt}
\end{small}

The implementation is rather simplistic and is not particularly noteworthy.  Line 23 computes the sign of the result using the ``?''
operator from the C programming language.  Line 47 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$.

\section{Squaring}
\label{sec:basesquare}

Squaring is a special case of multiplication where both multiplicands are equal.  At first it may seem like there is no significant optimization
available but in fact there is.  Consider the multiplication of $576$ against $241$.  In total there will be nine single precision multiplications
performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot  6$, $2 \cdot 7$ and $2 \cdot 5$.  Now consider
the multiplication of $123$ against $123$.  The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$,
$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$.  On closer inspection some of the products are equivalent.  For example, $3 \cdot 2 = 2 \cdot 3$
and $3 \cdot 1 = 1 \cdot 3$.

For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$
required for multiplication.  The following diagram gives an example of the operations required.

\begin{figure}[here]
\begin{center}
\begin{tabular}{ccccc|c}
&&1&2&3&\\
$\times$ &&1&2&3&\\
\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\
       & $2 \cdot 1$  & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\
         $1 \cdot 1$  & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\
\end{tabular}
\end{center}
\caption{Squaring Optimization Diagram}
\end{figure}

Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious.  For the purposes of this discussion let $x$
represent the number being squared.  The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it.

The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product.  Every non-square term of a column will
appear twice hence the name ``double product''.  Every odd column is made up entirely of double products.  In fact every column is made up of double
products and at most one square (\textit{see the exercise section}).

The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row,
occurs at column $2k + 1$.  For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero.
Column two of row one is a square and column three is the first unique column.

\subsection{The Baseline Squaring Algorithm}
The baseline squaring algorithm is meant to be a catch-all squaring algorithm.  It will handle any of the input sizes that the faster routines
will not handle.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_sqr}. \\
\textbf{Input}.   mp\_int $a$ \\
3289
3290
3291
3292
3293
3294
3295
3296
3297
3298
3299
3300
3301
3302
3303
3304
3305
3306
3307
3308
3309
3310
3311
3312
3313

































































3314
3315
3316
3317
3318
3319
3320
3321
3322
3323
3324
3325
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3327
3328
3329
3330
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3332
3333
3334
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3337
3338
3339
3340
3341
3342
3343
3344
\end{center}
\end{small}
\caption{Algorithm s\_mp\_sqr}
\end{figure}

\textbf{Algorithm s\_mp\_sqr.}
This algorithm computes the square of an input using the three observations on squaring.  It is based fairly faithfully on  algorithm 14.16 of HAC
\cite[pp.596-597]{HAC}.  Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring.  This allows the 
destination mp\_int to be the same as the source mp\_int.

The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while
the inner loop computes the columns of the partial result.  Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate
the carry and compute the double products.  

The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this
very algorithm.  The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that
when it is multiplied by two, it can be properly represented by a mp\_word.

Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial 
results calculated so far.  This involves expensive carry propagation which will be eliminated in the next algorithm.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sqr.c
\vspace{-3mm}
\begin{alltt}

































































\end{alltt}
\end{small}

Inside the outer loop (line 34) the square term is calculated on line 37.  The carry (line 44) has been
extracted from the mp\_word accumulator using a right shift.  Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized 
(lines 47 and 50) to simplify the inner loop.  The doubling is performed using two
additions (line 59) since it is usually faster than shifting, if not at least as fast.  

The important observation is that the inner loop does not begin at $iy = 0$ like for multiplication.  As such the inner loops
get progressively shorter as the algorithm proceeds.  This is what leads to the savings compared to using a multiplication to
square a number. 

\subsection{Faster Squaring by the ``Comba'' Method}
A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop.  Squaring has an additional
drawback that it must double the product inside the inner loop as well.  As for multiplication, the Comba technique can be used to eliminate these
performance hazards.

The first obvious solution is to make an array of mp\_words which will hold all of the columns.  This will indeed eliminate all of the carry
propagation operations from the inner loop.  However, the inner product must still be doubled $O(n^2)$ times.  The solution stems from the simple fact
that $2a + 2b + 2c = 2(a + b + c)$.  That is the sum of all of the double products is equal to double the sum of all the products.  For example,
$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$.  

However, we cannot simply double all of the columns, since the squares appear only once per row.  The most practical solution is to have two 
mp\_word arrays.  One array will hold the squares and the other array will hold the double products.  With both arrays the doubling and 
carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level.  In this case, we have an even simpler solution in mind.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\







|




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\end{center}
\end{small}
\caption{Algorithm s\_mp\_sqr}
\end{figure}

\textbf{Algorithm s\_mp\_sqr.}
This algorithm computes the square of an input using the three observations on squaring.  It is based fairly faithfully on  algorithm 14.16 of HAC
\cite[pp.596-597]{HAC}.  Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring.  This allows the
destination mp\_int to be the same as the source mp\_int.

The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while
the inner loop computes the columns of the partial result.  Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate
the carry and compute the double products.

The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this
very algorithm.  The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that
when it is multiplied by two, it can be properly represented by a mp\_word.

Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial
results calculated so far.  This involves expensive carry propagation which will be eliminated in the next algorithm.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sqr.c
\vspace{-3mm}
\begin{alltt}
016
017   /* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */
018   int s_mp_sqr (mp_int * a, mp_int * b)
019   \{
020     mp_int  t;
021     int     res, ix, iy, pa;
022     mp_word r;
023     mp_digit u, tmpx, *tmpt;
024
025     pa = a->used;
026     if ((res = mp_init_size (&t, (2 * pa) + 1)) != MP_OKAY) \{
027       return res;
028     \}
029
030     /* default used is maximum possible size */
031     t.used = (2 * pa) + 1;
032
033     for (ix = 0; ix < pa; ix++) \{
034       /* first calculate the digit at 2*ix */
035       /* calculate double precision result */
036       r = (mp_word)t.dp[2*ix] +
037           ((mp_word)a->dp[ix] * (mp_word)a->dp[ix]);
038
039       /* store lower part in result */
040       t.dp[ix+ix] = (mp_digit) (r & ((mp_word) MP_MASK));
041
042       /* get the carry */
043       u           = (mp_digit)(r >> ((mp_word) DIGIT_BIT));
044
045       /* left hand side of A[ix] * A[iy] */
046       tmpx        = a->dp[ix];
047
048       /* alias for where to store the results */
049       tmpt        = t.dp + ((2 * ix) + 1);
050
051       for (iy = ix + 1; iy < pa; iy++) \{
052         /* first calculate the product */
053         r       = ((mp_word)tmpx) * ((mp_word)a->dp[iy]);
054
055         /* now calculate the double precision result, note we use
056          * addition instead of *2 since it's easier to optimize
057          */
058         r       = ((mp_word) *tmpt) + r + r + ((mp_word) u);
059
060         /* store lower part */
061         *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
062
063         /* get carry */
064         u       = (mp_digit)(r >> ((mp_word) DIGIT_BIT));
065       \}
066       /* propagate upwards */
067       while (u != ((mp_digit) 0)) \{
068         r       = ((mp_word) *tmpt) + ((mp_word) u);
069         *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
070         u       = (mp_digit)(r >> ((mp_word) DIGIT_BIT));
071       \}
072     \}
073
074     mp_clamp (&t);
075     mp_exch (&t, b);
076     mp_clear (&t);
077     return MP_OKAY;
078   \}
079   #endif
080
\end{alltt}
\end{small}

Inside the outer loop (line 33) the square term is calculated on line 36.  The carry (line 43) has been
extracted from the mp\_word accumulator using a right shift.  Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized
(lines 46 and 49) to simplify the inner loop.  The doubling is performed using two
additions (line 58) since it is usually faster than shifting, if not at least as fast.

The important observation is that the inner loop does not begin at $iy = 0$ like for multiplication.  As such the inner loops
get progressively shorter as the algorithm proceeds.  This is what leads to the savings compared to using a multiplication to
square a number.

\subsection{Faster Squaring by the ``Comba'' Method}
A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop.  Squaring has an additional
drawback that it must double the product inside the inner loop as well.  As for multiplication, the Comba technique can be used to eliminate these
performance hazards.

The first obvious solution is to make an array of mp\_words which will hold all of the columns.  This will indeed eliminate all of the carry
propagation operations from the inner loop.  However, the inner product must still be doubled $O(n^2)$ times.  The solution stems from the simple fact
that $2a + 2b + 2c = 2(a + b + c)$.  That is the sum of all of the double products is equal to double the sum of all the products.  For example,
$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$.

However, we cannot simply double all of the columns, since the squares appear only once per row.  The most practical solution is to have two
mp\_word arrays.  One array will hold the squares and the other array will hold the double products.  With both arrays the doubling and
carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level.  In this case, we have an even simpler solution in mind.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\
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6.  $oldused \leftarrow b.used$ \\
7.  $b.used \leftarrow 2 \cdot a.used$ \\
8.  for $ix$ from $0$ to $pa - 1$ do \\
\hspace{3mm}8.1  $b_{ix} \leftarrow W_{ix}$ \\
9.  for $ix$ from $pa$ to $oldused - 1$ do \\
\hspace{3mm}9.1  $b_{ix} \leftarrow 0$ \\
10.  Clamp excess digits from $b$.  (\textit{mp\_clamp}) \\
11.  Return(\textit{MP\_OKAY}). \\ 
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm fast\_s\_mp\_sqr}
\end{figure}

\textbf{Algorithm fast\_s\_mp\_sqr.}
This algorithm computes the square of an input using the Comba technique.  It is designed to be a replacement for algorithm 
s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$.  
This algorithm is very similar to the Comba multiplier except with a few key differences we shall make note of.

First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively.  This is because the inner loop
products are to be doubled.  If we had added the previous carry in we would be doubling too much.  Next we perform an
addition MIN condition on $iy$ (step 5.5) to prevent overlapping digits.  For example, $a_3 \cdot a_5$ is equal
$a_5 \cdot a_3$.  Whereas in the multiplication case we would have $5 < a.used$ and $3 \ge 0$ is maintained since we double the sum
of the products just outside the inner loop we have to avoid doing this.  This is also a good thing since we perform
fewer multiplications and the routine ends up being faster.

Finally the last difference is the addition of the ``square'' term outside the inner loop (step 5.8).  We add in the square
only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rfloor$ position.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_sqr.c
\vspace{-3mm}
\begin{alltt}































































































\end{alltt}
\end{small}

This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for 
the special case of squaring.  

\subsection{Polynomial Basis Squaring}
The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring.  The minor exception
is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$.  Instead of performing $2n + 1$
multiplications to find the $\zeta$ relations, squaring operations are performed instead.  

\subsection{Karatsuba Squaring}
Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square.  
Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial.  The Karatsuba equation can be modified to square a 
number with the following equation.

\begin{equation}
h(x) = a^2x^2 + \left ((a + b)^2 - (a^2 + b^2) \right )x + b^2
\end{equation}

Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a + b)^2$.  As in 
Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of 
$O \left ( n^{lg(3)} \right )$.

If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm 
instead?  The answer to this arises from the cutoff point for squaring.  As in multiplication there exists a cutoff point, at which the 
time required for a Comba based squaring and a Karatsuba based squaring meet.  Due to the overhead inherent in the Karatsuba method, the cutoff 
point is fairly high.  For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits.  

Consider squaring a 200 digit number with this technique.  It will be split into two 100 digit halves which are subsequently squared.  
The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm.  If Karatsuba multiplication
were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\
\textbf{Input}.   mp\_int $a$ \\







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6.  $oldused \leftarrow b.used$ \\
7.  $b.used \leftarrow 2 \cdot a.used$ \\
8.  for $ix$ from $0$ to $pa - 1$ do \\
\hspace{3mm}8.1  $b_{ix} \leftarrow W_{ix}$ \\
9.  for $ix$ from $pa$ to $oldused - 1$ do \\
\hspace{3mm}9.1  $b_{ix} \leftarrow 0$ \\
10.  Clamp excess digits from $b$.  (\textit{mp\_clamp}) \\
11.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm fast\_s\_mp\_sqr}
\end{figure}

\textbf{Algorithm fast\_s\_mp\_sqr.}
This algorithm computes the square of an input using the Comba technique.  It is designed to be a replacement for algorithm
s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$.
This algorithm is very similar to the Comba multiplier except with a few key differences we shall make note of.

First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively.  This is because the inner loop
products are to be doubled.  If we had added the previous carry in we would be doubling too much.  Next we perform an
addition MIN condition on $iy$ (step 5.5) to prevent overlapping digits.  For example, $a_3 \cdot a_5$ is equal
$a_5 \cdot a_3$.  Whereas in the multiplication case we would have $5 < a.used$ and $3 \ge 0$ is maintained since we double the sum
of the products just outside the inner loop we have to avoid doing this.  This is also a good thing since we perform
fewer multiplications and the routine ends up being faster.

Finally the last difference is the addition of the ``square'' term outside the inner loop (step 5.8).  We add in the square
only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rfloor$ position.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_sqr.c
\vspace{-3mm}
\begin{alltt}
016
017   /* the jist of squaring...
018    * you do like mult except the offset of the tmpx [one that
019    * starts closer to zero] can't equal the offset of tmpy.
020    * So basically you set up iy like before then you min it with
021    * (ty-tx) so that it never happens.  You double all those
022    * you add in the inner loop
023
024   After that loop you do the squares and add them in.
025   */
026
027   int fast_s_mp_sqr (mp_int * a, mp_int * b)
028   \{
029     int       olduse, res, pa, ix, iz;
030     mp_digit   W[MP_WARRAY], *tmpx;
031     mp_word   W1;
032
033     /* grow the destination as required */
034     pa = a->used + a->used;
035     if (b->alloc < pa) \{
036       if ((res = mp_grow (b, pa)) != MP_OKAY) \{
037         return res;
038       \}
039     \}
040
041     /* number of output digits to produce */
042     W1 = 0;
043     for (ix = 0; ix < pa; ix++) \{
044         int      tx, ty, iy;
045         mp_word  _W;
046         mp_digit *tmpy;
047
048         /* clear counter */
049         _W = 0;
050
051         /* get offsets into the two bignums */
052         ty = MIN(a->used-1, ix);
053         tx = ix - ty;
054
055         /* setup temp aliases */
056         tmpx = a->dp + tx;
057         tmpy = a->dp + ty;
058
059         /* this is the number of times the loop will iterrate, essentially
060            while (tx++ < a->used && ty-- >= 0) \{ ... \}
061          */
062         iy = MIN(a->used-tx, ty+1);
063
064         /* now for squaring tx can never equal ty
065          * we halve the distance since they approach at a rate of 2x
066          * and we have to round because odd cases need to be executed
067          */
068         iy = MIN(iy, ((ty-tx)+1)>>1);
069
070         /* execute loop */
071         for (iz = 0; iz < iy; iz++) \{
072            _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--);
073         \}
074
075         /* double the inner product and add carry */
076         _W = _W + _W + W1;
077
078         /* even columns have the square term in them */
079         if ((ix&1) == 0) \{
080            _W += ((mp_word)a->dp[ix>>1])*((mp_word)a->dp[ix>>1]);
081         \}
082
083         /* store it */
084         W[ix] = (mp_digit)(_W & MP_MASK);
085
086         /* make next carry */
087         W1 = _W >> ((mp_word)DIGIT_BIT);
088     \}
089
090     /* setup dest */
091     olduse  = b->used;
092     b->used = a->used+a->used;
093
094     \{
095       mp_digit *tmpb;
096       tmpb = b->dp;
097       for (ix = 0; ix < pa; ix++) \{
098         *tmpb++ = W[ix] & MP_MASK;
099       \}
100
101       /* clear unused digits [that existed in the old copy of c] */
102       for (; ix < olduse; ix++) \{
103         *tmpb++ = 0;
104       \}
105     \}
106     mp_clamp (b);
107     return MP_OKAY;
108   \}
109   #endif
110
\end{alltt}
\end{small}

This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for
the special case of squaring.

\subsection{Polynomial Basis Squaring}
The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring.  The minor exception
is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$.  Instead of performing $2n + 1$
multiplications to find the $\zeta$ relations, squaring operations are performed instead.

\subsection{Karatsuba Squaring}
Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square.
Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial.  The Karatsuba equation can be modified to square a
number with the following equation.

\begin{equation}
h(x) = a^2x^2 + \left ((a + b)^2 - (a^2 + b^2) \right )x + b^2
\end{equation}

Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a + b)^2$.  As in
Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of
$O \left ( n^{lg(3)} \right )$.

If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm
instead?  The answer to this arises from the cutoff point for squaring.  As in multiplication there exists a cutoff point, at which the
time required for a Comba based squaring and a Karatsuba based squaring meet.  Due to the overhead inherent in the Karatsuba method, the cutoff
point is fairly high.  For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits.

Consider squaring a 200 digit number with this technique.  It will be split into two 100 digit halves which are subsequently squared.
The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm.  If Karatsuba multiplication
were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\
\textbf{Input}.   mp\_int $a$ \\
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as the radix point.  The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form.

By expanding $\left (x1 + x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $(x0 - x1)^2 - (x1^2 + x0^2)  = 2 \cdot x0 \cdot x1$.
Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then
this method is faster.  Assuming no further recursions occur, the difference can be estimated with the following inequality.

Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or
machine clock cycles.}. 

\begin{equation}
5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2
\end{equation}

For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$.  This implies that the following inequality should hold.
\begin{center}
\begin{tabular}{rcl}
${5n \over 3} + 3n^2 + 3n$     & $<$ & ${n \over 3} + 6n^2$ \\
${5 \over 3} + 3n + 3$     & $<$ & ${1 \over 3} + 6n$ \\
${13 \over 9}$     & $<$ & $n$ \\
\end{tabular}
\end{center}

This results in a cutoff point around $n = 2$.  As a consequence it is actually faster to compute the middle term the ``long way'' on processors
where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication.  On
the Intel P4 processor this ratio is 1:29 making this method even more beneficial.  The only common exception is the ARMv4 processor which has a
ratio of 1:7.  } than simpler operations such as addition.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_sqr.c
\vspace{-3mm}
\begin{alltt}






































































































\end{alltt}
\end{small}

This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul.  It uses the same inline style to copy and 
shift the input into the two halves.  The loop from line 54 to line 70 has been modified since only one input exists.  The \textbf{used}
count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin.  At this point $x1$ and $x0$ are valid equivalents
to the respective halves as if mp\_rshd and mp\_mod\_2d had been used.  

By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered.  On the Athlon the cutoff point
is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}).  On slower processors such as the Intel P4
it is actually below the Comba limit (\textit{at 110 digits}).

This routine uses the same error trap coding style as mp\_karatsuba\_sqr.  As the temporary variables are initialized errors are 
redirected to the error trap higher up.  If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and 
mp\_clears are executed normally.

\subsection{Toom-Cook Squaring}
The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used
instead of multiplication to find the five relations.  The reader is encouraged to read the description of the latter algorithm and try to 
derive their own Toom-Cook squaring algorithm.  

\subsection{High Level Squaring}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_sqr}. \\







|

















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as the radix point.  The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form.

By expanding $\left (x1 + x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $(x0 - x1)^2 - (x1^2 + x0^2)  = 2 \cdot x0 \cdot x1$.
Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then
this method is faster.  Assuming no further recursions occur, the difference can be estimated with the following inequality.

Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or
machine clock cycles.}.

\begin{equation}
5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2
\end{equation}

For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$.  This implies that the following inequality should hold.
\begin{center}
\begin{tabular}{rcl}
${5n \over 3} + 3n^2 + 3n$     & $<$ & ${n \over 3} + 6n^2$ \\
${5 \over 3} + 3n + 3$     & $<$ & ${1 \over 3} + 6n$ \\
${13 \over 9}$     & $<$ & $n$ \\
\end{tabular}
\end{center}

This results in a cutoff point around $n = 2$.  As a consequence it is actually faster to compute the middle term the ``long way'' on processors
where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication.  On
the Intel P4 processor this ratio is 1:29 making this method even more beneficial.  The only common exception is the ARMv4 processor which has a
ratio of 1:7.  } than simpler operations such as addition.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_sqr.c
\vspace{-3mm}
\begin{alltt}
016
017   /* Karatsuba squaring, computes b = a*a using three
018    * half size squarings
019    *
020    * See comments of karatsuba_mul for details.  It
021    * is essentially the same algorithm but merely
022    * tuned to perform recursive squarings.
023    */
024   int mp_karatsuba_sqr (mp_int * a, mp_int * b)
025   \{
026     mp_int  x0, x1, t1, t2, x0x0, x1x1;
027     int     B, err;
028
029     err = MP_MEM;
030
031     /* min # of digits */
032     B = a->used;
033
034     /* now divide in two */
035     B = B >> 1;
036
037     /* init copy all the temps */
038     if (mp_init_size (&x0, B) != MP_OKAY)
039       goto ERR;
040     if (mp_init_size (&x1, a->used - B) != MP_OKAY)
041       goto X0;
042
043     /* init temps */
044     if (mp_init_size (&t1, a->used * 2) != MP_OKAY)
045       goto X1;
046     if (mp_init_size (&t2, a->used * 2) != MP_OKAY)
047       goto T1;
048     if (mp_init_size (&x0x0, B * 2) != MP_OKAY)
049       goto T2;
050     if (mp_init_size (&x1x1, (a->used - B) * 2) != MP_OKAY)
051       goto X0X0;
052
053     \{
054       int x;
055       mp_digit *dst, *src;
056
057       src = a->dp;
058
059       /* now shift the digits */
060       dst = x0.dp;
061       for (x = 0; x < B; x++) \{
062         *dst++ = *src++;
063       \}
064
065       dst = x1.dp;
066       for (x = B; x < a->used; x++) \{
067         *dst++ = *src++;
068       \}
069     \}
070
071     x0.used = B;
072     x1.used = a->used - B;
073
074     mp_clamp (&x0);
075
076     /* now calc the products x0*x0 and x1*x1 */
077     if (mp_sqr (&x0, &x0x0) != MP_OKAY)
078       goto X1X1;           /* x0x0 = x0*x0 */
079     if (mp_sqr (&x1, &x1x1) != MP_OKAY)
080       goto X1X1;           /* x1x1 = x1*x1 */
081
082     /* now calc (x1+x0)**2 */
083     if (s_mp_add (&x1, &x0, &t1) != MP_OKAY)
084       goto X1X1;           /* t1 = x1 - x0 */
085     if (mp_sqr (&t1, &t1) != MP_OKAY)
086       goto X1X1;           /* t1 = (x1 - x0) * (x1 - x0) */
087
088     /* add x0y0 */
089     if (s_mp_add (&x0x0, &x1x1, &t2) != MP_OKAY)
090       goto X1X1;           /* t2 = x0x0 + x1x1 */
091     if (s_mp_sub (&t1, &t2, &t1) != MP_OKAY)
092       goto X1X1;           /* t1 = (x1+x0)**2 - (x0x0 + x1x1) */
093
094     /* shift by B */
095     if (mp_lshd (&t1, B) != MP_OKAY)
096       goto X1X1;           /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))<<B */
097     if (mp_lshd (&x1x1, B * 2) != MP_OKAY)
098       goto X1X1;           /* x1x1 = x1x1 << 2*B */
099
100     if (mp_add (&x0x0, &t1, &t1) != MP_OKAY)
101       goto X1X1;           /* t1 = x0x0 + t1 */
102     if (mp_add (&t1, &x1x1, b) != MP_OKAY)
103       goto X1X1;           /* t1 = x0x0 + t1 + x1x1 */
104
105     err = MP_OKAY;
106
107   X1X1:mp_clear (&x1x1);
108   X0X0:mp_clear (&x0x0);
109   T2:mp_clear (&t2);
110   T1:mp_clear (&t1);
111   X1:mp_clear (&x1);
112   X0:mp_clear (&x0);
113   ERR:
114     return err;
115   \}
116   #endif
117
\end{alltt}
\end{small}

This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul.  It uses the same inline style to copy and
shift the input into the two halves.  The loop from line 53 to line 69 has been modified since only one input exists.  The \textbf{used}
count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin.  At this point $x1$ and $x0$ are valid equivalents
to the respective halves as if mp\_rshd and mp\_mod\_2d had been used.

By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered.  On the Athlon the cutoff point
is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}).  On slower processors such as the Intel P4
it is actually below the Comba limit (\textit{at 110 digits}).

This routine uses the same error trap coding style as mp\_karatsuba\_sqr.  As the temporary variables are initialized errors are
redirected to the error trap higher up.  If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and
mp\_clears are executed normally.

\subsection{Toom-Cook Squaring}
The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used
instead of multiplication to find the five relations.  The reader is encouraged to read the description of the latter algorithm and try to
derive their own Toom-Cook squaring algorithm.

\subsection{High Level Squaring}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_sqr}. \\
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\end{small}
\caption{Algorithm mp\_sqr}
\end{figure}

\textbf{Algorithm mp\_sqr.}
This algorithm computes the square of the input using one of four different algorithms.  If the input is very large and has at least
\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used.  If
neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_sqr.c
\vspace{-3mm}
\begin{alltt}









































\end{alltt}
\end{small}

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\
                      & that have different number of digits in Karatsuba multiplication. \\
                      & \\
$\left [ 2 \right ] $ & In section 5.3 the fact that every column of a squaring is made up \\
                      & of double products and at most one square is stated.  Prove this statement. \\
                      & \\                      
$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\
                      & \\
$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\
                      & \\ 
$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\
                      & required for equation $6.7$ to be true.  \\
                      & \\
$\left [ 3 \right ] $ & Implement a threaded version of Comba multiplication (and squaring) where you \\
                      & compute subsets of the columns in each thread.  Determine a cutoff point where \\
                      & it is effective and add the logic to mp\_mul() and mp\_sqr(). \\
                      &\\
$\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and Toom-Cook.  You must \\
                      & increase the throughput of mp\_exptmod() for random odd moduli in the range \\
                      & $512 \ldots 4096$ bits significantly ($> 2x$) to complete this challenge. \\
                      & \\
\end{tabular}

\chapter{Modular Reduction}
\section{Basics of Modular Reduction}
\index{modular residue}
Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms, 
such as factoring.  Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set.  A number $a$ is said to be \textit{reduced}
modulo another number $b$ by finding the remainder of the division $a/b$.  Full integer division with remainder is a topic to be covered 
in~\ref{sec:division}.

Modular reduction is equivalent to solving for $r$ in the following equation.  $a = bq + r$ where $q = \lfloor a/b \rfloor$.  The result 
$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$.  In other vernacular $r$ is known as the 
``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and
other forms of residues.  

Modular reductions are normally used to create either finite groups, rings or fields.  The most common usage for performance driven modular reductions 
is in modular exponentiation algorithms.  That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible.  This operation is used in the 
RSA and Diffie-Hellman public key algorithms, for example.  Modular multiplication and squaring also appears as a fundamental operation in 
elliptic curve cryptographic algorithms.  As will be discussed in the subsequent chapter there exist fast algorithms for computing modular 
exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications.  These algorithms will produce partial results in the 
range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms.   They have also been used to create redundancy check 
algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems.  

\section{The Barrett Reduction}
The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate
division.  Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to 

\begin{equation}
c = a - b \cdot \lfloor a/b \rfloor
\end{equation}

Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper 
targeted the DSP56K processor.}  intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal.  However, 
DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types.  
It would take another common optimization to optimize the algorithm.

\subsection{Fixed Point Arithmetic}
The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers.  Fixed
point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were 
fairly slow if not unavailable.   The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit 
integer and a $q$-bit fraction part (\textit{where $p+q = k$}).  

In this system a $k$-bit integer $n$ would actually represent $n/2^q$.  For example, with $q = 4$ the integer $n = 37$ would actually represent the
value $2.3125$.  To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by 
moving the implied decimal point back to where it should be.  For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted 
to fixed point first by multiplying by $2^q$.  Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the 
fixed point representation of $5$.  The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$.  

This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication
of two fixed point numbers.  Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal.  If $2^q$ is 
equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic.  Using this fact dividing an integer 
$a$ by another integer $b$ can be achieved with the following expression.

\begin{equation}
\lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor
\end{equation}

The precision of the division is proportional to the value of $q$.  If the divisor $b$ is used frequently as is the case with 
modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift.  Both operations
are considerably faster than division on most processors.  

Consider dividing $19$ by $5$.  The correct result is $\lfloor 19/5 \rfloor = 3$.  With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which
leads to a product of $19$ which when divided by $2^q$ produces $2$.  However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and
the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct.  The value of $2^q$ must be close to or ideally
larger than the dividend.  In effect if $a$ is the dividend then $q$ should allow $0 \le \lfloor a/2^q \rfloor \le 1$ in order for this approach
to work correctly.  Plugging this form of divison into the original equation the following modular residue equation arises.

\begin{equation}
c = a - b \cdot \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor
\end{equation}

Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol.  Using the $\mu$
variable also helps re-inforce the idea that it is meant to be computed once and re-used.

\begin{equation}
c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor
\end{equation}

Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one.  In the context of Barrett
reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough
precision.  

Let $n$ represent the number of digits in $b$.  This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and 
another $n^2$ single precision multiplications to find the residue.  In total $3n^2$ single precision multiplications are required to 
reduce the number.  

For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$.  Consider reducing
$a = 180388626447$ modulo $b$ using the above reduction equation.  The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$.
By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found.

\subsection{Choosing a Radix Point}
Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications.  If that were the best
that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$.  
See~\ref{sec:division} for further details.} might as well be used in its place.  The key to optimizing the reduction is to reduce the precision of
the initial multiplication that finds the quotient.  

Let $a$ represent the number of which the residue is sought.  Let $b$ represent the modulus used to find the residue.  Let $m$ represent
the number of digits in $b$.  For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if 
two $m$-digit numbers have been multiplied.  Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer.  Digits below the 
$m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$.  Another way to
express this is by re-writing $a$ as two parts.  If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then 
${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$.  Since $a'$ is bound to be less than $b$ the quotient
is bound by $0 \le {a' \over b} < 1$.

Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero.  However, if the digits 
``might as well be zero'' they might as well not be there in the first place.  Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input
with the irrelevant digits trimmed.  Now the modular reduction is trimmed to the almost equivalent equation

\begin{equation}
c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor
\end{equation}

Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the 
exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$.  If the optimization had not been performed the divisor 
would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient 
$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two.  The original fixed point quotient can be off
by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient
can be off by an additional value of one for a total of at most two.  This implies that 
$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$.  By first subtracting $b$ times the quotient and then conditionally subtracting 
$b$ once or twice the residue is found.

The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single
precision multiplications, ignoring the subtractions required.  In total $2m^2 + m$ single precision multiplications are required to find the residue.  
This is considerably faster than the original attempt.

For example, let $\beta = 10$ represent the radix of the digits.  Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ 
represent the value of which the residue is desired.  In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$.  
With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$.  The quotient is then 
$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$.  Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$ 
is found.  

\subsection{Trimming the Quotient}
So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications.  As 
it stands now the algorithm is already fairly fast compared to a full integer division algorithm.  However, there is still room for
optimization.  

After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower
half of the product.  It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision 
multiplications.  If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly.  
In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed.  

The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number.  Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision
multiplications would be required.  Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number
of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications.  

\subsection{Trimming the Residue}
After the quotient has been calculated it is used to reduce the input.  As previously noted the algorithm is not exact and it can be off by a small
multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$.  If $b$ is $m$ digits than the 
result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are
implicitly zero.  

The next optimization arises from this very fact.  Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full
$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed.  Similarly the value of $a$ can
be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well.  A multiplication that produces 
only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications.  

With both optimizations in place the algorithm is the algorithm Barrett proposed.  It requires $m^2 + 2m - 1$ single precision multiplications which
is considerably faster than the straightforward $3m^2$ method.  

\subsection{The Barrett Algorithm}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce}. \\







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\end{small}
\caption{Algorithm mp\_sqr}
\end{figure}

\textbf{Algorithm mp\_sqr.}
This algorithm computes the square of the input using one of four different algorithms.  If the input is very large and has at least
\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used.  If
neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_sqr.c
\vspace{-3mm}
\begin{alltt}
016
017   /* computes b = a*a */
018   int
019   mp_sqr (mp_int * a, mp_int * b)
020   \{
021     int     res;
022
023   #ifdef BN_MP_TOOM_SQR_C
024     /* use Toom-Cook? */
025     if (a->used >= TOOM_SQR_CUTOFF) \{
026       res = mp_toom_sqr(a, b);
027     /* Karatsuba? */
028     \} else
029   #endif
030   #ifdef BN_MP_KARATSUBA_SQR_C
031     if (a->used >= KARATSUBA_SQR_CUTOFF) \{
032       res = mp_karatsuba_sqr (a, b);
033     \} else
034   #endif
035     \{
036   #ifdef BN_FAST_S_MP_SQR_C
037       /* can we use the fast comba multiplier? */
038       if ((((a->used * 2) + 1) < MP_WARRAY) &&
039            (a->used <
040            (1 << (((sizeof(mp_word) * CHAR_BIT) - (2 * DIGIT_BIT)) - 1)))) \{
041         res = fast_s_mp_sqr (a, b);
042       \} else
043   #endif
044       \{
045   #ifdef BN_S_MP_SQR_C
046         res = s_mp_sqr (a, b);
047   #else
048         res = MP_VAL;
049   #endif
050       \}
051     \}
052     b->sign = MP_ZPOS;
053     return res;
054   \}
055   #endif
056
\end{alltt}
\end{small}

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\
                      & that have different number of digits in Karatsuba multiplication. \\
                      & \\
$\left [ 2 \right ] $ & In section 5.3 the fact that every column of a squaring is made up \\
                      & of double products and at most one square is stated.  Prove this statement. \\
                      & \\
$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\
                      & \\
$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\
                      & \\
$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\
                      & required for equation $6.7$ to be true.  \\
                      & \\
$\left [ 3 \right ] $ & Implement a threaded version of Comba multiplication (and squaring) where you \\
                      & compute subsets of the columns in each thread.  Determine a cutoff point where \\
                      & it is effective and add the logic to mp\_mul() and mp\_sqr(). \\
                      &\\
$\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and Toom-Cook.  You must \\
                      & increase the throughput of mp\_exptmod() for random odd moduli in the range \\
                      & $512 \ldots 4096$ bits significantly ($> 2x$) to complete this challenge. \\
                      & \\
\end{tabular}

\chapter{Modular Reduction}
\section{Basics of Modular Reduction}
\index{modular residue}
Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms,
such as factoring.  Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set.  A number $a$ is said to be \textit{reduced}
modulo another number $b$ by finding the remainder of the division $a/b$.  Full integer division with remainder is a topic to be covered
in~\ref{sec:division}.

Modular reduction is equivalent to solving for $r$ in the following equation.  $a = bq + r$ where $q = \lfloor a/b \rfloor$.  The result
$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$.  In other vernacular $r$ is known as the
``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and
other forms of residues.

Modular reductions are normally used to create either finite groups, rings or fields.  The most common usage for performance driven modular reductions
is in modular exponentiation algorithms.  That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible.  This operation is used in the
RSA and Diffie-Hellman public key algorithms, for example.  Modular multiplication and squaring also appears as a fundamental operation in
elliptic curve cryptographic algorithms.  As will be discussed in the subsequent chapter there exist fast algorithms for computing modular
exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications.  These algorithms will produce partial results in the
range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms.   They have also been used to create redundancy check
algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems.

\section{The Barrett Reduction}
The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate
division.  Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to

\begin{equation}
c = a - b \cdot \lfloor a/b \rfloor
\end{equation}

Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper
targeted the DSP56K processor.}  intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal.  However,
DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types.
It would take another common optimization to optimize the algorithm.

\subsection{Fixed Point Arithmetic}
The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers.  Fixed
point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were
fairly slow if not unavailable.   The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit
integer and a $q$-bit fraction part (\textit{where $p+q = k$}).

In this system a $k$-bit integer $n$ would actually represent $n/2^q$.  For example, with $q = 4$ the integer $n = 37$ would actually represent the
value $2.3125$.  To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by
moving the implied decimal point back to where it should be.  For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted
to fixed point first by multiplying by $2^q$.  Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the
fixed point representation of $5$.  The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$.

This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication
of two fixed point numbers.  Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal.  If $2^q$ is
equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic.  Using this fact dividing an integer
$a$ by another integer $b$ can be achieved with the following expression.

\begin{equation}
\lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor
\end{equation}

The precision of the division is proportional to the value of $q$.  If the divisor $b$ is used frequently as is the case with
modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift.  Both operations
are considerably faster than division on most processors.

Consider dividing $19$ by $5$.  The correct result is $\lfloor 19/5 \rfloor = 3$.  With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which
leads to a product of $19$ which when divided by $2^q$ produces $2$.  However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and
the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct.  The value of $2^q$ must be close to or ideally
larger than the dividend.  In effect if $a$ is the dividend then $q$ should allow $0 \le \lfloor a/2^q \rfloor \le 1$ in order for this approach
to work correctly.  Plugging this form of divison into the original equation the following modular residue equation arises.

\begin{equation}
c = a - b \cdot \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor
\end{equation}

Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol.  Using the $\mu$
variable also helps re-inforce the idea that it is meant to be computed once and re-used.

\begin{equation}
c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor
\end{equation}

Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one.  In the context of Barrett
reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough
precision.

