Many hyperlinks are disabled.
Use anonymous login
to enable hyperlinks.
Overview
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: |
d1210bac0ba7064f80757bb3b3ba0d29 |
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
Changes to generic/tclStubInit.c.
︙ | ︙ | |||
745 746 747 748 749 750 751 752 753 754 755 756 757 758 | 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 }; | > | 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 | 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.
︙ | ︙ | |||
228 229 230 231 232 233 234 235 236 237 238 | } 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: | > > > > > | 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 | } 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.
︙ | ︙ | |||
69 70 71 72 73 74 75 76 77 78 79 80 81 82 | #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 | > | 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | #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 |
︙ | ︙ | |||
283 284 285 286 287 288 289 290 291 292 293 294 295 296 | 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 */ | > > > | 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 | 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 */ |
︙ | ︙ | |||
355 356 357 358 359 360 361 362 363 364 365 366 367 368 | 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 | > | 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 | 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 |
︙ | ︙ | |||
503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 | (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 */ | > > | 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 | (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.
|
| > | > > > > > | > > > > > > > > > > > > > > > > > > > | 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 | 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.
|
| | | | | | 1 2 3 4 5 6 | 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. |
Changes to libtommath/bn.ind.
1 2 | \begin{theindex} | | | | | | | | | | | | | | | | | | > | | | | > > | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | > > | | > | | | | | | | | | 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 | \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.
︙ | ︙ | |||
45 46 47 48 49 50 51 | \newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}} \def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}} \def\gap{\vspace{0.5ex}} \makeindex \begin{document} \frontmatter \pagestyle{empty} | | | | | 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 | \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{ } |
︙ | ︙ | |||
70 71 72 73 74 75 76 | \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 | | | | | | | | | | | | | | | | | | | | | | | | | | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 | \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|} |
︙ | ︙ | |||
228 229 230 231 232 233 234 | \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 \\ | | | | | | 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 | \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!). |
︙ | ︙ | |||
273 274 275 276 277 278 279 | \hline \end{tabular} \end{center} \end{small} \caption{LibTomMath Valuation} \end{figure} | | | | | | 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 | \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!] |
︙ | ︙ | |||
323 324 325 326 327 328 329 | to a string use the following function. \index{mp\_error\_to\_string} \begin{alltt} char *mp_error_to_string(int code); \end{alltt} | | | | | 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 | 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} |
︙ | ︙ | |||
370 371 372 373 374 375 376 | 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} | | | | | | | | | | | 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 | 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; \} |
︙ | ︙ | |||
447 448 449 450 451 452 453 | \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 | | | | | | | | | | | | | | | | | | | | | | | | | 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 | \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} |
︙ | ︙ | |||
659 660 661 662 663 664 665 | \begin{small} \begin{alltt} int main(void) \{ mp_int number; int result; if ((result = mp_init(&number)) != MP_OKAY) \{ | | | | | | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 | \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)); |
︙ | ︙ | |||
797 798 799 800 801 802 803 | \hline \end{tabular} \end{center} \caption{Comparison Codes for $a, b$} \label{fig:CMP} \end{figure} | | | | | | | | | | | | 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 | \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 $5 < 6$. |
︙ | ︙ | |||
878 879 880 881 882 883 884 | \begin{small} \begin{alltt} int main(void) \{ mp_int number1, number2; int result; if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{ | | | | | | | | | | | | | | | | | > > | | | > | > | | | | 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 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 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 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 1110 1111 1112 1113 1114 1115 | \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); |
︙ | ︙ | |||
1093 1094 1095 1096 1097 1098 1099 | \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} | | | 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 | \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} |
︙ | ︙ | |||
1118 1119 1120 1121 1122 1123 1124 | 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} | | | | | | | | | | | | | | | | | | | | | 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 1240 1241 1242 1243 1244 1245 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 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 | 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 |
︙ | ︙ | |||
1269 1270 1271 1272 1273 1274 1275 | 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} | | | | | | | | 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 | 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. |
︙ | ︙ | |||
1321 1322 1323 1324 1325 1326 1327 | \begin{alltt} int main(void) \{ mp_int a, b, c, mu; int result; | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | > > > > > > > > > > > > > > > > > > | < | | | | | | | | | | | | | | | | | | | | | | | | 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 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 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 1559 1560 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 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 1648 1649 1650 1651 1652 1653 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 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 | \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} \\ |
︙ | ︙ | |||
1713 1714 1715 1716 1717 1718 1719 | 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} | | | | 1770 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 | 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 |
︙ | ︙ | |||
1760 1761 1762 1763 1764 1765 1766 | \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 | | | | | 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 | \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 $c$. |
︙ | ︙ | |||
1800 1801 1802 1803 1804 1805 1806 | \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 $p$ | | > > > > > > > > > > > > > > > > > > > > > | 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 | \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.
|
| | | | | > > > > | 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 | #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.
|
| | | | | 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 | #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; } |
︙ | ︙ | |||
53 54 55 56 57 58 59 | if ((res = mp_copy (&y, &v)) != MP_OKAY) { goto LBL_ERR; } mp_set (&D, 1); top: /* 4. while u is even do */ | | | | | | 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 | 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) { |
︙ | ︙ | |||
111 112 113 114 115 116 117 | if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) { goto LBL_ERR; } } /* if not zero goto step 4 */ | | | 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | 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) { |
︙ | ︙ | |||
138 139 140 141 142 143 144 | c->sign = neg; res = MP_OKAY; LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &B, &D, NULL); return res; } #endif | > > > > | 138 139 140 141 142 143 144 145 146 147 148 | 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.
|
| | | | | | | | | | | | 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 | #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; |
︙ | ︙ | |||
111 112 113 114 115 116 117 | } /* now we have to propagate the carries and * shift the words downward [all those least * significant digits we zeroed]. */ { | | | | | | 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 | } /* 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++) { |
︙ | ︙ | |||
162 163 164 165 166 167 168 | /* 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 | > > > > | 162 163 164 165 166 167 168 169 170 171 172 | /* 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 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_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 |
︙ | ︙ | |||
31 32 33 34 35 36 37 | * 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]; | | | 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | * 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; } } |
︙ | ︙ | |||
74 75 76 77 78 79 80 | } /* store term */ W[ix] = ((mp_digit)_W) & MP_MASK; /* make next carry */ _W = _W >> ((mp_word)DIGIT_BIT); | | | | > > > > | 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 | } /* 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.
|
| | | | 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_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 |
︙ | ︙ | |||
71 72 73 74 75 76 77 | } /* setup dest */ olduse = c->used; c->used = pa; { | | > > > > | 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 | } /* 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.
|
| | | | 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_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 |
︙ | ︙ | |||
62 63 64 65 66 67 68 | */ 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 */ | | | 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | */ 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 */ |
︙ | ︙ | |||
104 105 106 107 108 109 110 | *tmpb++ = 0; } } mp_clamp (b); return MP_OKAY; } #endif | > > > > | 104 105 106 107 108 109 110 111 112 113 114 | *tmpb++ = 0; } } mp_clamp (b); return MP_OKAY; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_2expt.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 43 44 45 46 47 48 | #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.
|
| | | | 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_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 |
︙ | ︙ | |||
33 34 35 36 37 38 39 | /* force the sign of b to positive */ b->sign = MP_ZPOS; return MP_OKAY; } #endif | > > > > | 33 34 35 36 37 38 39 40 41 42 43 | /* 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.