Let $n$ represent the number of digits in $b$.  This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and
another $n^2$ single precision multiplications to find the residue.  In total $3n^2$ single precision multiplications are required to
reduce the number.

For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$.  Consider reducing
$a = 180388626447$ modulo $b$ using the above reduction equation.  The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$.
By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found.

\subsection{Choosing a Radix Point}
Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications.  If that were the best
that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$.
See~\ref{sec:division} for further details.} might as well be used in its place.  The key to optimizing the reduction is to reduce the precision of
the initial multiplication that finds the quotient.

Let $a$ represent the number of which the residue is sought.  Let $b$ represent the modulus used to find the residue.  Let $m$ represent
the number of digits in $b$.  For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if
two $m$-digit numbers have been multiplied.  Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer.  Digits below the
$m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$.  Another way to
express this is by re-writing $a$ as two parts.  If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then
${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$.  Since $a'$ is bound to be less than $b$ the quotient
is bound by $0 \le {a' \over b} < 1$.

Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero.  However, if the digits
``might as well be zero'' they might as well not be there in the first place.  Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input
with the irrelevant digits trimmed.  Now the modular reduction is trimmed to the almost equivalent equation

\begin{equation}
c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor
\end{equation}

Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the
exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$.  If the optimization had not been performed the divisor
would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient
$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two.  The original fixed point quotient can be off
by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient
can be off by an additional value of one for a total of at most two.  This implies that
$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$.  By first subtracting $b$ times the quotient and then conditionally subtracting
$b$ once or twice the residue is found.

The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single
precision multiplications, ignoring the subtractions required.  In total $2m^2 + m$ single precision multiplications are required to find the residue.
This is considerably faster than the original attempt.

For example, let $\beta = 10$ represent the radix of the digits.  Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$
represent the value of which the residue is desired.  In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$.
With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$.  The quotient is then
$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$.  Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$
is found.

\subsection{Trimming the Quotient}
So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications.  As
it stands now the algorithm is already fairly fast compared to a full integer division algorithm.  However, there is still room for
optimization.

After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower
half of the product.  It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision
multiplications.  If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly.
In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed.

The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number.  Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision
multiplications would be required.  Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number
of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications.

\subsection{Trimming the Residue}
After the quotient has been calculated it is used to reduce the input.  As previously noted the algorithm is not exact and it can be off by a small
multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$.  If $b$ is $m$ digits than the
result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are
implicitly zero.

The next optimization arises from this very fact.  Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full
$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed.  Similarly the value of $a$ can
be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well.  A multiplication that produces
only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications.

With both optimizations in place the algorithm is the algorithm Barrett proposed.  It requires $m^2 + 2m - 1$ single precision multiplications which
is considerably faster than the straightforward $3m^2$ method.

\subsection{The Barrett Algorithm}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce}. \\
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\end{center}
\end{small}
\caption{Algorithm mp\_reduce}
\end{figure}

\textbf{Algorithm mp\_reduce.}
This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm.  It is loosely based on algorithm 14.42 of HAC
\cite[pp.  602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}.  The algorithm has several restrictions and assumptions which must 
be adhered to for the algorithm to work.

First the modulus $b$ is assumed to be positive and greater than one.  If the modulus were less than or equal to one than subtracting
a multiple of it would either accomplish nothing or actually enlarge the input.  The input $a$ must be in the range $0 \le a < b^2$ in order
for the quotient to have enough precision.  If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem.
Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish.  The value of $\mu$ is passed as an argument to this 
algorithm and is assumed to be calculated and stored before the algorithm is used.  

Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position.  An algorithm called 
$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task.  The algorithm is based on $s\_mp\_mul\_digs$ except that
instead of stopping at a given level of precision it starts at a given level of precision.  This optimal algorithm can only be used if the number
of digits in $b$ is very much smaller than $\beta$.  

While it is known that 
$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied 
``borrow'' from the higher digits might leave a negative result.  After the multiple of the modulus has been subtracted from $a$ the residue must be 
fixed up in case it is negative.  The invariant $\beta^{m+1}$ must be added to the residue to make it positive again.  

The while loop at step 9 will subtract $b$ until the residue is less than $b$.  If the algorithm is performed correctly this step is 
performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce.c
\vspace{-3mm}
\begin{alltt}

















































































\end{alltt}
\end{small}

The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up.  This essentially halves
the number of single precision multiplications required.  However, the optimization is only safe if $\beta$ is much larger than the number of digits
in the modulus.  In the source code this is evaluated on lines 36 to 44 where algorithm s\_mp\_mul\_high\_digs is used when it is
safe to do so.  

\subsection{The Barrett Setup Algorithm}
In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance.  Ideally this value should be computed once and stored for
future use so that the Barrett algorithm can be used without delay.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce\_setup}. \\
\textbf{Input}.   mp\_int $a$ ($a > 1$)  \\







|





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>
>
>
>
>
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>
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>
>
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>
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|







5742
5743
5744
5745
5746
5747
5748
5749
5750
5751
5752
5753
5754
5755
5756
5757
5758
5759
5760
5761
5762
5763
5764
5765
5766
5767
5768
5769
5770
5771
5772
5773
5774
5775
5776
5777
5778
5779
5780
5781
5782
5783
5784
5785
5786
5787
5788
5789
5790
5791
5792
5793
5794
5795
5796
5797
5798
5799
5800
5801
5802
5803
5804
5805
5806
5807
5808
5809
5810
5811
5812
5813
5814
5815
5816
5817
5818
5819
5820
5821
5822
5823
5824
5825
5826
5827
5828
5829
5830
5831
5832
5833
5834
5835
5836
5837
5838
5839
5840
5841
5842
5843
5844
5845
5846
5847
5848
5849
5850
5851
5852
5853
5854
5855
5856
5857
5858
5859
5860
5861
5862
5863
5864
5865
5866
5867
5868
5869
5870
5871
5872
5873
\end{center}
\end{small}
\caption{Algorithm mp\_reduce}
\end{figure}

\textbf{Algorithm mp\_reduce.}
This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm.  It is loosely based on algorithm 14.42 of HAC
\cite[pp.  602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}.  The algorithm has several restrictions and assumptions which must
be adhered to for the algorithm to work.

First the modulus $b$ is assumed to be positive and greater than one.  If the modulus were less than or equal to one than subtracting
a multiple of it would either accomplish nothing or actually enlarge the input.  The input $a$ must be in the range $0 \le a < b^2$ in order
for the quotient to have enough precision.  If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem.
Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish.  The value of $\mu$ is passed as an argument to this
algorithm and is assumed to be calculated and stored before the algorithm is used.

Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position.  An algorithm called
$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task.  The algorithm is based on $s\_mp\_mul\_digs$ except that
instead of stopping at a given level of precision it starts at a given level of precision.  This optimal algorithm can only be used if the number
of digits in $b$ is very much smaller than $\beta$.

While it is known that
$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied
``borrow'' from the higher digits might leave a negative result.  After the multiple of the modulus has been subtracted from $a$ the residue must be
fixed up in case it is negative.  The invariant $\beta^{m+1}$ must be added to the residue to make it positive again.

The while loop at step 9 will subtract $b$ until the residue is less than $b$.  If the algorithm is performed correctly this step is
performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce.c
\vspace{-3mm}
\begin{alltt}
016
017   /* reduces x mod m, assumes 0 < x < m**2, mu is
018    * precomputed via mp_reduce_setup.
019    * From HAC pp.604 Algorithm 14.42
020    */
021   int mp_reduce (mp_int * x, mp_int * m, mp_int * mu)
022   \{
023     mp_int  q;
024     int     res, um = m->used;
025
026     /* q = x */
027     if ((res = mp_init_copy (&q, x)) != MP_OKAY) \{
028       return res;
029     \}
030
031     /* q1 = x / b**(k-1)  */
032     mp_rshd (&q, um - 1);
033
034     /* according to HAC this optimization is ok */
035     if (((mp_digit) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) \{
036       if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) \{
037         goto CLEANUP;
038       \}
039     \} else \{
040   #ifdef BN_S_MP_MUL_HIGH_DIGS_C
041       if ((res = s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) \{
042         goto CLEANUP;
043       \}
044   #elif defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C)
045       if ((res = fast_s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) \{
046         goto CLEANUP;
047       \}
048   #else
049       \{
050         res = MP_VAL;
051         goto CLEANUP;
052       \}
053   #endif
054     \}
055
056     /* q3 = q2 / b**(k+1) */
057     mp_rshd (&q, um + 1);
058
059     /* x = x mod b**(k+1), quick (no division) */
060     if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) \{
061       goto CLEANUP;
062     \}
063
064     /* q = q * m mod b**(k+1), quick (no division) */
065     if ((res = s_mp_mul_digs (&q, m, &q, um + 1)) != MP_OKAY) \{
066       goto CLEANUP;
067     \}
068
069     /* x = x - q */
070     if ((res = mp_sub (x, &q, x)) != MP_OKAY) \{
071       goto CLEANUP;
072     \}
073
074     /* If x < 0, add b**(k+1) to it */
075     if (mp_cmp_d (x, 0) == MP_LT) \{
076       mp_set (&q, 1);
077       if ((res = mp_lshd (&q, um + 1)) != MP_OKAY)
078         goto CLEANUP;
079       if ((res = mp_add (x, &q, x)) != MP_OKAY)
080         goto CLEANUP;
081     \}
082
083     /* Back off if it's too big */
084     while (mp_cmp (x, m) != MP_LT) \{
085       if ((res = s_mp_sub (x, m, x)) != MP_OKAY) \{
086         goto CLEANUP;
087       \}
088     \}
089
090   CLEANUP:
091     mp_clear (&q);
092
093     return res;
094   \}
095   #endif
096
\end{alltt}
\end{small}

The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up.  This essentially halves
the number of single precision multiplications required.  However, the optimization is only safe if $\beta$ is much larger than the number of digits
in the modulus.  In the source code this is evaluated on lines 36 to 43 where algorithm s\_mp\_mul\_high\_digs is used when it is
safe to do so.

\subsection{The Barrett Setup Algorithm}
In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance.  Ideally this value should be computed once and stored for
future use so that the Barrett algorithm can be used without delay.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce\_setup}. \\
\textbf{Input}.   mp\_int $a$ ($a > 1$)  \\
3853
3854
3855
3856
3857
3858
3859















3860
3861
3862
3863
3864
3865
3866
3867
3868
3869
3870
3871
3872
3873
3874
3875
3876
3877
3878
3879
3880
3881
3882
3883
3884
3885
3886
3887
3888
This algorithm computes the reciprocal $\mu$ required for Barrett reduction.  First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot  m}$ which
is equivalent and much faster.  The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_setup.c
\vspace{-3mm}
\begin{alltt}















\end{alltt}
\end{small}

This simple routine calculates the reciprocal $\mu$ required by Barrett reduction.  Note the extended usage of algorithm mp\_div where the variable
which would received the remainder is passed as NULL.  As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the 
remainder to be passed as NULL meaning to ignore the value.  

\section{The Montgomery Reduction}
Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting 
form of reduction in common use.  It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a 
residue times a constant.  However, as perplexing as this may sound the algorithm is relatively simple and very efficient.  

Throughout this entire section the variable $n$ will represent the modulus used to form the residue.  As will be discussed shortly the value of
$n$ must be odd.  The variable $x$ will represent the quantity of which the residue is sought.  Similar to the Barrett algorithm the input
is restricted to $0 \le x < n^2$.  To begin the description some simple number theory facts must be established.

\textbf{Fact 1.}  Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$.  Another way
to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$.  Adding zero will not change the value of the residue.  

\textbf{Fact 2.}  If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$.  Actually
this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to 
multiplication by $k^{-1}$ modulo $n$.  

From these two simple facts the following simple algorithm can be derived.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}







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>
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>




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5887
5888
5889
5890
5891
5892
5893
5894
5895
5896
5897
5898
5899
5900
5901
5902
5903
5904
5905
5906
5907
5908
5909
5910
5911
5912
5913
5914
5915
5916
5917
5918
5919
5920
5921
5922
5923
5924
5925
5926
5927
5928
5929
5930
5931
5932
5933
5934
5935
5936
5937
This algorithm computes the reciprocal $\mu$ required for Barrett reduction.  First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot  m}$ which
is equivalent and much faster.  The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_setup.c
\vspace{-3mm}
\begin{alltt}
016
017   /* pre-calculate the value required for Barrett reduction
018    * For a given modulus "b" it calulates the value required in "a"
019    */
020   int mp_reduce_setup (mp_int * a, mp_int * b)
021   \{
022     int     res;
023
024     if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) \{
025       return res;
026     \}
027     return mp_div (a, b, a, NULL);
028   \}
029   #endif
030
\end{alltt}
\end{small}

This simple routine calculates the reciprocal $\mu$ required by Barrett reduction.  Note the extended usage of algorithm mp\_div where the variable
which would received the remainder is passed as NULL.  As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the
remainder to be passed as NULL meaning to ignore the value.

\section{The Montgomery Reduction}
Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting
form of reduction in common use.  It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a
residue times a constant.  However, as perplexing as this may sound the algorithm is relatively simple and very efficient.

Throughout this entire section the variable $n$ will represent the modulus used to form the residue.  As will be discussed shortly the value of
$n$ must be odd.  The variable $x$ will represent the quantity of which the residue is sought.  Similar to the Barrett algorithm the input
is restricted to $0 \le x < n^2$.  To begin the description some simple number theory facts must be established.

\textbf{Fact 1.}  Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$.  Another way
to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$.  Adding zero will not change the value of the residue.

\textbf{Fact 2.}  If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$.  Actually
this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to
multiplication by $k^{-1}$ modulo $n$.

From these two simple facts the following simple algorithm can be derived.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
3900
3901
3902
3903
3904
3905
3906
3907
3908
3909
3910
3911
3912
3913
3914
3915
\end{center}
\end{small}
\caption{Algorithm Montgomery Reduction}
\end{figure}

The algorithm reduces the input one bit at a time using the two congruencies stated previously.  Inside the loop $n$, which is odd, is
added to $x$ if $x$ is odd.  This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two.  Since
$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$.  Let $r$ represent the 
final result of the Montgomery algorithm.  If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to 
$0 \le r < \lfloor x/2^k \rfloor + n$.  As a result at most a single subtraction is required to get the residue desired.

\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|l|}
\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\







|
|







5949
5950
5951
5952
5953
5954
5955
5956
5957
5958
5959
5960
5961
5962
5963
5964
\end{center}
\end{small}
\caption{Algorithm Montgomery Reduction}
\end{figure}

The algorithm reduces the input one bit at a time using the two congruencies stated previously.  Inside the loop $n$, which is odd, is
added to $x$ if $x$ is odd.  This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two.  Since
$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$.  Let $r$ represent the
final result of the Montgomery algorithm.  If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to
$0 \le r < \lfloor x/2^k \rfloor + n$.  As a result at most a single subtraction is required to get the residue desired.

\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|l|}
\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\
3926
3927
3928
3929
3930
3931
3932
3933
3934
3935
3936
3937
3938
3939
3940
3941
3942
3943
3944
3945
\end{tabular}
\end{center}
\end{small}
\caption{Example of Montgomery Reduction (I)}
\label{fig:MONT1}
\end{figure}

Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 9$ (note $\beta^k = 512$ which is larger than $n$).  The result of 
the algorithm $r = 178$ is congruent to the value of $2^{-9} \cdot 5555 \mbox{ (mod }257\mbox{)}$.  When $r$ is multiplied by $2^9$ modulo $257$ the correct residue 
$r \equiv 158$ is produced.  

Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$.  The current algorithm requires $2k^2$ single precision shifts
and $k^2$ single precision additions.  At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful.  
Fortunately there exists an alternative representation of the algorithm.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Montgomery Reduction} (modified I). \\







|
|
|


|







5975
5976
5977
5978
5979
5980
5981
5982
5983
5984
5985
5986
5987
5988
5989
5990
5991
5992
5993
5994
\end{tabular}
\end{center}
\end{small}
\caption{Example of Montgomery Reduction (I)}
\label{fig:MONT1}
\end{figure}

Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 9$ (note $\beta^k = 512$ which is larger than $n$).  The result of
the algorithm $r = 178$ is congruent to the value of $2^{-9} \cdot 5555 \mbox{ (mod }257\mbox{)}$.  When $r$ is multiplied by $2^9$ modulo $257$ the correct residue
$r \equiv 158$ is produced.

Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$.  The current algorithm requires $2k^2$ single precision shifts
and $k^2$ single precision additions.  At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful.
Fortunately there exists an alternative representation of the algorithm.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Montgomery Reduction} (modified I). \\
3980
3981
3982
3983
3984
3985
3986
3987
3988
3989
3990
3991
3992
3993
3994
3995
3996
3997
\end{tabular}
\end{center}
\end{small}
\caption{Example of Montgomery Reduction (II)}
\label{fig:MONT2}
\end{figure}

Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 9$. 
With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the 
loop.  Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed.  In those iterations the $t$'th bit of $x$ is 
zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero.  

\subsection{Digit Based Montgomery Reduction}
Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis.  Consider the
previous algorithm re-written to compute the Montgomery reduction in this new fashion.

\begin{figure}[!here]
\begin{small}







|
|
|
|







6029
6030
6031
6032
6033
6034
6035
6036
6037
6038
6039
6040
6041
6042
6043
6044
6045
6046
\end{tabular}
\end{center}
\end{small}
\caption{Example of Montgomery Reduction (II)}
\label{fig:MONT2}
\end{figure}

Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 9$.
With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the
loop.  Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed.  In those iterations the $t$'th bit of $x$ is
zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero.

\subsection{Digit Based Montgomery Reduction}
Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis.  Consider the
previous algorithm re-written to compute the Montgomery reduction in this new fashion.

\begin{figure}[!here]
\begin{small}
4007
4008
4009
4010
4011
4012
4013
4014
4015
4016
4017
4018
4019
4020
4021
4022
4023
4024
4025
4026
4027
4028
4029
4030
4031
4032
4033
4034
4035
4036
4037
4038
4039
4040
4041
4042
4043
4044
4045
4046
4047
4048
4049
4050
4051
4052
4053
4054
4055
4056
4057
4058
4059
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Montgomery Reduction (modified II)}
\end{figure}

The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue.  If the first digit of 
the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit.  This
problem breaks down to solving the following congruency.  

\begin{center}
\begin{tabular}{rcl}
$x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\
$\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\
$\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
\end{tabular}
\end{center}

In each iteration of the loop on step 1 a new value of $\mu$ must be calculated.  The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used 
extensively in this algorithm and should be precomputed.  Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$.  

For example, let $\beta = 10$ represent the radix.  Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$.  Let $x = 33$ 
represent the value to reduce.

\newpage\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\
\hline --                 & $33$ & --\\
\hline $0$                 & $33 + \mu n = 50$ & $1$ \\
\hline $1$                 & $50 + \mu n \beta = 900$ & $5$ \\
\hline
\end{tabular}
\end{center}
\caption{Example of Montgomery Reduction}
\end{figure}

The final result $900$ is then divided by $\beta^k$ to produce the final result $9$.  The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ 
which implies the result is not the modular residue of $x$ modulo $n$.  However, recall that the residue is actually multiplied by $\beta^{-k}$ in
the algorithm.  To get the true residue the value must be multiplied by $\beta^k$.  In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and
the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$.  

\subsection{Baseline Montgomery Reduction}
The baseline Montgomery reduction algorithm will produce the residue for any size input.  It is designed to be a catch-all algororithm for 
Montgomery reductions.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_montgomery\_reduce}. \\
\textbf{Input}.   mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\







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\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Montgomery Reduction (modified II)}
\end{figure}

The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue.  If the first digit of
the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit.  This
problem breaks down to solving the following congruency.

\begin{center}
\begin{tabular}{rcl}
$x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\
$\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\
$\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
\end{tabular}
\end{center}

In each iteration of the loop on step 1 a new value of $\mu$ must be calculated.  The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used
extensively in this algorithm and should be precomputed.  Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$.

For example, let $\beta = 10$ represent the radix.  Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$.  Let $x = 33$
represent the value to reduce.

\newpage\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\
\hline --                 & $33$ & --\\
\hline $0$                 & $33 + \mu n = 50$ & $1$ \\
\hline $1$                 & $50 + \mu n \beta = 900$ & $5$ \\
\hline
\end{tabular}
\end{center}
\caption{Example of Montgomery Reduction}
\end{figure}

The final result $900$ is then divided by $\beta^k$ to produce the final result $9$.  The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$
which implies the result is not the modular residue of $x$ modulo $n$.  However, recall that the residue is actually multiplied by $\beta^{-k}$ in
the algorithm.  To get the true residue the value must be multiplied by $\beta^k$.  In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and
the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$.

\subsection{Baseline Montgomery Reduction}
The baseline Montgomery reduction algorithm will produce the residue for any size input.  It is designed to be a catch-all algororithm for
Montgomery reductions.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_montgomery\_reduce}. \\
\textbf{Input}.   mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\
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\end{small}
\caption{Algorithm mp\_montgomery\_reduce}
\end{figure}

\textbf{Algorithm mp\_montgomery\_reduce.}
This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm.  The algorithm is loosely based
on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop.  The
restrictions on this algorithm are fairly easy to adapt to.  First $0 \le x < n^2$ bounds the input to numbers in the same range as 
for the Barrett algorithm.  Additionally if $n > 1$ and $n$ is odd there will exist a modular inverse $\rho$.  $\rho$ must be calculated in
advance of this algorithm.  Finally the variable $k$ is fixed and a pseudonym for $n.used$.  

Step 2 decides whether a faster Montgomery algorithm can be used.  It is based on the Comba technique meaning that there are limits on
the size of the input.  This algorithm is discussed in sub-section 6.3.3.

Step 5 is the main reduction loop of the algorithm.  The value of $\mu$ is calculated once per iteration in the outer loop.  The inner loop
calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits.  Both the addition and
multiplication are performed in the same loop to save time and memory.  Step 5.4 will handle any additional carries that escape the inner loop.

Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications 
in the inner loop.  In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision
multiplications.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_reduce.c
\vspace{-3mm}
\begin{alltt}



































































































\end{alltt}
\end{small}

This is the baseline implementation of the Montgomery reduction algorithm.  Lines 31 to 36 determine if the Comba based
routine can be used instead.  Line 47 computes the value of $\mu$ for that particular iteration of the outer loop.  

The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop.  The alias $tmpx$ refers to the $ix$'th digit of $x$ and
the alias $tmpn$ refers to the modulus $n$.  

\subsection{Faster ``Comba'' Montgomery Reduction}

The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial
nature of the inner loop.  The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba
technique.  The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates
a $k \times 1$ product $k$ times. 

The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$.  This means the 
carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit.  The solution as it turns out is very simple.  
Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry.  

With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases
the speed of the algorithm.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\
\textbf{Input}.   mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\







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\end{small}
\caption{Algorithm mp\_montgomery\_reduce}
\end{figure}

\textbf{Algorithm mp\_montgomery\_reduce.}
This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm.  The algorithm is loosely based
on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop.  The
restrictions on this algorithm are fairly easy to adapt to.  First $0 \le x < n^2$ bounds the input to numbers in the same range as
for the Barrett algorithm.  Additionally if $n > 1$ and $n$ is odd there will exist a modular inverse $\rho$.  $\rho$ must be calculated in
advance of this algorithm.  Finally the variable $k$ is fixed and a pseudonym for $n.used$.

Step 2 decides whether a faster Montgomery algorithm can be used.  It is based on the Comba technique meaning that there are limits on
the size of the input.  This algorithm is discussed in sub-section 6.3.3.

Step 5 is the main reduction loop of the algorithm.  The value of $\mu$ is calculated once per iteration in the outer loop.  The inner loop
calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits.  Both the addition and
multiplication are performed in the same loop to save time and memory.  Step 5.4 will handle any additional carries that escape the inner loop.

Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications
in the inner loop.  In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision
multiplications.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_reduce.c
\vspace{-3mm}
\begin{alltt}
016
017   /* computes xR**-1 == x (mod N) via Montgomery Reduction */
018   int
019   mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
020   \{
021     int     ix, res, digs;
022     mp_digit mu;
023
024     /* can the fast reduction [comba] method be used?
025      *
026      * Note that unlike in mul you're safely allowed *less*
027      * than the available columns [255 per default] since carries
028      * are fixed up in the inner loop.
029      */
030     digs = (n->used * 2) + 1;
031     if ((digs < MP_WARRAY) &&
032         (n->used <
033         (1 << ((CHAR_BIT * sizeof(mp_word)) - (2 * DIGIT_BIT))))) \{
034       return fast_mp_montgomery_reduce (x, n, rho);
035     \}
036
037     /* grow the input as required */
038     if (x->alloc < digs) \{
039       if ((res = mp_grow (x, digs)) != MP_OKAY) \{
040         return res;
041       \}
042     \}
043     x->used = digs;
044
045     for (ix = 0; ix < n->used; ix++) \{
046       /* mu = ai * rho mod b
047        *
048        * The value of rho must be precalculated via
049        * montgomery_setup() such that
050        * it equals -1/n0 mod b this allows the
051        * following inner loop to reduce the
052        * input one digit at a time
053        */
054       mu = (mp_digit) (((mp_word)x->dp[ix] * (mp_word)rho) & MP_MASK);
055
056       /* a = a + mu * m * b**i */
057       \{
058         int iy;
059         mp_digit *tmpn, *tmpx, u;
060         mp_word r;
061
062         /* alias for digits of the modulus */
063         tmpn = n->dp;
064
065         /* alias for the digits of x [the input] */
066         tmpx = x->dp + ix;
067
068         /* set the carry to zero */
069         u = 0;
070
071         /* Multiply and add in place */
072         for (iy = 0; iy < n->used; iy++) \{
073           /* compute product and sum */
074           r       = ((mp_word)mu * (mp_word)*tmpn++) +
075                      (mp_word) u + (mp_word) *tmpx;
076
077           /* get carry */
078           u       = (mp_digit)(r >> ((mp_word) DIGIT_BIT));
079
080           /* fix digit */
081           *tmpx++ = (mp_digit)(r & ((mp_word) MP_MASK));
082         \}
083         /* At this point the ix'th digit of x should be zero */
084
085
086         /* propagate carries upwards as required*/
087         while (u != 0) \{
088           *tmpx   += u;
089           u        = *tmpx >> DIGIT_BIT;
090           *tmpx++ &= MP_MASK;
091         \}
092       \}
093     \}
094
095     /* at this point the n.used'th least
096      * significant digits of x are all zero
097      * which means we can shift x to the
098      * right by n.used digits and the
099      * residue is unchanged.
100      */
101
102     /* x = x/b**n.used */
103     mp_clamp(x);
104     mp_rshd (x, n->used);
105
106     /* if x >= n then x = x - n */
107     if (mp_cmp_mag (x, n) != MP_LT) \{
108       return s_mp_sub (x, n, x);
109     \}
110
111     return MP_OKAY;
112   \}
113   #endif
114
\end{alltt}
\end{small}

This is the baseline implementation of the Montgomery reduction algorithm.  Lines 30 to 35 determine if the Comba based
routine can be used instead.  Line 48 computes the value of $\mu$ for that particular iteration of the outer loop.

The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop.  The alias $tmpx$ refers to the $ix$'th digit of $x$ and
the alias $tmpn$ refers to the modulus $n$.

\subsection{Faster ``Comba'' Montgomery Reduction}

The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial
nature of the inner loop.  The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba
technique.  The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates
a $k \times 1$ product $k$ times.

The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$.  This means the
carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit.  The solution as it turns out is very simple.
Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry.

With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases
the speed of the algorithm.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\
\textbf{Input}.   mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\
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\end{small}
\caption{Algorithm fast\_mp\_montgomery\_reduce}
\end{figure}

\textbf{Algorithm fast\_mp\_montgomery\_reduce.}
This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique.  It is on most computer platforms significantly
faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}).  The algorithm has the same restrictions
on the input as the baseline reduction algorithm.  An additional two restrictions are imposed on this algorithm.  The number of digits $k$ in the 
the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$.   When $\beta = 2^{28}$ this algorithm can be used to reduce modulo
a modulus of at most $3,556$ bits in length.  

As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product.  It is initially filled with the
contents of $x$ with the excess digits zeroed.  The reduction loop is very similar the to the baseline loop at heart.  The multiplication on step
4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$.  Some multipliers such
as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce.  By performing
a single precision multiplication instead half the amount of time is spent.

Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work.  That is what step
4.3 will do.  In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards.  Note
how the upper bits of those same words are not reduced modulo $\beta$.  This is because those values will be discarded shortly and there is no
point.

Step 5 will propagate the remainder of the carries upwards.  On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are
stored in the destination $x$.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_fast\_mp\_montgomery\_reduce.c
\vspace{-3mm}
\begin{alltt}

























































































































































\end{alltt}
\end{small}

The $\hat W$ array is first filled with digits of $x$ on line 48 then the rest of the digits are zeroed on line 55.  Both loops share
the same alias variables to make the code easier to read.  

The value of $\mu$ is calculated in an interesting fashion.  First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit.  This
forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision.   Line 110 fixes the carry 
for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$.

The for loop on line 109 propagates the rest of the carries upwards through the columns.  The for loop on line 126 reduces the columns
modulo $\beta$ and shifts them $k$ places at the same time.  The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th
digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$.  

\subsection{Montgomery Setup}
To calculate the variable $\rho$ a relatively simple algorithm will be required.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_montgomery\_setup}. \\
\textbf{Input}.   mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\
\textbf{Output}.  $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
\hline \\
1.  $b \leftarrow n_0$ \\
2.  If $b$ is even return(\textit{MP\_VAL}) \\
3.  $x \leftarrow (((b + 2) \mbox{ AND } 4) << 1) + b$ \\
4.  for $k$ from 0 to $\lceil lg(lg(\beta)) \rceil - 2$ do \\
\hspace{3mm}4.1  $x \leftarrow x \cdot (2 - bx)$ \\
5.  $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\
6.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_montgomery\_setup} 
\end{figure}

\textbf{Algorithm mp\_montgomery\_setup.}
This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms.  It uses a very interesting trick 
to calculate $1/n_0$ when $\beta$ is a power of two.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_setup.c
\vspace{-3mm}
\begin{alltt}








































\end{alltt}
\end{small}

This source code computes the value of $\rho$ required to perform Montgomery reduction.  It has been modified to avoid performing excess
multiplications when $\beta$ is not the default 28-bits.  

\section{The Diminished Radix Algorithm}
The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett
or Montgomery methods for certain forms of moduli.  The technique is based on the following simple congruence.

\begin{equation}
(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)}
\end{equation}

This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive.  It used the fact that if $n = 2^{31}$ and $k=1$ that 
then a x86 multiplier could produce the 62-bit product and use  the ``shrd'' instruction to perform a double-precision right shift.  The proof
of the above equation is very simple.  First write $x$ in the product form.

\begin{equation}
x = qn + r
\end{equation}

Now reduce both sides modulo $(n - k)$.

\begin{equation}
x \equiv qk + r  \mbox{ (mod }(n-k)\mbox{)}
\end{equation}

The variable $n$ reduces modulo $n - k$ to $k$.  By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ 
into the equation the original congruence is reproduced, thus concluding the proof.  The following algorithm is based on this observation.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Diminished Radix Reduction}. \\







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\end{small}
\caption{Algorithm fast\_mp\_montgomery\_reduce}
\end{figure}

\textbf{Algorithm fast\_mp\_montgomery\_reduce.}
This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique.  It is on most computer platforms significantly
faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}).  The algorithm has the same restrictions
on the input as the baseline reduction algorithm.  An additional two restrictions are imposed on this algorithm.  The number of digits $k$ in the
the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$.   When $\beta = 2^{28}$ this algorithm can be used to reduce modulo
a modulus of at most $3,556$ bits in length.

As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product.  It is initially filled with the
contents of $x$ with the excess digits zeroed.  The reduction loop is very similar the to the baseline loop at heart.  The multiplication on step
4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$.  Some multipliers such
as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce.  By performing
a single precision multiplication instead half the amount of time is spent.

Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work.  That is what step
4.3 will do.  In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards.  Note
how the upper bits of those same words are not reduced modulo $\beta$.  This is because those values will be discarded shortly and there is no
point.

Step 5 will propagate the remainder of the carries upwards.  On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are
stored in the destination $x$.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_fast\_mp\_montgomery\_reduce.c
\vspace{-3mm}
\begin{alltt}
016
017   /* computes xR**-1 == x (mod N) via Montgomery Reduction
018    *
019    * This is an optimized implementation of montgomery_reduce
020    * which uses the comba method to quickly calculate the columns of the
021    * reduction.
022    *
023    * Based on Algorithm 14.32 on pp.601 of HAC.
024   */
025   int fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
026   \{
027     int     ix, res, olduse;
028     mp_word W[MP_WARRAY];
029
030     /* get old used count */
031     olduse = x->used;
032
033     /* grow a as required */
034     if (x->alloc < (n->used + 1)) \{
035       if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) \{
036         return res;
037       \}
038     \}
039
040     /* first we have to get the digits of the input into
041      * an array of double precision words W[...]
042      */
043     \{
044       mp_word *_W;
045       mp_digit *tmpx;
046
047       /* alias for the W[] array */
048       _W   = W;
049
050       /* alias for the digits of  x*/
051       tmpx = x->dp;
052
053       /* copy the digits of a into W[0..a->used-1] */
054       for (ix = 0; ix < x->used; ix++) \{
055         *_W++ = *tmpx++;
056       \}
057
058       /* zero the high words of W[a->used..m->used*2] */
059       for (; ix < ((n->used * 2) + 1); ix++) \{
060         *_W++ = 0;
061       \}
062     \}
063
064     /* now we proceed to zero successive digits
065      * from the least significant upwards
066      */
067     for (ix = 0; ix < n->used; ix++) \{
068       /* mu = ai * m' mod b
069        *
070        * We avoid a double precision multiplication (which isn't required)
071        * by casting the value down to a mp_digit.  Note this requires
072        * that W[ix-1] have  the carry cleared (see after the inner loop)
073        */
074       mp_digit mu;
075       mu = (mp_digit) (((W[ix] & MP_MASK) * rho) & MP_MASK);
076
077       /* a = a + mu * m * b**i
078        *
079        * This is computed in place and on the fly.  The multiplication
080        * by b**i is handled by offseting which columns the results
081        * are added to.
082        *
083        * Note the comba method normally doesn't handle carries in the
084        * inner loop In this case we fix the carry from the previous
085        * column since the Montgomery reduction requires digits of the
086        * result (so far) [see above] to work.  This is
087        * handled by fixing up one carry after the inner loop.  The
088        * carry fixups are done in order so after these loops the
089        * first m->used words of W[] have the carries fixed
090        */
091       \{
092         int iy;
093         mp_digit *tmpn;
094         mp_word *_W;
095
096         /* alias for the digits of the modulus */
097         tmpn = n->dp;
098
099         /* Alias for the columns set by an offset of ix */
100         _W = W + ix;
101
102         /* inner loop */
103         for (iy = 0; iy < n->used; iy++) \{
104             *_W++ += ((mp_word)mu) * ((mp_word)*tmpn++);
105         \}
106       \}
107
108       /* now fix carry for next digit, W[ix+1] */
109       W[ix + 1] += W[ix] >> ((mp_word) DIGIT_BIT);
110     \}
111
112     /* now we have to propagate the carries and
113      * shift the words downward [all those least
114      * significant digits we zeroed].
115      */
116     \{
117       mp_digit *tmpx;
118       mp_word *_W, *_W1;
119
120       /* nox fix rest of carries */
121
122       /* alias for current word */
123       _W1 = W + ix;
124
125       /* alias for next word, where the carry goes */
126       _W = W + ++ix;
127
128       for (; ix <= ((n->used * 2) + 1); ix++) \{
129         *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT);
130       \}
131
132       /* copy out, A = A/b**n
133        *
134        * The result is A/b**n but instead of converting from an
135        * array of mp_word to mp_digit than calling mp_rshd
136        * we just copy them in the right order
137        */
138
139       /* alias for destination word */
140       tmpx = x->dp;
141
142       /* alias for shifted double precision result */
143       _W = W + n->used;
144
145       for (ix = 0; ix < (n->used + 1); ix++) \{
146         *tmpx++ = (mp_digit)(*_W++ & ((mp_word) MP_MASK));
147       \}
148
149       /* zero oldused digits, if the input a was larger than
150        * m->used+1 we'll have to clear the digits
151        */
152       for (; ix < olduse; ix++) \{
153         *tmpx++ = 0;
154       \}
155     \}
156
157     /* set the max used and clamp */
158     x->used = n->used + 1;
159     mp_clamp (x);
160
161     /* if A >= m then A = A - m */
162     if (mp_cmp_mag (x, n) != MP_LT) \{
163       return s_mp_sub (x, n, x);
164     \}
165     return MP_OKAY;
166   \}
167   #endif
168
\end{alltt}
\end{small}

The $\hat W$ array is first filled with digits of $x$ on line 50 then the rest of the digits are zeroed on line 54.  Both loops share
the same alias variables to make the code easier to read.