|
| | | | 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_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; |
︙ | ︙ | |||
43 44 45 46 47 48 49 | res = s_mp_sub (a, b, c); } } return res; } #endif | > > > > | 43 44 45 46 47 48 49 50 51 52 53 | res = s_mp_sub (a, b, c); } } return res; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_add_d.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 | #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 */ |
︙ | ︙ | |||
103 104 105 106 107 108 109 | } mp_clamp(c); return MP_OKAY; } #endif | > > > > | 103 104 105 106 107 108 109 110 111 112 113 | } mp_clamp(c); return MP_OKAY; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_addmod.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_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; |
︙ | ︙ | |||
31 32 33 34 35 36 37 | return res; } res = mp_mod (&t, c, d); mp_clear (&t); return res; } #endif | > > > > | 31 32 33 34 35 36 37 38 39 40 41 | return res; } res = mp_mod (&t, c, d); mp_clear (&t); return res; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_and.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_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; |
︙ | ︙ | |||
47 48 49 50 51 52 53 | mp_clamp (&t); mp_exch (c, &t); mp_clear (&t); return MP_OKAY; } #endif | > > > > | 47 48 49 50 51 52 53 54 55 56 57 | mp_clamp (&t); mp_exch (c, &t); mp_clear (&t); return MP_OKAY; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_clamp.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 43 44 | #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.
|
| | | | 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_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; |
︙ | ︙ | |||
34 35 36 37 38 39 40 | /* reset members to make debugging easier */ a->dp = NULL; a->alloc = a->used = 0; a->sign = MP_ZPOS; } } #endif | > > > > | 34 35 36 37 38 39 40 41 42 43 44 | /* 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.
|
| | | > > > > | 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 | #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.
|
| | | | 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_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 */ |
︙ | ︙ | |||
33 34 35 36 37 38 39 | /* if negative compare opposite direction */ return mp_cmp_mag(b, a); } else { return mp_cmp_mag(a, b); } } #endif | > > > > | 33 34 35 36 37 38 39 40 41 42 43 | /* 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.
|
| | | | 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_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) { |
︙ | ︙ | |||
34 35 36 37 38 39 40 | } else if (a->dp[0] < b) { return MP_LT; } else { return MP_EQ; } } #endif | > > > > | 34 35 36 37 38 39 40 41 42 43 44 | } 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.
|
| | | | 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_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; |
︙ | ︙ | |||
45 46 47 48 49 50 51 | if (*tmpa < *tmpb) { return MP_LT; } } return MP_EQ; } #endif | > > > > | 45 46 47 48 49 50 51 52 53 54 55 | if (*tmpa < *tmpb) { return MP_LT; } } return MP_EQ; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_cnt_lsb.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 43 44 45 46 47 48 49 50 51 52 53 | #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.
|
| | | | 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_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; |
︙ | ︙ | |||
31 32 33 34 35 36 37 | if ((res = mp_grow (b, a->used)) != MP_OKAY) { return res; } } /* zero b and copy the parameters over */ { | | | 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | 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 */ |
︙ | ︙ | |||
58 59 60 61 62 63 64 | /* copy used count and sign */ b->used = a->used; b->sign = a->sign; return MP_OKAY; } #endif | > > > > | 58 59 60 61 62 63 64 65 66 67 68 | /* 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.
|
| | | | 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_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; |
︙ | ︙ | |||
35 36 37 38 39 40 41 | while (q > ((mp_digit) 0)) { ++r; q >>= ((mp_digit) 1); } return r; } #endif | > > > > | 35 36 37 38 39 40 41 42 43 44 45 | while (q > ((mp_digit) 0)) { ++r; q >>= ((mp_digit) 1); } return r; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_div.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 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 | #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); |
︙ | ︙ | |||
183 184 185 186 187 188 189 | /* step 3. for i from n down to (t + 1) */ for (i = n; i >= (t + 1); i--) { if (i > x.used) { continue; } | | | | > | | | | | | | | | | | | | | | | | | | | > > > > > > | 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 | /* 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.
|
| | | | | 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 | #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; |
︙ | ︙ | |||
58 59 60 61 62 63 64 | } } b->sign = a->sign; mp_clamp (b); return MP_OKAY; } #endif | > > > > | 58 59 60 61 62 63 64 65 66 67 68 | } } b->sign = a->sign; mp_clamp (b); return MP_OKAY; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_div_2d.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_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; |
︙ | ︙ | |||
54 55 56 57 58 59 60 | 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) { | | | 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | 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; |
︙ | ︙ | |||
87 88 89 90 91 92 93 | if (d != NULL) { mp_exch (&t, d); } mp_clear (&t); return MP_OKAY; } #endif | > > > > | 87 88 89 90 91 92 93 94 95 96 97 | 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.
|
| | | | 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_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; |
︙ | ︙ | |||
69 70 71 72 73 74 75 | } mp_clear(&q); return res; } #endif | > > > > | 69 70 71 72 73 74 75 76 77 78 79 | } mp_clear(&q); return res; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_div_d.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 | #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 43 44 45 46 47 48 | /* cannot divide by zero */ if (b == 0) { return MP_VAL; } /* quick outs */ | | | 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | /* 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; |
︙ | ︙ | |||
104 105 106 107 108 109 110 | } mp_clear(&q); return res; } #endif | > > > > | 104 105 106 107 108 109 110 111 112 113 114 | } mp_clear(&q); return res; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_dr_is_modulus.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_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 34 35 36 37 38 39 | return 0; } } return 1; } #endif | > > > > | 33 34 35 36 37 38 39 40 41 42 43 | return 0; } } return 1; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_dr_reduce.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_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 37 38 39 40 41 42 | mp_word r; mp_digit mu, *tmpx1, *tmpx2; /* m = digits in modulus */ m = n->used; /* ensure that "x" has at least 2m digits */ | | | 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 | 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. |
︙ | ︙ | |||
58 59 60 61 62 63 64 | 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++) { | | | > > > > > > | 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 | 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.
|
| | | > > > > | 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 | #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 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 | #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 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 | #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.
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 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 | #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.
|
| | | | 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_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) |
︙ | ︙ | |||
85 86 87 88 89 90 91 | 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 | | > > > > | 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 | 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.
|
| | | | 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_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. * |
︙ | ︙ | |||
92 93 94 95 96 97 98 | #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 | | | | 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | #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; |
︙ | ︙ | |||
215 216 217 218 219 220 221 | 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 */ | | | | 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 | 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; |
︙ | ︙ | |||
262 263 264 265 266 267 268 | bitcpy = 0; bitbuf = 0; mode = 1; } } /* if bits remain then square/multiply */ | | | 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 | 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; |
︙ | ︙ | |||
310 311 312 313 314 315 316 | mp_clear(&M[1]); for (x = 1<<(winsize-1); x < (1 << winsize); x++) { mp_clear (&M[x]); } return err; } #endif | > > > > > | 310 311 312 313 314 315 316 317 318 319 320 321 | 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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 | #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; |
︙ | ︙ | |||
57 58 59 60 61 62 63 | 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) { | | | | > > > > | 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 | 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 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_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; |
︙ | ︙ | |||
57 58 59 60 61 62 63 | a->sign = neg; } return MP_OKAY; } #endif | > > > > | 57 58 59 60 61 62 63 64 65 66 67 | a->sign = neg; } return MP_OKAY; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_fwrite.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_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; |
︙ | ︙ | |||
42 43 44 45 46 47 48 | } XFREE (buf); return MP_OKAY; } #endif | > > > > | 42 43 44 45 46 47 48 49 50 51 52 | } XFREE (buf); return MP_OKAY; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_gcd.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_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; |
︙ | ︙ | |||
66 67 68 69 70 71 72 | if (v_lsb != k) { if ((res = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) { goto LBL_V; } } | | | 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | 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 */ |
︙ | ︙ | |||
95 96 97 98 99 100 101 | c->sign = MP_ZPOS; res = MP_OKAY; LBL_V:mp_clear (&u); LBL_U:mp_clear (&v); return res; } #endif | > > > > | 95 96 97 98 99 100 101 102 103 104 105 | 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.
|
| | | | | | | > > > > | 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 | #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.
|
| | | | 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_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; |
︙ | ︙ | |||
47 48 49 50 51 52 53 | for (; i < a->alloc; i++) { a->dp[i] = 0; } } return MP_OKAY; } #endif | > > > > | 47 48 49 50 51 52 53 54 55 56 57 | for (; i < a->alloc; i++) { a->dp[i] = 0; } } return MP_OKAY; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_init.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_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; |
︙ | ︙ | |||
36 37 38 39 40 41 42 | a->used = 0; a->alloc = MP_PREC; a->sign = MP_ZPOS; return MP_OKAY; } #endif | > > > > | 36 37 38 39 40 41 42 43 44 45 46 | 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.
|
| | | | > > > > | 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 | #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.
|
| | | | 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_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 */ |
︙ | ︙ | |||
33 34 35 36 37 38 39 | /* end the current list */ va_end(args); /* now start cleaning up */ cur_arg = mp; va_start(clean_args, mp); | | > > > > | 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 | /* 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.
|
| | | > > > > | 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 | #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.
|
| | | > > > > | 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 | #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.
|
| | | | 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_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 | > > > > | 38 39 40 41 42 43 44 45 46 47 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 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 | #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 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 | #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; } |
︙ | ︙ | |||
133 134 135 136 137 138 139 | if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) { goto LBL_ERR; } } /* if not zero goto step 4 */ | | | 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) { goto LBL_ERR; } } /* if not zero goto step 4 */ if (mp_iszero (&u) == 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 | > > > > | 165 166 167 168 169 170 171 172 173 174 175 | /* 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 17 18 19 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 79 80 81 82 83 84 | 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 */ | | | | | | | | > > > > | 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 | 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 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 | } /* 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) */ | | | | | | > > > > | 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 | } /* 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_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 | /* now shift the digits */ x0.used = y0.used = B; x1.used = a->used - B; y1.used = b->used - B; { | | | | 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; { 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 | > > > > | 157 158 159 160 161 162 163 164 165 166 167 | 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 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_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 |
︙ | ︙ | |||
48 49 50 51 52 53 54 | 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; { | | | | 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | 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++; |
︙ | ︙ | |||
111 112 113 114 115 116 117 | T1:mp_clear (&t1); X1:mp_clear (&x1); X0:mp_clear (&x0); ERR: return err; } #endif | > > > > | 111 112 113 114 115 116 117 118 119 120 121 | 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.
|
| | | | 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_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; |
︙ | ︙ | |||
50 51 52 53 54 55 56 | c->sign = MP_ZPOS; LBL_T: mp_clear_multi (&t1, &t2, NULL); return res; } #endif | > > > > | 50 51 52 53 54 55 56 57 58 59 60 | 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 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 | #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 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 | #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 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_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 40 41 | /* copy */ if ((res = mp_copy (a, c)) != MP_OKAY) { return res; } /* zero digits above the last digit of the modulus */ | | > > > > | 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 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.
|
| | | > > > > | 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 | #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.
|
| | | | | 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 | #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; } |
︙ | ︙ | |||
49 50 51 52 53 54 55 | } } } return MP_OKAY; } #endif | > > > > | 49 50 51 52 53 54 55 56 57 58 59 | } } } return MP_OKAY; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_montgomery_reduce.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 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 | #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; } } } |
︙ | ︙ | |||
108 109 110 111 112 113 114 | if (mp_cmp_mag (x, n) != MP_LT) { return s_mp_sub (x, n, x); } return MP_OKAY; } #endif | > > > > | 108 109 110 111 112 113 114 115 116 117 118 | 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.
|
| | | | 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_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; |
︙ | ︙ | |||
32 33 34 35 36 37 38 | b = n->dp[0]; if ((b & 1) == 0) { return MP_VAL; } x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */ | | | | | | > > > > | 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 | 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.
|
| | | | 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_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; |
︙ | ︙ | |||
40 41 42 43 44 45 46 | * 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) && | | | > | > > > > | 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 | * 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.
|
| | | | | | 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_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; |
︙ | ︙ | |||
72 73 74 75 76 77 78 | *tmpb++ = 0; } } b->sign = a->sign; return MP_OKAY; } #endif | > > > > | 72 73 74 75 76 77 78 79 80 81 82 | *tmpb++ = 0; } } b->sign = a->sign; return MP_OKAY; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_mul_2d.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 43 44 45 46 47 48 49 50 51 52 53 54 55 | #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; |
︙ | ︙ | |||
75 76 77 78 79 80 81 | c->dp[(c->used)++] = r; } } mp_clamp (c); return MP_OKAY; } #endif | > > > > | 75 76 77 78 79 80 81 82 83 84 85 | c->dp[(c->used)++] = r; } } mp_clamp (c); return MP_OKAY; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_mul_d.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 | #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; |
︙ | ︙ | |||
44 45 46 47 48 49 50 | /* zero carry */ u = 0; /* compute columns */ for (ix = 0; ix < a->used; ix++) { /* compute product and carry sum for this term */ | | | 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | /* 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)); } |
︙ | ︙ | |||
69 70 71 72 73 74 75 | /* set used count */ c->used = a->used + 1; mp_clamp(c); return MP_OKAY; } #endif | > > > > | 69 70 71 72 73 74 75 76 77 78 79 | /* 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.
|
| | | | 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_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; |
︙ | ︙ | |||
30 31 32 33 34 35 36 | return res; } res = mp_mod (&t, c, d); mp_clear (&t); return res; } #endif | > > > > | 30 31 32 33 34 35 36 37 38 39 40 | 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.
|
| | | | < | < < < < < < < < < < < < < < < | | < < < | < < | | < < < | < < | < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < | 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 | #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.
|
| | | | 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_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) { |
︙ | ︙ | |||
30 31 32 33 34 35 36 | } else { b->sign = MP_ZPOS; } return MP_OKAY; } #endif | > > > > | 30 31 32 33 34 35 36 37 38 39 40 | } else { b->sign = MP_ZPOS; } return MP_OKAY; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_or.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_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; |
︙ | ︙ | |||
40 41 42 43 44 45 46 | } 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 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_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_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 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_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 | goto LBL_R; } if ((err = mp_exptmod (b, &r, a, &y)) != MP_OKAY) { goto LBL_Y; } /* if y != 1 and y != n1 do */ | | | | 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; |
︙ | ︙ | |||
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 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 82 83 84 85 86 | 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 { | | | 80 81 82 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) == MP_YES) { /* force odd */ if ((err = mp_sub_d(a, 1, a)) != MP_OKAY) { return err; } } } |
︙ | ︙ | |||
125 126 127 128 129 130 131 | } /* set flag if zero */ if (res_tab[x] == 0) { y = 1; } } | | | | 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | } /* 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) { |
︙ | ︙ | |||
160 161 162 163 164 165 166 | err = MP_OKAY; LBL_ERR: mp_clear(&b); return err; } #endif | > > > > | 160 161 162 163 164 165 166 167 168 169 170 | 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 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_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 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 | #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; |
︙ | ︙ | |||
91 92 93 94 95 96 97 | /* is it prime? */ if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { goto error; } if (res == MP_NO) { continue; } | | | > > > > | 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 | /* 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.