The value of $\mu$ is calculated in an interesting fashion.  First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit.  This
forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision.   Line 109 fixes the carry
for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$.

The for loop on line 108 propagates the rest of the carries upwards through the columns.  The for loop on line 125 reduces the columns
modulo $\beta$ and shifts them $k$ places at the same time.  The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th
digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$.

\subsection{Montgomery Setup}
To calculate the variable $\rho$ a relatively simple algorithm will be required.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_montgomery\_setup}. \\
\textbf{Input}.   mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\
\textbf{Output}.  $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
\hline \\
1.  $b \leftarrow n_0$ \\
2.  If $b$ is even return(\textit{MP\_VAL}) \\
3.  $x \leftarrow (((b + 2) \mbox{ AND } 4) << 1) + b$ \\
4.  for $k$ from 0 to $\lceil lg(lg(\beta)) \rceil - 2$ do \\
\hspace{3mm}4.1  $x \leftarrow x \cdot (2 - bx)$ \\
5.  $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\
6.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_montgomery\_setup}
\end{figure}

\textbf{Algorithm mp\_montgomery\_setup.}
This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms.  It uses a very interesting trick
to calculate $1/n_0$ when $\beta$ is a power of two.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_setup.c
\vspace{-3mm}
\begin{alltt}
016
017   /* setups the montgomery reduction stuff */
018   int
019   mp_montgomery_setup (mp_int * n, mp_digit * rho)
020   \{
021     mp_digit x, b;
022
023   /* fast inversion mod 2**k
024    *
025    * Based on the fact that
026    *
027    * XA = 1 (mod 2**n)  =>  (X(2-XA)) A = 1 (mod 2**2n)
028    *                    =>  2*X*A - X*X*A*A = 1
029    *                    =>  2*(1) - (1)     = 1
030    */
031     b = n->dp[0];
032
033     if ((b & 1) == 0) \{
034       return MP_VAL;
035     \}
036
037     x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */
038     x *= 2 - (b * x);             /* here x*a==1 mod 2**8 */
039   #if !defined(MP_8BIT)
040     x *= 2 - (b * x);             /* here x*a==1 mod 2**16 */
041   #endif
042   #if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT))
043     x *= 2 - (b * x);             /* here x*a==1 mod 2**32 */
044   #endif
045   #ifdef MP_64BIT
046     x *= 2 - (b * x);             /* here x*a==1 mod 2**64 */
047   #endif
048
049     /* rho = -1/m mod b */
050     *rho = (mp_digit)(((mp_word)1 << ((mp_word) DIGIT_BIT)) - x) & MP_MASK;
051
052     return MP_OKAY;
053   \}
054   #endif
055
\end{alltt}
\end{small}

This source code computes the value of $\rho$ required to perform Montgomery reduction.  It has been modified to avoid performing excess
multiplications when $\beta$ is not the default 28-bits.

\section{The Diminished Radix Algorithm}
The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett
or Montgomery methods for certain forms of moduli.  The technique is based on the following simple congruence.

\begin{equation}
(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)}
\end{equation}

This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive.  It used the fact that if $n = 2^{31}$ and $k=1$ that
then a x86 multiplier could produce the 62-bit product and use  the ``shrd'' instruction to perform a double-precision right shift.  The proof
of the above equation is very simple.  First write $x$ in the product form.

\begin{equation}
x = qn + r
\end{equation}

Now reduce both sides modulo $(n - k)$.

\begin{equation}
x \equiv qk + r  \mbox{ (mod }(n-k)\mbox{)}
\end{equation}

The variable $n$ reduces modulo $n - k$ to $k$.  By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$
into the equation the original congruence is reproduced, thus concluding the proof.  The following algorithm is based on this observation.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Diminished Radix Reduction}. \\
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\caption{Algorithm Diminished Radix Reduction}
\label{fig:DR}
\end{figure}

This algorithm will reduce $x$ modulo $n - k$ and return the residue.  If $0 \le x < (n - k)^2$ then the algorithm will loop almost always
once or twice and occasionally three times.  For simplicity sake the value of $x$ is bounded by the following simple polynomial.

\begin{equation} 
0 \le x < n^2 + k^2 - 2nk
\end{equation}

The true bound is  $0 \le x < (n - k - 1)^2$ but this has quite a few more terms.  The value of $q$ after step 1 is bounded by the following.

\begin{equation}
q < n - 2k - k^2/n
\end{equation}

Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero.  The value of $x$ after step 3 is bounded trivially as
$0 \le x < n$.  By step four the sum $x + q$ is bounded by 

\begin{equation}
0 \le q + x < (k + 1)n - 2k^2 - 1
\end{equation}

With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3.  After the second pass it is highly unlike that the
sum in step 4 will exceed $n - k$.  In practice fewer than three passes of the algorithm are required to reduce virtually every input in the 
range $0 \le x < (n - k - 1)^2$.  

\begin{figure}
\begin{small}
\begin{center}
\begin{tabular}{|l|}
\hline
$x = 123456789, n = 256, k = 3$ \\
\hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\
$q \leftarrow q*k = 1446759$ \\
$x \leftarrow x \mbox{ mod } n = 21$ \\
$x \leftarrow x + q = 1446780$ \\
$x \leftarrow x - (n - k) = 1446527$ \\
\hline 
$q \leftarrow \lfloor x/n \rfloor = 5650$ \\
$q \leftarrow q*k = 16950$ \\
$x \leftarrow x \mbox{ mod } n = 127$ \\
$x \leftarrow x + q = 17077$ \\
$x \leftarrow x - (n - k) = 16824$ \\
\hline 
$q \leftarrow \lfloor x/n \rfloor = 65$ \\
$q \leftarrow q*k = 195$ \\
$x \leftarrow x \mbox{ mod } n = 184$ \\
$x \leftarrow x + q = 379$ \\
$x \leftarrow x - (n - k) = 126$ \\
\hline
\end{tabular}







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\caption{Algorithm Diminished Radix Reduction}
\label{fig:DR}
\end{figure}

This algorithm will reduce $x$ modulo $n - k$ and return the residue.  If $0 \le x < (n - k)^2$ then the algorithm will loop almost always
once or twice and occasionally three times.  For simplicity sake the value of $x$ is bounded by the following simple polynomial.

\begin{equation}
0 \le x < n^2 + k^2 - 2nk
\end{equation}

The true bound is  $0 \le x < (n - k - 1)^2$ but this has quite a few more terms.  The value of $q$ after step 1 is bounded by the following.

\begin{equation}
q < n - 2k - k^2/n
\end{equation}

Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero.  The value of $x$ after step 3 is bounded trivially as
$0 \le x < n$.  By step four the sum $x + q$ is bounded by

\begin{equation}
0 \le q + x < (k + 1)n - 2k^2 - 1
\end{equation}

With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3.  After the second pass it is highly unlike that the
sum in step 4 will exceed $n - k$.  In practice fewer than three passes of the algorithm are required to reduce virtually every input in the
range $0 \le x < (n - k - 1)^2$.

\begin{figure}
\begin{small}
\begin{center}
\begin{tabular}{|l|}
\hline
$x = 123456789, n = 256, k = 3$ \\
\hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\
$q \leftarrow q*k = 1446759$ \\
$x \leftarrow x \mbox{ mod } n = 21$ \\
$x \leftarrow x + q = 1446780$ \\
$x \leftarrow x - (n - k) = 1446527$ \\
\hline
$q \leftarrow \lfloor x/n \rfloor = 5650$ \\
$q \leftarrow q*k = 16950$ \\
$x \leftarrow x \mbox{ mod } n = 127$ \\
$x \leftarrow x + q = 17077$ \\
$x \leftarrow x - (n - k) = 16824$ \\
\hline
$q \leftarrow \lfloor x/n \rfloor = 65$ \\
$q \leftarrow q*k = 195$ \\
$x \leftarrow x \mbox{ mod } n = 184$ \\
$x \leftarrow x + q = 379$ \\
$x \leftarrow x - (n - k) = 126$ \\
\hline
\end{tabular}
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three passes were required to find the residue $x \equiv 126$.


\subsection{Choice of Moduli}
On the surface this algorithm looks like a very expensive algorithm.  It requires a couple of subtractions followed by multiplication and other
modular reductions.  The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen.

Division in general is a very expensive operation to perform.  The one exception is when the division is by a power of the radix of representation used.  
Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right.  Similarly division 
by two (\textit{or powers of two}) is very simple for binary computers to perform.  It would therefore seem logical to choose $n$ of the form $2^p$ 
which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits.  

However, there is one operation related to division of power of twos that is even faster than this.  If $n = \beta^p$ then the division may be 
performed by moving whole digits to the right $p$ places.  In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$.  
Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$.  

Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted
modulus'' will refer to a modulus of the form $2^p - k$.  The word ``restricted'' in this case refers to the fact that it is based on the 
$2^p$ logic except $p$ must be a multiple of $lg(\beta)$.  

\subsection{Choice of $k$}
Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$
in step 2 is the most expensive operation.  Fortunately the choice of $k$ is not terribly limited.  For all intents and purposes it might
as well be a single digit.  The smaller the value of $k$ is the faster the algorithm will be.  

\subsection{Restricted Diminished Radix Reduction}
The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$.  This algorithm can reduce 
an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}.  The implementation
of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition 
of $x$ and $q$.  The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular 
exponentiations are performed.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_dr\_reduce}. \\







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6711
6712
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6747
three passes were required to find the residue $x \equiv 126$.


\subsection{Choice of Moduli}
On the surface this algorithm looks like a very expensive algorithm.  It requires a couple of subtractions followed by multiplication and other
modular reductions.  The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen.

Division in general is a very expensive operation to perform.  The one exception is when the division is by a power of the radix of representation used.
Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right.  Similarly division
by two (\textit{or powers of two}) is very simple for binary computers to perform.  It would therefore seem logical to choose $n$ of the form $2^p$
which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits.

However, there is one operation related to division of power of twos that is even faster than this.  If $n = \beta^p$ then the division may be
performed by moving whole digits to the right $p$ places.  In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$.
Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$.

Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted
modulus'' will refer to a modulus of the form $2^p - k$.  The word ``restricted'' in this case refers to the fact that it is based on the
$2^p$ logic except $p$ must be a multiple of $lg(\beta)$.

\subsection{Choice of $k$}
Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$
in step 2 is the most expensive operation.  Fortunately the choice of $k$ is not terribly limited.  For all intents and purposes it might
as well be a single digit.  The smaller the value of $k$ is the faster the algorithm will be.

\subsection{Restricted Diminished Radix Reduction}
The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$.  This algorithm can reduce
an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}.  The implementation
of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition
of $x$ and $q$.  The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular
exponentiations are performed.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_dr\_reduce}. \\
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4433
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4450
4451
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4454
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4457
4458
4459
4460
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4462
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4464
4465
4466
4467
4468
4469
4470
4471
4472
\end{center}
\end{small}
\caption{Algorithm mp\_dr\_reduce}
\end{figure}

\textbf{Algorithm mp\_dr\_reduce.}
This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$.  It has similar restrictions to that of the Barrett reduction
with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$.  

This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization.  The division by $\beta^m$, multiplication by $k$
and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4.  The division by $\beta^m$ is emulated by accessing
the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position.  After the loop the $m$'th
digit is set to the carry and the upper digits are zeroed.  Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to 
$x$ before the addition of the multiple of the upper half.  

At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required.  First $n$ is subtracted from $x$ and then the algorithm resumes
at step 3.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_reduce.c
\vspace{-3mm}
\begin{alltt}













































































\end{alltt}
\end{small}

The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$.  The label on line 52 is where
the algorithm will resume if further reduction passes are required.  In theory it could be placed at the top of the function however, the size of
the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time.  

The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits.  By reading digits from $x$ offset by $m$ digits
a division by $\beta^m$ can be simulated virtually for free.  The loop on line 64 performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11})
in this algorithm.

By line 67 the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed.  Similarly by line 74 the 
same pointer will point to the $m+1$'th digit where the zeroes will be placed.  

Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required.  
With the same logic at line 81 the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used
as well.  Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code
does not need to be checked.

\subsubsection{Setup}
To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required.  This algorithm is not really complicated but provided for
completeness.








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6769
6770
6771
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6777
6778
6779
6780
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6783
6784
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6802
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6872
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6874
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6876
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6878
6879
6880
6881
6882
6883
6884
6885
6886
6887
6888
6889
6890
\end{center}
\end{small}
\caption{Algorithm mp\_dr\_reduce}
\end{figure}

\textbf{Algorithm mp\_dr\_reduce.}
This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$.  It has similar restrictions to that of the Barrett reduction
with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$.

This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization.  The division by $\beta^m$, multiplication by $k$
and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4.  The division by $\beta^m$ is emulated by accessing
the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position.  After the loop the $m$'th
digit is set to the carry and the upper digits are zeroed.  Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to
$x$ before the addition of the multiple of the upper half.

At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required.  First $n$ is subtracted from $x$ and then the algorithm resumes
at step 3.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_reduce.c
\vspace{-3mm}
\begin{alltt}
016
017   /* reduce "x" in place modulo "n" using the Diminished Radix algorithm.
018    *
019    * Based on algorithm from the paper
020    *
021    * "Generating Efficient Primes for Discrete Log Cryptosystems"
022    *                 Chae Hoon Lim, Pil Joong Lee,
023    *          POSTECH Information Research Laboratories
024    *
025    * The modulus must be of a special format [see manual]
026    *
027    * Has been modified to use algorithm 7.10 from the LTM book instead
028    *
029    * Input x must be in the range 0 <= x <= (n-1)**2
030    */
031   int
032   mp_dr_reduce (mp_int * x, mp_int * n, mp_digit k)
033   \{
034     int      err, i, m;
035     mp_word  r;
036     mp_digit mu, *tmpx1, *tmpx2;
037
038     /* m = digits in modulus */
039     m = n->used;
040
041     /* ensure that "x" has at least 2m digits */
042     if (x->alloc < (m + m)) \{
043       if ((err = mp_grow (x, m + m)) != MP_OKAY) \{
044         return err;
045       \}
046     \}
047
048   /* top of loop, this is where the code resumes if
049    * another reduction pass is required.
050    */
051   top:
052     /* aliases for digits */
053     /* alias for lower half of x */
054     tmpx1 = x->dp;
055
056     /* alias for upper half of x, or x/B**m */
057     tmpx2 = x->dp + m;
058
059     /* set carry to zero */
060     mu = 0;
061
062     /* compute (x mod B**m) + k * [x/B**m] inline and inplace */
063     for (i = 0; i < m; i++) \{
064         r         = (((mp_word)*tmpx2++) * (mp_word)k) + *tmpx1 + mu;
065         *tmpx1++  = (mp_digit)(r & MP_MASK);
066         mu        = (mp_digit)(r >> ((mp_word)DIGIT_BIT));
067     \}
068
069     /* set final carry */
070     *tmpx1++ = mu;
071
072     /* zero words above m */
073     for (i = m + 1; i < x->used; i++) \{
074         *tmpx1++ = 0;
075     \}
076
077     /* clamp, sub and return */
078     mp_clamp (x);
079
080     /* if x >= n then subtract and reduce again
081      * Each successive "recursion" makes the input smaller and smaller.
082      */
083     if (mp_cmp_mag (x, n) != MP_LT) \{
084       if ((err = s_mp_sub(x, n, x)) != MP_OKAY) \{
085         return err;
086       \}
087       goto top;
088     \}
089     return MP_OKAY;
090   \}
091   #endif
092
\end{alltt}
\end{small}

The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$.  The label on line 51 is where
the algorithm will resume if further reduction passes are required.  In theory it could be placed at the top of the function however, the size of
the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time.

The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits.  By reading digits from $x$ offset by $m$ digits
a division by $\beta^m$ can be simulated virtually for free.  The loop on line 63 performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11})
in this algorithm.

By line 70 the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed.  Similarly by line 73 the
same pointer will point to the $m+1$'th digit where the zeroes will be placed.

Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required.
With the same logic at line 84 the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used
as well.  Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code
does not need to be checked.

\subsubsection{Setup}
To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required.  This algorithm is not really complicated but provided for
completeness.

4486
4487
4488
4489
4490
4491
4492













4493
4494
4495
4496
4497
4498
4499
\caption{Algorithm mp\_dr\_setup}
\end{figure}

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_setup.c
\vspace{-3mm}
\begin{alltt}













\end{alltt}
\end{small}

\subsubsection{Modulus Detection}
Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus.  An integer is said to be
of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value.








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>







6904
6905
6906
6907
6908
6909
6910
6911
6912
6913
6914
6915
6916
6917
6918
6919
6920
6921
6922
6923
6924
6925
6926
6927
6928
6929
6930
\caption{Algorithm mp\_dr\_setup}
\end{figure}

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_setup.c
\vspace{-3mm}
\begin{alltt}
016
017   /* determines the setup value */
018   void mp_dr_setup(mp_int *a, mp_digit *d)
019   \{
020      /* the casts are required if DIGIT_BIT is one less than
021       * the number of bits in a mp_digit [e.g. DIGIT_BIT==31]
022       */
023      *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) -
024           ((mp_word)a->dp[0]));
025   \}
026
027   #endif
028
\end{alltt}
\end{small}

\subsubsection{Modulus Detection}
Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus.  An integer is said to be
of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value.

4521
4522
4523
4524
4525
4526
4527
























4528
4529
4530
4531
4532
4533
4534
in the mp\_int.  Step 2 tests all but the first digit to see if they are equal to $\beta - 1$.  If the algorithm manages to get to
step 3 then $n$ must be of Diminished Radix form.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_is\_modulus.c
\vspace{-3mm}
\begin{alltt}
























\end{alltt}
\end{small}

\subsection{Unrestricted Diminished Radix Reduction}
The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$.  This algorithm
is a straightforward adaptation of algorithm~\ref{fig:DR}.








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6952
6953
6954
6955
6956
6957
6958
6959
6960
6961
6962
6963
6964
6965
6966
6967
6968
6969
6970
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6972
6973
6974
6975
6976
6977
6978
6979
6980
6981
6982
6983
6984
6985
6986
6987
6988
6989
in the mp\_int.  Step 2 tests all but the first digit to see if they are equal to $\beta - 1$.  If the algorithm manages to get to
step 3 then $n$ must be of Diminished Radix form.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_is\_modulus.c
\vspace{-3mm}
\begin{alltt}
016
017   /* determines if a number is a valid DR modulus */
018   int mp_dr_is_modulus(mp_int *a)
019   \{
020      int ix;
021
022      /* must be at least two digits */
023      if (a->used < 2) \{
024         return 0;
025      \}
026
027      /* must be of the form b**k - a [a <= b] so all
028       * but the first digit must be equal to -1 (mod b).
029       */
030      for (ix = 1; ix < a->used; ix++) \{
031          if (a->dp[ix] != MP_MASK) \{
032             return 0;
033          \}
034      \}
035      return 1;
036   \}
037
038   #endif
039
\end{alltt}
\end{small}

\subsection{Unrestricted Diminished Radix Reduction}
The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$.  This algorithm
is a straightforward adaptation of algorithm~\ref{fig:DR}.

4558
4559
4560
4561
4562
4563
4564
4565
4566
4567
4568
4569
4570












































4571
4572
4573
4574
4575
4576
4577
4578
4579
4580
4581
4582
4583
4584
4585
4586
4587
4588
4589
4590
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_2k}
\end{figure}

\textbf{Algorithm mp\_reduce\_2k.}
This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$.  Division by $2^p$ is emulated with a right
shift which makes the algorithm fairly inexpensive to use.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k.c
\vspace{-3mm}
\begin{alltt}












































\end{alltt}
\end{small}

The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$.  The call to mp\_div\_2d
on line 31 calculates both the quotient $q$ and the remainder $a$ required.  By doing both in a single function call the code size
is kept fairly small.  The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without
any multiplications.  

The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are 
positive.  By using the unsigned versions the overhead is kept to a minimum.  

\subsubsection{Unrestricted Setup}
To setup this reduction algorithm the value of $k = 2^p - n$ is required.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\
\textbf{Input}.   mp\_int $n$   \\







|





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7013
7014
7015
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7089
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_2k}
\end{figure}

\textbf{Algorithm mp\_reduce\_2k.}
This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$.  Division by $2^p$ is emulated with a right
shift which makes the algorithm fairly inexpensive to use.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k.c
\vspace{-3mm}
\begin{alltt}
016
017   /* reduces a modulo n where n is of the form 2**p - d */
018   int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d)
019   \{
020      mp_int q;
021      int    p, res;
022
023      if ((res = mp_init(&q)) != MP_OKAY) \{
024         return res;
025      \}
026
027      p = mp_count_bits(n);
028   top:
029      /* q = a/2**p, a = a mod 2**p */
030      if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) \{
031         goto ERR;
032      \}
033
034      if (d != 1) \{
035         /* q = q * d */
036         if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) \{
037            goto ERR;
038         \}
039      \}
040
041      /* a = a + q */
042      if ((res = s_mp_add(a, &q, a)) != MP_OKAY) \{
043         goto ERR;
044      \}
045
046      if (mp_cmp_mag(a, n) != MP_LT) \{
047         if ((res = s_mp_sub(a, n, a)) != MP_OKAY) \{
048            goto ERR;
049         \}
050         goto top;
051      \}
052
053   ERR:
054      mp_clear(&q);
055      return res;
056   \}
057
058   #endif
059
\end{alltt}
\end{small}

The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$.  The call to mp\_div\_2d
on line 30 calculates both the quotient $q$ and the remainder $a$ required.  By doing both in a single function call the code size
is kept fairly small.  The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without
any multiplications.

The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are
positive.  By using the unsigned versions the overhead is kept to a minimum.

\subsubsection{Unrestricted Setup}
To setup this reduction algorithm the value of $k = 2^p - n$ is required.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\
\textbf{Input}.   mp\_int $n$   \\
4600
4601
4602
4603
4604
4605
4606
4607
4608
4609
4610
4611
4612




























4613
4614
4615
4616
4617
4618
4619
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_2k\_setup}
\end{figure}

\textbf{Algorithm mp\_reduce\_2k\_setup.}
This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k.  By making a temporary variable $x$ equal to $2^p$ a subtraction
is sufficient to solve for $k$.  Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k\_setup.c
\vspace{-3mm}
\begin{alltt}




























\end{alltt}
\end{small}

\subsubsection{Unrestricted Detection}
An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true.

\begin{enumerate}







|





>
>
>
>
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>
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>
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>
>
>
>
>
>
>
>
>
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>
>
>
>
>
>
>







7099
7100
7101
7102
7103
7104
7105
7106
7107
7108
7109
7110
7111
7112
7113
7114
7115
7116
7117
7118
7119
7120
7121
7122
7123
7124
7125
7126
7127
7128
7129
7130
7131
7132
7133
7134
7135
7136
7137
7138
7139
7140
7141
7142
7143
7144
7145
7146
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_2k\_setup}
\end{figure}

\textbf{Algorithm mp\_reduce\_2k\_setup.}
This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k.  By making a temporary variable $x$ equal to $2^p$ a subtraction
is sufficient to solve for $k$.  Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k\_setup.c
\vspace{-3mm}
\begin{alltt}
016
017   /* determines the setup value */
018   int mp_reduce_2k_setup(mp_int *a, mp_digit *d)
019   \{
020      int res, p;
021      mp_int tmp;
022
023      if ((res = mp_init(&tmp)) != MP_OKAY) \{
024         return res;
025      \}
026
027      p = mp_count_bits(a);
028      if ((res = mp_2expt(&tmp, p)) != MP_OKAY) \{
029         mp_clear(&tmp);
030         return res;
031      \}
032
033      if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) \{
034         mp_clear(&tmp);
035         return res;
036      \}
037
038      *d = tmp.dp[0];
039      mp_clear(&tmp);
040      return MP_OKAY;
041   \}
042   #endif
043
\end{alltt}
\end{small}

\subsubsection{Unrestricted Detection}
An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true.

\begin{enumerate}
4644
4645
4646
4647
4648
4649
4650
4651
4652
4653
4654
4655
4656

































4657
4658
4659
4660
4661
4662
4663
4664
4665
4666
4667
4668
4669
4670
4671
4672
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_is\_2k}
\end{figure}

\textbf{Algorithm mp\_reduce\_is\_2k.}
This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_is\_2k.c
\vspace{-3mm}
\begin{alltt}

































\end{alltt}
\end{small}



\section{Algorithm Comparison}
So far three very different algorithms for modular reduction have been discussed.  Each of the algorithms have their own strengths and weaknesses
that makes having such a selection very useful.  The following table sumarizes the three algorithms along with comparisons of work factors.  Since
all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table.  

\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\
\hline Barrett    & $m^2 + 2m - 1$ & None              & $79$ & $1087$ & $4223$ \\
\hline Montgomery & $m^2 + m$      & $n$ must be odd   & $72$ & $1056$ & $4160$ \\







|





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|







7171
7172
7173
7174
7175
7176
7177
7178
7179
7180
7181
7182
7183
7184
7185
7186
7187
7188
7189
7190
7191
7192
7193
7194
7195
7196
7197
7198
7199
7200
7201
7202
7203
7204
7205
7206
7207
7208
7209
7210
7211
7212
7213
7214
7215
7216
7217
7218
7219
7220
7221
7222
7223
7224
7225
7226
7227
7228
7229
7230
7231
7232
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_is\_2k}
\end{figure}

\textbf{Algorithm mp\_reduce\_is\_2k.}
This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_is\_2k.c
\vspace{-3mm}
\begin{alltt}
016
017   /* determines if mp_reduce_2k can be used */
018   int mp_reduce_is_2k(mp_int *a)
019   \{
020      int ix, iy, iw;
021      mp_digit iz;
022
023      if (a->used == 0) \{
024         return MP_NO;
025      \} else if (a->used == 1) \{
026         return MP_YES;
027      \} else if (a->used > 1) \{
028         iy = mp_count_bits(a);
029         iz = 1;
030         iw = 1;
031
032         /* Test every bit from the second digit up, must be 1 */
033         for (ix = DIGIT_BIT; ix < iy; ix++) \{
034             if ((a->dp[iw] & iz) == 0) \{
035                return MP_NO;
036             \}
037             iz <<= 1;
038             if (iz > (mp_digit)MP_MASK) \{
039                ++iw;
040                iz = 1;
041             \}
042         \}
043      \}
044      return MP_YES;
045   \}
046
047   #endif
048
\end{alltt}
\end{small}



\section{Algorithm Comparison}
So far three very different algorithms for modular reduction have been discussed.  Each of the algorithms have their own strengths and weaknesses
that makes having such a selection very useful.  The following table sumarizes the three algorithms along with comparisons of work factors.  Since
all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table.

\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\
\hline Barrett    & $m^2 + 2m - 1$ & None              & $79$ & $1087$ & $4223$ \\
\hline Montgomery & $m^2 + m$      & $n$ must be odd   & $72$ & $1056$ & $4160$ \\
4694
4695
4696
4697
4698
4699
4700
4701
4702
4703
4704
4705
4706
4707
4708
4709
4710
4711
4712
4713
4714
4715
4716
4717
4718
4719
4720
4721
4722
4723
4724
4725
4726
4727
4728
4729
4730
4731
4732
4733
4734
4735
4736
4737
4738
4739
4740
                     & \\
$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly.  \\
                     & \\
$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\
                     & (\textit{figure~\ref{fig:DR}}) terminates.  Also prove the probability that it will \\
                     & terminate within $1 \le k \le 10$ iterations. \\
                     & \\
\end{tabular}                     


\chapter{Exponentiation}
Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$.  A variant of exponentiation, computed
in a finite field or ring, is called modular exponentiation.  This latter style of operation is typically used in public key 
cryptosystems such as RSA and Diffie-Hellman.  The ability to quickly compute modular exponentiations is of great benefit to any
such cryptosystem and many methods have been sought to speed it up.

\section{Exponentiation Basics}
A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired.  However, as $b$ grows in size
the number of multiplications becomes prohibitive.  Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature
with a $1024$-bit key.  Such a calculation could never be completed as it would take simply far too long.

Fortunately there is a very simple algorithm based on the laws of exponents.  Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which
are two trivial relationships between the base and the exponent.  Let $b_i$ represent the $i$'th bit of $b$ starting from the least 
significant bit.  If $b$ is a $k$-bit integer than the following equation is true.

\begin{equation}
a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i}
\end{equation}

By taking the base $a$ logarithm of both sides of the equation the following equation is the result.

\begin{equation}
b = \sum_{i=0}^{k-1}2^i \cdot b_i
\end{equation}

The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to
$a^{2^{i+1}}$.  This observation forms the basis of essentially all fast exponentiation algorithms.  It requires $k$ squarings and on average
$k \over 2$ multiplications to compute the result.  This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times.

While this current method is a considerable speed up there are further improvements to be made.  For example, the $a^{2^i}$ term does not need to 
be computed in an auxilary variable.  Consider the following equivalent algorithm.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Left to Right Exponentiation}. \\







|




|









|
















|







7254
7255
7256
7257
7258
7259
7260
7261
7262
7263
7264
7265
7266
7267
7268
7269
7270
7271
7272
7273
7274
7275
7276
7277
7278
7279
7280
7281
7282
7283
7284
7285
7286
7287
7288
7289
7290
7291
7292
7293
7294
7295
7296
7297
7298
7299
7300
                     & \\
$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly.  \\
                     & \\
$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\
                     & (\textit{figure~\ref{fig:DR}}) terminates.  Also prove the probability that it will \\
                     & terminate within $1 \le k \le 10$ iterations. \\
                     & \\
\end{tabular}


\chapter{Exponentiation}
Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$.  A variant of exponentiation, computed
in a finite field or ring, is called modular exponentiation.  This latter style of operation is typically used in public key
cryptosystems such as RSA and Diffie-Hellman.  The ability to quickly compute modular exponentiations is of great benefit to any
such cryptosystem and many methods have been sought to speed it up.

\section{Exponentiation Basics}
A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired.  However, as $b$ grows in size
the number of multiplications becomes prohibitive.  Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature
with a $1024$-bit key.  Such a calculation could never be completed as it would take simply far too long.

Fortunately there is a very simple algorithm based on the laws of exponents.  Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which
are two trivial relationships between the base and the exponent.  Let $b_i$ represent the $i$'th bit of $b$ starting from the least
significant bit.  If $b$ is a $k$-bit integer than the following equation is true.

\begin{equation}
a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i}
\end{equation}

By taking the base $a$ logarithm of both sides of the equation the following equation is the result.

\begin{equation}
b = \sum_{i=0}^{k-1}2^i \cdot b_i
\end{equation}

The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to
$a^{2^{i+1}}$.  This observation forms the basis of essentially all fast exponentiation algorithms.  It requires $k$ squarings and on average
$k \over 2$ multiplications to compute the result.  This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times.

While this current method is a considerable speed up there are further improvements to be made.  For example, the $a^{2^i}$ term does not need to
be computed in an auxilary variable.  Consider the following equivalent algorithm.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Left to Right Exponentiation}. \\
4752
4753
4754
4755
4756
4757
4758
4759
4760
4761
4762
4763
4764
4765
4766
4767
4768
4769
4770
4771
4772
4773
4774
4775
4776
4777
4778
4779
4780
4781
4782
4783
4784
4785
4786
4787
4788
4789
4790
4791
4792
4793
\end{small}
\caption{Left to Right Exponentiation}
\label{fig:LTOR}
\end{figure}

This algorithm starts from the most significant bit and works towards the least significant bit.  When the $i$'th bit of $b$ is set $a$ is
multiplied against the current product.  In each iteration the product is squared which doubles the exponent of the individual terms of the
product.  

For example, let $b = 101100_2 \equiv 44_{10}$.  The following chart demonstrates the actions of the algorithm.

\newpage\begin{figure}
\begin{center}
\begin{tabular}{|c|c|}
\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\
\hline - & $1$ \\
\hline $5$ & $a$ \\
\hline $4$ & $a^2$ \\
\hline $3$ & $a^4 \cdot a$ \\
\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\
\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\
\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\
\hline
\end{tabular}
\end{center}
\caption{Example of Left to Right Exponentiation}
\end{figure}

When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation.  This particular algorithm is 
called ``Left to Right'' because it reads the exponent in that order.  All of the exponentiation algorithms that will be presented are of this nature.  

\subsection{Single Digit Exponentiation}
The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit.  It is intended 
to be used when a small power of an input is required (\textit{e.g. $a^5$}).  It is faster than simply multiplying $b - 1$ times for all values of 
$b$ that are greater than three.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_expt\_d}. \\
\textbf{Input}.   mp\_int $a$ and mp\_digit $b$ \\







|




















|
|


|
|
|







7312
7313
7314
7315
7316
7317
7318
7319
7320
7321
7322
7323
7324
7325
7326
7327
7328
7329
7330
7331
7332
7333
7334
7335
7336
7337
7338
7339
7340
7341
7342
7343
7344
7345
7346
7347
7348
7349
7350
7351
7352
7353
\end{small}
\caption{Left to Right Exponentiation}
\label{fig:LTOR}
\end{figure}

This algorithm starts from the most significant bit and works towards the least significant bit.  When the $i$'th bit of $b$ is set $a$ is
multiplied against the current product.  In each iteration the product is squared which doubles the exponent of the individual terms of the
product.

For example, let $b = 101100_2 \equiv 44_{10}$.  The following chart demonstrates the actions of the algorithm.

\newpage\begin{figure}
\begin{center}
\begin{tabular}{|c|c|}
\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\
\hline - & $1$ \\
\hline $5$ & $a$ \\
\hline $4$ & $a^2$ \\
\hline $3$ & $a^4 \cdot a$ \\
\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\
\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\
\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\
\hline
\end{tabular}
\end{center}
\caption{Example of Left to Right Exponentiation}
\end{figure}

When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation.  This particular algorithm is
called ``Left to Right'' because it reads the exponent in that order.  All of the exponentiation algorithms that will be presented are of this nature.

\subsection{Single Digit Exponentiation}
The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit.  It is intended
to be used when a small power of an input is required (\textit{e.g. $a^5$}).  It is faster than simply multiplying $b - 1$ times for all values of
$b$ that are greater than three.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_expt\_d}. \\
\textbf{Input}.   mp\_int $a$ and mp\_digit $b$ \\
4807
4808
4809
4810
4811
4812
4813
4814
4815
4816
4817
4818
4819
4820
4821
4822
4823
4824
4825
4826
4827
4828
































































4829
4830
4831
4832
4833
4834
4835
4836
4837
4838
4839
4840
4841
4842
\end{center}
\end{small}
\caption{Algorithm mp\_expt\_d}
\end{figure}

\textbf{Algorithm mp\_expt\_d.}
This algorithm computes the value of $a$ raised to the power of a single digit $b$.  It uses the left to right exponentiation algorithm to
quickly compute the exponentiation.  It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the 
exponent is a fixed width.  

A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$.  The result is set to the initial value of 
$1$ in the subsequent step.

Inside the loop the exponent is read from the most significant bit first down to the least significant bit.  First $c$ is invariably squared
on step 3.1.  In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$.  The value
of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit.  In effect each
iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_expt\_d.c
\vspace{-3mm}
\begin{alltt}
































































\end{alltt}
\end{small}

Line 29 sets the initial value of the result to $1$.  Next the loop on line 31 steps through each bit of the exponent starting from
the most significant down towards the least significant. The invariant squaring operation placed on line 33 is performed first.  After 
the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set.  The shift on line
47 moves all of the bits of the exponent upwards towards the most significant location.  

\section{$k$-ary Exponentiation}
When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor
slower than squaring.  Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$.  Suppose instead it referred to
the $i$'th $k$-bit digit of the exponent of $b$.  For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY}
computes the same exponentiation.  A group of $k$ bits from the exponent is called a \textit{window}.  That is it is a small window on only a
portion of the entire exponent.  Consider the following modification to the basic left to right exponentiation algorithm.







|
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7367
7368
7369
7370
7371
7372
7373
7374
7375
7376
7377
7378
7379
7380
7381
7382
7383
7384
7385
7386
7387
7388
7389
7390
7391
7392
7393
7394
7395
7396
7397
7398
7399
7400
7401
7402
7403
7404
7405
7406
7407
7408
7409
7410
7411
7412
7413
7414
7415
7416
7417
7418
7419
7420
7421
7422
7423
7424
7425
7426
7427
7428
7429
7430
7431
7432
7433
7434
7435
7436
7437
7438
7439
7440
7441
7442
7443
7444
7445
7446
7447
7448
7449
7450
7451
7452
7453
7454
7455
7456
7457
7458
7459
7460
7461
7462
7463
7464
7465
7466
\end{center}
\end{small}
\caption{Algorithm mp\_expt\_d}
\end{figure}

\textbf{Algorithm mp\_expt\_d.}
This algorithm computes the value of $a$ raised to the power of a single digit $b$.  It uses the left to right exponentiation algorithm to
quickly compute the exponentiation.  It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the
exponent is a fixed width.

A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$.  The result is set to the initial value of
$1$ in the subsequent step.

Inside the loop the exponent is read from the most significant bit first down to the least significant bit.  First $c$ is invariably squared
on step 3.1.  In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$.  The value
of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit.  In effect each
iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_expt\_d\_ex.c
\vspace{-3mm}
\begin{alltt}
016
017   /* calculate c = a**b  using a square-multiply algorithm */
018   int mp_expt_d_ex (mp_int * a, mp_digit b, mp_int * c, int fast)
019   \{
020     int     res;
021     unsigned int x;
022
023     mp_int  g;
024
025     if ((res = mp_init_copy (&g, a)) != MP_OKAY) \{
026       return res;
027     \}
028
029     /* set initial result */
030     mp_set (c, 1);
031
032     if (fast != 0) \{
033       while (b > 0) \{
034         /* if the bit is set multiply */
035         if ((b & 1) != 0) \{
036           if ((res = mp_mul (c, &g, c)) != MP_OKAY) \{
037             mp_clear (&g);
038             return res;
039           \}
040         \}
041
042         /* square */
043         if (b > 1) \{
044           if ((res = mp_sqr (&g, &g)) != MP_OKAY) \{
045             mp_clear (&g);
046             return res;
047           \}
048         \}
049
050         /* shift to next bit */
051         b >>= 1;
052       \}
053     \}
054     else \{
055       for (x = 0; x < DIGIT_BIT; x++) \{
056         /* square */
057         if ((res = mp_sqr (c, c)) != MP_OKAY) \{
058           mp_clear (&g);
059           return res;
060         \}
061
062         /* if the bit is set multiply */
063         if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) \{
064           if ((res = mp_mul (c, &g, c)) != MP_OKAY) \{
065              mp_clear (&g);
066              return res;
067           \}
068         \}
069
070         /* shift to next bit */
071         b <<= 1;
072       \}
073     \} /* if ... else */
074
075     mp_clear (&g);
076     return MP_OKAY;
077   \}
078   #endif
079
\end{alltt}
\end{small}

This describes only the algorithm that is used when the parameter $fast$ is $0$.  Line 30 sets the initial value of the result to $1$.  Next the loop on line 55 steps through each bit of the exponent starting from
the most significant down towards the least significant. The invariant squaring operation placed on line 57 is performed first.  After
the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set.  The shift on line
71 moves all of the bits of the exponent upwards towards the most significant location.

\section{$k$-ary Exponentiation}
When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor
slower than squaring.  Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$.  Suppose instead it referred to
the $i$'th $k$-bit digit of the exponent of $b$.  For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY}
computes the same exponentiation.  A group of $k$ bits from the exponent is called a \textit{window}.  That is it is a small window on only a
portion of the entire exponent.  Consider the following modification to the basic left to right exponentiation algorithm.
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4875
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\end{small}
\caption{$k$-ary Exponentiation}
\label{fig:KARY}
\end{figure}

The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times.  If the values of $a^g$ for $0 < g < 2^k$ have been
precomputed this algorithm requires only $t$ multiplications and $tk$ squarings.  The table can be generated with $2^{k - 1} - 1$ squarings and
$2^{k - 1} + 1$ multiplications.  This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$.  
However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}.

Suppose $k = 4$ and $t = 100$.  This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation.  The
original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value.  The total number of squarings
has increased slightly but the number of multiplications has nearly halved.

\subsection{Optimal Values of $k$}
An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$.  The simplest
approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result.  Table~\ref{fig:OPTK} lists optimal values of $k$
for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}.  

\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\
\hline $16$ & $2$ & $27$ & $24$ \\







|









|







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\end{small}
\caption{$k$-ary Exponentiation}
\label{fig:KARY}
\end{figure}

The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times.  If the values of $a^g$ for $0 < g < 2^k$ have been
precomputed this algorithm requires only $t$ multiplications and $tk$ squarings.  The table can be generated with $2^{k - 1} - 1$ squarings and
$2^{k - 1} + 1$ multiplications.  This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$.
However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}.

Suppose $k = 4$ and $t = 100$.  This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation.  The
original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value.  The total number of squarings
has increased slightly but the number of multiplications has nearly halved.

\subsection{Optimal Values of $k$}
An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$.  The simplest
approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result.  Table~\ref{fig:OPTK} lists optimal values of $k$
for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}.

\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\
\hline $16$ & $2$ & $27$ & $24$ \\
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4902
4903
4904
4905
4906
4907
4908
4909
4910
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4912
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\end{center}
\caption{Optimal Values of $k$ for $k$-ary Exponentiation}
\label{fig:OPTK}
\end{figure}

\subsection{Sliding-Window Exponentiation}
A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$.  Essentially
this is a table for all values of $g$ where the most significant bit of $g$ is a one.  However, in order for this to be allowed in the 
algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided.  

Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}.  

\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\
\hline $16$ & $3$ & $24$ & $27$ \\







|
|

|







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\end{center}
\caption{Optimal Values of $k$ for $k$-ary Exponentiation}
\label{fig:OPTK}
\end{figure}

\subsection{Sliding-Window Exponentiation}
A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$.  Essentially
this is a table for all values of $g$ where the most significant bit of $g$ is a one.  However, in order for this to be allowed in the
algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided.

Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm {\ref{fig:KARY}}.

\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\
\hline $16$ & $3$ & $24$ & $27$ \\
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4985
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\end{center}
\end{small}
\caption{Sliding Window $k$-ary Exponentiation}
\end{figure}

Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent.  While this
algorithm requires the same number of squarings it can potentially have fewer multiplications.  The pre-computed table $a^g$ is also half
the size as the previous table.  

Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms.  The first algorithm will divide the exponent up as 
the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$.  The second algorithm will break the 
exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$.  The single digit $0$ in the second representation are where
a single squaring took place instead of a squaring and multiplication.  In total the first method requires $10$ multiplications and $18$ 
squarings.  The second method requires $8$ multiplications and $18$ squarings.  

In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster.  

\section{Modular Exponentiation}

Modular exponentiation is essentially computing the power of a base within a finite field or ring.  For example, computing 
$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation.  Instead of first computing $a^b$ and then reducing it 
modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation.  

This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using
one of the algorithms presented in chapter six.  

Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first.  This algorithm
will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The
value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}).  If no inverse exists the algorithm
terminates with an error.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_exptmod}. \\
\textbf{Input}.   mp\_int $a$, $b$ and $c$ \\







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\end{center}
\end{small}
\caption{Sliding Window $k$-ary Exponentiation}
\end{figure}

Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent.  While this
algorithm requires the same number of squarings it can potentially have fewer multiplications.  The pre-computed table $a^g$ is also half
the size as the previous table.

Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms.  The first algorithm will divide the exponent up as
the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$.  The second algorithm will break the
exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$.  The single digit $0$ in the second representation are where
a single squaring took place instead of a squaring and multiplication.  In total the first method requires $10$ multiplications and $18$
squarings.  The second method requires $8$ multiplications and $18$ squarings.

In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster.

\section{Modular Exponentiation}

Modular exponentiation is essentially computing the power of a base within a finite field or ring.  For example, computing
$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation.  Instead of first computing $a^b$ and then reducing it
modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation.

This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using
one of the algorithms presented in chapter six.

Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first.  This algorithm
will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The
value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}).  If no inverse exists the algorithm
terminates with an error.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_exptmod}. \\
\textbf{Input}.   mp\_int $a$, $b$ and $c$ \\
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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_exptmod}
\end{figure}

\textbf{Algorithm mp\_exptmod.}
The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod.  It is a sliding window $k$-ary algorithm 
which uses Barrett reduction to reduce the product modulo $p$.  The second algorithm mp\_exptmod\_fast performs the same operation 
except it uses either Montgomery or Diminished Radix reduction.  The two latter reduction algorithms are clumped in the same exponentiation
algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}).  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_exptmod.c
\vspace{-3mm}
\begin{alltt}






























































































\end{alltt}
\end{small}

In order to keep the algorithms in a known state the first step on line 29 is to reject any negative modulus as input.  If the exponent is
negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$.  The temporary variable $tmpG$ is assigned
the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$.  The algorithm will recuse with these new values with a positive
exponent.

If the exponent is positive the algorithm resumes the exponentiation.  Line 77 determines if the modulus is of the restricted Diminished Radix 
form.  If it is not line 70 attempts to determine if it is of a unrestricted Diminished Radix form.  The integer $dr$ will take on one
of three values.

\begin{enumerate}
\item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form.
\item $dr = 1$ means that the modulus is of restricted Diminished Radix form.
\item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form.
\end{enumerate}

Line 69 determines if the fast modular exponentiation algorithm can be used.  It is allowed if $dr \ne 0$ or if the modulus is odd.  Otherwise,
the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction.  

\subsection{Barrett Modular Exponentiation}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}







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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_exptmod}
\end{figure}

\textbf{Algorithm mp\_exptmod.}
The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod.  It is a sliding window $k$-ary algorithm
which uses Barrett reduction to reduce the product modulo $p$.  The second algorithm mp\_exptmod\_fast performs the same operation
except it uses either Montgomery or Diminished Radix reduction.  The two latter reduction algorithms are clumped in the same exponentiation
algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}).

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_exptmod.c
\vspace{-3mm}
\begin{alltt}
016
017
018   /* this is a shell function that calls either the normal or Montgomery
019    * exptmod functions.  Originally the call to the montgomery code was
020    * embedded in the normal function but that wasted alot of stack space
021    * for nothing (since 99% of the time the Montgomery code would be called)
022    */
023   int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
024   \{
025     int dr;
026
027     /* modulus P must be positive */
028     if (P->sign == MP_NEG) \{
029        return MP_VAL;
030     \}
031
032     /* if exponent X is negative we have to recurse */
033     if (X->sign == MP_NEG) \{
034   #ifdef BN_MP_INVMOD_C
035        mp_int tmpG, tmpX;
036        int err;
037
038        /* first compute 1/G mod P */
039        if ((err = mp_init(&tmpG)) != MP_OKAY) \{
040           return err;
041        \}
042        if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) \{
043           mp_clear(&tmpG);
044           return err;
045        \}
046
047        /* now get |X| */
048        if ((err = mp_init(&tmpX)) != MP_OKAY) \{
049           mp_clear(&tmpG);
050           return err;
051        \}
052        if ((err = mp_abs(X, &tmpX)) != MP_OKAY) \{
053           mp_clear_multi(&tmpG, &tmpX, NULL);
054           return err;
055        \}
056
057        /* and now compute (1/G)**|X| instead of G**X [X < 0] */
058        err = mp_exptmod(&tmpG, &tmpX, P, Y);
059        mp_clear_multi(&tmpG, &tmpX, NULL);
060        return err;
061   #else
062        /* no invmod */
063        return MP_VAL;
064   #endif
065     \}
066
067   /* modified diminished radix reduction */
068   #if defined(BN_MP_REDUCE_IS_2K_L_C) && defined(BN_MP_REDUCE_2K_L_C) && defin
      ed(BN_S_MP_EXPTMOD_C)
069     if (mp_reduce_is_2k_l(P) == MP_YES) \{
070        return s_mp_exptmod(G, X, P, Y, 1);
071     \}
072   #endif
073
074   #ifdef BN_MP_DR_IS_MODULUS_C
075     /* is it a DR modulus? */
076     dr = mp_dr_is_modulus(P);
077   #else
078     /* default to no */
079     dr = 0;
080   #endif
081
082   #ifdef BN_MP_REDUCE_IS_2K_C
083     /* if not, is it a unrestricted DR modulus? */
084     if (dr == 0) \{
085        dr = mp_reduce_is_2k(P) << 1;
086     \}
087   #endif
088
089     /* if the modulus is odd or dr != 0 use the montgomery method */
090   #ifdef BN_MP_EXPTMOD_FAST_C
091     if ((mp_isodd (P) == MP_YES) || (dr !=  0)) \{
092       return mp_exptmod_fast (G, X, P, Y, dr);
093     \} else \{
094   #endif
095   #ifdef BN_S_MP_EXPTMOD_C
096       /* otherwise use the generic Barrett reduction technique */
097       return s_mp_exptmod (G, X, P, Y, 0);
098   #else
099       /* no exptmod for evens */
100       return MP_VAL;
101   #endif
102   #ifdef BN_MP_EXPTMOD_FAST_C
103     \}
104   #endif
105   \}
106
107   #endif
108
\end{alltt}
\end{small}

In order to keep the algorithms in a known state the first step on line 28 is to reject any negative modulus as input.  If the exponent is
negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$.  The temporary variable $tmpG$ is assigned
the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$.  The algorithm will recuse with these new values with a positive
exponent.

If the exponent is positive the algorithm resumes the exponentiation.  Line 76 determines if the modulus is of the restricted Diminished Radix
form.  If it is not line 69 attempts to determine if it is of a unrestricted Diminished Radix form.  The integer $dr$ will take on one
of three values.

\begin{enumerate}
\item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form.
\item $dr = 1$ means that the modulus is of restricted Diminished Radix form.
\item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form.
\end{enumerate}

Line 69 determines if the fast modular exponentiation algorithm can be used.  It is allowed if $dr \ne 0$ or if the modulus is odd.  Otherwise,
the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction.

\subsection{Barrett Modular Exponentiation}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
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\caption{Algorithm s\_mp\_exptmod (continued)}
\end{figure}

\textbf{Algorithm s\_mp\_exptmod.}
This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$.  It takes advantage of the Barrett reduction
algorithm to keep the product small throughout the algorithm.

The first two steps determine the optimal window size based on the number of bits in the exponent.  The larger the exponent the 
larger the window size becomes.  After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated.  This
table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$.  

After the table is allocated the first power of $g$ is found.  Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make
the rest of the algorithm more efficient.  The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$
times.  The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$.

Now that the table is available the sliding window may begin.  The following list describes the functions of all the variables in the window.
\begin{enumerate}
\item The variable $mode$ dictates how the bits of the exponent are interpreted.  
\begin{enumerate}
   \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet.  For example, if the exponent were simply 
         $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit.  In this case bits are ignored until a non-zero bit is found.  
   \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet.  In this mode leading $0$ bits 
         are read and a single squaring is performed.  If a non-zero bit is read a new window is created.  
   \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit
         downwards.
\end{enumerate}
\item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read.  When it reaches zero a new digit
      is fetched from the exponent.
\item The variable $buf$ holds the currently read digit of the exponent. 
\item The variable $digidx$ is an index into the exponents digits.  It starts at the leading digit $x.used - 1$ and moves towards the trailing digit.
\item The variable $bitcpy$ indicates how many bits are in the currently formed window.  When it reaches $winsize$ the window is flushed and
      the appropriate operations performed.
\item The variable $bitbuf$ holds the current bits of the window being formed.  
\end{enumerate}

All of step 12 is the window processing loop.  It will iterate while there are digits available form the exponent to read.  The first step
inside this loop is to extract a new digit if no more bits are available in the current digit.  If there are no bits left a new digit is
read and if there are no digits left than the loop terminates.  

After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit
upwards.  In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to 
trailing edges the entire exponent is read from most significant bit to least significant bit.

At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read.  This prevents the 
algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read.  Step 12.6 and 12.7-10 handle
the two cases of $mode = 1$ and $mode = 2$ respectively.  

\begin{center}
\begin{figure}[here]
\includegraphics{pics/expt_state.ps}
\caption{Sliding Window State Diagram}
\label{pic:expt_state}
\end{figure}
\end{center}

By step 13 there are no more digits left in the exponent.  However, there may be partial bits in the window left.  If $mode = 2$ then 
a Left-to-Right algorithm is used to process the remaining few bits.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_exptmod.c
\vspace{-3mm}
\begin{alltt}










































































































































































































































\end{alltt}
\end{small}

Lines 32 through 46 determine the optimal window size based on the length of the exponent in bits.  The window divisions are sorted
from smallest to greatest so that in each \textbf{if} statement only one condition must be tested.  For example, by the \textbf{if} statement 
on line 38 the value of $x$ is already known to be greater than $140$.  

The conditional piece of code beginning on line 48 allows the window size to be restricted to five bits.  This logic is used to ensure
the table of precomputed powers of $G$ remains relatively small.  

The for loop on line 61 initializes the $M$ array while lines 72 and 77 through 86 initialize the reduction
function that will be used for this modulus.

-- More later.

\section{Quick Power of Two}
Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms.  Recall that a logical shift left $m << k$ is
equivalent to $m \cdot 2^k$.  By this logic when $m = 1$ a quick power of two can be achieved.







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\caption{Algorithm s\_mp\_exptmod (continued)}
\end{figure}

\textbf{Algorithm s\_mp\_exptmod.}
This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$.  It takes advantage of the Barrett reduction
algorithm to keep the product small throughout the algorithm.

The first two steps determine the optimal window size based on the number of bits in the exponent.  The larger the exponent the
larger the window size becomes.  After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated.  This
table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$.

After the table is allocated the first power of $g$ is found.  Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make
the rest of the algorithm more efficient.  The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$
times.  The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$.

Now that the table is available the sliding window may begin.  The following list describes the functions of all the variables in the window.
\begin{enumerate}
\item The variable $mode$ dictates how the bits of the exponent are interpreted.
\begin{enumerate}
   \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet.  For example, if the exponent were simply
         $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit.  In this case bits are ignored until a non-zero bit is found.
   \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet.  In this mode leading $0$ bits
         are read and a single squaring is performed.  If a non-zero bit is read a new window is created.
   \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit
         downwards.
\end{enumerate}
\item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read.  When it reaches zero a new digit
      is fetched from the exponent.
\item The variable $buf$ holds the currently read digit of the exponent.
\item The variable $digidx$ is an index into the exponents digits.  It starts at the leading digit $x.used - 1$ and moves towards the trailing digit.
\item The variable $bitcpy$ indicates how many bits are in the currently formed window.  When it reaches $winsize$ the window is flushed and
      the appropriate operations performed.
\item The variable $bitbuf$ holds the current bits of the window being formed.
\end{enumerate}

All of step 12 is the window processing loop.  It will iterate while there are digits available form the exponent to read.  The first step
inside this loop is to extract a new digit if no more bits are available in the current digit.  If there are no bits left a new digit is
read and if there are no digits left than the loop terminates.

After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit
upwards.  In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to
trailing edges the entire exponent is read from most significant bit to least significant bit.

At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read.  This prevents the
algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read.  Step 12.6 and 12.7-10 handle
the two cases of $mode = 1$ and $mode = 2$ respectively.

\begin{center}
\begin{figure}[here]
\includegraphics{pics/expt_state.ps}
\caption{Sliding Window State Diagram}
\label{pic:expt_state}
\end{figure}
\end{center}

By step 13 there are no more digits left in the exponent.  However, there may be partial bits in the window left.  If $mode = 2$ then
a Left-to-Right algorithm is used to process the remaining few bits.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_exptmod.c
\vspace{-3mm}
\begin{alltt}
016   #ifdef MP_LOW_MEM
017      #define TAB_SIZE 32
018   #else
019      #define TAB_SIZE 256
020   #endif
021
022   int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmod
      e)
023   \{
024     mp_int  M[TAB_SIZE], res, mu;
025     mp_digit buf;
026     int     err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize;
027     int (*redux)(mp_int*,mp_int*,mp_int*);
028
029     /* find window size */
030     x = mp_count_bits (X);
031     if (x <= 7) \{
032       winsize = 2;
033     \} else if (x <= 36) \{
034       winsize = 3;
035     \} else if (x <= 140) \{
036       winsize = 4;
037     \} else if (x <= 450) \{
038       winsize = 5;
039     \} else if (x <= 1303) \{
040       winsize = 6;
041     \} else if (x <= 3529) \{
042       winsize = 7;
043     \} else \{
044       winsize = 8;
045     \}
046
047   #ifdef MP_LOW_MEM
048       if (winsize > 5) \{
049          winsize = 5;
050       \}
051   #endif
052
053     /* init M array */
054     /* init first cell */
055     if ((err = mp_init(&M[1])) != MP_OKAY) \{
056        return err;
057     \}
058
059     /* now init the second half of the array */
060     for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{
061       if ((err = mp_init(&M[x])) != MP_OKAY) \{
062         for (y = 1<<(winsize-1); y < x; y++) \{
063           mp_clear (&M[y]);
064         \}
065         mp_clear(&M[1]);
066         return err;
067       \}
068     \}
069
070     /* create mu, used for Barrett reduction */
071     if ((err = mp_init (&mu)) != MP_OKAY) \{
072       goto LBL_M;
073     \}
074
075     if (redmode == 0) \{
076        if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) \{
077           goto LBL_MU;
078        \}
079        redux = mp_reduce;
080     \} else \{
081        if ((err = mp_reduce_2k_setup_l (P, &mu)) != MP_OKAY) \{
082           goto LBL_MU;
083        \}
084        redux = mp_reduce_2k_l;
085     \}
086
087     /* create M table
088      *
089      * The M table contains powers of the base,
090      * e.g. M[x] = G**x mod P
091      *
092      * The first half of the table is not
093      * computed though accept for M[0] and M[1]
094      */
095     if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) \{
096       goto LBL_MU;
097     \}
098
099     /* compute the value at M[1<<(winsize-1)] by squaring
100      * M[1] (winsize-1) times
101      */
102     if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) \{
103       goto LBL_MU;
104     \}
105
106     for (x = 0; x < (winsize - 1); x++) \{
107       /* square it */
108       if ((err = mp_sqr (&M[1 << (winsize - 1)],
109                          &M[1 << (winsize - 1)])) != MP_OKAY) \{
110         goto LBL_MU;
111       \}
112
113       /* reduce modulo P */
114       if ((err = redux (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) \{
115         goto LBL_MU;
116       \}
117     \}
118
119     /* create upper table, that is M[x] = M[x-1] * M[1] (mod P)
120      * for x = (2**(winsize - 1) + 1) to (2**winsize - 1)
121      */
122     for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) \{
123       if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) \{
124         goto LBL_MU;
125       \}
126       if ((err = redux (&M[x], P, &mu)) != MP_OKAY) \{
127         goto LBL_MU;
128       \}
129     \}
130
131     /* setup result */
132     if ((err = mp_init (&res)) != MP_OKAY) \{
133       goto LBL_MU;
134     \}
135     mp_set (&res, 1);
136
137     /* set initial mode and bit cnt */
138     mode   = 0;
139     bitcnt = 1;
140     buf    = 0;
141     digidx = X->used - 1;
142     bitcpy = 0;
143     bitbuf = 0;
144
145     for (;;) \{
146       /* grab next digit as required */
147       if (--bitcnt == 0) \{
148         /* if digidx == -1 we are out of digits */
149         if (digidx == -1) \{
150           break;
151         \}
152         /* read next digit and reset the bitcnt */
153         buf    = X->dp[digidx--];
154         bitcnt = (int) DIGIT_BIT;
155       \}
156
157       /* grab the next msb from the exponent */
158       y     = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1;
159       buf <<= (mp_digit)1;
160
161       /* if the bit is zero and mode == 0 then we ignore it
162        * These represent the leading zero bits before the first 1 bit
163        * in the exponent.  Technically this opt is not required but it
164        * does lower the # of trivial squaring/reductions used
165        */
166       if ((mode == 0) && (y == 0)) \{
167         continue;
168       \}
169
170       /* if the bit is zero and mode == 1 then we square */
171       if ((mode == 1) && (y == 0)) \{
172         if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
173           goto LBL_RES;
174         \}
175         if ((err = redux (&res, P, &mu)) != MP_OKAY) \{
176           goto LBL_RES;
177         \}
178         continue;
179       \}
180
181       /* else we add it to the window */
182       bitbuf |= (y << (winsize - ++bitcpy));
183       mode    = 2;
184
185       if (bitcpy == winsize) \{
186         /* ok window is filled so square as required and multiply  */
187         /* square first */
188         for (x = 0; x < winsize; x++) \{
189           if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
190             goto LBL_RES;
191           \}
192           if ((err = redux (&res, P, &mu)) != MP_OKAY) \{
193             goto LBL_RES;
194           \}
195         \}
196
197         /* then multiply */
198         if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) \{
199           goto LBL_RES;
200         \}
201         if ((err = redux (&res, P, &mu)) != MP_OKAY) \{
202           goto LBL_RES;
203         \}
204
205         /* empty window and reset */
206         bitcpy = 0;
207         bitbuf = 0;
208         mode   = 1;
209       \}
210     \}
211
212     /* if bits remain then square/multiply */
213     if ((mode == 2) && (bitcpy > 0)) \{
214       /* square then multiply if the bit is set */
215       for (x = 0; x < bitcpy; x++) \{
216         if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
217           goto LBL_RES;
218         \}
219         if ((err = redux (&res, P, &mu)) != MP_OKAY) \{
220           goto LBL_RES;
221         \}
222
223         bitbuf <<= 1;
224         if ((bitbuf & (1 << winsize)) != 0) \{
225           /* then multiply */
226           if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) \{
227             goto LBL_RES;
228           \}
229           if ((err = redux (&res, P, &mu)) != MP_OKAY) \{
230             goto LBL_RES;
231           \}
232         \}
233       \}
234     \}
235
236     mp_exch (&res, Y);
237     err = MP_OKAY;
238   LBL_RES:mp_clear (&res);
239   LBL_MU:mp_clear (&mu);
240   LBL_M:
241     mp_clear(&M[1]);
242     for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{
243       mp_clear (&M[x]);
244     \}
245     return err;
246   \}
247   #endif
248
\end{alltt}
\end{small}

Lines 31 through 45 determine the optimal window size based on the length of the exponent in bits.  The window divisions are sorted
from smallest to greatest so that in each \textbf{if} statement only one condition must be tested.  For example, by the \textbf{if} statement
on line 37 the value of $x$ is already known to be greater than $140$.

The conditional piece of code beginning on line 47 allows the window size to be restricted to five bits.  This logic is used to ensure
the table of precomputed powers of $G$ remains relatively small.

The for loop on line 60 initializes the $M$ array while lines 71 and 76 through 85 initialize the reduction
function that will be used for this modulus.

-- More later.

\section{Quick Power of Two}
Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms.  Recall that a logical shift left $m << k$ is
equivalent to $m \cdot 2^k$.  By this logic when $m = 1$ a quick power of two can be achieved.
5233
5234
5235
5236
5237
5238
5239





























5240
5241
5242
5243
5244
5245
5246
5247
5248
5249
5250
5251
5252
5253
5254
5255
5256
5257
5258
5259
5260
5261
5262
5263
5264
5265

\textbf{Algorithm mp\_2expt.}

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_2expt.c
\vspace{-3mm}
\begin{alltt}





























\end{alltt}
\end{small}

\chapter{Higher Level Algorithms}

This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package.  These
routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important.  

The first section describes a method of integer division with remainder that is universally well known.  It provides the signed division logic
for the package.  The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations.  
These algorithms serve mostly to simplify other algorithms where small constants are required.  The last two sections discuss how to manipulate 
various representations of integers.  For example, converting from an mp\_int to a string of character.

\section{Integer Division with Remainder}
\label{sec:division}

Integer division aside from modular exponentiation is the most intensive algorithm to compute.  Like addition, subtraction and multiplication
the basis of this algorithm is the long-hand division algorithm taught to school children.  Throughout this discussion several common variables
will be used.  Let $x$ represent the divisor and $y$ represent the dividend.  Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and 
let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$.  The following simple algorithm will be used to start the discussion.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Radix-$\beta$ Integer Division}. \\







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8185
8186
8187
8188
8189
8190
8191
8192
8193
8194
8195
8196
8197
8198
8199
8200
8201
8202
8203
8204
8205
8206
8207
8208
8209
8210
8211
8212
8213
8214
8215
8216
8217
8218
8219
8220
8221
8222
8223
8224
8225
8226
8227
8228
8229
8230
8231
8232
8233
8234
8235
8236
8237
8238
8239
8240
8241
8242
8243
8244
8245
8246

\textbf{Algorithm mp\_2expt.}

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_2expt.c
\vspace{-3mm}
\begin{alltt}
016
017   /* computes a = 2**b
018    *
019    * Simple algorithm which zeroes the int, grows it then just sets one bit
020    * as required.
021    */
022   int
023   mp_2expt (mp_int * a, int b)
024   \{
025     int     res;
026
027     /* zero a as per default */
028     mp_zero (a);
029
030     /* grow a to accomodate the single bit */
031     if ((res = mp_grow (a, (b / DIGIT_BIT) + 1)) != MP_OKAY) \{
032       return res;
033     \}
034
035     /* set the used count of where the bit will go */
036     a->used = (b / DIGIT_BIT) + 1;
037
038     /* put the single bit in its place */
039     a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT);
040
041     return MP_OKAY;
042   \}
043   #endif
044
\end{alltt}
\end{small}

\chapter{Higher Level Algorithms}

This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package.  These
routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important.

The first section describes a method of integer division with remainder that is universally well known.  It provides the signed division logic
for the package.  The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations.
These algorithms serve mostly to simplify other algorithms where small constants are required.  The last two sections discuss how to manipulate
various representations of integers.  For example, converting from an mp\_int to a string of character.

\section{Integer Division with Remainder}
\label{sec:division}

Integer division aside from modular exponentiation is the most intensive algorithm to compute.  Like addition, subtraction and multiplication
the basis of this algorithm is the long-hand division algorithm taught to school children.  Throughout this discussion several common variables
will be used.  Let $x$ represent the divisor and $y$ represent the dividend.  Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and
let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$.  The following simple algorithm will be used to start the discussion.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Radix-$\beta$ Integer Division}. \\
5281
5282
5283
5284
5285
5286
5287
5288
5289
5290
5291
5292
5293
5294
5295
5296
5297
5298
5299
5300
5301
5302
5303
5304
5305
5306
5307
5308
5309
5310
5311
5312
5313
5314
5315
5316
5317
5318
5319
5320
5321
5322
5323
5324
5325
5326
5327
5328
5329
5330
\caption{Algorithm Radix-$\beta$ Integer Division}
\label{fig:raddiv}
\end{figure}

As children we are taught this very simple algorithm for the case of $\beta = 10$.  Almost instinctively several optimizations are taught for which
their reason of existing are never explained.  For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor.

To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and 
simultaneously $(k + 1)x\beta^t$ is greater than $y$.  Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have.  The habitual method
used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient.  By only using leading
digits a much simpler division may be used to form an educated guess at what the value must be.  In this case $k = \lfloor 54/23\rfloor = 2$ quickly 
arises as a possible  solution.  Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$.  
As a  result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$.

Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder 
$y = 841 - 3x\beta = 181$.  Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the
remainder $y = 181 - 7x = 20$.  The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since 
$237 \cdot 23 + 20 = 5471$ is true.  

\subsection{Quotient Estimation}
\label{sec:divest}
As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend.  When $p$ leading
digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows.  Technically
speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the
dividend and divisor are zero.  

The value of the estimation may off by a few values in either direction and in general is fairly correct.  A simplification \cite[pp. 271]{TAOCPV2}
of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$.  The estimate 
using this technique is never too small.  For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$ 
represent the most significant digits of the dividend and divisor respectively.

\textbf{Proof.}\textit{  The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to 
$k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. }
The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger.  For all other 
cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$.  The latter portion of the inequalility
$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values.  Next a series of 
inequalities will prove the hypothesis.

\begin{equation}
y - \hat k x \le y - \hat k x_s\beta^s
\end{equation}

This is trivially true since $x \ge x_s\beta^s$.  Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$.  

\begin{equation}
y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s)
\end{equation}

By simplifying the previous inequality the following inequality is formed.








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|







8262
8263
8264
8265
8266
8267
8268
8269
8270
8271
8272
8273
8274
8275
8276
8277
8278
8279
8280
8281
8282
8283
8284
8285
8286
8287
8288
8289
8290
8291
8292
8293
8294
8295
8296
8297
8298
8299
8300
8301
8302
8303
8304
8305
8306
8307
8308
8309
8310
8311
\caption{Algorithm Radix-$\beta$ Integer Division}
\label{fig:raddiv}
\end{figure}

As children we are taught this very simple algorithm for the case of $\beta = 10$.  Almost instinctively several optimizations are taught for which
their reason of existing are never explained.  For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor.

To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and
simultaneously $(k + 1)x\beta^t$ is greater than $y$.  Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have.  The habitual method
used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient.  By only using leading
digits a much simpler division may be used to form an educated guess at what the value must be.  In this case $k = \lfloor 54/23\rfloor = 2$ quickly
arises as a possible  solution.  Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$.
As a  result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$.

Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder
$y = 841 - 3x\beta = 181$.  Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the
remainder $y = 181 - 7x = 20$.  The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since
$237 \cdot 23 + 20 = 5471$ is true.

\subsection{Quotient Estimation}
\label{sec:divest}
As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend.  When $p$ leading
digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows.  Technically
speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the
dividend and divisor are zero.

The value of the estimation may off by a few values in either direction and in general is fairly correct.  A simplification \cite[pp. 271]{TAOCPV2}
of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$.  The estimate
using this technique is never too small.  For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$
represent the most significant digits of the dividend and divisor respectively.

\textbf{Proof.}\textit{  The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to
$k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. }
The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger.  For all other
cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$.  The latter portion of the inequalility
$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values.  Next a series of
inequalities will prove the hypothesis.

\begin{equation}
y - \hat k x \le y - \hat k x_s\beta^s
\end{equation}

This is trivially true since $x \ge x_s\beta^s$.  Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$.

\begin{equation}
y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s)
\end{equation}

By simplifying the previous inequality the following inequality is formed.

5341
5342
5343
5344
5345
5346
5347
5348
5349
5350
5351
5352
5353
5354
5355
5356
5357
5358
5359
5360
5361
Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which concludes the proof.  \textbf{QED}


\subsection{Normalized Integers}
For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$.  By multiplying both
$x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original
remainder.  The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will
lie in the domain of a single digit.  Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$.  

\begin{equation} 
{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta} 
\end{equation}

At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small.  

\subsection{Radix-$\beta$ Division with Remainder}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_div}. \\







|

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|







8322
8323
8324
8325
8326
8327
8328
8329
8330
8331
8332
8333
8334
8335
8336
8337
8338
8339
8340
8341
8342
Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which concludes the proof.  \textbf{QED}


\subsection{Normalized Integers}
For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$.  By multiplying both
$x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original
remainder.  The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will
lie in the domain of a single digit.  Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$.

\begin{equation}
{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta}
\end{equation}

At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small.

\subsection{Radix-$\beta$ Division with Remainder}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_div}. \\
5445
5446
5447
5448
5449
5450
5451
5452
5453
5454
5455
5456
5457
5458
5459
5460
5461
5462
5463
5464
5465
5466
5467
5468
5469
5470
5471
5472
5473
5474
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5478
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5481
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5487




















































































































































































































































































5488
5489
5490
5491
5492
5493
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5500
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5503
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5509
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5522
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5525
5526
5527
5528
5529
5530
5531
5532
\end{small}
\caption{Algorithm mp\_div (continued)}
\end{figure}
\textbf{Algorithm mp\_div.}
This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor.  The algorithm is a signed
division and will produce a fully qualified quotient and remainder.

First the divisor $b$ must be non-zero which is enforced in step one.  If the divisor is larger than the dividend than the quotient is implicitly 
zero and the remainder is the dividend.  

After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient.  Two unsigned copies of the
divisor $y$ and dividend $x$ are made as well.  The core of the division algorithm is an unsigned division and will only work if the values are
positive.  Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$.  
This is performed by shifting both to the left by enough bits to get the desired normalization.  

At this point the division algorithm can begin producing digits of the quotient.  Recall that maximum value of the estimation used is 
$2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means.  In this case $y$ is shifted
to the left (\textit{step ten}) so that it has the same number of digits as $x$.  The loop on step eleven will subtract multiples of the 
shifted copy of $y$ until $x$ is smaller.  Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two
times to produce the desired leading digit of the quotient.  

Now the remainder of the digits can be produced.  The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly
accurately approximate the true quotient digit.  The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by
induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$.  

Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high.  The next step of the estimation process is
to refine the estimation.  The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher
order approximation to adjust the quotient digit.

After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced
by optimizing Barrett reduction.}.  Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of
algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large.  

Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the 
remainder.  An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC}
is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie 
outside their respective boundaries.  For example, if $t = 0$ or $i \le 1$ then the digits would be undefined.  In those cases the digits should
respectively be replaced with a zero.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_div.c
\vspace{-3mm}
\begin{alltt}




















































































































































































































































































\end{alltt}
\end{small}

The implementation of this algorithm differs slightly from the pseudo code presented previously.  In this algorithm either of the quotient $c$ or
remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired.  For example, the C code to call the division
algorithm with only the quotient is 

\begin{verbatim}
mp_div(&a, &b, &c, NULL);  /* c = [a/b] */
\end{verbatim}

Lines 109 and 113 handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor 
respectively.  After the two trivial cases all of the temporary variables are initialized.  Line 148 determines the sign of 
the quotient and line 148 ensures that both $x$ and $y$ are positive.  

The number of bits in the leading digit is calculated on line 151.  Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits
of precision which when reduced modulo $lg(\beta)$ produces the value of $k$.  In this case $k$ is the number of bits in the leading digit which is
exactly what is required.  For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting
them to the left by $lg(\beta) - 1 - k$ bits.

Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively.  These are first used to produce the 
leading digit of the quotient.  The loop beginning on line 184 will produce the remainder of the quotient digits.