|
| | | < < < < < < | > > > > > > | 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 | #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; |
︙ | ︙ | |||
77 78 79 80 81 82 83 | } else { *size = 3; } return MP_OKAY; } #endif | > > > > | 77 78 79 80 81 82 83 84 85 86 87 | } else { *size = 3; } return MP_OKAY; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_radix_smap.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 | #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.
|
| | | | | > > > > | 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 | #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.
|
| | | | | | | | 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 | #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 |
︙ | ︙ | |||
76 77 78 79 80 81 82 | if ( *str != '\0' ) { mp_zero( a ); return MP_VAL; } /* set the sign only if a != 0 */ | | > > > > | 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | 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.
|
| | | | 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_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; |
︙ | ︙ | |||
31 32 33 34 35 36 37 | } else { a->sign = MP_NEG; } return MP_OKAY; } #endif | > > > > | 31 32 33 34 35 36 37 38 39 40 41 | } else { a->sign = MP_NEG; } return MP_OKAY; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_read_unsigned_bin.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_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 34 35 36 37 38 39 | /* read the bytes in */ while (c-- > 0) { if ((res = mp_mul_2d (a, 8, a)) != MP_OKAY) { return res; } #ifndef MP_8BIT | | | | | | > > > > | 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 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 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 | #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 84 85 86 87 88 89 | /* 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; } } | | > > > > | 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 | /* 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 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 | #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 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 | #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 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_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 | > > > > | 37 38 39 40 41 42 43 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 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_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 | > > > > | 34 35 36 37 38 39 40 41 42 43 44 | } ERR: mp_clear(&tmp); return res; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_reduce_is_2k.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_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; |
︙ | ︙ | |||
42 43 44 45 46 47 48 | } } } return MP_YES; } #endif | > > > > | 42 43 44 45 46 47 48 49 50 51 52 | } } } return MP_YES; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_reduce_is_2k_l.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_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; |
︙ | ︙ | |||
34 35 36 37 38 39 40 | return (iy >= (a->used/2)) ? MP_YES : MP_NO; } return MP_NO; } #endif | > > > > | 34 35 36 37 38 39 40 41 42 43 44 | return (iy >= (a->used/2)) ? MP_YES : MP_NO; } return MP_NO; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_reduce_setup.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 | #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.
|
| | | | | 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_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] */ |
︙ | ︙ | |||
62 63 64 65 66 67 68 | } } /* remove excess digits */ a->used -= b; } #endif | > > > > | 62 63 64 65 66 67 68 69 70 71 72 | } } /* remove excess digits */ a->used -= b; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_set.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 | #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.
|
| | | | 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_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; |
︙ | ︙ | |||
38 39 40 41 42 43 44 | /* ensure that digits are not clamped off */ a->used += 1; } mp_clamp (a); return MP_OKAY; } #endif | > > > > | 38 39 40 41 42 43 44 45 46 47 48 | /* 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.
|
| | | | > > > > > | 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_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.
|
| | | > > > > | 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 | #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.
|
| | | | | | | > > > > > > | 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 | #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.
|
| | | | 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_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; |
︙ | ︙ | |||
31 32 33 34 35 36 37 | return res; } res = mp_mod (&t, b, c); mp_clear (&t); return res; } #endif | > > > > | 31 32 33 34 35 36 37 38 39 40 41 | return res; } res = mp_mod (&t, b, c); mp_clear (&t); return res; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_sqrt.c.
|
| | | | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 | #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 */ |
︙ | ︙ | |||
136 137 138 139 140 141 142 | E1: mp_clear(&t2); E2: mp_clear(&t1); return res; } #endif | > > > > | 136 137 138 139 140 141 142 143 144 145 146 | E1: mp_clear(&t2); E2: mp_clear(&t1); return res; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_sub.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_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; |
︙ | ︙ | |||
49 50 51 52 53 54 55 | res = s_mp_sub (b, a, c); } } return res; } #endif | > > > > | 49 50 51 52 53 54 55 56 57 58 59 | res = s_mp_sub (b, a, c); } } return res; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_sub_d.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 | #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] |
︙ | ︙ | |||
45 46 47 48 49 50 51 | /* setup regs */ oldused = c->used; tmpa = a->dp; tmpc = c->dp; /* if a <= b simply fix the single digit */ | | | | > > > > | 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 | /* 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.
|
| | | | 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_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; |
︙ | ︙ | |||
32 33 34 35 36 37 38 | return res; } res = mp_mod (&t, c, d); mp_clear (&t); return res; } #endif | > > > > | 32 33 34 35 36 37 38 39 40 41 42 | 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.
|
| | | | > > > > | 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 | #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.
|
| | | > > > > | 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 | #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.
|
| | | | > > > > | 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 | #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.
|
| | | > > > > | 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 | #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.
|
| | | | | | | | | | | | | > > | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | > > > > | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 | #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 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_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 38 39 40 41 | goto ERR; } if ((res = mp_copy(a, &a1)) != MP_OKAY) { goto ERR; } mp_rshd(&a1, B); | | > > | 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 | 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 */ |
︙ | ︙ | |||
111 112 113 114 115 116 117 | 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. */ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | > > > > | 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 | 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 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 | #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 67 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 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 | #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 53 54 55 56 57 58 | t.sign = MP_ZPOS; /* subtract a char */ --maxlen; } digs = 0; | | | 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | 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; |
︙ | ︙ | |||
78 79 80 81 82 83 84 | *str = '\0'; mp_clear (&t); return MP_OKAY; } #endif | > > > > | 78 79 80 81 82 83 84 85 86 87 88 | *str = '\0'; mp_clear (&t); return MP_OKAY; } #endif /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/bn_mp_unsigned_bin_size.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 | #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.
|
| | | | 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_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 42 43 44 45 46 47 | } mp_clamp (&t); mp_exch (c, &t); mp_clear (&t); return MP_OKAY; } #endif | > > > > | 41 42 43 44 45 46 47 48 49 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.
|
| | | > > > > | 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 | #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 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_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 52 53 54 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 54 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 19 20 21 22 23 24 25 26 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_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 | } else { min = a->used; max = b->used; x = b; } /* init result */ | | | | | 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_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 | 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 */ | | | | 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 | bitcpy = 0; bitbuf = 0; mode = 1; } } /* if bits remain then square/multiply */ | | | 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 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | #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; |
︙ | ︙ | |||
57 58 59 60 61 62 63 | /* 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 */ | | | | | > > > > | 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 | /* 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.
|
| | | | | 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 | #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; } |
︙ | ︙ | |||
53 54 55 56 57 58 59 | 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 */ | | | | > > > > | 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 | 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.
|
| | | | | | | | | 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 | #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 |
︙ | ︙ | |||
74 75 76 77 78 79 80 | mp_clamp (&t); mp_exch (&t, b); mp_clear (&t); return MP_OKAY; } #endif | > > > > | 74 75 76 77 78 79 80 81 82 83 84 | 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.
|
| | | | 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_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; |
︙ | ︙ | |||
31 32 33 34 35 36 37 | return res; } } olduse = c->used; c->used = max; { | | | | | | > > > > | 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 | 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.
|
| | | > > > > | 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 | #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.
︙ | ︙ | |||
11 12 13 14 15 16 17 | #Tom St Denis #get graphics type if (shift =~ /PDF/) { $graph = ""; } else { $graph = ".ps"; | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 | #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) { $str = "chapter two"; |
︙ | ︙ | |||
224 225 226 227 228 229 230 | } elsif ($a == 10) { $str = "chapter ten"; } } else { $str = "chapter " . $str; } } else { | | | | | | 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 | } elsif ($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] . "~"; } } |
︙ | ︙ | |||
259 260 261 262 263 264 265 | ++$wroteline; } } print "Read $readline lines, wrote $wroteline lines\n"; close (OUT); close (IN); | > > | 259 260 261 262 263 264 265 266 267 | ++$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.