The conditional ``continue'' on line 187 is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the
algorithm eliminates multiple non-zero digits in a single iteration.  This ensures that $x_i$ is always non-zero since by definition the digits
above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}.  

Lines 214, 216 and 223 through 225 manually construct the high accuracy estimations by setting the digits of the two mp\_int 
variables directly.  

\section{Single Digit Helpers}

This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants.  All of 
the helper functions assume the single digit input is positive and will treat them as such.

\subsection{Single Digit Addition and Subtraction}

Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction 
algorithms.   As a result these algorithms are subtantially simpler with a slight cost in performance.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_add\_d}. \\







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\end{small}
\caption{Algorithm mp\_div (continued)}
\end{figure}
\textbf{Algorithm mp\_div.}
This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor.  The algorithm is a signed
division and will produce a fully qualified quotient and remainder.

First the divisor $b$ must be non-zero which is enforced in step one.  If the divisor is larger than the dividend than the quotient is implicitly
zero and the remainder is the dividend.

After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient.  Two unsigned copies of the
divisor $y$ and dividend $x$ are made as well.  The core of the division algorithm is an unsigned division and will only work if the values are
positive.  Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$.
This is performed by shifting both to the left by enough bits to get the desired normalization.

At this point the division algorithm can begin producing digits of the quotient.  Recall that maximum value of the estimation used is
$2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means.  In this case $y$ is shifted
to the left (\textit{step ten}) so that it has the same number of digits as $x$.  The loop on step eleven will subtract multiples of the
shifted copy of $y$ until $x$ is smaller.  Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two
times to produce the desired leading digit of the quotient.

Now the remainder of the digits can be produced.  The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly
accurately approximate the true quotient digit.  The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by
induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$.

Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high.  The next step of the estimation process is
to refine the estimation.  The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher
order approximation to adjust the quotient digit.

After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced
by optimizing Barrett reduction.}.  Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of
algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large.

Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the
remainder.  An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC}
is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie
outside their respective boundaries.  For example, if $t = 0$ or $i \le 1$ then the digits would be undefined.  In those cases the digits should
respectively be replaced with a zero.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_div.c
\vspace{-3mm}
\begin{alltt}
016
017   #ifdef BN_MP_DIV_SMALL
018
019   /* slower bit-bang division... also smaller */
020   int mp_div(mp_int * a, mp_int * b, mp_int * c, mp_int * d)
021   \{
022      mp_int ta, tb, tq, q;
023      int    res, n, n2;
024
025     /* is divisor zero ? */
026     if (mp_iszero (b) == MP_YES) \{
027       return MP_VAL;
028     \}
029
030     /* if a < b then q=0, r = a */
031     if (mp_cmp_mag (a, b) == MP_LT) \{
032       if (d != NULL) \{
033         res = mp_copy (a, d);
034       \} else \{
035         res = MP_OKAY;
036       \}
037       if (c != NULL) \{
038         mp_zero (c);
039       \}
040       return res;
041     \}
042
043     /* init our temps */
044     if ((res = mp_init_multi(&ta, &tb, &tq, &q, NULL)) != MP_OKAY) \{
045        return res;
046     \}
047
048
049     mp_set(&tq, 1);
050     n = mp_count_bits(a) - mp_count_bits(b);
051     if (((res = mp_abs(a, &ta)) != MP_OKAY) ||
052         ((res = mp_abs(b, &tb)) != MP_OKAY) ||
053         ((res = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) ||
054         ((res = mp_mul_2d(&tq, n, &tq)) != MP_OKAY)) \{
055         goto LBL_ERR;
056     \}
057
058     while (n-- >= 0) \{
059        if (mp_cmp(&tb, &ta) != MP_GT) \{
060           if (((res = mp_sub(&ta, &tb, &ta)) != MP_OKAY) ||
061               ((res = mp_add(&q, &tq, &q)) != MP_OKAY)) \{
062              goto LBL_ERR;
063           \}
064        \}
065        if (((res = mp_div_2d(&tb, 1, &tb, NULL)) != MP_OKAY) ||
066            ((res = mp_div_2d(&tq, 1, &tq, NULL)) != MP_OKAY)) \{
067              goto LBL_ERR;
068        \}
069     \}
070
071     /* now q == quotient and ta == remainder */
072     n  = a->sign;
073     n2 = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;
074     if (c != NULL) \{
075        mp_exch(c, &q);
076        c->sign  = (mp_iszero(c) == MP_YES) ? MP_ZPOS : n2;
077     \}
078     if (d != NULL) \{
079        mp_exch(d, &ta);
080        d->sign = (mp_iszero(d) == MP_YES) ? MP_ZPOS : n;
081     \}
082   LBL_ERR:
083      mp_clear_multi(&ta, &tb, &tq, &q, NULL);
084      return res;
085   \}
086
087   #else
088
089   /* integer signed division.
090    * c*b + d == a [e.g. a/b, c=quotient, d=remainder]
091    * HAC pp.598 Algorithm 14.20
092    *
093    * Note that the description in HAC is horribly
094    * incomplete.  For example, it doesn't consider
095    * the case where digits are removed from 'x' in
096    * the inner loop.  It also doesn't consider the
097    * case that y has fewer than three digits, etc..
098    *
099    * The overall algorithm is as described as
100    * 14.20 from HAC but fixed to treat these cases.
101   */
102   int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
103   \{
104     mp_int  q, x, y, t1, t2;
105     int     res, n, t, i, norm, neg;
106
107     /* is divisor zero ? */
108     if (mp_iszero (b) == MP_YES) \{
109       return MP_VAL;
110     \}
111
112     /* if a < b then q=0, r = a */
113     if (mp_cmp_mag (a, b) == MP_LT) \{
114       if (d != NULL) \{
115         res = mp_copy (a, d);
116       \} else \{
117         res = MP_OKAY;
118       \}
119       if (c != NULL) \{
120         mp_zero (c);
121       \}
122       return res;
123     \}
124
125     if ((res = mp_init_size (&q, a->used + 2)) != MP_OKAY) \{
126       return res;
127     \}
128     q.used = a->used + 2;
129
130     if ((res = mp_init (&t1)) != MP_OKAY) \{
131       goto LBL_Q;
132     \}
133
134     if ((res = mp_init (&t2)) != MP_OKAY) \{
135       goto LBL_T1;
136     \}
137
138     if ((res = mp_init_copy (&x, a)) != MP_OKAY) \{
139       goto LBL_T2;
140     \}
141
142     if ((res = mp_init_copy (&y, b)) != MP_OKAY) \{
143       goto LBL_X;
144     \}
145
146     /* fix the sign */
147     neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;
148     x.sign = y.sign = MP_ZPOS;
149
150     /* normalize both x and y, ensure that y >= b/2, [b == 2**DIGIT_BIT] */
151     norm = mp_count_bits(&y) % DIGIT_BIT;
152     if (norm < (int)(DIGIT_BIT-1)) \{
153        norm = (DIGIT_BIT-1) - norm;
154        if ((res = mp_mul_2d (&x, norm, &x)) != MP_OKAY) \{
155          goto LBL_Y;
156        \}
157        if ((res = mp_mul_2d (&y, norm, &y)) != MP_OKAY) \{
158          goto LBL_Y;
159        \}
160     \} else \{
161        norm = 0;
162     \}
163
164     /* note hac does 0 based, so if used==5 then its 0,1,2,3,4, e.g. use 4 */
165     n = x.used - 1;
166     t = y.used - 1;
167
168     /* while (x >= y*b**n-t) do \{ q[n-t] += 1; x -= y*b**\{n-t\} \} */
169     if ((res = mp_lshd (&y, n - t)) != MP_OKAY) \{ /* y = y*b**\{n-t\} */
170       goto LBL_Y;
171     \}
172
173     while (mp_cmp (&x, &y) != MP_LT) \{
174       ++(q.dp[n - t]);
175       if ((res = mp_sub (&x, &y, &x)) != MP_OKAY) \{
176         goto LBL_Y;
177       \}
178     \}
179
180     /* reset y by shifting it back down */
181     mp_rshd (&y, n - t);
182
183     /* step 3. for i from n down to (t + 1) */
184     for (i = n; i >= (t + 1); i--) \{
185       if (i > x.used) \{
186         continue;
187       \}
188
189       /* step 3.1 if xi == yt then set q\{i-t-1\} to b-1,
190        * otherwise set q\{i-t-1\} to (xi*b + x\{i-1\})/yt */
191       if (x.dp[i] == y.dp[t]) \{
192         q.dp[(i - t) - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1);
193       \} else \{
194         mp_word tmp;
195         tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT);
196         tmp |= ((mp_word) x.dp[i - 1]);
197         tmp /= ((mp_word) y.dp[t]);
198         if (tmp > (mp_word) MP_MASK) \{
199           tmp = MP_MASK;
200         \}
201         q.dp[(i - t) - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK));
202       \}
203
204       /* while (q\{i-t-1\} * (yt * b + y\{t-1\})) >
205                xi * b**2 + xi-1 * b + xi-2
206
207          do q\{i-t-1\} -= 1;
208       */
209       q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] + 1) & MP_MASK;
210       do \{
211         q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] - 1) & MP_MASK;
212
213         /* find left hand */
214         mp_zero (&t1);
215         t1.dp[0] = ((t - 1) < 0) ? 0 : y.dp[t - 1];
216         t1.dp[1] = y.dp[t];
217         t1.used = 2;
218         if ((res = mp_mul_d (&t1, q.dp[(i - t) - 1], &t1)) != MP_OKAY) \{
219           goto LBL_Y;
220         \}
221
222         /* find right hand */
223         t2.dp[0] = ((i - 2) < 0) ? 0 : x.dp[i - 2];
224         t2.dp[1] = ((i - 1) < 0) ? 0 : x.dp[i - 1];
225         t2.dp[2] = x.dp[i];
226         t2.used = 3;
227       \} while (mp_cmp_mag(&t1, &t2) == MP_GT);
228
229       /* step 3.3 x = x - q\{i-t-1\} * y * b**\{i-t-1\} */
230       if ((res = mp_mul_d (&y, q.dp[(i - t) - 1], &t1)) != MP_OKAY) \{
231         goto LBL_Y;
232       \}
233
234       if ((res = mp_lshd (&t1, (i - t) - 1)) != MP_OKAY) \{
235         goto LBL_Y;
236       \}
237
238       if ((res = mp_sub (&x, &t1, &x)) != MP_OKAY) \{
239         goto LBL_Y;
240       \}
241
242       /* if x < 0 then \{ x = x + y*b**\{i-t-1\}; q\{i-t-1\} -= 1; \} */
243       if (x.sign == MP_NEG) \{
244         if ((res = mp_copy (&y, &t1)) != MP_OKAY) \{
245           goto LBL_Y;
246         \}
247         if ((res = mp_lshd (&t1, (i - t) - 1)) != MP_OKAY) \{
248           goto LBL_Y;
249         \}
250         if ((res = mp_add (&x, &t1, &x)) != MP_OKAY) \{
251           goto LBL_Y;
252         \}
253
254         q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] - 1UL) & MP_MASK;
255       \}
256     \}
257
258     /* now q is the quotient and x is the remainder
259      * [which we have to normalize]
260      */
261
262     /* get sign before writing to c */
263     x.sign = (x.used == 0) ? MP_ZPOS : a->sign;
264
265     if (c != NULL) \{
266       mp_clamp (&q);
267       mp_exch (&q, c);
268       c->sign = neg;
269     \}
270
271     if (d != NULL) \{
272       if ((res = mp_div_2d (&x, norm, &x, NULL)) != MP_OKAY) \{
273         goto LBL_Y;
274       \}
275       mp_exch (&x, d);
276     \}
277
278     res = MP_OKAY;
279
280   LBL_Y:mp_clear (&y);
281   LBL_X:mp_clear (&x);
282   LBL_T2:mp_clear (&t2);
283   LBL_T1:mp_clear (&t1);
284   LBL_Q:mp_clear (&q);
285     return res;
286   \}
287
288   #endif
289
290   #endif
291
\end{alltt}
\end{small}

The implementation of this algorithm differs slightly from the pseudo code presented previously.  In this algorithm either of the quotient $c$ or
remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired.  For example, the C code to call the division
algorithm with only the quotient is

\begin{verbatim}
mp_div(&a, &b, &c, NULL);  /* c = [a/b] */
\end{verbatim}

Lines 108 and 113 handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor
respectively.  After the two trivial cases all of the temporary variables are initialized.  Line 147 determines the sign of
the quotient and line 148 ensures that both $x$ and $y$ are positive.

The number of bits in the leading digit is calculated on line 151.  Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits
of precision which when reduced modulo $lg(\beta)$ produces the value of $k$.  In this case $k$ is the number of bits in the leading digit which is
exactly what is required.  For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting
them to the left by $lg(\beta) - 1 - k$ bits.

Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively.  These are first used to produce the
leading digit of the quotient.  The loop beginning on line 184 will produce the remainder of the quotient digits.

The conditional ``continue'' on line 186 is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the
algorithm eliminates multiple non-zero digits in a single iteration.  This ensures that $x_i$ is always non-zero since by definition the digits
above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}.

Lines 214, 216 and 223 through 225 manually construct the high accuracy estimations by setting the digits of the two mp\_int
variables directly.

\section{Single Digit Helpers}

This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants.  All of
the helper functions assume the single digit input is positive and will treat them as such.

\subsection{Single Digit Addition and Subtraction}

Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction
algorithms.   As a result these algorithms are subtantially simpler with a slight cost in performance.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_add\_d}. \\
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\textbf{Algorithm mp\_add\_d.}
This algorithm initiates a temporary mp\_int with the value of the single digit and uses algorithm mp\_add to add the two values together.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_add\_d.c
\vspace{-3mm}
\begin{alltt}





























































































\end{alltt}
\end{small}

Clever use of the letter 't'.

\subsubsection{Subtraction}
The single digit subtraction algorithm mp\_sub\_d is essentially the same except it uses mp\_sub to subtract the digit from the mp\_int.







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\textbf{Algorithm mp\_add\_d.}
This algorithm initiates a temporary mp\_int with the value of the single digit and uses algorithm mp\_add to add the two values together.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_add\_d.c
\vspace{-3mm}
\begin{alltt}
016
017   /* single digit addition */
018   int
019   mp_add_d (mp_int * a, mp_digit b, mp_int * c)
020   \{
021     int     res, ix, oldused;
022     mp_digit *tmpa, *tmpc, mu;
023
024     /* grow c as required */
025     if (c->alloc < (a->used + 1)) \{
026        if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) \{
027           return res;
028        \}
029     \}
030
031     /* if a is negative and |a| >= b, call c = |a| - b */
032     if ((a->sign == MP_NEG) && ((a->used > 1) || (a->dp[0] >= b))) \{
033        /* temporarily fix sign of a */
034        a->sign = MP_ZPOS;
035
036        /* c = |a| - b */
037        res = mp_sub_d(a, b, c);
038
039        /* fix sign  */
040        a->sign = c->sign = MP_NEG;
041
042        /* clamp */
043        mp_clamp(c);
044
045        return res;
046     \}
047
048     /* old number of used digits in c */
049     oldused = c->used;
050
051     /* sign always positive */
052     c->sign = MP_ZPOS;
053
054     /* source alias */
055     tmpa    = a->dp;
056
057     /* destination alias */
058     tmpc    = c->dp;
059
060     /* if a is positive */
061     if (a->sign == MP_ZPOS) \{
062        /* add digit, after this we're propagating
063         * the carry.
064         */
065        *tmpc   = *tmpa++ + b;
066        mu      = *tmpc >> DIGIT_BIT;
067        *tmpc++ &= MP_MASK;
068
069        /* now handle rest of the digits */
070        for (ix = 1; ix < a->used; ix++) \{
071           *tmpc   = *tmpa++ + mu;
072           mu      = *tmpc >> DIGIT_BIT;
073           *tmpc++ &= MP_MASK;
074        \}
075        /* set final carry */
076        ix++;
077        *tmpc++  = mu;
078
079        /* setup size */
080        c->used = a->used + 1;
081     \} else \{
082        /* a was negative and |a| < b */
083        c->used  = 1;
084
085        /* the result is a single digit */
086        if (a->used == 1) \{
087           *tmpc++  =  b - a->dp[0];
088        \} else \{
089           *tmpc++  =  b;
090        \}
091
092        /* setup count so the clearing of oldused
093         * can fall through correctly
094         */
095        ix       = 1;
096     \}
097
098     /* now zero to oldused */
099     while (ix++ < oldused) \{
100        *tmpc++ = 0;
101     \}
102     mp_clamp(c);
103
104     return MP_OKAY;
105   \}
106
107   #endif
108
\end{alltt}
\end{small}

Clever use of the letter 't'.

\subsubsection{Subtraction}
The single digit subtraction algorithm mp\_sub\_d is essentially the same except it uses mp\_sub to subtract the digit from the mp\_int.
5589
5590
5591
5592
5593
5594
5595
5596
5597
5598
5599
5600
5601
5602




























































5603
5604
5605
5606
5607
5608
5609
5610
5611
5612
5613
5614
5615
5616
5617
5618
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul\_d}
\end{figure}
\textbf{Algorithm mp\_mul\_d.}
This algorithm quickly multiplies an mp\_int by a small single digit value.  It is specially tailored to the job and has a minimal of overhead.  
Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_d.c
\vspace{-3mm}
\begin{alltt}




























































\end{alltt}
\end{small}

In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is 
read from the source.  This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively.  

\subsection{Single Digit Division}
Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion.  Since the
divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_div\_d}. \\
\textbf{Input}.   mp\_int $a$ and a mp\_digit $b$ \\







|
|





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8939
8940
8941
8942
8943
8944
8945
8946
8947
8948
8949
8950
8951
8952
8953
8954
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9026
9027
9028
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul\_d}
\end{figure}
\textbf{Algorithm mp\_mul\_d.}
This algorithm quickly multiplies an mp\_int by a small single digit value.  It is specially tailored to the job and has a minimal of overhead.
Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_d.c
\vspace{-3mm}
\begin{alltt}
016
017   /* multiply by a digit */
018   int
019   mp_mul_d (mp_int * a, mp_digit b, mp_int * c)
020   \{
021     mp_digit u, *tmpa, *tmpc;
022     mp_word  r;
023     int      ix, res, olduse;
024
025     /* make sure c is big enough to hold a*b */
026     if (c->alloc < (a->used + 1)) \{
027       if ((res = mp_grow (c, a->used + 1)) != MP_OKAY) \{
028         return res;
029       \}
030     \}
031
032     /* get the original destinations used count */
033     olduse = c->used;
034
035     /* set the sign */
036     c->sign = a->sign;
037
038     /* alias for a->dp [source] */
039     tmpa = a->dp;
040
041     /* alias for c->dp [dest] */
042     tmpc = c->dp;
043
044     /* zero carry */
045     u = 0;
046
047     /* compute columns */
048     for (ix = 0; ix < a->used; ix++) \{
049       /* compute product and carry sum for this term */
050       r       = (mp_word)u + ((mp_word)*tmpa++ * (mp_word)b);
051
052       /* mask off higher bits to get a single digit */
053       *tmpc++ = (mp_digit) (r & ((mp_word) MP_MASK));
054
055       /* send carry into next iteration */
056       u       = (mp_digit) (r >> ((mp_word) DIGIT_BIT));
057     \}
058
059     /* store final carry [if any] and increment ix offset  */
060     *tmpc++ = u;
061     ++ix;
062
063     /* now zero digits above the top */
064     while (ix++ < olduse) \{
065        *tmpc++ = 0;
066     \}
067
068     /* set used count */
069     c->used = a->used + 1;
070     mp_clamp(c);
071
072     return MP_OKAY;
073   \}
074   #endif
075
\end{alltt}
\end{small}

In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is
read from the source.  This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively.

\subsection{Single Digit Division}
Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion.  Since the
divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_div\_d}. \\
\textbf{Input}.   mp\_int $a$ and a mp\_digit $b$ \\
5641
5642
5643
5644
5645
5646
5647
5648
5649
5650
5651
5652
5653
5654
5655
5656
5657
































































































5658
5659
5660
5661
5662
5663
5664
5665
5666
5667
5668
5669
5670
5671
5672
5673
5674
5675
5676
5677
5678
5679
5680
5681
5682
5683
5684
5685
5686
5687
5688
\end{center}
\end{small}
\caption{Algorithm mp\_div\_d}
\end{figure}
\textbf{Algorithm mp\_div\_d.}
This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach.  Essentially in every iteration of the
algorithm another digit of the dividend is reduced and another digit of quotient produced.  Provided $b < \beta$ the value of $\hat w$
after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$.  

If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3.  It replaces the division by three with
a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup.  In essence it is much like the Barrett reduction
from chapter seven.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_d.c
\vspace{-3mm}
\begin{alltt}
































































































\end{alltt}
\end{small}

Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to
indicate the respective value is not required.  This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created.

The division and remainder on lines 44 and @45,%@ can be replaced often by a single division on most processors.  For example, the 32-bit x86 based 
processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously.  Unfortunately the GCC 
compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively.  

\subsection{Single Digit Root Extraction}

Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned.  Algorithms such as the Newton-Raphson approximation 
(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$.  

\begin{equation}
x_{i+1} = x_i - {f(x_i) \over f'(x_i)}
\label{eqn:newton}
\end{equation}

In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired.  The derivative of $f(x)$ is 
simply $f'(x) = nx^{n - 1}$.  Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain
such as the real numbers.  As a result the root found can be above the true root by few and must be manually adjusted.  Ideally at the end of the 
algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_n\_root}. \\
\textbf{Input}.   mp\_int $a$ and a mp\_digit $b$ \\







|



|





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9052
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9184
9185
9186
9187
9188
9189
9190
9191
9192
9193
9194
\end{center}
\end{small}
\caption{Algorithm mp\_div\_d}
\end{figure}
\textbf{Algorithm mp\_div\_d.}
This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach.  Essentially in every iteration of the
algorithm another digit of the dividend is reduced and another digit of quotient produced.  Provided $b < \beta$ the value of $\hat w$
after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$.

If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3.  It replaces the division by three with
a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup.  In essence it is much like the Barrett reduction
from chapter seven.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_d.c
\vspace{-3mm}
\begin{alltt}
016
017   static int s_is_power_of_two(mp_digit b, int *p)
018   \{
019      int x;
020
021      /* fast return if no power of two */
022      if ((b == 0) || ((b & (b-1)) != 0)) \{
023         return 0;
024      \}
025
026      for (x = 0; x < DIGIT_BIT; x++) \{
027         if (b == (((mp_digit)1)<<x)) \{
028            *p = x;
029            return 1;
030         \}
031      \}
032      return 0;
033   \}
034
035   /* single digit division (based on routine from MPI) */
036   int mp_div_d (mp_int * a, mp_digit b, mp_int * c, mp_digit * d)
037   \{
038     mp_int  q;
039     mp_word w;
040     mp_digit t;
041     int     res, ix;
042
043     /* cannot divide by zero */
044     if (b == 0) \{
045        return MP_VAL;
046     \}
047
048     /* quick outs */
049     if ((b == 1) || (mp_iszero(a) == MP_YES)) \{
050        if (d != NULL) \{
051           *d = 0;
052        \}
053        if (c != NULL) \{
054           return mp_copy(a, c);
055        \}
056        return MP_OKAY;
057     \}
058
059     /* power of two ? */
060     if (s_is_power_of_two(b, &ix) == 1) \{
061        if (d != NULL) \{
062           *d = a->dp[0] & ((((mp_digit)1)<<ix) - 1);
063        \}
064        if (c != NULL) \{
065           return mp_div_2d(a, ix, c, NULL);
066        \}
067        return MP_OKAY;
068     \}
069
070   #ifdef BN_MP_DIV_3_C
071     /* three? */
072     if (b == 3) \{
073        return mp_div_3(a, c, d);
074     \}
075   #endif
076
077     /* no easy answer [c'est la vie].  Just division */
078     if ((res = mp_init_size(&q, a->used)) != MP_OKAY) \{
079        return res;
080     \}
081
082     q.used = a->used;
083     q.sign = a->sign;
084     w = 0;
085     for (ix = a->used - 1; ix >= 0; ix--) \{
086        w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]);
087
088        if (w >= b) \{
089           t = (mp_digit)(w / b);
090           w -= ((mp_word)t) * ((mp_word)b);
091         \} else \{
092           t = 0;
093         \}
094         q.dp[ix] = (mp_digit)t;
095     \}
096
097     if (d != NULL) \{
098        *d = (mp_digit)w;
099     \}
100
101     if (c != NULL) \{
102        mp_clamp(&q);
103        mp_exch(&q, c);
104     \}
105     mp_clear(&q);
106
107     return res;
108   \}
109
110   #endif
111
\end{alltt}
\end{small}

Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to
indicate the respective value is not required.  This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created.

The division and remainder on lines 89 and 90 can be replaced often by a single division on most processors.  For example, the 32-bit x86 based
processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously.  Unfortunately the GCC
compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively.

\subsection{Single Digit Root Extraction}

Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned.  Algorithms such as the Newton-Raphson approximation
(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$.

\begin{equation}
x_{i+1} = x_i - {f(x_i) \over f'(x_i)}
\label{eqn:newton}
\end{equation}

In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired.  The derivative of $f(x)$ is
simply $f'(x) = nx^{n - 1}$.  Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain
such as the real numbers.  As a result the root found can be above the true root by few and must be manually adjusted.  Ideally at the end of the
algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_n\_root}. \\
\textbf{Input}.   mp\_int $a$ and a mp\_digit $b$ \\
5715
5716
5717
5718
5719
5720
5721
5722
5723
5724
5725
5726
5727
5728
5729
5730
5731











5732
5733
5734
5735
5736
5737
5738
5739
5740
5741
5742
5743
5744
5745
5746
\end{center}
\end{small}
\caption{Algorithm mp\_n\_root}
\end{figure}
\textbf{Algorithm mp\_n\_root.}
This algorithm finds the integer $n$'th root of an input using the Newton-Raphson approach.  It is partially optimized based on the observation
that the numerator of ${f(x) \over f'(x)}$ can be derived from a partial denominator.  That is at first the denominator is calculated by finding
$x^{b - 1}$.  This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator.  This saves a total of $b - 1$ 
multiplications by t$1$ inside the loop.  

The initial value of the approximation is t$2 = 2$ which allows the algorithm to start with very small values and quickly converge on the
root.  Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_n\_root.c
\vspace{-3mm}
\begin{alltt}











\end{alltt}
\end{small}

\section{Random Number Generation}

Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms.  Pollard-Rho 
factoring for example, can make use of random values as starting points to find factors of a composite integer.  In this case the algorithm presented
is solely for simulations and not intended for cryptographic use.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_rand}. \\
\textbf{Input}.   An integer $b$ \\







|
|


|





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>
>
>
>
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>
>





|

|







9221
9222
9223
9224
9225
9226
9227
9228
9229
9230
9231
9232
9233
9234
9235
9236
9237
9238
9239
9240
9241
9242
9243
9244
9245
9246
9247
9248
9249
9250
9251
9252
9253
9254
9255
9256
9257
9258
9259
9260
9261
9262
9263
\end{center}
\end{small}
\caption{Algorithm mp\_n\_root}
\end{figure}
\textbf{Algorithm mp\_n\_root.}
This algorithm finds the integer $n$'th root of an input using the Newton-Raphson approach.  It is partially optimized based on the observation
that the numerator of ${f(x) \over f'(x)}$ can be derived from a partial denominator.  That is at first the denominator is calculated by finding
$x^{b - 1}$.  This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator.  This saves a total of $b - 1$
multiplications by t$1$ inside the loop.

The initial value of the approximation is t$2 = 2$ which allows the algorithm to start with very small values and quickly converge on the
root.  Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_n\_root.c
\vspace{-3mm}
\begin{alltt}
016
017   /* wrapper function for mp_n_root_ex()
018    * computes c = (a)**(1/b) such that (c)**b <= a and (c+1)**b > a
019    */
020   int mp_n_root (mp_int * a, mp_digit b, mp_int * c)
021   \{
022     return mp_n_root_ex(a, b, c, 0);
023   \}
024
025   #endif
026
\end{alltt}
\end{small}

\section{Random Number Generation}

Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms.  Pollard-Rho
factoring for example, can make use of random values as starting points to find factors of a composite integer.  In this case the algorithm presented
is solely for simulations and not intended for cryptographic use.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_rand}. \\
\textbf{Input}.   An integer $b$ \\
5760
5761
5762
5763
5764
5765
5766
5767
5768
5769
5770
5771
5772





































5773
5774
5775
5776
5777
5778
5779
5780
5781
5782
5783
5784
5785
5786
5787
5788
5789
5790
5791
5792
5793
5794
5795
5796
5797
5798
5799
\end{center}
\end{small}
\caption{Algorithm mp\_rand}
\end{figure}
\textbf{Algorithm mp\_rand.}
This algorithm produces a pseudo-random integer of $b$ digits.  By ensuring that the first digit is non-zero the algorithm also guarantees that the
final result has at least $b$ digits.  It relies heavily on a third-part random number generator which should ideally generate uniformly all of
the integers from $0$ to $\beta - 1$.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_rand.c
\vspace{-3mm}
\begin{alltt}





































\end{alltt}
\end{small}

\section{Formatted Representations}
The ability to emit a radix-$n$ textual representation of an integer is useful for interacting with human parties.  For example, the ability to
be given a string of characters such as ``114585'' and turn it into the radix-$\beta$ equivalent would make it easier to enter numbers
into a program.

\subsection{Reading Radix-n Input}
For the purposes of this text we will assume that a simple lower ASCII map (\ref{fig:ASC}) is used for the values of from $0$ to $63$ to 
printable characters.  For example, when the character ``N'' is read it represents the integer $23$.  The first $16$ characters of the
map are for the common representations up to hexadecimal.  After that they match the ``base64'' encoding scheme which are suitable chosen
such that they are printable.  While outputting as base64 may not be too helpful for human operators it does allow communication via non binary
mediums.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{cc|cc|cc|cc}
\hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} &  \textbf{Value} & \textbf{Char} \\
\hline 
0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 \\
4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\
8 & 8 & 9 & 9 & 10 & A & 11 & B \\
12 & C & 13 & D & 14 & E & 15 & F \\
16 & G & 17 & H & 18 & I & 19 & J \\
20 & K & 21 & L & 22 & M & 23 & N \\
24 & O & 25 & P & 26 & Q & 27 & R \\







|





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9277
9278
9279
9280
9281
9282
9283
9284
9285
9286
9287
9288
9289
9290
9291
9292
9293
9294
9295
9296
9297
9298
9299
9300
9301
9302
9303
9304
9305
9306
9307
9308
9309
9310
9311
9312
9313
9314
9315
9316
9317
9318
9319
9320
9321
9322
9323
9324
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9327
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9330
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9332
9333
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9336
9337
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9340
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9342
9343
9344
9345
9346
9347
9348
9349
9350
9351
9352
9353
\end{center}
\end{small}
\caption{Algorithm mp\_rand}
\end{figure}
\textbf{Algorithm mp\_rand.}
This algorithm produces a pseudo-random integer of $b$ digits.  By ensuring that the first digit is non-zero the algorithm also guarantees that the
final result has at least $b$ digits.  It relies heavily on a third-part random number generator which should ideally generate uniformly all of
the integers from $0$ to $\beta - 1$.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_rand.c
\vspace{-3mm}
\begin{alltt}
016
017   /* makes a pseudo-random int of a given size */
018   int
019   mp_rand (mp_int * a, int digits)
020   \{
021     int     res;
022     mp_digit d;
023
024     mp_zero (a);
025     if (digits <= 0) \{
026       return MP_OKAY;
027     \}
028
029     /* first place a random non-zero digit */
030     do \{
031       d = ((mp_digit) abs (MP_GEN_RANDOM())) & MP_MASK;
032     \} while (d == 0);
033
034     if ((res = mp_add_d (a, d, a)) != MP_OKAY) \{
035       return res;
036     \}
037
038     while (--digits > 0) \{
039       if ((res = mp_lshd (a, 1)) != MP_OKAY) \{
040         return res;
041       \}
042
043       if ((res = mp_add_d (a, ((mp_digit) abs (MP_GEN_RANDOM())), a)) != MP_OK
      AY) \{
044         return res;
045       \}
046     \}
047
048     return MP_OKAY;
049   \}
050   #endif
051
\end{alltt}
\end{small}

\section{Formatted Representations}
The ability to emit a radix-$n$ textual representation of an integer is useful for interacting with human parties.  For example, the ability to
be given a string of characters such as ``114585'' and turn it into the radix-$\beta$ equivalent would make it easier to enter numbers
into a program.

\subsection{Reading Radix-n Input}
For the purposes of this text we will assume that a simple lower ASCII map (\ref{fig:ASC}) is used for the values of from $0$ to $63$ to
printable characters.  For example, when the character ``N'' is read it represents the integer $23$.  The first $16$ characters of the
map are for the common representations up to hexadecimal.  After that they match the ``base64'' encoding scheme which are suitable chosen
such that they are printable.  While outputting as base64 may not be too helpful for human operators it does allow communication via non binary
mediums.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{cc|cc|cc|cc}
\hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} &  \textbf{Value} & \textbf{Char} \\
\hline
0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 \\
4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\
8 & 8 & 9 & 9 & 10 & A & 11 & B \\
12 & C & 13 & D & 14 & E & 15 & F \\
16 & G & 17 & H & 18 & I & 19 & J \\
20 & K & 21 & L & 22 & M & 23 & N \\
24 & O & 25 & P & 26 & Q & 27 & R \\
5839
5840
5841
5842
5843
5844
5845
5846
5847
5848
5849
5850
5851
5852
5853
5854


































































5855
5856
5857
5858
5859
5860
5861
5862
5863
5864
5865
5866
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_read\_radix}
\end{figure}
\textbf{Algorithm mp\_read\_radix.}
This algorithm will read an ASCII string and produce the radix-$\beta$ mp\_int representation of the same integer.  A minus symbol ``-'' may precede the 
string  to indicate the value is negative, otherwise it is assumed to be positive.  The algorithm will read up to $sn$ characters from the input
and will stop when it reads a character it cannot map the algorithm stops reading characters from the string.  This allows numbers to be embedded
as part of larger input without any significant problem.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_read\_radix.c
\vspace{-3mm}
\begin{alltt}


































































\end{alltt}
\end{small}

\subsection{Generating Radix-$n$ Output}
Generating radix-$n$ output is fairly trivial with a division and remainder algorithm.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_toradix}. \\
\textbf{Input}.   A mp\_int $a$ and an integer $r$\\







|








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9393
9394
9395
9396
9397
9398
9399
9400
9401
9402
9403
9404
9405
9406
9407
9408
9409
9410
9411
9412
9413
9414
9415
9416
9417
9418
9419
9420
9421
9422
9423
9424
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9427
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9429
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9436
9437
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9440
9441
9442
9443
9444
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9446
9447
9448
9449
9450
9451
9452
9453
9454
9455
9456
9457
9458
9459
9460
9461
9462
9463
9464
9465
9466
9467
9468
9469
9470
9471
9472
9473
9474
9475
9476
9477
9478
9479
9480
9481
9482
9483
9484
9485
9486
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_read\_radix}
\end{figure}
\textbf{Algorithm mp\_read\_radix.}
This algorithm will read an ASCII string and produce the radix-$\beta$ mp\_int representation of the same integer.  A minus symbol ``-'' may precede the
string  to indicate the value is negative, otherwise it is assumed to be positive.  The algorithm will read up to $sn$ characters from the input
and will stop when it reads a character it cannot map the algorithm stops reading characters from the string.  This allows numbers to be embedded
as part of larger input without any significant problem.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_read\_radix.c
\vspace{-3mm}
\begin{alltt}
016
017   /* read a string [ASCII] in a given radix */
018   int mp_read_radix (mp_int * a, const char *str, int radix)
019   \{
020     int     y, res, neg;
021     char    ch;
022
023     /* zero the digit bignum */
024     mp_zero(a);
025
026     /* make sure the radix is ok */
027     if ((radix < 2) || (radix > 64)) \{
028       return MP_VAL;
029     \}
030
031     /* if the leading digit is a
032      * minus set the sign to negative.
033      */
034     if (*str == '-') \{
035       ++str;
036       neg = MP_NEG;
037     \} else \{
038       neg = MP_ZPOS;
039     \}
040
041     /* set the integer to the default of zero */
042     mp_zero (a);
043
044     /* process each digit of the string */
045     while (*str != '\symbol{92}0') \{
046       /* if the radix <= 36 the conversion is case insensitive
047        * this allows numbers like 1AB and 1ab to represent the same  value
048        * [e.g. in hex]
049        */
050       ch = (radix <= 36) ? (char)toupper((int)*str) : *str;
051       for (y = 0; y < 64; y++) \{
052         if (ch == mp_s_rmap[y]) \{
053            break;
054         \}
055       \}
056
057       /* if the char was found in the map
058        * and is less than the given radix add it
059        * to the number, otherwise exit the loop.
060        */
061       if (y < radix) \{
062         if ((res = mp_mul_d (a, (mp_digit) radix, a)) != MP_OKAY) \{
063            return res;
064         \}
065         if ((res = mp_add_d (a, (mp_digit) y, a)) != MP_OKAY) \{
066            return res;
067         \}
068       \} else \{
069         break;
070       \}
071       ++str;
072     \}
073
074     /* set the sign only if a != 0 */
075     if (mp_iszero(a) != MP_YES) \{
076        a->sign = neg;
077     \}
078     return MP_OKAY;
079   \}
080   #endif
081
\end{alltt}
\end{small}

\subsection{Generating Radix-$n$ Output}
Generating radix-$n$ output is fairly trivial with a division and remainder algorithm.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_toradix}. \\
\textbf{Input}.   A mp\_int $a$ and an integer $r$\\
5886
5887
5888
5889
5890
5891
5892
5893
5894
5895
5896
5897
5898
5899
5900
5901
5902
5903
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_toradix}
\end{figure}
\textbf{Algorithm mp\_toradix.}
This algorithm computes the radix-$r$ representation of an mp\_int $a$.  The ``digits'' of the representation are extracted by reducing 
successive powers of $\lfloor a / r^k \rfloor$ the input modulo $r$ until $r^k > a$.  Note that instead of actually dividing by $r^k$ in
each iteration the quotient $\lfloor a / r \rfloor$ is saved for the next iteration.  As a result a series of trivial $n \times 1$ divisions
are required instead of a series of $n \times k$ divisions.  One design flaw of this approach is that the digits are produced in the reverse order 
(see~\ref{fig:mpradix}).  To remedy this flaw the digits must be swapped or simply ``reversed''.

\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline \textbf{Value of $a$} & \textbf{Value of $d$} & \textbf{Value of $str$} \\
\hline $1234$ & -- & -- \\







|


|







9506
9507
9508
9509
9510
9511
9512
9513
9514
9515
9516
9517
9518
9519
9520
9521
9522
9523
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_toradix}
\end{figure}
\textbf{Algorithm mp\_toradix.}
This algorithm computes the radix-$r$ representation of an mp\_int $a$.  The ``digits'' of the representation are extracted by reducing
successive powers of $\lfloor a / r^k \rfloor$ the input modulo $r$ until $r^k > a$.  Note that instead of actually dividing by $r^k$ in
each iteration the quotient $\lfloor a / r \rfloor$ is saved for the next iteration.  As a result a series of trivial $n \times 1$ divisions
are required instead of a series of $n \times k$ divisions.  One design flaw of this approach is that the digits are produced in the reverse order
(see~\ref{fig:mpradix}).  To remedy this flaw the digits must be swapped or simply ``reversed''.

\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline \textbf{Value of $a$} & \textbf{Value of $d$} & \textbf{Value of $str$} \\
\hline $1234$ & -- & -- \\
5912
5913
5914
5915
5916
5917
5918
























































5919
5920
5921
5922
5923
5924
5925
5926
5927
5928
5929
5930
5931
5932
5933
5934
5935
5936
5937
5938
5939
5940
\label{fig:mpradix}
\end{figure}

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_toradix.c
\vspace{-3mm}
\begin{alltt}
























































\end{alltt}
\end{small}

\chapter{Number Theoretic Algorithms}
This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi 
symbol computation.  These algorithms arise as essential components in several key cryptographic algorithms such as the RSA public key algorithm and
various Sieve based factoring algorithms.

\section{Greatest Common Divisor}
The greatest common divisor of two integers $a$ and $b$, often denoted as $(a, b)$ is the largest integer $k$ that is a proper divisor of
both $a$ and $b$.  That is, $k$ is the largest integer such that $0 \equiv a \mbox{ (mod }k\mbox{)}$ and $0 \equiv b \mbox{ (mod }k\mbox{)}$ occur
simultaneously.

The most common approach (cite) is to reduce one input modulo another.  That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then
$r$ is also divisible by $k$.  The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Greatest Common Divisor (I)}. \\
\textbf{Input}.   Two positive integers $a$ and $b$ greater than zero. \\







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9532
9533
9534
9535
9536
9537
9538
9539
9540
9541
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9544
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9601
9602
9603
9604
9605
9606
9607
9608
9609
9610
9611
9612
9613
9614
9615
9616
\label{fig:mpradix}
\end{figure}

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_toradix.c
\vspace{-3mm}
\begin{alltt}
016
017   /* stores a bignum as a ASCII string in a given radix (2..64) */
018   int mp_toradix (mp_int * a, char *str, int radix)
019   \{
020     int     res, digs;
021     mp_int  t;
022     mp_digit d;
023     char   *_s = str;
024
025     /* check range of the radix */
026     if ((radix < 2) || (radix > 64)) \{
027       return MP_VAL;
028     \}
029
030     /* quick out if its zero */
031     if (mp_iszero(a) == MP_YES) \{
032        *str++ = '0';
033        *str = '\symbol{92}0';
034        return MP_OKAY;
035     \}
036
037     if ((res = mp_init_copy (&t, a)) != MP_OKAY) \{
038       return res;
039     \}
040
041     /* if it is negative output a - */
042     if (t.sign == MP_NEG) \{
043       ++_s;
044       *str++ = '-';
045       t.sign = MP_ZPOS;
046     \}
047
048     digs = 0;
049     while (mp_iszero (&t) == MP_NO) \{
050       if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) \{
051         mp_clear (&t);
052         return res;
053       \}
054       *str++ = mp_s_rmap[d];
055       ++digs;
056     \}
057
058     /* reverse the digits of the string.  In this case _s points
059      * to the first digit [exluding the sign] of the number]
060      */
061     bn_reverse ((unsigned char *)_s, digs);
062
063     /* append a NULL so the string is properly terminated */
064     *str = '\symbol{92}0';
065
066     mp_clear (&t);
067     return MP_OKAY;
068   \}
069
070   #endif
071
\end{alltt}
\end{small}

\chapter{Number Theoretic Algorithms}
This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi
symbol computation.  These algorithms arise as essential components in several key cryptographic algorithms such as the RSA public key algorithm and
various Sieve based factoring algorithms.

\section{Greatest Common Divisor}
The greatest common divisor of two integers $a$ and $b$, often denoted as $(a, b)$ is the largest integer $k$ that is a proper divisor of
both $a$ and $b$.  That is, $k$ is the largest integer such that $0 \equiv a \mbox{ (mod }k\mbox{)}$ and $0 \equiv b \mbox{ (mod }k\mbox{)}$ occur
simultaneously.

The most common approach (cite) is to reduce one input modulo another.  That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then
$r$ is also divisible by $k$.  The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Greatest Common Divisor (I)}. \\
\textbf{Input}.   Two positive integers $a$ and $b$ greater than zero. \\
5950
5951
5952
5953
5954
5955
5956
5957
5958
5959
5960
5961
5962
5963
5964
5965
5966
\end{center}
\end{small}
\caption{Algorithm Greatest Common Divisor (I)}
\label{fig:gcd1}
\end{figure}

This algorithm will quickly converge on the greatest common divisor since the residue $r$ tends diminish rapidly.  However, divisions are
relatively expensive operations to perform and should ideally be avoided.  There is another approach based on a similar relationship of 
greatest common divisors.  The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$.  
In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Greatest Common Divisor (II)}. \\
\textbf{Input}.   Two positive integers $a$ and $b$ greater than zero. \\







|
|
|







9626
9627
9628
9629
9630
9631
9632
9633
9634
9635
9636
9637
9638
9639
9640
9641
9642
\end{center}
\end{small}
\caption{Algorithm Greatest Common Divisor (I)}
\label{fig:gcd1}
\end{figure}

This algorithm will quickly converge on the greatest common divisor since the residue $r$ tends diminish rapidly.  However, divisions are
relatively expensive operations to perform and should ideally be avoided.  There is another approach based on a similar relationship of
greatest common divisors.  The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$.
In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Greatest Common Divisor (II)}. \\
\textbf{Input}.   Two positive integers $a$ and $b$ greater than zero. \\
5976
5977
5978
5979
5980
5981
5982
5983
5984
5985
5986
5987
5988
5989
5990
5991
5992
5993
5994
5995
5996
5997
5998
5999
6000
\end{small}
\caption{Algorithm Greatest Common Divisor (II)}
\label{fig:gcd2}
\end{figure}

\textbf{Proof} \textit{Algorithm~\ref{fig:gcd2} will return the greatest common divisor of $a$ and $b$.}
The algorithm in figure~\ref{fig:gcd2} will eventually terminate since $b \ge a$ the subtraction in step 1.2 will be a value less than $b$.  In other
words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$.  Since both $a$ and $b$ are always 
divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the 
second to last iteration of the algorithm $b = a$ and clearly $(a, a) = a$ which concludes the proof.  \textbf{QED}.

As a matter of practicality algorithm \ref{fig:gcd1} decreases far too slowly to be useful.  Specially if $b$ is much larger than $a$ such that 
$b - a$ is still very much larger than $a$.  A simple addition to the algorithm is to divide $b - a$ by a power of some integer $p$ which does
not divide the greatest common divisor but will divide $b - a$.  In this case ${b - a} \over p$ is also an integer and still divisible by
the greatest common divisor.

However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first.  
Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Greatest Common Divisor (III)}. \\
\textbf{Input}.   Two positive integers $a$ and $b$ greater than zero. \\







|
|


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9658
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9671
9672
9673
9674
9675
9676
\end{small}
\caption{Algorithm Greatest Common Divisor (II)}
\label{fig:gcd2}
\end{figure}

\textbf{Proof} \textit{Algorithm~\ref{fig:gcd2} will return the greatest common divisor of $a$ and $b$.}
The algorithm in figure~\ref{fig:gcd2} will eventually terminate since $b \ge a$ the subtraction in step 1.2 will be a value less than $b$.  In other
words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$.  Since both $a$ and $b$ are always
divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the
second to last iteration of the algorithm $b = a$ and clearly $(a, a) = a$ which concludes the proof.  \textbf{QED}.

As a matter of practicality algorithm \ref{fig:gcd1} decreases far too slowly to be useful.  Specially if $b$ is much larger than $a$ such that
$b - a$ is still very much larger than $a$.  A simple addition to the algorithm is to divide $b - a$ by a power of some integer $p$ which does
not divide the greatest common divisor but will divide $b - a$.  In this case ${b - a} \over p$ is also an integer and still divisible by
the greatest common divisor.

However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first.
Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Greatest Common Divisor (III)}. \\
\textbf{Input}.   Two positive integers $a$ and $b$ greater than zero. \\
6019
6020
6021
6022
6023
6024
6025
6026
6027
6028
6029
6030
6031
6032
6033
6034
6035
6036
6037
6038
6039
6040
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Greatest Common Divisor (III)}
\label{fig:gcd3}
\end{figure}

This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$ 
decreases more rapidly.  The first loop on step two removes powers of $p$ that are in common.  A count, $k$, is kept which will present a common
divisor of $p^k$.  After step two the remaining common divisor of $a$ and $b$ cannot be divisible by $p$.  This means that $p$ can be safely 
divided out of the difference $b - a$ so long as the division leaves no remainder.  

In particular the value of $p$ should be chosen such that the division on step 5.3.1 occur often.  It also helps that division by $p$ be easy
to compute.  The ideal choice of $p$ is two since division by two amounts to a right logical shift.  Another important observation is that by
step five both $a$ and $b$ are odd.  Therefore, the diffrence $b - a$ must be even which means that each iteration removes one bit from the 
largest of the pair.

\subsection{Complete Greatest Common Divisor}
The algorithms presented so far cannot handle inputs which are zero or negative.  The following algorithm can handle all input cases properly
and will produce the greatest common divisor.

\newpage\begin{figure}[!here]







|

|
|



|







9695
9696
9697
9698
9699
9700
9701
9702
9703
9704
9705
9706
9707
9708
9709
9710
9711
9712
9713
9714
9715
9716
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Greatest Common Divisor (III)}
\label{fig:gcd3}
\end{figure}

This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$
decreases more rapidly.  The first loop on step two removes powers of $p$ that are in common.  A count, $k$, is kept which will present a common
divisor of $p^k$.  After step two the remaining common divisor of $a$ and $b$ cannot be divisible by $p$.  This means that $p$ can be safely
divided out of the difference $b - a$ so long as the division leaves no remainder.

In particular the value of $p$ should be chosen such that the division on step 5.3.1 occur often.  It also helps that division by $p$ be easy
to compute.  The ideal choice of $p$ is two since division by two amounts to a right logical shift.  Another important observation is that by
step five both $a$ and $b$ are odd.  Therefore, the diffrence $b - a$ must be even which means that each iteration removes one bit from the
largest of the pair.

\subsection{Complete Greatest Common Divisor}
The algorithms presented so far cannot handle inputs which are zero or negative.  The following algorithm can handle all input cases properly
and will produce the greatest common divisor.

\newpage\begin{figure}[!here]
6074
6075
6076
6077
6078
6079
6080
6081
6082
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6084
6085
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6102






















































































6103
6104
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6125
6126
6127
6128
6129
6130
6131
6132
6133
6134
6135
6136
\end{center}
\end{small}
\caption{Algorithm mp\_gcd}
\end{figure}
\textbf{Algorithm mp\_gcd.}
This algorithm will produce the greatest common divisor of two mp\_ints $a$ and $b$.  The algorithm was originally based on Algorithm B of
Knuth \cite[pp. 338]{TAOCPV2} but has been modified to be simpler to explain.  In theory it achieves the same asymptotic working time as
Algorithm B and in practice this appears to be true.  

The first two steps handle the cases where either one of or both inputs are zero.  If either input is zero the greatest common divisor is the 
largest input or zero if they are both zero.  If the inputs are not trivial than $u$ and $v$ are assigned the absolute values of 
$a$ and $b$ respectively and the algorithm will proceed to reduce the pair.

Step five will divide out any common factors of two and keep track of the count in the variable $k$.  After this step, two is no longer a
factor of the remaining greatest common divisor between $u$ and $v$ and can be safely evenly divided out of either whenever they are even.  Step 
six and seven ensure that the $u$ and $v$ respectively have no more factors of two.  At most only one of the while--loops will iterate since 
they cannot both be even.

By step eight both of $u$ and $v$ are odd which is required for the inner logic.  First the pair are swapped such that $v$ is equal to
or greater than $u$.  This ensures that the subtraction on step 8.2 will always produce a positive and even result.  Step 8.3 removes any
factors of two from the difference $u$ to ensure that in the next iteration of the loop both are once again odd.

After $v = 0$ occurs the variable $u$ has the greatest common divisor of the pair $\left < u, v \right >$ just after step six.  The result
must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_gcd.c
\vspace{-3mm}
\begin{alltt}






















































































\end{alltt}
\end{small}

This function makes use of the macros mp\_iszero and mp\_iseven.  The former evaluates to $1$ if the input mp\_int is equivalent to the 
integer zero otherwise it evaluates to $0$.  The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise
it evaluates to $0$.  Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero.  The three 
trivial cases of inputs are handled on lines 24 through 30.  After those lines the inputs are assumed to be non-zero.

Lines 32 and 37 make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively.  At this point the common factors of two 
must be divided out of the two inputs.  The block starting at line 44 removes common factors of two by first counting the number of trailing
zero bits in both.  The local integer $k$ is used to keep track of how many factors of $2$ are pulled out of both values.  It is assumed that 
the number of factors will not exceed the maximum value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than 
entries than are accessible by an ``int'' so this is not a limitation.}.  

At this point there are no more common factors of two in the two values.  The divisions by a power of two on lines 62 and 68 remove 
any independent factors of two such that both $u$ and $v$ are guaranteed to be an odd integer before hitting the main body of the algorithm.  The while loop
on line 73 performs the reduction of the pair until $v$ is equal to zero.  The unsigned comparison and subtraction algorithms are used in
place of the full signed routines since both values are guaranteed to be positive and the result of the subtraction is guaranteed to be non-negative.

\section{Least Common Multiple}
The least common multiple of a pair of integers is their product divided by their greatest common divisor.  For two integers $a$ and $b$ the
least common multiple is normally denoted as $[ a, b ]$ and numerically equivalent to ${ab} \over {(a, b)}$.  For example, if $a = 2 \cdot 2 \cdot 3 = 12$
and $b = 2 \cdot 3 \cdot 3 \cdot 7 = 126$ the least common multiple is ${126 \over {(12, 126)}} = {126 \over 6} = 21$.

The least common multiple arises often in coding theory as well as number theory.  If two functions have periods of $a$ and $b$ respectively they will
collide, that is be in synchronous states, after only $[ a, b ]$ iterations.  This is why, for example, random number generators based on 
Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}).  
Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_lcm}. \\







|

|
|



|
|







|





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9750
9751
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9891
9892
9893
9894
9895
9896
9897
9898
\end{center}
\end{small}
\caption{Algorithm mp\_gcd}
\end{figure}
\textbf{Algorithm mp\_gcd.}
This algorithm will produce the greatest common divisor of two mp\_ints $a$ and $b$.  The algorithm was originally based on Algorithm B of
Knuth \cite[pp. 338]{TAOCPV2} but has been modified to be simpler to explain.  In theory it achieves the same asymptotic working time as
Algorithm B and in practice this appears to be true.

The first two steps handle the cases where either one of or both inputs are zero.  If either input is zero the greatest common divisor is the
largest input or zero if they are both zero.  If the inputs are not trivial than $u$ and $v$ are assigned the absolute values of
$a$ and $b$ respectively and the algorithm will proceed to reduce the pair.

Step five will divide out any common factors of two and keep track of the count in the variable $k$.  After this step, two is no longer a
factor of the remaining greatest common divisor between $u$ and $v$ and can be safely evenly divided out of either whenever they are even.  Step
six and seven ensure that the $u$ and $v$ respectively have no more factors of two.  At most only one of the while--loops will iterate since
they cannot both be even.

By step eight both of $u$ and $v$ are odd which is required for the inner logic.  First the pair are swapped such that $v$ is equal to
or greater than $u$.  This ensures that the subtraction on step 8.2 will always produce a positive and even result.  Step 8.3 removes any
factors of two from the difference $u$ to ensure that in the next iteration of the loop both are once again odd.

After $v = 0$ occurs the variable $u$ has the greatest common divisor of the pair $\left < u, v \right >$ just after step six.  The result
must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_gcd.c
\vspace{-3mm}
\begin{alltt}
016
017   /* Greatest Common Divisor using the binary method */
018   int mp_gcd (mp_int * a, mp_int * b, mp_int * c)
019   \{
020     mp_int  u, v;
021     int     k, u_lsb, v_lsb, res;
022
023     /* either zero than gcd is the largest */
024     if (mp_iszero (a) == MP_YES) \{
025       return mp_abs (b, c);
026     \}
027     if (mp_iszero (b) == MP_YES) \{
028       return mp_abs (a, c);
029     \}
030
031     /* get copies of a and b we can modify */
032     if ((res = mp_init_copy (&u, a)) != MP_OKAY) \{
033       return res;
034     \}
035
036     if ((res = mp_init_copy (&v, b)) != MP_OKAY) \{
037       goto LBL_U;
038     \}
039
040     /* must be positive for the remainder of the algorithm */
041     u.sign = v.sign = MP_ZPOS;
042
043     /* B1.  Find the common power of two for u and v */
044     u_lsb = mp_cnt_lsb(&u);
045     v_lsb = mp_cnt_lsb(&v);
046     k     = MIN(u_lsb, v_lsb);
047
048     if (k > 0) \{
049        /* divide the power of two out */
050        if ((res = mp_div_2d(&u, k, &u, NULL)) != MP_OKAY) \{
051           goto LBL_V;
052        \}
053
054        if ((res = mp_div_2d(&v, k, &v, NULL)) != MP_OKAY) \{
055           goto LBL_V;
056        \}
057     \}
058
059     /* divide any remaining factors of two out */
060     if (u_lsb != k) \{
061        if ((res = mp_div_2d(&u, u_lsb - k, &u, NULL)) != MP_OKAY) \{
062           goto LBL_V;
063        \}
064     \}
065
066     if (v_lsb != k) \{
067        if ((res = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) \{
068           goto LBL_V;
069        \}
070     \}
071
072     while (mp_iszero(&v) == MP_NO) \{
073        /* make sure v is the largest */
074        if (mp_cmp_mag(&u, &v) == MP_GT) \{
075           /* swap u and v to make sure v is >= u */
076           mp_exch(&u, &v);
077        \}
078
079        /* subtract smallest from largest */
080        if ((res = s_mp_sub(&v, &u, &v)) != MP_OKAY) \{
081           goto LBL_V;
082        \}
083
084        /* Divide out all factors of two */
085        if ((res = mp_div_2d(&v, mp_cnt_lsb(&v), &v, NULL)) != MP_OKAY) \{
086           goto LBL_V;
087        \}
088     \}
089
090     /* multiply by 2**k which we divided out at the beginning */
091     if ((res = mp_mul_2d (&u, k, c)) != MP_OKAY) \{
092        goto LBL_V;
093     \}
094     c->sign = MP_ZPOS;
095     res = MP_OKAY;
096   LBL_V:mp_clear (&u);
097   LBL_U:mp_clear (&v);
098     return res;
099   \}
100   #endif
101
\end{alltt}
\end{small}

This function makes use of the macros mp\_iszero and mp\_iseven.  The former evaluates to $1$ if the input mp\_int is equivalent to the
integer zero otherwise it evaluates to $0$.  The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise
it evaluates to $0$.  Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero.  The three
trivial cases of inputs are handled on lines 23 through 29.  After those lines the inputs are assumed to be non-zero.

Lines 32 and 36 make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively.  At this point the common factors of two
must be divided out of the two inputs.  The block starting at line 43 removes common factors of two by first counting the number of trailing
zero bits in both.  The local integer $k$ is used to keep track of how many factors of $2$ are pulled out of both values.  It is assumed that
the number of factors will not exceed the maximum value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than
entries than are accessible by an ``int'' so this is not a limitation.}.

At this point there are no more common factors of two in the two values.  The divisions by a power of two on lines 61 and 67 remove
any independent factors of two such that both $u$ and $v$ are guaranteed to be an odd integer before hitting the main body of the algorithm.  The while loop
on line 72 performs the reduction of the pair until $v$ is equal to zero.  The unsigned comparison and subtraction algorithms are used in
place of the full signed routines since both values are guaranteed to be positive and the result of the subtraction is guaranteed to be non-negative.

\section{Least Common Multiple}
The least common multiple of a pair of integers is their product divided by their greatest common divisor.  For two integers $a$ and $b$ the
least common multiple is normally denoted as $[ a, b ]$ and numerically equivalent to ${ab} \over {(a, b)}$.  For example, if $a = 2 \cdot 2 \cdot 3 = 12$
and $b = 2 \cdot 3 \cdot 3 \cdot 7 = 126$ the least common multiple is ${126 \over {(12, 126)}} = {126 \over 6} = 21$.

The least common multiple arises often in coding theory as well as number theory.  If two functions have periods of $a$ and $b$ respectively they will
collide, that is be in synchronous states, after only $[ a, b ]$ iterations.  This is why, for example, random number generators based on
Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}).
Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_lcm}. \\
6151
6152
6153
6154
6155
6156
6157









































6158
6159
6160
6161
6162
6163
6164
6165
6166
6167
6168
6169
6170
6171
6172
6173
6174
6175
6176
6177
6178
6179
6180
6181
This algorithm computes the least common multiple of two mp\_int inputs $a$ and $b$.  It computes the least common multiple directly by
dividing the product of the two inputs by their greatest common divisor.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_lcm.c
\vspace{-3mm}
\begin{alltt}









































\end{alltt}
\end{small}

\section{Jacobi Symbol Computation}
To explain the Jacobi Symbol we shall first discuss the Legendre function\footnote{Arrg.  What is the name of this?} off which the Jacobi symbol is 
defined.  The Legendre function computes whether or not an integer $a$ is a quadratic residue modulo an odd prime $p$.  Numerically it is
equivalent to equation \ref{eqn:legendre}.

\textit{-- Tom, don't be an ass, cite your source here...!}

\begin{equation}
a^{(p-1)/2} \equiv \begin{array}{rl}
                              -1 &  \mbox{if }a\mbox{ is a quadratic non-residue.} \\
                              0  &  \mbox{if }a\mbox{ divides }p\mbox{.} \\
                              1  &  \mbox{if }a\mbox{ is a quadratic residue}. 
                              \end{array} \mbox{ (mod }p\mbox{)}
\label{eqn:legendre}                              
\end{equation}

\textbf{Proof.} \textit{Equation \ref{eqn:legendre} correctly identifies the residue status of an integer $a$ modulo a prime $p$.}
An integer $a$ is a quadratic residue if the following equation has a solution.

\begin{equation}
x^2 \equiv a \mbox{ (mod }p\mbox{)}







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9913
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This algorithm computes the least common multiple of two mp\_int inputs $a$ and $b$.  It computes the least common multiple directly by
dividing the product of the two inputs by their greatest common divisor.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_lcm.c
\vspace{-3mm}
\begin{alltt}
016
017   /* computes least common multiple as |a*b|/(a, b) */
018   int mp_lcm (mp_int * a, mp_int * b, mp_int * c)
019   \{
020     int     res;
021     mp_int  t1, t2;
022
023
024     if ((res = mp_init_multi (&t1, &t2, NULL)) != MP_OKAY) \{
025       return res;
026     \}
027
028     /* t1 = get the GCD of the two inputs */
029     if ((res = mp_gcd (a, b, &t1)) != MP_OKAY) \{
030       goto LBL_T;
031     \}
032
033     /* divide the smallest by the GCD */
034     if (mp_cmp_mag(a, b) == MP_LT) \{
035        /* store quotient in t2 such that t2 * b is the LCM */
036        if ((res = mp_div(a, &t1, &t2, NULL)) != MP_OKAY) \{
037           goto LBL_T;
038        \}
039        res = mp_mul(b, &t2, c);
040     \} else \{
041        /* store quotient in t2 such that t2 * a is the LCM */
042        if ((res = mp_div(b, &t1, &t2, NULL)) != MP_OKAY) \{
043           goto LBL_T;
044        \}
045        res = mp_mul(a, &t2, c);
046     \}
047
048     /* fix the sign to positive */
049     c->sign = MP_ZPOS;
050
051   LBL_T:
052     mp_clear_multi (&t1, &t2, NULL);
053     return res;
054   \}
055   #endif
056
\end{alltt}
\end{small}

\section{Jacobi Symbol Computation}
To explain the Jacobi Symbol we shall first discuss the Legendre function\footnote{Arrg.  What is the name of this?} off which the Jacobi symbol is
defined.  The Legendre function computes whether or not an integer $a$ is a quadratic residue modulo an odd prime $p$.  Numerically it is
equivalent to equation \ref{eqn:legendre}.

\textit{-- Tom, don't be an ass, cite your source here...!}

\begin{equation}
a^{(p-1)/2} \equiv \begin{array}{rl}
                              -1 &  \mbox{if }a\mbox{ is a quadratic non-residue.} \\
                              0  &  \mbox{if }a\mbox{ divides }p\mbox{.} \\
                              1  &  \mbox{if }a\mbox{ is a quadratic residue}.
                              \end{array} \mbox{ (mod }p\mbox{)}
\label{eqn:legendre}
\end{equation}

\textbf{Proof.} \textit{Equation \ref{eqn:legendre} correctly identifies the residue status of an integer $a$ modulo a prime $p$.}
An integer $a$ is a quadratic residue if the following equation has a solution.

\begin{equation}
x^2 \equiv a \mbox{ (mod }p\mbox{)}
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Whether equation \ref{eqn:root} has a solution or not equation \ref{eqn:rooti} is always true.  If $a^{(p-1)/2} - 1 \equiv 0 \mbox{ (mod }p\mbox{)}$
then the quantity in the braces must be zero.  By reduction,

\begin{eqnarray}
\left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \equiv 0  \nonumber \\
\left (x^2 \right )^{(p-1)/2} \equiv a^{(p-1)/2} \nonumber \\
x^2 \equiv a \mbox{ (mod }p\mbox{)} 
\end{eqnarray}

As a result there must be a solution to the quadratic equation and in turn $a$ must be a quadratic residue.  If $a$ does not divide $p$ and $a$
is not a quadratic residue then the only other value $a^{(p-1)/2}$ may be congruent to is $-1$ since
\begin{equation}
0 \equiv a^{p - 1} - 1 \equiv (a^{(p-1)/2} + 1)(a^{(p-1)/2} - 1) \mbox{ (mod }p\mbox{)}
\end{equation}
One of the terms on the right hand side must be zero.  \textbf{QED}

\subsection{Jacobi Symbol}
The Jacobi symbol is a generalization of the Legendre function for any odd non prime moduli $p$ greater than 2.  If $p = \prod_{i=0}^n p_i$ then
the Jacobi symbol $\left ( { a \over p } \right )$ is equal to the following equation.

\begin{equation}
\left ( { a \over p } \right ) = \left ( { a \over p_0} \right ) \left ( { a \over p_1} \right ) \ldots \left ( { a \over p_n} \right )
\end{equation}

By inspection if $p$ is prime the Jacobi symbol is equivalent to the Legendre function.  The following facts\footnote{See HAC \cite[pp. 72-74]{HAC} for
further details.} will be used to derive an efficient Jacobi symbol algorithm.  Where $p$ is an odd integer greater than two and $a, b \in \Z$ the
following are true.  

\begin{enumerate}
\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$. 
\item $\left ( { ab \over p} \right ) = \left ( { a \over p} \right )\left ( { b \over p} \right )$.
\item If $a \equiv b$ then $\left ( { a \over p} \right ) = \left ( { b \over p} \right )$.
\item $\left ( { 2 \over p} \right )$ equals $1$ if $p \equiv 1$ or $7 \mbox{ (mod }8\mbox{)}$.  Otherwise, it equals $-1$.
\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$.  More specifically 
$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$.  
\end{enumerate}

Using these facts if $a = 2^k \cdot a'$ then

\begin{eqnarray}
\left ( { a \over p } \right ) = \left ( {{2^k} \over p } \right ) \left ( {a' \over p} \right ) \nonumber \\
                               = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right ) 
\label{eqn:jacobi}
\end{eqnarray}

By fact five, 

\begin{equation}
\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} 
\end{equation}

Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then 

\begin{equation}
\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} 
\end{equation}

By putting both observations into equation \ref{eqn:jacobi} the following simplified equation is formed.

\begin{equation}
\left ( { a \over p } \right ) = \left ( {2 \over p } \right )^k \left ( {{p\mbox{ mod }a'} \over a'} \right )  \cdot (-1)^{(p-1)(a'-1)/4} 
\end{equation}

The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively.  The value of 
$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$.  Using this approach the 
factors of $p$ do not have to be known.  Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the 
Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_jacobi}. \\
\textbf{Input}.   mp\_int $a$ and $p$, $a \ge 0$, $p \ge 3$, $p \equiv 1 \mbox{ (mod }2\mbox{)}$ \\







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Whether equation \ref{eqn:root} has a solution or not equation \ref{eqn:rooti} is always true.  If $a^{(p-1)/2} - 1 \equiv 0 \mbox{ (mod }p\mbox{)}$
then the quantity in the braces must be zero.  By reduction,

\begin{eqnarray}
\left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \equiv 0  \nonumber \\
\left (x^2 \right )^{(p-1)/2} \equiv a^{(p-1)/2} \nonumber \\
x^2 \equiv a \mbox{ (mod }p\mbox{)}
\end{eqnarray}

As a result there must be a solution to the quadratic equation and in turn $a$ must be a quadratic residue.  If $a$ does not divide $p$ and $a$
is not a quadratic residue then the only other value $a^{(p-1)/2}$ may be congruent to is $-1$ since
\begin{equation}
0 \equiv a^{p - 1} - 1 \equiv (a^{(p-1)/2} + 1)(a^{(p-1)/2} - 1) \mbox{ (mod }p\mbox{)}
\end{equation}
One of the terms on the right hand side must be zero.  \textbf{QED}

\subsection{Jacobi Symbol}
The Jacobi symbol is a generalization of the Legendre function for any odd non prime moduli $p$ greater than 2.  If $p = \prod_{i=0}^n p_i$ then
the Jacobi symbol $\left ( { a \over p } \right )$ is equal to the following equation.

\begin{equation}
\left ( { a \over p } \right ) = \left ( { a \over p_0} \right ) \left ( { a \over p_1} \right ) \ldots \left ( { a \over p_n} \right )
\end{equation}

By inspection if $p$ is prime the Jacobi symbol is equivalent to the Legendre function.  The following facts\footnote{See HAC \cite[pp. 72-74]{HAC} for
further details.} will be used to derive an efficient Jacobi symbol algorithm.  Where $p$ is an odd integer greater than two and $a, b \in \Z$ the
following are true.

\begin{enumerate}
\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$.
\item $\left ( { ab \over p} \right ) = \left ( { a \over p} \right )\left ( { b \over p} \right )$.
\item If $a \equiv b$ then $\left ( { a \over p} \right ) = \left ( { b \over p} \right )$.
\item $\left ( { 2 \over p} \right )$ equals $1$ if $p \equiv 1$ or $7 \mbox{ (mod }8\mbox{)}$.  Otherwise, it equals $-1$.
\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$.  More specifically
$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$.
\end{enumerate}

Using these facts if $a = 2^k \cdot a'$ then

\begin{eqnarray}
\left ( { a \over p } \right ) = \left ( {{2^k} \over p } \right ) \left ( {a' \over p} \right ) \nonumber \\
                               = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right )
\label{eqn:jacobi}
\end{eqnarray}

By fact five,

\begin{equation}
\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4}
\end{equation}

Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then

\begin{equation}
\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4}
\end{equation}

By putting both observations into equation \ref{eqn:jacobi} the following simplified equation is formed.

\begin{equation}
\left ( { a \over p } \right ) = \left ( {2 \over p } \right )^k \left ( {{p\mbox{ mod }a'} \over a'} \right )  \cdot (-1)^{(p-1)(a'-1)/4}
\end{equation}

The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively.  The value of
$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$.  Using this approach the
factors of $p$ do not have to be known.  Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the
Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_jacobi}. \\
\textbf{Input}.   mp\_int $a$ and $p$, $a \ge 0$, $p \ge 3$, $p \equiv 1 \mbox{ (mod }2\mbox{)}$ \\
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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_jacobi}
\end{figure}
\textbf{Algorithm mp\_jacobi.}
This algorithm computes the Jacobi symbol for an arbitrary positive integer $a$ with respect to an odd integer $p$ greater than three.  The algorithm
is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}.  

Step numbers one and two handle the trivial cases of $a = 0$ and $a = 1$ respectively.  Step five determines the number of two factors in the
input $a$.  If $k$ is even than the term $\left ( { 2 \over p } \right )^k$ must always evaluate to one.  If $k$ is odd than the term evaluates to one 
if $p_0$ is congruent to one or seven modulo eight, otherwise it evaluates to $-1$. After the the $\left ( { 2 \over p } \right )^k$ term is handled 
the $(-1)^{(p-1)(a'-1)/4}$ is computed and multiplied against the current product $s$.  The latter term evaluates to one if both $p$ and $a'$ 
are congruent to one modulo four, otherwise it evaluates to negative one.

By step nine if $a'$ does not equal one a recursion is required.  Step 9.1 computes $p' \equiv p \mbox{ (mod }a'\mbox{)}$ and will recurse to compute
$\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi product.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_jacobi.c
\vspace{-3mm}
\begin{alltt}


































































































\end{alltt}
\end{small}

As a matter of practicality the variable $a'$ as per the pseudo-code is reprensented by the variable $a1$ since the $'$ symbol is not valid for a C 
variable name character. 

The two simple cases of $a = 0$ and $a = 1$ are handled at the very beginning to simplify the algorithm.  If the input is non-trivial the algorithm
has to proceed compute the Jacobi.  The variable $s$ is used to hold the current Jacobi product.  Note that $s$ is merely a C ``int'' data type since
the values it may obtain are merely $-1$, $0$ and $1$.  

After a local copy of $a$ is made all of the factors of two are divided out and the total stored in $k$.  Technically only the least significant
bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same 
processor requirements and neither is faster than the other.

Line 58 through 71 determines the value of $\left ( { 2 \over p } \right )^k$.  If the least significant bit of $k$ is zero than
$k$ is even and the value is one.  Otherwise, the value of $s$ depends on which residue class $p$ belongs to modulo eight.  The value of
$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines 71 through 74.  

Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$.  

\textit{-- Comment about default $s$ and such...}

\section{Modular Inverse}
\label{sec:modinv}
The modular inverse of a number actually refers to the modular multiplicative inverse.  Essentially for any integer $a$ such that $(a, p) = 1$ there
exist another integer $b$ such that $ab \equiv 1 \mbox{ (mod }p\mbox{)}$.  The integer $b$ is called the multiplicative inverse of $a$ which is
denoted as $b = a^{-1}$.  Technically speaking modular inversion is a well defined operation for any finite ring or field not just for rings and 
fields of integers.  However, the former will be the matter of discussion.