1 2 3 4 | 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 | > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | 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 | 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() |
︙ | ︙ | |||
109 110 111 112 113 114 115 | 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. | | | | 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 | 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.
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 <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; | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | > > > > | > > | > > > > > | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | > | > > | < > > > > > > | > > > > > > > > > > > | > > > > > > > > > > > > > > > > | > > > > > > > > > > > > > > > > > > > > | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | > > > | > > > > > > | > > > > > > > > > > > > > > > > > > > > > > > > > > | | | > > | > | | | < < < < | < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < | < < < < | < < < < < < < < < < < < | < < < < < < < < < < < < < < < | < < < < < < < < < | < < < | < < < < < < < < < < < < < < < < < < < < < < < < < | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 | #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); |
︙ | ︙ | |||
235 236 237 238 239 240 241 | 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 */ | | | | | | | | | < | | | | | | | | | | | | | | < > | | | > | > > | | | | | | | | | | | | | | | | | > | | | < | | | > < | | < | > > | | | | | > > | 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 | 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; |
︙ | ︙ | |||
424 425 426 427 428 429 430 | } 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); | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | > | | | | | | | | | | | | | | | > > > > > > > > | 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 | } 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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | #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); | > > > > > > > | 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 | #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); |
︙ | ︙ | |||
40 41 42 43 44 45 46 | } /* RDTSC from Scott Duplichan */ static ulong64 TIMFUNC(void) { #if defined __GNUC__ #if defined(__i386__) || defined(__x86_64__) | > > > | | < | | 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | } /* 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"); |
︙ | ︙ | |||
73 74 75 76 77 78 79 80 81 82 83 84 85 | #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; | > > > > > > > > > > > > | < | < < < | | | | | | | | < > > > | | | | > > | | | | | | | | < | 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 | #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", |
︙ | ︙ | |||
231 232 233 234 235 236 237 | "347743159439876626079252796797422223177535447388206607607181663903045907591201940478223621722118173270898487582987137708656414344685816179420855160986340457973820182883508387588163122354089264395604796675278966117567294812714812796820596564876450716066283126720010859041484786529056457896367683122960411136319", "47266428956356393164697365098120418976400602706072312735924071745438532218237979333351774907308168340693326687317443721193266215155735814510792148768576498491199122744351399489453533553203833318691678263241941706256996197460424029012419012634671862283532342656309677173602509498417976091509154360039893165037637034737020327399910409885798185771003505320583967737293415979917317338985837385734747478364242020380416892056650841470869294527543597349250299539682430605173321029026555546832473048600327036845781970289288898317888427517364945316709081173840186150794397479045034008257793436817683392375274635794835245695887", "436463808505957768574894870394349739623346440601945961161254440072143298152040105676491048248110146278752857839930515766167441407021501229924721335644557342265864606569000117714935185566842453630868849121480179691838399545644365571106757731317371758557990781880691336695584799313313687287468894148823761785582982549586183756806449017542622267874275103877481475534991201849912222670102069951687572917937634467778042874315463238062009202992087620963771759666448266532858079402669920025224220613419441069718482837399612644978839925207109870840278194042158748845445131729137117098529028886770063736487420613144045836803985635654192482395882603511950547826439092832800532152534003936926017612446606135655146445620623395788978726744728503058670046885876251527122350275750995227", "11424167473351836398078306042624362277956429440521137061889702611766348760692206243140413411077394583180726863277012016602279290144126785129569474909173584789822341986742719230331946072730319555984484911716797058875905400999504305877245849119687509023232790273637466821052576859232452982061831009770786031785669030271542286603956118755585683996118896215213488875253101894663403069677745948305893849505434201763745232895780711972432011344857521691017896316861403206449421332243658855453435784006517202894181640562433575390821384210960117518650374602256601091379644034244332285065935413233557998331562749140202965844219336298970011513882564935538704289446968322281451907487362046511461221329799897350993370560697505809686438782036235372137015731304779072430260986460269894522159103008260495503005267165927542949439526272736586626709581721032189532726389643625590680105784844246152702670169304203783072275089194754889511973916207", "1214855636816562637502584060163403830270705000634713483015101384881871978446801224798536155406895823305035467591632531067547890948695117172076954220727075688048751022421198712032848890056357845974246560748347918630050853933697792254955890439720297560693579400297062396904306270145886830719309296352765295712183040773146419022875165382778007040109957609739589875590885701126197906063620133954893216612678838507540777138437797705602453719559017633986486649523611975865005712371194067612263330335590526176087004421363598470302731349138773205901447704682181517904064735636518462452242791676541725292378925568296858010151852326316777511935037531017413910506921922450666933202278489024521263798482237150056835746454842662048692127173834433089016107854491097456725016327709663199738238442164843147132789153725513257167915555162094970853584447993125488607696008169807374736711297007473812256272245489405898470297178738029484459690836250560495461579533254473316340608217876781986188705928270735695752830825527963838355419762516246028680280988020401914551825487349990306976304093109384451438813251211051597392127491464898797406789175453067960072008590614886532333015881171367104445044718144312416815712216611576221546455968770801413440778423979", NULL }; | | | | | | 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 | "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(); |
︙ | ︙ | |||
267 268 269 270 271 272 273 | 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); | | | | | | | | | 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 | 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); |
︙ | ︙ | |||
303 304 305 306 307 308 309 | 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); | | | > > > > | 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 | 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 65 66 67 68 69 70 | # 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 | | | 64 65 66 67 68 69 70 71 72 73 74 75 76 77 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 70 71 72 73 74 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; } | > > > > > > > > > | 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | 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 54 55 56 57 58 59 | fclose(out); mp_clear(&a); mp_clear(&b); return 0; } | > > > > > | 53 54 55 56 57 58 59 60 61 62 63 64 | fclose(out); mp_clear(&a); mp_clear(&b); return 0; } /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/etc/mersenne.c.
︙ | ︙ | |||
134 135 136 137 138 139 140 | /* but make sure its prime */ while (isprime (k) == 0) { k += 2; } } return 0; } | > > > > | 134 135 136 137 138 139 140 141 142 143 144 | /* but make sure its prime */ while (isprime (k) == 0) { k += 2; } } return 0; } /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/etc/mont.c.
︙ | ︙ | |||
35 36 37 38 39 40 41 | } } printf("PASSED\n"); } return 0; } | > > > > > > > > > | 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 | } } printf("PASSED\n"); } return 0; } /* $Source$ */ /* $Revision$ */ /* $Date$ */ |
Changes to libtommath/etc/pprime.c.
︙ | ︙ | |||
390 391 392 393 394 395 396 | mp_toradix (&p, buf, 10); printf ("P == %s\n", buf); mp_toradix (&q, buf, 10); printf ("Q == %s\n", buf); return 0; } | > > > > | 390 391 392 393 394 395 396 397 398 399 400 | 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.
1 2 3 4 5 6 7 8 | /* 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 | | > > > > > | | | | 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 | /* 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 |
︙ | ︙ | |||
38 39 40 41 42 43 44 | return __getReg (3116); #else #error need rdtsc function for this build #endif } | < < | 43 44 45 46 47 48 49 50 51 52 53 54 55 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 64 65 66 67 68 69 | mp_init (&a); mp_init (&b); mp_init (&c); mp_rand (&a, size); mp_rand (&b, size); | | | 66 67 68 69 70 71 72 73 74 75 76 77 78 79 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 92 93 94 95 96 97 | ulong64 t1; mp_init (&a); mp_init (&b); mp_rand (&a, size); | | | 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | 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++) { |
︙ | ︙ | |||
113 114 115 116 117 118 119 | int main (void) { ulong64 t1, t2; int x, y; | | | > > > > | 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 | 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 2 3 4 | #Makefile for GCC # #Tom St Denis | < < | < | | | < | < < | < < | < < < < < < | < < < < < < < < < < < < | | < < | < < < < > | > | | | < < > > > > | < | < | < < < < | | | | | | | | | < > > | | > | | | | | | | | | | | > > > > | | | | > > > | > > > > > > > > > > > | | > > > > > > > > > > > | < < < < < < < | | < > > | > > > > | > > | > | > | > | > > > | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 | #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.