The simplest approach is to compute the algebraic inverse of the input.  That is to compute $b \equiv a^{\Phi(p) - 1}$.  If $\Phi(p)$ is the 
order of the multiplicative subgroup modulo $p$ then $b$ must be the multiplicative inverse of $a$.  The proof of which is trivial.

\begin{equation}
ab \equiv a \left (a^{\Phi(p) - 1} \right ) \equiv a^{\Phi(p)} \equiv a^0 \equiv 1 \mbox{ (mod }p\mbox{)}
\end{equation}

However, as simple as this approach may be it has two serious flaws.  It requires that the value of $\Phi(p)$ be known which if $p$ is composite 
requires all of the prime factors.  This approach also is very slow as the size of $p$ grows.  

A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear 
Diophantine\footnote{See LeVeque \cite[pp. 40-43]{LeVeque} for more information.} equation.

\begin{equation}
ab + pq = 1
\end{equation}

Where $a$, $b$, $p$ and $q$ are all integers.  If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of 
$a$ modulo $p$.  The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$.  
However, instead of using that algorithm directly a variant known as the binary Extended Euclidean algorithm will be used in its place.  The
binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine 
equation.  

\subsection{General Case}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_invmod}. \\







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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_jacobi}
\end{figure}
\textbf{Algorithm mp\_jacobi.}
This algorithm computes the Jacobi symbol for an arbitrary positive integer $a$ with respect to an odd integer $p$ greater than three.  The algorithm
is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}.

Step numbers one and two handle the trivial cases of $a = 0$ and $a = 1$ respectively.  Step five determines the number of two factors in the
input $a$.  If $k$ is even than the term $\left ( { 2 \over p } \right )^k$ must always evaluate to one.  If $k$ is odd than the term evaluates to one
if $p_0$ is congruent to one or seven modulo eight, otherwise it evaluates to $-1$. After the the $\left ( { 2 \over p } \right )^k$ term is handled
the $(-1)^{(p-1)(a'-1)/4}$ is computed and multiplied against the current product $s$.  The latter term evaluates to one if both $p$ and $a'$
are congruent to one modulo four, otherwise it evaluates to negative one.

By step nine if $a'$ does not equal one a recursion is required.  Step 9.1 computes $p' \equiv p \mbox{ (mod }a'\mbox{)}$ and will recurse to compute
$\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi product.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_jacobi.c
\vspace{-3mm}
\begin{alltt}
016
017   /* computes the jacobi c = (a | n) (or Legendre if n is prime)
018    * HAC pp. 73 Algorithm 2.149
019    * HAC is wrong here, as the special case of (0 | 1) is not
020    * handled correctly.
021    */
022   int mp_jacobi (mp_int * a, mp_int * n, int *c)
023   \{
024     mp_int  a1, p1;
025     int     k, s, r, res;
026     mp_digit residue;
027
028     /* if a < 0 return MP_VAL */
029     if (mp_isneg(a) == MP_YES) \{
030        return MP_VAL;
031     \}
032
033     /* if n <= 0 return MP_VAL */
034     if (mp_cmp_d(n, 0) != MP_GT) \{
035        return MP_VAL;
036     \}
037
038     /* step 1. handle case of a == 0 */
039     if (mp_iszero (a) == MP_YES) \{
040        /* special case of a == 0 and n == 1 */
041        if (mp_cmp_d (n, 1) == MP_EQ) \{
042          *c = 1;
043        \} else \{
044          *c = 0;
045        \}
046        return MP_OKAY;
047     \}
048
049     /* step 2.  if a == 1, return 1 */
050     if (mp_cmp_d (a, 1) == MP_EQ) \{
051       *c = 1;
052       return MP_OKAY;
053     \}
054
055     /* default */
056     s = 0;
057
058     /* step 3.  write a = a1 * 2**k  */
059     if ((res = mp_init_copy (&a1, a)) != MP_OKAY) \{
060       return res;
061     \}
062
063     if ((res = mp_init (&p1)) != MP_OKAY) \{
064       goto LBL_A1;
065     \}
066
067     /* divide out larger power of two */
068     k = mp_cnt_lsb(&a1);
069     if ((res = mp_div_2d(&a1, k, &a1, NULL)) != MP_OKAY) \{
070        goto LBL_P1;
071     \}
072
073     /* step 4.  if e is even set s=1 */
074     if ((k & 1) == 0) \{
075       s = 1;
076     \} else \{
077       /* else set s=1 if p = 1/7 (mod 8) or s=-1 if p = 3/5 (mod 8) */
078       residue = n->dp[0] & 7;
079
080       if ((residue == 1) || (residue == 7)) \{
081         s = 1;
082       \} else if ((residue == 3) || (residue == 5)) \{
083         s = -1;
084       \}
085     \}
086
087     /* step 5.  if p == 3 (mod 4) *and* a1 == 3 (mod 4) then s = -s */
088     if ( ((n->dp[0] & 3) == 3) && ((a1.dp[0] & 3) == 3)) \{
089       s = -s;
090     \}
091
092     /* if a1 == 1 we're done */
093     if (mp_cmp_d (&a1, 1) == MP_EQ) \{
094       *c = s;
095     \} else \{
096       /* n1 = n mod a1 */
097       if ((res = mp_mod (n, &a1, &p1)) != MP_OKAY) \{
098         goto LBL_P1;
099       \}
100       if ((res = mp_jacobi (&p1, &a1, &r)) != MP_OKAY) \{
101         goto LBL_P1;
102       \}
103       *c = s * r;
104     \}
105
106     /* done */
107     res = MP_OKAY;
108   LBL_P1:mp_clear (&p1);
109   LBL_A1:mp_clear (&a1);
110     return res;
111   \}
112   #endif
113
\end{alltt}
\end{small}

As a matter of practicality the variable $a'$ as per the pseudo-code is reprensented by the variable $a1$ since the $'$ symbol is not valid for a C
variable name character.

The two simple cases of $a = 0$ and $a = 1$ are handled at the very beginning to simplify the algorithm.  If the input is non-trivial the algorithm
has to proceed compute the Jacobi.  The variable $s$ is used to hold the current Jacobi product.  Note that $s$ is merely a C ``int'' data type since
the values it may obtain are merely $-1$, $0$ and $1$.

After a local copy of $a$ is made all of the factors of two are divided out and the total stored in $k$.  Technically only the least significant
bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same
processor requirements and neither is faster than the other.

Line 59 through 71 determines the value of $\left ( { 2 \over p } \right )^k$.  If the least significant bit of $k$ is zero than
$k$ is even and the value is one.  Otherwise, the value of $s$ depends on which residue class $p$ belongs to modulo eight.  The value of
$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines 73 through 76.

Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$.

\textit{-- Comment about default $s$ and such...}

\section{Modular Inverse}
\label{sec:modinv}
The modular inverse of a number actually refers to the modular multiplicative inverse.  Essentially for any integer $a$ such that $(a, p) = 1$ there
exist another integer $b$ such that $ab \equiv 1 \mbox{ (mod }p\mbox{)}$.  The integer $b$ is called the multiplicative inverse of $a$ which is
denoted as $b = a^{-1}$.  Technically speaking modular inversion is a well defined operation for any finite ring or field not just for rings and
fields of integers.  However, the former will be the matter of discussion.

The simplest approach is to compute the algebraic inverse of the input.  That is to compute $b \equiv a^{\Phi(p) - 1}$.  If $\Phi(p)$ is the
order of the multiplicative subgroup modulo $p$ then $b$ must be the multiplicative inverse of $a$.  The proof of which is trivial.

\begin{equation}
ab \equiv a \left (a^{\Phi(p) - 1} \right ) \equiv a^{\Phi(p)} \equiv a^0 \equiv 1 \mbox{ (mod }p\mbox{)}
\end{equation}

However, as simple as this approach may be it has two serious flaws.  It requires that the value of $\Phi(p)$ be known which if $p$ is composite
requires all of the prime factors.  This approach also is very slow as the size of $p$ grows.

A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear
Diophantine\footnote{See LeVeque \cite[pp. 40-43]{LeVeque} for more information.} equation.

\begin{equation}
ab + pq = 1
\end{equation}

Where $a$, $b$, $p$ and $q$ are all integers.  If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of
$a$ modulo $p$.  The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$.
However, instead of using that algorithm directly a variant known as the binary Extended Euclidean algorithm will be used in its place.  The
binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine
equation.

\subsection{General Case}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_invmod}. \\
6412
6413
6414
6415
6416
6417
6418
6419
6420
6421
6422
6423
6424
6425
6426
6427
6428
6429
6430
6431
6432
6433
6434
6435
6436
6437
6438
6439
6440
6441
























6442
6443
6444
6445
6446
6447
6448
6449
6450
6451
6452
6453
6454
6455
6456
6457
6458
6459
6460
6461
6462
6463
6464
6465
6466
6467
6468
6469
6470
6471
6472
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6474
6475
6476
6477
6478
6479
6480
6481
6482
6483
6484
6485
6486
6487
6488
6489
6490
15.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\end{figure}
\textbf{Algorithm mp\_invmod.}
This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$.  This algorithm is a variation of the 
extended binary Euclidean algorithm from HAC \cite[pp. 608]{HAC}.  It has been modified to only compute the modular inverse and not a complete
Diophantine solution.  

If $b \le 0$ than the modulus is invalid and MP\_VAL is returned.  Similarly if both $a$ and $b$ are even then there cannot be a multiplicative
inverse for $a$ and the error is reported.  

The astute reader will observe that steps seven through nine are very similar to the binary greatest common divisor algorithm mp\_gcd.  In this case
the other variables to the Diophantine equation are solved.  The algorithm terminates when $u = 0$ in which case the solution is

\begin{equation}
Ca + Db = v
\end{equation}

If $v$, the greatest common divisor of $a$ and $b$ is not equal to one then the algorithm will report an error as no inverse exists.  Otherwise, $C$
is the modular inverse of $a$.  The actual value of $C$ is congruent to, but not necessarily equal to, the ideal modular inverse which should lie 
within $1 \le a^{-1} < b$.  Step numbers twelve and thirteen adjust the inverse until it is in range.  If the original input $a$ is within $0 < a < p$ 
then only a couple of additions or subtractions will be required to adjust the inverse.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_invmod.c
\vspace{-3mm}
\begin{alltt}
























\end{alltt}
\end{small}

\subsubsection{Odd Moduli}

When the modulus $b$ is odd the variables $A$ and $C$ are fixed and are not required to compute the inverse.  In particular by attempting to solve
the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$.  

The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed.  This 
optimization will halve the time required to compute the modular inverse.

\section{Primality Tests}

A non-zero integer $a$ is said to be prime if it is not divisible by any other integer excluding one and itself.  For example, $a = 7$ is prime 
since the integers $2 \ldots 6$ do not evenly divide $a$.  By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$. 

Prime numbers arise in cryptography considerably as they allow finite fields to be formed.  The ability to determine whether an integer is prime or
not quickly has been a viable subject in cryptography and number theory for considerable time.  The algorithms that will be presented are all
probablistic algorithms in that when they report an integer is composite it must be composite.  However, when the algorithms report an integer is
prime the algorithm may be incorrect.  

As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as 
well be zero.  For the purposes of these discussions let $n$ represent the candidate integer of which the primality is in question.

\subsection{Trial Division}

Trial division means to attempt to evenly divide a candidate integer by small prime integers.  If the candidate can be evenly divided it obviously
cannot be prime.  By dividing by all primes $1 < p \le \sqrt{n}$ this test can actually prove whether an integer is prime.  However, such a test
would require a prohibitive amount of time as $n$ grows.

Instead of dividing by every prime, a smaller, more mangeable set of primes may be used instead.  By performing trial division with only a subset
of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate is prime.  However, often it can prove a candidate is not prime.

The benefit of this test is that trial division by small values is fairly efficient.  Specially compared to the other algorithms that will be
discussed shortly.  The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by
$1 - {1.12 \over ln(q)}$.  The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range 
$3 \le q \le 100$.  

At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly.  At $q = 90$ further testing is generally not going to 
be of any practical use.  In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate 
approximately $80\%$ of all candidate integers.  The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base.  The 
array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_prime\_is\_divisible}. \\
\textbf{Input}.   mp\_int $a$ \\







|

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10313
10314
10315
10316
10317
10318
10319
10320
10321
10322
10323
10324
10325
10326
10327
10328
10329
10330
10331
10332
10333
10334
10335
10336
10337
10338
10339
10340
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10361
10362
10363
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10366
10367
10368
10369
10370
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10372
10373
10374
10375
10376
10377
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10379
10380
10381
10382
10383
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10386
10387
10388
10389
10390
10391
10392
10393
10394
10395
10396
10397
10398
10399
10400
10401
10402
10403
10404
10405
10406
10407
10408
10409
10410
10411
10412
10413
10414
10415
15.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\end{figure}
\textbf{Algorithm mp\_invmod.}
This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$.  This algorithm is a variation of the
extended binary Euclidean algorithm from HAC \cite[pp. 608]{HAC}.  It has been modified to only compute the modular inverse and not a complete
Diophantine solution.

If $b \le 0$ than the modulus is invalid and MP\_VAL is returned.  Similarly if both $a$ and $b$ are even then there cannot be a multiplicative
inverse for $a$ and the error is reported.

The astute reader will observe that steps seven through nine are very similar to the binary greatest common divisor algorithm mp\_gcd.  In this case
the other variables to the Diophantine equation are solved.  The algorithm terminates when $u = 0$ in which case the solution is

\begin{equation}
Ca + Db = v
\end{equation}

If $v$, the greatest common divisor of $a$ and $b$ is not equal to one then the algorithm will report an error as no inverse exists.  Otherwise, $C$
is the modular inverse of $a$.  The actual value of $C$ is congruent to, but not necessarily equal to, the ideal modular inverse which should lie
within $1 \le a^{-1} < b$.  Step numbers twelve and thirteen adjust the inverse until it is in range.  If the original input $a$ is within $0 < a < p$
then only a couple of additions or subtractions will be required to adjust the inverse.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_invmod.c
\vspace{-3mm}
\begin{alltt}
016
017   /* hac 14.61, pp608 */
018   int mp_invmod (mp_int * a, mp_int * b, mp_int * c)
019   \{
020     /* b cannot be negative */
021     if ((b->sign == MP_NEG) || (mp_iszero(b) == MP_YES)) \{
022       return MP_VAL;
023     \}
024
025   #ifdef BN_FAST_MP_INVMOD_C
026     /* if the modulus is odd we can use a faster routine instead */
027     if (mp_isodd (b) == MP_YES) \{
028       return fast_mp_invmod (a, b, c);
029     \}
030   #endif
031
032   #ifdef BN_MP_INVMOD_SLOW_C
033     return mp_invmod_slow(a, b, c);
034   #else
035     return MP_VAL;
036   #endif
037   \}
038   #endif
039
\end{alltt}
\end{small}

\subsubsection{Odd Moduli}

When the modulus $b$ is odd the variables $A$ and $C$ are fixed and are not required to compute the inverse.  In particular by attempting to solve
the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$.

The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed.  This
optimization will halve the time required to compute the modular inverse.

\section{Primality Tests}

A non-zero integer $a$ is said to be prime if it is not divisible by any other integer excluding one and itself.  For example, $a = 7$ is prime
since the integers $2 \ldots 6$ do not evenly divide $a$.  By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$.

Prime numbers arise in cryptography considerably as they allow finite fields to be formed.  The ability to determine whether an integer is prime or
not quickly has been a viable subject in cryptography and number theory for considerable time.  The algorithms that will be presented are all
probablistic algorithms in that when they report an integer is composite it must be composite.  However, when the algorithms report an integer is
prime the algorithm may be incorrect.

As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as
well be zero.  For the purposes of these discussions let $n$ represent the candidate integer of which the primality is in question.

\subsection{Trial Division}

Trial division means to attempt to evenly divide a candidate integer by small prime integers.  If the candidate can be evenly divided it obviously
cannot be prime.  By dividing by all primes $1 < p \le \sqrt{n}$ this test can actually prove whether an integer is prime.  However, such a test
would require a prohibitive amount of time as $n$ grows.

Instead of dividing by every prime, a smaller, more mangeable set of primes may be used instead.  By performing trial division with only a subset
of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate is prime.  However, often it can prove a candidate is not prime.

The benefit of this test is that trial division by small values is fairly efficient.  Specially compared to the other algorithms that will be
discussed shortly.  The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by
$1 - {1.12 \over ln(q)}$.  The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range
$3 \le q \le 100$.

At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly.  At $q = 90$ further testing is generally not going to
be of any practical use.  In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate
approximately $80\%$ of all candidate integers.  The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base.  The
array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_prime\_is\_divisible}. \\
\textbf{Input}.   mp\_int $a$ \\
6500
6501
6502
6503
6504
6505
6506
6507
6508
6509
6510
6511
6512































6513
6514
6515
6516
6517
6518
6519
6520
6521
6522










































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6524
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6526
6527
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6530
6531
6532
6533
6534
6535
6536
6537
6538
6539
6540
6541
6542
6543
6544
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_prime\_is\_divisible}
\end{figure}
\textbf{Algorithm mp\_prime\_is\_divisible.}
This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_is\_divisible.c
\vspace{-3mm}
\begin{alltt}































\end{alltt}
\end{small}

The algorithm defaults to a return of $0$ in case an error occurs.  The values in the prime table are all specified to be in the range of a 
mp\_digit.  The table \_\_prime\_tab is defined in the following file.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_prime\_tab.c
\vspace{-3mm}
\begin{alltt}










































\end{alltt}
\end{small}

Note that there are two possible tables.  When an mp\_digit is 7-bits long only the primes upto $127$ may be included, otherwise the primes
upto $1619$ are used.  Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit. 

\subsection{The Fermat Test}
The Fermat test is probably one the oldest tests to have a non-trivial probability of success.  It is based on the fact that if $n$ is in 
fact prime then $a^{n} \equiv a \mbox{ (mod }n\mbox{)}$ for all $0 < a < n$.  The reason being that if $n$ is prime than the order of
the multiplicative sub group is $n - 1$.  Any base $a$ must have an order which divides $n - 1$ and as such $a^n$ is equivalent to 
$a^1 = a$.  

If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$.  In which case 
it is possible that $a^n \nequiv a \mbox{ (mod }n\mbox{)}$.  However, this test is not absolute as it is possible that the order
of a base will divide $n - 1$ which would then be reported as prime.  Such a base yields what is known as a Fermat pseudo-prime.  Several 
integers known as Carmichael numbers will be a pseudo-prime to all valid bases.  Fortunately such numbers are extremely rare as $n$ grows
in size.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}







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10509
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10529
10530
10531
10532
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10534
10535
10536
10537
10538
10539
10540
10541
10542
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_prime\_is\_divisible}
\end{figure}
\textbf{Algorithm mp\_prime\_is\_divisible.}
This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_is\_divisible.c
\vspace{-3mm}
\begin{alltt}
016
017   /* determines if an integers is divisible by one
018    * of the first PRIME_SIZE primes or not
019    *
020    * sets result to 0 if not, 1 if yes
021    */
022   int mp_prime_is_divisible (mp_int * a, int *result)
023   \{
024     int     err, ix;
025     mp_digit res;
026
027     /* default to not */
028     *result = MP_NO;
029
030     for (ix = 0; ix < PRIME_SIZE; ix++) \{
031       /* what is a mod LBL_prime_tab[ix] */
032       if ((err = mp_mod_d (a, ltm_prime_tab[ix], &res)) != MP_OKAY) \{
033         return err;
034       \}
035
036       /* is the residue zero? */
037       if (res == 0) \{
038         *result = MP_YES;
039         return MP_OKAY;
040       \}
041     \}
042
043     return MP_OKAY;
044   \}
045   #endif
046
\end{alltt}
\end{small}

The algorithm defaults to a return of $0$ in case an error occurs.  The values in the prime table are all specified to be in the range of a
mp\_digit.  The table \_\_prime\_tab is defined in the following file.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_prime\_tab.c
\vspace{-3mm}
\begin{alltt}
016   const mp_digit ltm_prime_tab[] = \{
017     0x0002, 0x0003, 0x0005, 0x0007, 0x000B, 0x000D, 0x0011, 0x0013,
018     0x0017, 0x001D, 0x001F, 0x0025, 0x0029, 0x002B, 0x002F, 0x0035,
019     0x003B, 0x003D, 0x0043, 0x0047, 0x0049, 0x004F, 0x0053, 0x0059,
020     0x0061, 0x0065, 0x0067, 0x006B, 0x006D, 0x0071, 0x007F,
021   #ifndef MP_8BIT
022     0x0083,
023     0x0089, 0x008B, 0x0095, 0x0097, 0x009D, 0x00A3, 0x00A7, 0x00AD,
024     0x00B3, 0x00B5, 0x00BF, 0x00C1, 0x00C5, 0x00C7, 0x00D3, 0x00DF,
025     0x00E3, 0x00E5, 0x00E9, 0x00EF, 0x00F1, 0x00FB, 0x0101, 0x0107,
026     0x010D, 0x010F, 0x0115, 0x0119, 0x011B, 0x0125, 0x0133, 0x0137,
027
028     0x0139, 0x013D, 0x014B, 0x0151, 0x015B, 0x015D, 0x0161, 0x0167,
029     0x016F, 0x0175, 0x017B, 0x017F, 0x0185, 0x018D, 0x0191, 0x0199,
030     0x01A3, 0x01A5, 0x01AF, 0x01B1, 0x01B7, 0x01BB, 0x01C1, 0x01C9,
031     0x01CD, 0x01CF, 0x01D3, 0x01DF, 0x01E7, 0x01EB, 0x01F3, 0x01F7,
032     0x01FD, 0x0209, 0x020B, 0x021D, 0x0223, 0x022D, 0x0233, 0x0239,
033     0x023B, 0x0241, 0x024B, 0x0251, 0x0257, 0x0259, 0x025F, 0x0265,
034     0x0269, 0x026B, 0x0277, 0x0281, 0x0283, 0x0287, 0x028D, 0x0293,
035     0x0295, 0x02A1, 0x02A5, 0x02AB, 0x02B3, 0x02BD, 0x02C5, 0x02CF,
036
037     0x02D7, 0x02DD, 0x02E3, 0x02E7, 0x02EF, 0x02F5, 0x02F9, 0x0301,
038     0x0305, 0x0313, 0x031D, 0x0329, 0x032B, 0x0335, 0x0337, 0x033B,
039     0x033D, 0x0347, 0x0355, 0x0359, 0x035B, 0x035F, 0x036D, 0x0371,
040     0x0373, 0x0377, 0x038B, 0x038F, 0x0397, 0x03A1, 0x03A9, 0x03AD,
041     0x03B3, 0x03B9, 0x03C7, 0x03CB, 0x03D1, 0x03D7, 0x03DF, 0x03E5,
042     0x03F1, 0x03F5, 0x03FB, 0x03FD, 0x0407, 0x0409, 0x040F, 0x0419,
043     0x041B, 0x0425, 0x0427, 0x042D, 0x043F, 0x0443, 0x0445, 0x0449,
044     0x044F, 0x0455, 0x045D, 0x0463, 0x0469, 0x047F, 0x0481, 0x048B,
045
046     0x0493, 0x049D, 0x04A3, 0x04A9, 0x04B1, 0x04BD, 0x04C1, 0x04C7,
047     0x04CD, 0x04CF, 0x04D5, 0x04E1, 0x04EB, 0x04FD, 0x04FF, 0x0503,
048     0x0509, 0x050B, 0x0511, 0x0515, 0x0517, 0x051B, 0x0527, 0x0529,
049     0x052F, 0x0551, 0x0557, 0x055D, 0x0565, 0x0577, 0x0581, 0x058F,
050     0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3,
051     0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7,
052     0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623,
053     0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653
054   #endif
055   \};
056   #endif
057
\end{alltt}
\end{small}

Note that there are two possible tables.  When an mp\_digit is 7-bits long only the primes upto $127$ may be included, otherwise the primes
upto $1619$ are used.  Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit.

\subsection{The Fermat Test}
The Fermat test is probably one the oldest tests to have a non-trivial probability of success.  It is based on the fact that if $n$ is in
fact prime then $a^{n} \equiv a \mbox{ (mod }n\mbox{)}$ for all $0 < a < n$.  The reason being that if $n$ is prime than the order of
the multiplicative sub group is $n - 1$.  Any base $a$ must have an order which divides $n - 1$ and as such $a^n$ is equivalent to
$a^1 = a$.

If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$.  In which case
it is possible that $a^n \nequiv a \mbox{ (mod }n\mbox{)}$.  However, this test is not absolute as it is possible that the order
of a base will divide $n - 1$ which would then be reported as prime.  Such a base yields what is known as a Fermat pseudo-prime.  Several
integers known as Carmichael numbers will be a pseudo-prime to all valid bases.  Fortunately such numbers are extremely rare as $n$ grows
in size.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
6556
6557
6558
6559
6560
6561
6562
6563
6564
6565
6566
6567
6568











































6569
6570
6571
6572
6573
6574
6575
6576
6577
6578
6579
6580
6581
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_prime\_fermat}
\end{figure}
\textbf{Algorithm mp\_prime\_fermat.}
This algorithm determines whether an mp\_int $a$ is a Fermat prime to the base $b$ or not.  It uses a single modular exponentiation to
determine the result.  

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_fermat.c
\vspace{-3mm}
\begin{alltt}











































\end{alltt}
\end{small}

\subsection{The Miller-Rabin Test}
The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen 
candidate  integers.  The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the 
value must be equal to $-1$.  The squarings are stopped as soon as $-1$ is observed.  If the value of $1$ is observed first it means that
some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}







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10554
10555
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10560
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10564
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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_prime\_fermat}
\end{figure}
\textbf{Algorithm mp\_prime\_fermat.}
This algorithm determines whether an mp\_int $a$ is a Fermat prime to the base $b$ or not.  It uses a single modular exponentiation to
determine the result.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_fermat.c
\vspace{-3mm}
\begin{alltt}
016
017   /* performs one Fermat test.
018    *
019    * If "a" were prime then b**a == b (mod a) since the order of
020    * the multiplicative sub-group would be phi(a) = a-1.  That means
021    * it would be the same as b**(a mod (a-1)) == b**1 == b (mod a).
022    *
023    * Sets result to 1 if the congruence holds, or zero otherwise.
024    */
025   int mp_prime_fermat (mp_int * a, mp_int * b, int *result)
026   \{
027     mp_int  t;
028     int     err;
029
030     /* default to composite  */
031     *result = MP_NO;
032
033     /* ensure b > 1 */
034     if (mp_cmp_d(b, 1) != MP_GT) \{
035        return MP_VAL;
036     \}
037
038     /* init t */
039     if ((err = mp_init (&t)) != MP_OKAY) \{
040       return err;
041     \}
042
043     /* compute t = b**a mod a */
044     if ((err = mp_exptmod (b, a, a, &t)) != MP_OKAY) \{
045       goto LBL_T;
046     \}
047
048     /* is it equal to b? */
049     if (mp_cmp (&t, b) == MP_EQ) \{
050       *result = MP_YES;
051     \}
052
053     err = MP_OKAY;
054   LBL_T:mp_clear (&t);
055     return err;
056   \}
057   #endif
058
\end{alltt}
\end{small}

\subsection{The Miller-Rabin Test}
The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen
candidate  integers.  The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the
value must be equal to $-1$.  The squarings are stopped as soon as $-1$ is observed.  If the value of $1$ is observed first it means that
some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
6603
6604
6605
6606
6607
6608
6609
6610
6611
6612
6613
6614
6615
6616
6617
6618
6619
6620




















































































6621
6622
6623
6624
6625
6626
6627
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_prime\_miller\_rabin}
\end{figure}
\textbf{Algorithm mp\_prime\_miller\_rabin.}
This algorithm performs one trial round of the Miller-Rabin algorithm to the base $b$.  It will set $c = 1$ if the algorithm cannot determine
if $b$ is composite or $c = 0$ if $b$ is provably composite.  The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$.  

If the value $y \equiv b^r$ is congruent to $\pm 1$ then the algorithm cannot prove if $a$ is composite or not.  Otherwise, the algorithm will
square $y$ upto $s - 1$ times stopping only when $y \equiv -1$.  If $y^2 \equiv 1$ and $y \nequiv \pm 1$ then the algorithm can report that $a$
is provably composite.  If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite.  If $a$ is not provably 
composite then it is \textit{probably} prime.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_miller\_rabin.c
\vspace{-3mm}
\begin{alltt}




















































































\end{alltt}
\end{small}




\backmatter







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\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_prime\_miller\_rabin}
\end{figure}
\textbf{Algorithm mp\_prime\_miller\_rabin.}
This algorithm performs one trial round of the Miller-Rabin algorithm to the base $b$.  It will set $c = 1$ if the algorithm cannot determine
if $b$ is composite or $c = 0$ if $b$ is provably composite.  The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$.

If the value $y \equiv b^r$ is congruent to $\pm 1$ then the algorithm cannot prove if $a$ is composite or not.  Otherwise, the algorithm will
square $y$ upto $s - 1$ times stopping only when $y \equiv -1$.  If $y^2 \equiv 1$ and $y \nequiv \pm 1$ then the algorithm can report that $a$
is provably composite.  If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite.  If $a$ is not provably
composite then it is \textit{probably} prime.

\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_miller\_rabin.c
\vspace{-3mm}
\begin{alltt}
016
017   /* Miller-Rabin test of "a" to the base of "b" as described in
018    * HAC pp. 139 Algorithm 4.24
019    *
020    * Sets result to 0 if definitely composite or 1 if probably prime.
021    * Randomly the chance of error is no more than 1/4 and often
022    * very much lower.
023    */
024   int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result)
025   \{
026     mp_int  n1, y, r;
027     int     s, j, err;
028
029     /* default */
030     *result = MP_NO;
031
032     /* ensure b > 1 */
033     if (mp_cmp_d(b, 1) != MP_GT) \{
034        return MP_VAL;
035     \}
036
037     /* get n1 = a - 1 */
038     if ((err = mp_init_copy (&n1, a)) != MP_OKAY) \{
039       return err;
040     \}
041     if ((err = mp_sub_d (&n1, 1, &n1)) != MP_OKAY) \{
042       goto LBL_N1;
043     \}
044
045     /* set 2**s * r = n1 */
046     if ((err = mp_init_copy (&r, &n1)) != MP_OKAY) \{
047       goto LBL_N1;
048     \}
049
050     /* count the number of least significant bits
051      * which are zero
052      */
053     s = mp_cnt_lsb(&r);
054
055     /* now divide n - 1 by 2**s */
056     if ((err = mp_div_2d (&r, s, &r, NULL)) != MP_OKAY) \{
057       goto LBL_R;
058     \}
059
060     /* compute y = b**r mod a */
061     if ((err = mp_init (&y)) != MP_OKAY) \{
062       goto LBL_R;
063     \}
064     if ((err = mp_exptmod (b, &r, a, &y)) != MP_OKAY) \{
065       goto LBL_Y;
066     \}
067
068     /* if y != 1 and y != n1 do */
069     if ((mp_cmp_d (&y, 1) != MP_EQ) && (mp_cmp (&y, &n1) != MP_EQ)) \{
070       j = 1;
071       /* while j <= s-1 and y != n1 */
072       while ((j <= (s - 1)) && (mp_cmp (&y, &n1) != MP_EQ)) \{
073         if ((err = mp_sqrmod (&y, a, &y)) != MP_OKAY) \{
074            goto LBL_Y;
075         \}
076
077         /* if y == 1 then composite */
078         if (mp_cmp_d (&y, 1) == MP_EQ) \{
079            goto LBL_Y;
080         \}
081
082         ++j;
083       \}
084
085       /* if y != n1 then composite */
086       if (mp_cmp (&y, &n1) != MP_EQ) \{
087         goto LBL_Y;
088       \}
089     \}
090
091     /* probably prime now */
092     *result = MP_YES;
093   LBL_Y:mp_clear (&y);
094   LBL_R:mp_clear (&r);
095   LBL_N1:mp_clear (&n1);
096     return err;
097   \}
098   #endif
099
\end{alltt}
\end{small}




\backmatter
Changes to libtommath/tommath_class.h.
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#define BN_MP_DIV_2D_C
#define BN_MP_DIV_3_C
#define BN_MP_DIV_D_C
#define BN_MP_DR_IS_MODULUS_C
#define BN_MP_DR_REDUCE_C
#define BN_MP_DR_SETUP_C
#define BN_MP_EXCH_C

#define BN_MP_EXPT_D_C

#define BN_MP_EXPTMOD_C
#define BN_MP_EXPTMOD_FAST_C
#define BN_MP_EXTEUCLID_C
#define BN_MP_FREAD_C
#define BN_MP_FWRITE_C
#define BN_MP_GCD_C
#define BN_MP_GET_INT_C


#define BN_MP_GROW_C

#define BN_MP_INIT_C
#define BN_MP_INIT_COPY_C
#define BN_MP_INIT_MULTI_C
#define BN_MP_INIT_SET_C
#define BN_MP_INIT_SET_INT_C
#define BN_MP_INIT_SIZE_C
#define BN_MP_INVMOD_C







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#define BN_MP_DIV_2D_C
#define BN_MP_DIV_3_C
#define BN_MP_DIV_D_C
#define BN_MP_DR_IS_MODULUS_C
#define BN_MP_DR_REDUCE_C
#define BN_MP_DR_SETUP_C
#define BN_MP_EXCH_C
#define BN_MP_EXPORT_C
#define BN_MP_EXPT_D_C
#define BN_MP_EXPT_D_EX_C
#define BN_MP_EXPTMOD_C
#define BN_MP_EXPTMOD_FAST_C
#define BN_MP_EXTEUCLID_C
#define BN_MP_FREAD_C
#define BN_MP_FWRITE_C
#define BN_MP_GCD_C
#define BN_MP_GET_INT_C
#define BN_MP_GET_LONG_C
#define BN_MP_GET_LONG_LONG_C
#define BN_MP_GROW_C
#define BN_MP_IMPORT_C
#define BN_MP_INIT_C
#define BN_MP_INIT_COPY_C
#define BN_MP_INIT_MULTI_C
#define BN_MP_INIT_SET_C
#define BN_MP_INIT_SET_INT_C
#define BN_MP_INIT_SIZE_C
#define BN_MP_INVMOD_C
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#define BN_MP_MONTGOMERY_SETUP_C
#define BN_MP_MUL_C
#define BN_MP_MUL_2_C
#define BN_MP_MUL_2D_C
#define BN_MP_MUL_D_C
#define BN_MP_MULMOD_C
#define BN_MP_N_ROOT_C

#define BN_MP_NEG_C
#define BN_MP_OR_C
#define BN_MP_PRIME_FERMAT_C
#define BN_MP_PRIME_IS_DIVISIBLE_C
#define BN_MP_PRIME_IS_PRIME_C
#define BN_MP_PRIME_MILLER_RABIN_C
#define BN_MP_PRIME_NEXT_PRIME_C







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#define BN_MP_MONTGOMERY_SETUP_C
#define BN_MP_MUL_C
#define BN_MP_MUL_2_C
#define BN_MP_MUL_2D_C
#define BN_MP_MUL_D_C
#define BN_MP_MULMOD_C
#define BN_MP_N_ROOT_C
#define BN_MP_N_ROOT_EX_C
#define BN_MP_NEG_C
#define BN_MP_OR_C
#define BN_MP_PRIME_FERMAT_C
#define BN_MP_PRIME_IS_DIVISIBLE_C
#define BN_MP_PRIME_IS_PRIME_C
#define BN_MP_PRIME_MILLER_RABIN_C
#define BN_MP_PRIME_NEXT_PRIME_C
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#define BN_MP_REDUCE_2K_SETUP_L_C
#define BN_MP_REDUCE_IS_2K_C
#define BN_MP_REDUCE_IS_2K_L_C
#define BN_MP_REDUCE_SETUP_C
#define BN_MP_RSHD_C
#define BN_MP_SET_C
#define BN_MP_SET_INT_C


#define BN_MP_SHRINK_C
#define BN_MP_SIGNED_BIN_SIZE_C
#define BN_MP_SQR_C
#define BN_MP_SQRMOD_C
#define BN_MP_SQRT_C

#define BN_MP_SUB_C
#define BN_MP_SUB_D_C
#define BN_MP_SUBMOD_C
#define BN_MP_TO_SIGNED_BIN_C
#define BN_MP_TO_SIGNED_BIN_N_C
#define BN_MP_TO_UNSIGNED_BIN_C
#define BN_MP_TO_UNSIGNED_BIN_N_C







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#define BN_MP_REDUCE_2K_SETUP_L_C
#define BN_MP_REDUCE_IS_2K_C
#define BN_MP_REDUCE_IS_2K_L_C
#define BN_MP_REDUCE_SETUP_C
#define BN_MP_RSHD_C
#define BN_MP_SET_C
#define BN_MP_SET_INT_C
#define BN_MP_SET_LONG_C
#define BN_MP_SET_LONG_LONG_C
#define BN_MP_SHRINK_C
#define BN_MP_SIGNED_BIN_SIZE_C
#define BN_MP_SQR_C
#define BN_MP_SQRMOD_C
#define BN_MP_SQRT_C
#define BN_MP_SQRTMOD_PRIME_C
#define BN_MP_SUB_C
#define BN_MP_SUB_D_C
#define BN_MP_SUBMOD_C
#define BN_MP_TO_SIGNED_BIN_C
#define BN_MP_TO_SIGNED_BIN_N_C
#define BN_MP_TO_UNSIGNED_BIN_C
#define BN_MP_TO_UNSIGNED_BIN_N_C
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#endif

#if defined(BN_MP_DR_SETUP_C)
#endif

#if defined(BN_MP_EXCH_C)
#endif








#if defined(BN_MP_EXPT_D_C)




   #define BN_MP_INIT_COPY_C
   #define BN_MP_SET_C
   #define BN_MP_SQR_C
   #define BN_MP_CLEAR_C
   #define BN_MP_MUL_C
#endif

#if defined(BN_MP_EXPTMOD_C)
   #define BN_MP_INIT_C
   #define BN_MP_INVMOD_C
   #define BN_MP_CLEAR_C
   #define BN_MP_ABS_C








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#endif

#if defined(BN_MP_DR_SETUP_C)
#endif

#if defined(BN_MP_EXCH_C)
#endif

#if defined(BN_MP_EXPORT_C)
   #define BN_MP_INIT_COPY_C
   #define BN_MP_COUNT_BITS_C
   #define BN_MP_DIV_2D_C
   #define BN_MP_CLEAR_C
#endif

#if defined(BN_MP_EXPT_D_C)
   #define BN_MP_EXPT_D_EX_C
#endif

#if defined(BN_MP_EXPT_D_EX_C)
   #define BN_MP_INIT_COPY_C
   #define BN_MP_SET_C
   #define BN_MP_MUL_C
   #define BN_MP_CLEAR_C
   #define BN_MP_SQR_C
#endif

#if defined(BN_MP_EXPTMOD_C)
   #define BN_MP_INIT_C
   #define BN_MP_INVMOD_C
   #define BN_MP_CLEAR_C
   #define BN_MP_ABS_C
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   #define BN_MP_RADIX_SIZE_C
   #define BN_MP_TORADIX_C
#endif

#if defined(BN_MP_GCD_C)
   #define BN_MP_ISZERO_C
   #define BN_MP_ABS_C
   #define BN_MP_ZERO_C
   #define BN_MP_INIT_COPY_C
   #define BN_MP_CNT_LSB_C
   #define BN_MP_DIV_2D_C
   #define BN_MP_CMP_MAG_C
   #define BN_MP_EXCH_C
   #define BN_S_MP_SUB_C
   #define BN_MP_MUL_2D_C
   #define BN_MP_CLEAR_C
#endif

#if defined(BN_MP_GET_INT_C)
#endif







#if defined(BN_MP_GROW_C)
#endif







#if defined(BN_MP_INIT_C)
#endif

#if defined(BN_MP_INIT_COPY_C)

   #define BN_MP_COPY_C
#endif

#if defined(BN_MP_INIT_MULTI_C)
   #define BN_MP_ERR_C
   #define BN_MP_INIT_C
   #define BN_MP_CLEAR_C







<












>
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>
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>







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   #define BN_MP_RADIX_SIZE_C
   #define BN_MP_TORADIX_C
#endif

#if defined(BN_MP_GCD_C)
   #define BN_MP_ISZERO_C
   #define BN_MP_ABS_C

   #define BN_MP_INIT_COPY_C
   #define BN_MP_CNT_LSB_C
   #define BN_MP_DIV_2D_C
   #define BN_MP_CMP_MAG_C
   #define BN_MP_EXCH_C
   #define BN_S_MP_SUB_C
   #define BN_MP_MUL_2D_C
   #define BN_MP_CLEAR_C
#endif

#if defined(BN_MP_GET_INT_C)
#endif

#if defined(BN_MP_GET_LONG_C)
#endif

#if defined(BN_MP_GET_LONG_LONG_C)
#endif

#if defined(BN_MP_GROW_C)
#endif

#if defined(BN_MP_IMPORT_C)
   #define BN_MP_ZERO_C
   #define BN_MP_MUL_2D_C
   #define BN_MP_CLAMP_C
#endif

#if defined(BN_MP_INIT_C)
#endif

#if defined(BN_MP_INIT_COPY_C)
   #define BN_MP_INIT_SIZE_C
   #define BN_MP_COPY_C
#endif

#if defined(BN_MP_INIT_MULTI_C)
   #define BN_MP_ERR_C
   #define BN_MP_INIT_C
   #define BN_MP_CLEAR_C
477
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486
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   #define BN_MP_CLEAR_C
#endif

#if defined(BN_MP_KARATSUBA_MUL_C)
   #define BN_MP_MUL_C
   #define BN_MP_INIT_SIZE_C
   #define BN_MP_CLAMP_C
   #define BN_MP_SUB_C
   #define BN_MP_ADD_C

   #define BN_MP_LSHD_C
   #define BN_MP_CLEAR_C
#endif

#if defined(BN_MP_KARATSUBA_SQR_C)
   #define BN_MP_INIT_SIZE_C
   #define BN_MP_CLAMP_C
   #define BN_MP_SQR_C
   #define BN_MP_SUB_C
   #define BN_S_MP_ADD_C
   #define BN_MP_LSHD_C
   #define BN_MP_ADD_C
   #define BN_MP_CLEAR_C
#endif

#if defined(BN_MP_LCM_C)
   #define BN_MP_INIT_MULTI_C







|

>








|
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   #define BN_MP_CLEAR_C
#endif

#if defined(BN_MP_KARATSUBA_MUL_C)
   #define BN_MP_MUL_C
   #define BN_MP_INIT_SIZE_C
   #define BN_MP_CLAMP_C
   #define BN_S_MP_ADD_C
   #define BN_MP_ADD_C
   #define BN_S_MP_SUB_C
   #define BN_MP_LSHD_C
   #define BN_MP_CLEAR_C
#endif

#if defined(BN_MP_KARATSUBA_SQR_C)
   #define BN_MP_INIT_SIZE_C
   #define BN_MP_CLAMP_C
   #define BN_MP_SQR_C
   #define BN_S_MP_ADD_C
   #define BN_S_MP_SUB_C
   #define BN_MP_LSHD_C
   #define BN_MP_ADD_C
   #define BN_MP_CLEAR_C
#endif

#if defined(BN_MP_LCM_C)
   #define BN_MP_INIT_MULTI_C
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521
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   #define BN_MP_RSHD_C
#endif

#if defined(BN_MP_MOD_C)
   #define BN_MP_INIT_C
   #define BN_MP_DIV_C
   #define BN_MP_CLEAR_C
   #define BN_MP_ADD_C
   #define BN_MP_EXCH_C

#endif

#if defined(BN_MP_MOD_2D_C)
   #define BN_MP_ZERO_C
   #define BN_MP_COPY_C
   #define BN_MP_CLAMP_C
#endif







|

>







545
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   #define BN_MP_RSHD_C
#endif

#if defined(BN_MP_MOD_C)
   #define BN_MP_INIT_C
   #define BN_MP_DIV_C
   #define BN_MP_CLEAR_C
   #define BN_MP_ISZERO_C
   #define BN_MP_EXCH_C
   #define BN_MP_ADD_C
#endif

#if defined(BN_MP_MOD_2D_C)
   #define BN_MP_ZERO_C
   #define BN_MP_COPY_C
   #define BN_MP_CLAMP_C
#endif
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   #define BN_MP_INIT_C
   #define BN_MP_MUL_C
   #define BN_MP_CLEAR_C
   #define BN_MP_MOD_C
#endif

#if defined(BN_MP_N_ROOT_C)




   #define BN_MP_INIT_C
   #define BN_MP_SET_C
   #define BN_MP_COPY_C
   #define BN_MP_EXPT_D_C
   #define BN_MP_MUL_C
   #define BN_MP_SUB_C
   #define BN_MP_MUL_D_C
   #define BN_MP_DIV_C
   #define BN_MP_CMP_C
   #define BN_MP_SUB_D_C
   #define BN_MP_EXCH_C







>
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>



|







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   #define BN_MP_INIT_C
   #define BN_MP_MUL_C
   #define BN_MP_CLEAR_C
   #define BN_MP_MOD_C
#endif

#if defined(BN_MP_N_ROOT_C)
   #define BN_MP_N_ROOT_EX_C
#endif

#if defined(BN_MP_N_ROOT_EX_C)
   #define BN_MP_INIT_C
   #define BN_MP_SET_C
   #define BN_MP_COPY_C
   #define BN_MP_EXPT_D_EX_C
   #define BN_MP_MUL_C
   #define BN_MP_SUB_C
   #define BN_MP_MUL_D_C
   #define BN_MP_DIV_C
   #define BN_MP_CMP_C
   #define BN_MP_SUB_D_C
   #define BN_MP_EXCH_C
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   #define BN_MP_SUB_D_C
   #define BN_MP_DIV_2_C
   #define BN_MP_MUL_2_C
   #define BN_MP_ADD_D_C
#endif

#if defined(BN_MP_RADIX_SIZE_C)
   #define BN_MP_COUNT_BITS_C
   #define BN_MP_INIT_COPY_C
   #define BN_MP_ISZERO_C
   #define BN_MP_DIV_D_C
   #define BN_MP_CLEAR_C
#endif

#if defined(BN_MP_RADIX_SMAP_C)
   #define BN_MP_S_RMAP_C
#endif

#if defined(BN_MP_RAND_C)
   #define BN_MP_ZERO_C
   #define BN_MP_ADD_D_C
   #define BN_MP_LSHD_C
#endif

#if defined(BN_MP_READ_RADIX_C)
   #define BN_MP_ZERO_C
   #define BN_MP_S_RMAP_C
   #define BN_MP_RADIX_SMAP_C
   #define BN_MP_MUL_D_C
   #define BN_MP_ADD_D_C
   #define BN_MP_ISZERO_C
#endif

#if defined(BN_MP_READ_SIGNED_BIN_C)
   #define BN_MP_READ_UNSIGNED_BIN_C







|
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<







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   #define BN_MP_SUB_D_C
   #define BN_MP_DIV_2_C
   #define BN_MP_MUL_2_C
   #define BN_MP_ADD_D_C
#endif

#if defined(BN_MP_RADIX_SIZE_C)
   #define BN_MP_ISZERO_C
   #define BN_MP_COUNT_BITS_C
   #define BN_MP_INIT_COPY_C
   #define BN_MP_DIV_D_C
   #define BN_MP_CLEAR_C
#endif

#if defined(BN_MP_RADIX_SMAP_C)
   #define BN_MP_S_RMAP_C
#endif

#if defined(BN_MP_RAND_C)
   #define BN_MP_ZERO_C
   #define BN_MP_ADD_D_C
   #define BN_MP_LSHD_C
#endif

#if defined(BN_MP_READ_RADIX_C)
   #define BN_MP_ZERO_C
   #define BN_MP_S_RMAP_C

   #define BN_MP_MUL_D_C
   #define BN_MP_ADD_D_C
   #define BN_MP_ISZERO_C
#endif

#if defined(BN_MP_READ_SIGNED_BIN_C)
   #define BN_MP_READ_UNSIGNED_BIN_C
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#endif

#if defined(BN_MP_SET_INT_C)
   #define BN_MP_ZERO_C
   #define BN_MP_MUL_2D_C
   #define BN_MP_CLAMP_C
#endif







#if defined(BN_MP_SHRINK_C)
#endif

#if defined(BN_MP_SIGNED_BIN_SIZE_C)
   #define BN_MP_UNSIGNED_BIN_SIZE_C
#endif







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#endif

#if defined(BN_MP_SET_INT_C)
   #define BN_MP_ZERO_C
   #define BN_MP_MUL_2D_C
   #define BN_MP_CLAMP_C
#endif

#if defined(BN_MP_SET_LONG_C)
#endif

#if defined(BN_MP_SET_LONG_LONG_C)
#endif

#if defined(BN_MP_SHRINK_C)
#endif

#if defined(BN_MP_SIGNED_BIN_SIZE_C)
   #define BN_MP_UNSIGNED_BIN_SIZE_C
#endif
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   #define BN_MP_DIV_C
   #define BN_MP_ADD_C
   #define BN_MP_DIV_2_C
   #define BN_MP_CMP_MAG_C
   #define BN_MP_EXCH_C
   #define BN_MP_CLEAR_C
#endif




















#if defined(BN_MP_SUB_C)
   #define BN_S_MP_ADD_C
   #define BN_MP_CMP_MAG_C
   #define BN_S_MP_SUB_C
#endif








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   #define BN_MP_DIV_C
   #define BN_MP_ADD_C
   #define BN_MP_DIV_2_C
   #define BN_MP_CMP_MAG_C
   #define BN_MP_EXCH_C
   #define BN_MP_CLEAR_C
#endif

#if defined(BN_MP_SQRTMOD_PRIME_C)
   #define BN_MP_CMP_D_C
   #define BN_MP_ZERO_C
   #define BN_MP_JACOBI_C
   #define BN_MP_INIT_MULTI_C
   #define BN_MP_MOD_D_C
   #define BN_MP_ADD_D_C
   #define BN_MP_DIV_2_C
   #define BN_MP_EXPTMOD_C
   #define BN_MP_COPY_C
   #define BN_MP_SUB_D_C
   #define BN_MP_ISEVEN_C
   #define BN_MP_SET_INT_C
   #define BN_MP_SQRMOD_C
   #define BN_MP_MULMOD_C
   #define BN_MP_SET_C
   #define BN_MP_CLEAR_MULTI_C
#endif

#if defined(BN_MP_SUB_C)
   #define BN_S_MP_ADD_C
   #define BN_MP_CMP_MAG_C
   #define BN_S_MP_SUB_C
#endif

Added libtommath/tommath_private.h.






















































































































































































































































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/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 *
 * Tom St Denis, [email protected], http://math.libtomcrypt.com
 */
#ifndef TOMMATH_PRIV_H_
#define TOMMATH_PRIV_H_

#include <tommath.h>
#include <ctype.h>

#if 0

#define MIN(x,y) (((x) < (y)) ? (x) : (y))

#define MAX(x,y) (((x) > (y)) ? (x) : (y))

#ifdef __cplusplus
extern "C" {

/* C++ compilers don't like assigning void * to mp_digit * */
#define  OPT_CAST(x)  (x *)

#else

/* C on the other hand doesn't care */
#define  OPT_CAST(x)

#endif


/* define heap macros */
#ifndef XMALLOC
   /* default to libc stuff */
   #define XMALLOC  malloc
   #define XFREE    free
   #define XREALLOC realloc
   #define XCALLOC  calloc
#else
   /* prototypes for our heap functions */
   extern void *XMALLOC(size_t n);
   extern void *XREALLOC(void *p, size_t n);
   extern void *XCALLOC(size_t n, size_t s);
   extern void XFREE(void *p);
#endif

/* lowlevel functions, do not call! */
int s_mp_add(mp_int *a, mp_int *b, mp_int *c);
int s_mp_sub(mp_int *a, mp_int *b, mp_int *c);
#define s_mp_mul(a, b, c) s_mp_mul_digs(a, b, c, (a)->used + (b)->used + 1)
int fast_s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
int s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
int fast_s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
int s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
int fast_s_mp_sqr(mp_int *a, mp_int *b);
int s_mp_sqr(mp_int *a, mp_int *b);
int mp_karatsuba_mul(mp_int *a, mp_int *b, mp_int *c);
int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c);
int mp_karatsuba_sqr(mp_int *a, mp_int *b);
int mp_toom_sqr(mp_int *a, mp_int *b);
int fast_mp_invmod(mp_int *a, mp_int *b, mp_int *c);
int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c);
int fast_mp_montgomery_reduce(mp_int *x, mp_int *n, mp_digit rho);
int mp_exptmod_fast(mp_int *G, mp_int *X, mp_int *P, mp_int *Y, int redmode);
int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode);
void bn_reverse(unsigned char *s, int len);

extern const char *mp_s_rmap;

/* Fancy macro to set an MPI from another type.
 * There are several things assumed:
 *  x is the counter and unsigned
 *  a is the pointer to the MPI
 *  b is the original value that should be set in the MPI.
 */
#define MP_SET_XLONG(func_name, type)                    \
int func_name (mp_int * a, type b)                       \
{                                                        \
  unsigned int  x;                                       \
  int           res;                                     \
                                                         \
  mp_zero (a);                                           \
                                                         \
  /* set four bits at a time */                          \
  for (x = 0; x < (sizeof(type) * 2u); x++) {            \
    /* shift the number up four bits */                  \
    if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) {        \
      return res;                                        \
    }                                                    \
                                                         \
    /* OR in the top four bits of the source */          \
    a->dp[0] |= (b >> ((sizeof(type) * 8u) - 4u)) & 15u; \
                                                         \
    /* shift the source up to the next four bits */      \
    b <<= 4;                                             \
                                                         \
    /* ensure that digits are not clamped off */         \
    a->used += 1;                                        \
  }                                                      \
  mp_clamp (a);                                          \
  return MP_OKAY;                                        \
}
#endif

#ifdef __cplusplus
   }
#endif

#endif


/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to libtommath/tommath_superclass.h.
66
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70
71
72




    */
   #undef  BN_S_MP_MUL_DIGS_C
   #undef  BN_S_MP_SQR_C
   #undef  BN_MP_MONTGOMERY_REDUCE_C
#endif

#endif











>
>
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66
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    */
   #undef  BN_S_MP_MUL_DIGS_C
   #undef  BN_S_MP_SQR_C
   #undef  BN_MP_MONTGOMERY_REDUCE_C
#endif

#endif

/* $Source$ */
/* $Revision$ */
/* $Date$ */
Changes to macosx/Tcl.xcode/project.pbxproj.
162
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		F96D4AD408F272CA004A47F5 /* tclUnixThrd.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D446908F272B9004A47F5 /* tclUnixThrd.c */; };
		F96D4AD608F272CA004A47F5 /* tclUnixTime.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D446B08F272B9004A47F5 /* tclUnixTime.c */; };
		F9E61D28090A481F002B3151 /* bn_mp_cmp_d.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D427108F272B3004A47F5 /* bn_mp_cmp_d.c */; };
		F9E61D29090A486C002B3151 /* bn_mp_neg.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42A208F272B3004A47F5 /* bn_mp_neg.c */; };
		F9E61D2A090A4891002B3151 /* bn_mp_sqrt.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42C008F272B3004A47F5 /* bn_mp_sqrt.c */; };
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				F96D427608F272B3004A47F5 /* bn_mp_div.c */,
				F96D427708F272B3004A47F5 /* bn_mp_div_2.c */,
				F96D427808F272B3004A47F5 /* bn_mp_div_2d.c */,
				F96D427908F272B3004A47F5 /* bn_mp_div_3.c */,
				F96D427A08F272B3004A47F5 /* bn_mp_div_d.c */,
				F96D427E08F272B3004A47F5 /* bn_mp_exch.c */,
				F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */,

				F96D428708F272B3004A47F5 /* bn_mp_grow.c */,
				F96D428808F272B3004A47F5 /* bn_mp_init.c */,
				F96D428908F272B3004A47F5 /* bn_mp_init_copy.c */,
				F96D428A08F272B3004A47F5 /* bn_mp_init_multi.c */,
				F96D428B08F272B3004A47F5 /* bn_mp_init_set.c */,
				F96D428D08F272B3004A47F5 /* bn_mp_init_size.c */,
				F96D429208F272B3004A47F5 /* bn_mp_karatsuba_mul.c */,







>







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				F96D427608F272B3004A47F5 /* bn_mp_div.c */,
				F96D427708F272B3004A47F5 /* bn_mp_div_2.c */,
				F96D427808F272B3004A47F5 /* bn_mp_div_2d.c */,
				F96D427908F272B3004A47F5 /* bn_mp_div_3.c */,
				F96D427A08F272B3004A47F5 /* bn_mp_div_d.c */,
				F96D427E08F272B3004A47F5 /* bn_mp_exch.c */,
				F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */,
				F96D427F08F272B3004A47F5 /* bn_mp_expt_d_ex.c */,
				F96D428708F272B3004A47F5 /* bn_mp_grow.c */,
				F96D428808F272B3004A47F5 /* bn_mp_init.c */,
				F96D428908F272B3004A47F5 /* bn_mp_init_copy.c */,
				F96D428A08F272B3004A47F5 /* bn_mp_init_multi.c */,
				F96D428B08F272B3004A47F5 /* bn_mp_init_set.c */,
				F96D428D08F272B3004A47F5 /* bn_mp_init_size.c */,
				F96D429208F272B3004A47F5 /* bn_mp_karatsuba_mul.c */,
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				F96D48F408F272C3004A47F5 /* bn_mp_div.c in Sources */,
				F96D48F508F272C3004A47F5 /* bn_mp_div_2.c in Sources */,
				F96D48F608F272C3004A47F5 /* bn_mp_div_2d.c in Sources */,
				F96D48F708F272C3004A47F5 /* bn_mp_div_3.c in Sources */,
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				F96D490508F272C3004A47F5 /* bn_mp_grow.c in Sources */,
				F96D490608F272C3004A47F5 /* bn_mp_init.c in Sources */,
				F96D490708F272C3004A47F5 /* bn_mp_init_copy.c in Sources */,
				F96D490808F272C3004A47F5 /* bn_mp_init_multi.c in Sources */,
				F96D490908F272C3004A47F5 /* bn_mp_init_set.c in Sources */,
				F96D490B08F272C3004A47F5 /* bn_mp_init_size.c in Sources */,
				F96D491008F272C3004A47F5 /* bn_mp_karatsuba_mul.c in Sources */,







>







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				F96D48F408F272C3004A47F5 /* bn_mp_div.c in Sources */,
				F96D48F508F272C3004A47F5 /* bn_mp_div_2.c in Sources */,
				F96D48F608F272C3004A47F5 /* bn_mp_div_2d.c in Sources */,
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				F96D490508F272C3004A47F5 /* bn_mp_grow.c in Sources */,
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				F96D490708F272C3004A47F5 /* bn_mp_init_copy.c in Sources */,
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				F96D491008F272C3004A47F5 /* bn_mp_karatsuba_mul.c in Sources */,
Changes to macosx/Tcl.xcodeproj/project.pbxproj.
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		F96D428808F272B3004A47F5 /* bn_mp_init.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_init.c; sourceTree = "<group>"; };
		F96D428908F272B3004A47F5 /* bn_mp_init_copy.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_init_copy.c; sourceTree = "<group>"; };
		F96D428A08F272B3004A47F5 /* bn_mp_init_multi.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_init_multi.c; sourceTree = "<group>"; };
		F96D428B08F272B3004A47F5 /* bn_mp_init_set.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_init_set.c; sourceTree = "<group>"; };
		F96D428D08F272B3004A47F5 /* bn_mp_init_size.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_init_size.c; sourceTree = "<group>"; };
		F96D429208F272B3004A47F5 /* bn_mp_karatsuba_mul.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_karatsuba_mul.c; sourceTree = "<group>"; };
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				F96D427608F272B3004A47F5 /* bn_mp_div.c */,
				F96D427708F272B3004A47F5 /* bn_mp_div_2.c */,
				F96D427808F272B3004A47F5 /* bn_mp_div_2d.c */,
				F96D427908F272B3004A47F5 /* bn_mp_div_3.c */,
				F96D427A08F272B3004A47F5 /* bn_mp_div_d.c */,
				F96D427E08F272B3004A47F5 /* bn_mp_exch.c */,
				F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */,

				F96D428708F272B3004A47F5 /* bn_mp_grow.c */,
				F96D428808F272B3004A47F5 /* bn_mp_init.c */,
				F96D428908F272B3004A47F5 /* bn_mp_init_copy.c */,
				F96D428A08F272B3004A47F5 /* bn_mp_init_multi.c */,
				F96D428B08F272B3004A47F5 /* bn_mp_init_set.c */,
				F96D428D08F272B3004A47F5 /* bn_mp_init_size.c */,
				F96D429208F272B3004A47F5 /* bn_mp_karatsuba_mul.c */,







>







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				F96D427608F272B3004A47F5 /* bn_mp_div.c */,
				F96D427708F272B3004A47F5 /* bn_mp_div_2.c */,
				F96D427808F272B3004A47F5 /* bn_mp_div_2d.c */,
				F96D427908F272B3004A47F5 /* bn_mp_div_3.c */,
				F96D427A08F272B3004A47F5 /* bn_mp_div_d.c */,
				F96D427E08F272B3004A47F5 /* bn_mp_exch.c */,
				F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */,
				F96D427F08F272B3004A47F5 /* bn_mp_expt_d_ex.c */,
				F96D428708F272B3004A47F5 /* bn_mp_grow.c */,
				F96D428808F272B3004A47F5 /* bn_mp_init.c */,
				F96D428908F272B3004A47F5 /* bn_mp_init_copy.c */,
				F96D428A08F272B3004A47F5 /* bn_mp_init_multi.c */,
				F96D428B08F272B3004A47F5 /* bn_mp_init_set.c */,
				F96D428D08F272B3004A47F5 /* bn_mp_init_size.c */,
				F96D429208F272B3004A47F5 /* bn_mp_karatsuba_mul.c */,
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				F96D48F408F272C3004A47F5 /* bn_mp_div.c in Sources */,
				F96D48F508F272C3004A47F5 /* bn_mp_div_2.c in Sources */,
				F96D48F608F272C3004A47F5 /* bn_mp_div_2d.c in Sources */,
				F96D48F708F272C3004A47F5 /* bn_mp_div_3.c in Sources */,
				F96D48F808F272C3004A47F5 /* bn_mp_div_d.c in Sources */,
				F96D48FC08F272C3004A47F5 /* bn_mp_exch.c in Sources */,
				F9E61D2C090A48AC002B3151 /* bn_mp_expt_d.c in Sources */,

				F96D490508F272C3004A47F5 /* bn_mp_grow.c in Sources */,
				F96D490608F272C3004A47F5 /* bn_mp_init.c in Sources */,
				F96D490708F272C3004A47F5 /* bn_mp_init_copy.c in Sources */,
				F96D490808F272C3004A47F5 /* bn_mp_init_multi.c in Sources */,
				F96D490908F272C3004A47F5 /* bn_mp_init_set.c in Sources */,
				F96D490B08F272C3004A47F5 /* bn_mp_init_size.c in Sources */,
				F96D491008F272C3004A47F5 /* bn_mp_karatsuba_mul.c in Sources */,







>







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				F96D48F408F272C3004A47F5 /* bn_mp_div.c in Sources */,
				F96D48F508F272C3004A47F5 /* bn_mp_div_2.c in Sources */,
				F96D48F608F272C3004A47F5 /* bn_mp_div_2d.c in Sources */,
				F96D48F708F272C3004A47F5 /* bn_mp_div_3.c in Sources */,
				F96D48F808F272C3004A47F5 /* bn_mp_div_d.c in Sources */,
				F96D48FC08F272C3004A47F5 /* bn_mp_exch.c in Sources */,
				F9E61D2C090A48AC002B3151 /* bn_mp_expt_d.c in Sources */,
				F9E61D2C090A48AC002B3151 /* bn_mp_expt_d_ex.c in Sources */,
				F96D490508F272C3004A47F5 /* bn_mp_grow.c in Sources */,
				F96D490608F272C3004A47F5 /* bn_mp_init.c in Sources */,
				F96D490708F272C3004A47F5 /* bn_mp_init_copy.c in Sources */,
				F96D490808F272C3004A47F5 /* bn_mp_init_multi.c in Sources */,
				F96D490908F272C3004A47F5 /* bn_mp_init_set.c in Sources */,
				F96D490B08F272C3004A47F5 /* bn_mp_init_size.c in Sources */,
				F96D491008F272C3004A47F5 /* bn_mp_karatsuba_mul.c in Sources */,
Changes to unix/Makefile.in.
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TOMMATH_OBJS = bncore.o bn_reverse.o bn_fast_s_mp_mul_digs.o \
	bn_fast_s_mp_sqr.o bn_mp_add.o bn_mp_and.o \
        bn_mp_add_d.o bn_mp_clamp.o bn_mp_clear.o bn_mp_clear_multi.o \
        bn_mp_cmp.o bn_mp_cmp_d.o bn_mp_cmp_mag.o \
	bn_mp_cnt_lsb.o bn_mp_copy.o \
	bn_mp_count_bits.o bn_mp_div.o bn_mp_div_d.o bn_mp_div_2.o \
	bn_mp_div_2d.o bn_mp_div_3.o \
        bn_mp_exch.o bn_mp_expt_d.o bn_mp_grow.o bn_mp_init.o \
	bn_mp_init_copy.o bn_mp_init_multi.o bn_mp_init_set.o \
	bn_mp_init_set_int.o bn_mp_init_size.o bn_mp_karatsuba_mul.o \
	bn_mp_karatsuba_sqr.o \
        bn_mp_lshd.o bn_mp_mod.o bn_mp_mod_2d.o bn_mp_mul.o bn_mp_mul_2.o \
        bn_mp_mul_2d.o bn_mp_mul_d.o bn_mp_neg.o bn_mp_or.o \
	bn_mp_radix_size.o bn_mp_radix_smap.o \
        bn_mp_read_radix.o bn_mp_rshd.o bn_mp_set.o bn_mp_set_int.o \







|







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TOMMATH_OBJS = bncore.o bn_reverse.o bn_fast_s_mp_mul_digs.o \
	bn_fast_s_mp_sqr.o bn_mp_add.o bn_mp_and.o \
        bn_mp_add_d.o bn_mp_clamp.o bn_mp_clear.o bn_mp_clear_multi.o \
        bn_mp_cmp.o bn_mp_cmp_d.o bn_mp_cmp_mag.o \
	bn_mp_cnt_lsb.o bn_mp_copy.o \
	bn_mp_count_bits.o bn_mp_div.o bn_mp_div_d.o bn_mp_div_2.o \
	bn_mp_div_2d.o bn_mp_div_3.o \
        bn_mp_exch.o bn_mp_expt_d.o bn_mp_expt_d_ex.o bn_mp_grow.o bn_mp_init.o \
	bn_mp_init_copy.o bn_mp_init_multi.o bn_mp_init_set.o \
	bn_mp_init_set_int.o bn_mp_init_size.o bn_mp_karatsuba_mul.o \
	bn_mp_karatsuba_sqr.o \
        bn_mp_lshd.o bn_mp_mod.o bn_mp_mod_2d.o bn_mp_mul.o bn_mp_mul_2.o \
        bn_mp_mul_2d.o bn_mp_mul_d.o bn_mp_neg.o bn_mp_or.o \
	bn_mp_radix_size.o bn_mp_radix_smap.o \
        bn_mp_read_radix.o bn_mp_rshd.o bn_mp_set.o bn_mp_set_int.o \
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	$(TOMMATH_DIR)/bn_mp_div.c \
	$(TOMMATH_DIR)/bn_mp_div_d.c \
	$(TOMMATH_DIR)/bn_mp_div_2.c \
	$(TOMMATH_DIR)/bn_mp_div_2d.c \
	$(TOMMATH_DIR)/bn_mp_div_3.c \
	$(TOMMATH_DIR)/bn_mp_exch.c \
	$(TOMMATH_DIR)/bn_mp_expt_d.c \

	$(TOMMATH_DIR)/bn_mp_grow.c \
	$(TOMMATH_DIR)/bn_mp_init.c \
	$(TOMMATH_DIR)/bn_mp_init_copy.c \
	$(TOMMATH_DIR)/bn_mp_init_multi.c \
	$(TOMMATH_DIR)/bn_mp_init_set.c \
	$(TOMMATH_DIR)/bn_mp_init_set_int.c \
	$(TOMMATH_DIR)/bn_mp_init_size.c \







>







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	$(TOMMATH_DIR)/bn_mp_div.c \
	$(TOMMATH_DIR)/bn_mp_div_d.c \
	$(TOMMATH_DIR)/bn_mp_div_2.c \
	$(TOMMATH_DIR)/bn_mp_div_2d.c \
	$(TOMMATH_DIR)/bn_mp_div_3.c \
	$(TOMMATH_DIR)/bn_mp_exch.c \
	$(TOMMATH_DIR)/bn_mp_expt_d.c \
	$(TOMMATH_DIR)/bn_mp_expt_d_ex.c \
	$(TOMMATH_DIR)/bn_mp_grow.c \
	$(TOMMATH_DIR)/bn_mp_init.c \
	$(TOMMATH_DIR)/bn_mp_init_copy.c \
	$(TOMMATH_DIR)/bn_mp_init_multi.c \
	$(TOMMATH_DIR)/bn_mp_init_set.c \
	$(TOMMATH_DIR)/bn_mp_init_set_int.c \
	$(TOMMATH_DIR)/bn_mp_init_size.c \
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	$(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_3.c

bn_mp_exch.o: $(TOMMATH_DIR)/bn_mp_exch.c $(MATHHDRS)
	$(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_exch.c

bn_mp_expt_d.o: $(TOMMATH_DIR)/bn_mp_expt_d.c $(MATHHDRS)
	$(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_expt_d.c




bn_mp_grow.o: $(TOMMATH_DIR)/bn_mp_grow.c $(MATHHDRS)
	$(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_grow.c

bn_mp_init.o: $(TOMMATH_DIR)/bn_mp_init.c $(MATHHDRS)
	$(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init.c








>
>
>







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	$(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_3.c

bn_mp_exch.o: $(TOMMATH_DIR)/bn_mp_exch.c $(MATHHDRS)
	$(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_exch.c

bn_mp_expt_d.o: $(TOMMATH_DIR)/bn_mp_expt_d.c $(MATHHDRS)
	$(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_expt_d.c

bn_mp_expt_d_ex.o: $(TOMMATH_DIR)/bn_mp_expt_d_ex.c $(MATHHDRS)
	$(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_expt_d_ex.c

bn_mp_grow.o: $(TOMMATH_DIR)/bn_mp_grow.c $(MATHHDRS)
	$(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_grow.c

bn_mp_init.o: $(TOMMATH_DIR)/bn_mp_init.c $(MATHHDRS)
	$(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init.c

Changes to win/Makefile.in.
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	bn_mp_div.${OBJEXT} \
	bn_mp_div_d.${OBJEXT} \
	bn_mp_div_2.${OBJEXT} \
	bn_mp_div_2d.${OBJEXT} \
	bn_mp_div_3.${OBJEXT} \
	bn_mp_exch.${OBJEXT} \
	bn_mp_expt_d.${OBJEXT} \

	bn_mp_grow.${OBJEXT} \
	bn_mp_init.${OBJEXT} \
	bn_mp_init_copy.${OBJEXT} \
	bn_mp_init_multi.${OBJEXT} \
	bn_mp_init_set.${OBJEXT} \
	bn_mp_init_set_int.${OBJEXT} \
	bn_mp_init_size.${OBJEXT} \







>







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	bn_mp_div.${OBJEXT} \
	bn_mp_div_d.${OBJEXT} \
	bn_mp_div_2.${OBJEXT} \
	bn_mp_div_2d.${OBJEXT} \
	bn_mp_div_3.${OBJEXT} \
	bn_mp_exch.${OBJEXT} \
	bn_mp_expt_d.${OBJEXT} \
	bn_mp_expt_d_ex.${OBJEXT} \
	bn_mp_grow.${OBJEXT} \
	bn_mp_init.${OBJEXT} \
	bn_mp_init_copy.${OBJEXT} \
	bn_mp_init_multi.${OBJEXT} \
	bn_mp_init_set.${OBJEXT} \
	bn_mp_init_set_int.${OBJEXT} \
	bn_mp_init_size.${OBJEXT} \
Changes to win/makefile.vc.
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	$(TMP_DIR)\bn_mp_div.obj \
	$(TMP_DIR)\bn_mp_div_d.obj \
	$(TMP_DIR)\bn_mp_div_2.obj \
	$(TMP_DIR)\bn_mp_div_2d.obj \
	$(TMP_DIR)\bn_mp_div_3.obj \
	$(TMP_DIR)\bn_mp_exch.obj \
	$(TMP_DIR)\bn_mp_expt_d.obj \

	$(TMP_DIR)\bn_mp_grow.obj \
	$(TMP_DIR)\bn_mp_init.obj \
	$(TMP_DIR)\bn_mp_init_copy.obj \
	$(TMP_DIR)\bn_mp_init_multi.obj \
	$(TMP_DIR)\bn_mp_init_set.obj \
	$(TMP_DIR)\bn_mp_init_set_int.obj \
	$(TMP_DIR)\bn_mp_init_size.obj \







>







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	$(TMP_DIR)\bn_mp_div.obj \
	$(TMP_DIR)\bn_mp_div_d.obj \
	$(TMP_DIR)\bn_mp_div_2.obj \
	$(TMP_DIR)\bn_mp_div_2d.obj \
	$(TMP_DIR)\bn_mp_div_3.obj \
	$(TMP_DIR)\bn_mp_exch.obj \
	$(TMP_DIR)\bn_mp_expt_d.obj \
	$(TMP_DIR)\bn_mp_expt_d_ex.obj \
	$(TMP_DIR)\bn_mp_grow.obj \
	$(TMP_DIR)\bn_mp_init.obj \
	$(TMP_DIR)\bn_mp_init_copy.obj \
	$(TMP_DIR)\bn_mp_init_multi.obj \
	$(TMP_DIR)\bn_mp_init_set.obj \
	$(TMP_DIR)\bn_mp_init_set_int.obj \
	$(TMP_DIR)\bn_mp_init_size.obj \