1 2 3 4 5 6 7 8 9 | # # Borland C++Builder Makefile (makefile.bcc) # LIB = tlib CC = bcc32 CFLAGS = -c -O2 -I. | > | < < < < | | > | | | < | | | | > | | | | | | | | | > > | | > | > | 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 | # # 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.
1 2 3 4 5 6 7 8 9 10 | #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] | | > | < < < < | | > | | | < | | | | > | | | | | | | | | > > | | > | > | 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 | #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.
1 2 3 4 5 6 7 8 9 10 11 12 13 | #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] # | | < > | < < < < | | > | | | < | | | | > | | | | | | | | | > > | | > | > | | | | | | | 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 | #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) $(CFLAGS) -DTIMER demo/timing.c libtommath.a -o ltmtest clean: rm -f *.bat *.pdf *.o *.a *.obj *.lib *.exe *.dll etclib/*.o demo/demo.o test ltmtest mpitest mtest/mtest mtest/mtest.exe \ *.idx *.toc *.log *.aux *.dvi *.lof *.ind *.ilg *.ps *.log *.s mpi.c *.il etc/*.il *.dyn cd etc ; make clean cd pics ; make clean |
Changes to libtommath/makefile.msvc.
1 2 3 4 5 6 7 8 | #MSVC Makefile # #Tom St Denis CFLAGS = /I. /Ox /DWIN32 /W3 /Fo$@ default: library | > | < < < < | | > | | | < | | | | > | | | | | | | | | | > | | > | 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 | #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.
1 2 3 | #Makefile for GCC # #Tom St Denis | < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < | | < < | > | | < < < < > | > | | | > > > | < | < | < | < < < < | | | | | | | | | < > > | | > > > > | | > | | | > > > | | | | | | | | | 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 | #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.
︙ | ︙ | |||
13 14 15 16 17 18 19 | 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 }; | > > > > > | 13 14 15 16 17 18 19 20 21 22 23 24 | 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.
1 2 3 4 5 6 7 8 | /* Default configuration for MPI library */ #ifndef MPI_CONFIG_H_ #define MPI_CONFIG_H_ /* For boolean options, 0 = no | > | 1 2 3 4 5 6 7 8 9 | /* Default configuration for MPI library */ /* $Id$ */ #ifndef MPI_CONFIG_H_ #define MPI_CONFIG_H_ /* For boolean options, 0 = no |
︙ | ︙ | |||
79 80 81 82 83 84 85 | #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 */ | > > > > | 80 81 82 83 84 85 86 87 88 89 90 | #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.
︙ | ︙ | |||
9 10 11 12 13 14 15 | #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) | > > > > > | 9 10 11 12 13 14 15 16 17 18 19 20 | #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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | /* 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 | > > | | | | 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 | /* 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)] |
︙ | ︙ | |||
98 99 100 101 102 103 104 | "invalid input parameter", /* MP_BADARG */ "result is undefined" /* MP_UNDEF */ }; /* Value to digit maps for radix conversion */ /* s_dmap_1 - standard digits and letters */ | | | | 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 | "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. |
︙ | ︙ | |||
252 253 254 255 256 257 258 | if((res = mp_init(&mp[pos])) != MP_OKAY) goto CLEANUP; } return MP_OKAY; CLEANUP: | | | 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 | 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() */ /* }}} */ |
︙ | ︙ | |||
349 350 351 352 353 354 355 | 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)); | | | 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 | 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) { |
︙ | ︙ | |||
439 440 441 442 443 444 445 | /* {{{ mp_clear_array(mp[], count) */ void mp_clear_array(mp_int mp[], int count) { ARGCHK(mp != NULL && count > 0, MP_BADARG); | | | | 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 | 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; | | | 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 | 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) |
︙ | ︙ | |||
835 836 837 838 839 840 841 | mp_err res; ARGCHK(a != NULL && b != NULL, MP_BADARG); if((res = mp_copy(a, b)) != MP_OKAY) return res; | | | | 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 | 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() */ /* }}} */ |
︙ | ︙ | |||
864 865 866 867 868 869 870 | 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, | | | | | 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 | 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 | mp_clear(&tmp); return res; } s_mp_exch(&tmp, c); mp_clear(&tmp); } else { | | | 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 | } else { if((res = mp_copy(a, c)) != MP_OKAY) return res; if((res = s_mp_mul(c, b)) != MP_OKAY) return res; } | | | | 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 | */ if((cmp = s_mp_cmp(a, b)) < 0) { if(r) { if((res = mp_copy(a, r)) != MP_OKAY) return res; } | | | 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 | 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 */ | | | | 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 | */ mp_err mp_expt(mp_int *a, mp_int *b, mp_int *c) { mp_int s, x; mp_err res; mp_digit d; | | | 1260 1261 1262 1263 1264 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 | /* 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) { | | | | | 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 | 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 | | | | | 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 | 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 */ | | | | 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 | /* Copy result to output parameter */ mp_sub_d(&x, 1, &x); s_mp_exch(&x, b); CLEANUP: mp_clear(&x); X: | | | 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 | /* 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) | | | | | 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 1648 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 | */ int mp_cmp_int(mp_int *a, long z) { mp_int tmp; int out; ARGCHK(a != NULL, MP_EQ); | | | 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 | ++k; } /* Initialize t */ if(mp_isodd(&u)) { if((res = mp_copy(&v, &t)) != MP_OKAY) goto CLEANUP; | | | | 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 | /* 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; | | | 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 | /* }}} */ /*------------------------------------------------------------------------*/ /* {{{ More I/O Functions */ /* {{{ mp_read_signed_bin(mp, str, len) */ | | | 2251 2252 2253 2254 2255 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 | 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; } | | | | | 2328 2329 2330 2331 2332 2333 2334 2335 2336 2337 2338 2339 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 | if(dp == end && *dp == 0) { *str = '\0'; return MP_OKAY; } /* Generate digits in reverse order */ while(dp < end) { | | | 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 | while(d != 0) { ++len; d >>= 1; } return len; | | | 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 | 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; | | | | | 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 | /* }}} */ /* {{{ mp_value_radix_size(num, qty, radix) */ /* num = number of digits qty = number of bits per digit radix = target base | | | | 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 | /* Add trailing NUL to end the string */ str[pos--] = '\0'; /* Reverse the digits and sign indicator */ ix = 0; while(ix < pos) { | | | | | 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 | /* }}} */ /* {{{ Arithmetic helpers */ /* {{{ s_mp_lshd(mp, 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 | /* }}} */ /* {{{ s_mp_mul_2(mp) */ mp_err s_mp_mul_2(mp_int *mp) { | | | 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 | 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; | | | 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 | 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 | | | | 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 | 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) { | | | | 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 | 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 | | | 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 | w = CARRYOUT(w) ? 0 : 1; ++ix; } /* Clobber any leading zeroes we created */ s_mp_clamp(a); | | | 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 | /* 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); | | | 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 | /* 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) { | | | 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 | mp_word w, k = 0; mp_size ix, jx; mp_digit *pa, *pt; for(ix = 0; ix < len; ++ix, ++b) { if(*b == 0) continue; | | | 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 | 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; | | | | 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 | 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 | | | 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 | /* 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) { | | | | | | 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(", 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 | for any quotient we could ever possibly get, so we should not have to check for failures here */ DIGIT(", 0) = q; } /* Denormalize remainder */ | | | | 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(", 0) = q; } /* Denormalize remainder */ if(d != 0) s_mp_div_2d(&rem, d); s_mp_clamp("); s_mp_clamp(&rem); /* Copy quotient back to output */ s_mp_exch(", 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 | dig = k / DIGIT_BIT; bit = k % DIGIT_BIT; mp_zero(a); if((res = s_mp_pad(a, dig + 1)) != MP_OKAY) return res; | | | 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 | { mp_size ua = USED(a); mp_digit *ap = DIGITS(a); if(ua > 1) return MP_GT; | | | 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 | if(*dp) return -1; /* not a power of two */ --dp; --ix; } return ((uv - 1) * DIGIT_BIT) + extra; | | | 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 | 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; | | | | 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 | 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. */ | | | > > > > | 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 3960 3961 3962 3963 3964 3965 3966 3967 3968 3969 3970 3971 3972 3973 3974 3975 3976 3977 3978 3979 3980 3981 3982 3983 3984 3985 | 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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 | /* 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" | > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | /* 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" |
︙ | ︙ | |||
204 205 206 207 208 209 210 | #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); | | > > > > | 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 | #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.
︙ | ︙ | |||
35 36 37 38 39 40 41 42 43 44 45 | #endif #include <stdio.h> #include <stdlib.h> #include <time.h> #include "mpi.c" FILE *rng; void rand_num(mp_int *a) { | > > > > > | > | | > | > > > > > > > > > > | | > | | > | > > > > > > > > > > | | > > > > > > > > > > > > > > > > > > > > > > > > > | | > > > > > > | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 | #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"); |
︙ | ︙ | |||
176 177 178 179 180 181 182 | 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); | | | | 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 | 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); |
︙ | ︙ | |||
293 294 295 296 297 298 299 300 | 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); } | > > > | > > > > > > > | 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 | 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 39 40 41 42 43 44 45 46 47 48 49 50 51 | } /* 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 | > > > > | 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | } /* 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 |
︙ | ︙ | |||
59 60 61 62 63 64 65 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | | | | 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | * * 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; |
︙ | ︙ | |||
186 187 188 189 190 191 192 193 194 195 196 197 198 199 | 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 | > > > > | 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 | 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 |
︙ | ︙ | |||
358 359 360 361 362 363 364 365 366 367 368 369 370 371 | /* 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 | > > > > | 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 | /* 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 | * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ /* Fast (comba) multiplier * | | | | | | | | 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 | } /* number of output digits to produce */ pa = MIN(digs, a->used + b->used); /* clear the carry */ _W = 0; | | | | 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 | return res; } } /* number of output digits to produce */ pa = a->used + b->used; _W = 0; | | | | | 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 | * 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... | | | | | 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 | if ((res = mp_grow (b, pa)) != MP_OKAY) { return res; } } /* number of output digits to produce */ W1 = 0; | | | | 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 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | | 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 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | | 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 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | | 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 | * 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> | | > > > > | 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 | if (a->sign != b->sign) { if (a->sign == MP_NEG) { return MP_LT; } else { return MP_GT; } } | | > > > > | 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 | int n; mp_digit *tmpa, *tmpb; /* compare based on # of non-zero digits */ if (a->used > b->used) { return MP_GT; } | | | 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 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | | 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 | /* shortcut */ if (a->used == 0) { return 0; } /* get number of digits and add that */ r = (a->used - 1) * DIGIT_BIT; | | > > > > | 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 | return res; } mp_set(&tq, 1); n = mp_count_bits(a) - mp_count_bits(b); if (((res = mp_abs(a, &ta)) != MP_OKAY) || | | | 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 | LBL_ERR: mp_clear_multi(&ta, &tb, &tq, &q, NULL); return res; } #else | | | | | | | | 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 | /* step 3. for i from n down to (t + 1) */ for (i = n; i >= (t + 1); i--) { if (i > x.used) { continue; } | | | | | | | 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 | goto LBL_Y; } q.dp[i - t - 1] = (q.dp[i - t - 1] - 1UL) & MP_MASK; } } | | | | | 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 | 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; | | | | 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 | /* [optional] store the quotient */ if (c != NULL) { mp_clamp(&q); mp_exch(&q, c); } mp_clear(&q); | | > > > > | 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 | } #endif /* no easy answer [c'est la vie]. Just division */ if ((res = mp_init_size(&q, a->used)) != MP_OKAY) { return res; } | | | | | | > > > > | 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 | /* 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] */ | | > > > > | 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 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | > > > > | 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 | 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; | | | 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 | #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 | | > > > > | 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 | mp_clear(&M[1]); return err; } } /* determine and setup reduction code */ if (redmode == 0) { | | | | 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 | 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; } | | | 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 | 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 */ | > > > > > | | 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 | * 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; | | | | | | | > > > > | 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 | * 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; | | | | | > > > > | 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 | 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); } | | | | | > > > > | 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 | * 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 */ | | | > > > > | 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 | * 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> | | | | | > > > > | 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 | /* 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); | | > > > > | 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 | /* b cannot be negative */ if (b->sign == MP_NEG || mp_iszero(b) == 1) { return MP_VAL; } /* init temps */ | | | 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 | /* if its too low */ while (mp_cmp_d(&C, 0) == MP_LT) { if ((res = mp_add(&C, b, &C)) != MP_OKAY) { goto LBL_ERR; } } | | | > > > > | 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 | 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 */ | | | | 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 | } 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 | | | | 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 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | | | | | | | | | | | | | | | | 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 | tmpy = y1.dp; for (x = B; x < b->used; x++) { *tmpy++ = *tmpb++; } } | | | | 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 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | | | | 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 | 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); | | | | | | | > > > > | 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 | b->used = a->used; { register mp_digit r, rr, *tmpa, *tmpb; /* alias for source */ tmpa = a->dp; | | | | | | | | | | | > > > > | 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 | /* 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; } | | > > > > | 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 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | | | | | | | 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 | do { /* t1 = t2 */ if ((res = mp_copy (&t2, &t1)) != MP_OKAY) { goto LBL_T3; } /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */ | | | | | | | | 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 | * 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. | | | 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 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | | 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 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | | | | 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 | * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ /* makes a truly random prime of a given size (bits), * * Flags are as follows: | | | 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 | 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); | | | | | > > > > | 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 | /* init a copy of the input */ if ((res = mp_init_copy (&t, a)) != MP_OKAY) { return res; } /* force temp to positive */ | | > > > > | 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 | mp_zero(a); /* make sure the radix is ok */ if (radix < 2 || radix > 64) { return MP_VAL; } | | | | | | | | > > > > | 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 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | | | | | | 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 | /* 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; } } | | > > > > | 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 | */ /* 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; | | | | | | | | | > > > > | 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 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | | | | | | | | | > > > > | 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 | */ /* determines the setup value */ int mp_reduce_2k_setup(mp_int *a, mp_digit *d) { int res, p; mp_int tmp; | | | | | > > > > | 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 | */ /* determines the setup value */ int mp_reduce_2k_setup_l(mp_int *a, mp_int *d) { int res; mp_int tmp; | | | | | > > > > | 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 | */ /* determines if mp_reduce_2k can be used */ int mp_reduce_is_2k(mp_int *a) { int ix, iy, iw; mp_digit iz; | | | > > > > | 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 | * 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; | | | > > > > | 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 | /* 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; | | > > > > | 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 | /* bottom */ bottom = a->dp; /* top [offset into digits] */ top = a->dp + b; | | | | > > > > | 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 | /* set a 32-bit const */ int mp_set_int (mp_int * a, unsigned long b) { int x, res; mp_zero (a); | | > > > > | 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 | */ /* shrink a bignum */ int mp_shrink (mp_int * a) { mp_digit *tmp; int used = 1; | | | > > > > | 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 | int res; #ifdef BN_MP_TOOM_SQR_C /* use Toom-Cook? */ if (a->used >= TOOM_SQR_CUTOFF) { res = mp_toom_sqr(a, b); /* Karatsuba? */ | | | | | > > > > | 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 | * 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 */ | | | 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 | if ((res = mp_init(&t2)) != MP_OKAY) { goto E2; } /* First approx. (not very bad for large arg) */ mp_rshd (&t1,t1.used/2); | | | > > > > | 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 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | | | | | | | | | | | | | | | | | | | | | | | | | | | 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 | 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; } | | | | | | | | | | > > > > | 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 | * * The library is free for all purposes without any express * guarantee it works. * * Tom St Denis, [email protected], http://math.libtomcrypt.com */ | | | | 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 | 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; | | | 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 | /* U = carry bit of T[i] */ u = *tmpc >> ((mp_digit)DIGIT_BIT); /* take away carry bit from T[i] */ *tmpc++ &= MP_MASK; } | | | | 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 | winsize = 5; } #endif /* init M array */ /* init first cell */ if ((err = mp_init(&M[1])) != MP_OKAY) { | | | | | | | | | | 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 | * 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 | | | | 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 | /* 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]; | | | | 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 | 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); | | | 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 | /* 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 | | | | > > > > | 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 | /* 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. * | | < | < < < < < < < < < < < < < < < < < < | | | | | | > > > > | | > > > > | | > > | > | > > > > > | | < < < < | | < | | | | | | | | | < < | | < < | | < < < < < > > > > < > | | > > | 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 | /* 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 */ |
︙ | ︙ | |||
161 162 163 164 165 166 167 | TOOM_SQR_CUTOFF; /* define this to use lower memory usage routines (exptmods mostly) */ /* #define MP_LOW_MEM */ /* default precision */ #ifndef MP_PREC | | | | | | | | | 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 | 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); |
︙ | ︙ | |||
215 216 217 218 219 220 221 | 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) | | | > > > > > > > > > > > > > > > > > > > | | | | 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 | 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); |
︙ | ︙ | |||
343 344 345 346 347 348 349 350 351 352 353 354 355 356 | 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) */ | > | 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 | 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) */ |
︙ | ︙ | |||
378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 | 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 */ | > > > > | 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 | 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 */ |
︙ | ︙ | |||
449 450 451 452 453 454 455 | #ifdef MP_8BIT # define PRIME_SIZE 31 #else # define PRIME_SIZE 256 #endif /* table of first PRIME_SIZE primes */ | | | | | < | 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 | #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 * */ |
︙ | ︙ | |||
530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 | 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) | > > < < < < < < < < < < < < < < < < < < < < < < < > > > > > | 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 | 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.
|
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 | \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 |
Changes to libtommath/tommath.src.
︙ | ︙ | |||
62 63 64 65 66 67 68 | Greg Rose \\ QUALCOMM Australia \\ \end{tabular} %\end{small} } } \maketitle | | < < < < < < < < < < < | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 | 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 \\ |
︙ | ︙ | |||
400 401 402 403 404 405 406 | \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 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 | \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} |
︙ | ︙ | |||
603 604 605 606 607 608 609 | \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 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 | \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 |
︙ | ︙ | |||
750 751 752 753 754 755 756 | \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 | | | | | | | | | | | | | | | 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 | \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 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_clear} \end{figure} \textbf{Algorithm mp\_clear.} | | | | | | | | | | | 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 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_grow} \end{figure} \textbf{Algorithm mp\_grow.} | | | | | | | | | | | | 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 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_init\_size} \end{figure} \textbf{Algorithm mp\_init\_size.} | | | | | | | | | | | | | | | 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 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_init\_multi} \end{figure} \textbf{Algorithm mp\_init\_multi.} | | | | | | | | | | | | | | | | | | | | 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 | \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 | | | | | | 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 | % 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 | | | | | 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 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 | \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 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 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 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_zero} \end{figure} \textbf{Algorithm mp\_zero.} | | | | 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 1293 1294 1295 1296 1297 1298 | \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, | | | 1281 1282 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 | \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 | | | 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 1353 1354 1355 1356 1357 1358 | \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 \\ | | | | | | | | 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 | \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] |
︙ | ︙ | |||
1403 1404 1405 1406 1407 1408 1409 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_set\_int} \end{figure} \textbf{Algorithm mp\_set\_int.} | | | | | | | | | | | 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 | \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 \\ |
︙ | ︙ | |||
1466 1467 1468 1469 1470 1471 1472 | \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 | | | | | | | | | 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 | \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 $b$ \\ |
︙ | ︙ | |||
1506 1507 1508 1509 1510 1511 1512 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_cmp} \end{figure} \textbf{Algorithm mp\_cmp.} | | | | | | | | | | | | | | | | | | | | | | 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 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 1559 1560 1561 1562 1563 1564 1565 1566 | \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} |
︙ | ︙ | |||
1610 1611 1612 1613 1614 1615 1616 | \end{tabular} \end{small} \end{center} \caption{Algorithm s\_mp\_add} \end{figure} \textbf{Algorithm s\_mp\_add.} | | | | | | | | | | | | | | | | | 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 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 1648 1649 1650 1651 1652 1653 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 | \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 \\ |
︙ | ︙ | |||
1711 1712 1713 1714 1715 1716 1717 | \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. | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 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 $b$ \\ |
︙ | ︙ | |||
1786 1787 1788 1789 1790 1791 1792 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_add} \end{figure} \textbf{Algorithm mp\_add.} | | | | 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 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} \\ |
︙ | ︙ | |||
1817 1818 1819 1820 1821 1822 1823 | \end{tabular} \end{center} \end{small} \caption{Addition Guide Chart} \label{fig:AddChart} \end{figure} | | | | | | | | 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 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 1869 1870 1871 1872 1873 1874 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_sub} \end{figure} \textbf{Algorithm mp\_sub.} | | | 1857 1858 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 | \end{tabular} \end{center} \end{small} \caption{Subtraction Guide Chart} \label{fig:SubChart} \end{figure} | | | | | | | | | | | 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 | \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 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mul\_2} \end{figure} \textbf{Algorithm mp\_mul\_2.} | | | | | | | | | 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 | \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 | \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. | | | | | | | | | 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 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 $b$ \\ |
︙ | ︙ | |||
2060 2061 2062 2063 2064 2065 2066 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_lshd} \end{figure} \textbf{Algorithm mp\_lshd.} | | | | | | | | | | | 2049 2050 2051 2052 2053 2054 2055 2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067 2068 2069 2070 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085 2086 2087 2088 | \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 $b$ \\ |
︙ | ︙ | |||
2118 2119 2120 2121 2122 2123 2124 | \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 | | | | | | 2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130 2131 2132 2133 2134 2135 2136 2137 2138 2139 2140 2141 2142 | \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 | \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. | | | | | | | | | | | | | 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 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_div\_2d} \end{figure} \textbf{Algorithm mp\_div\_2d.} | | | | | | 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 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mod\_2d} \end{figure} \textbf{Algorithm mp\_mod\_2d.} | | | | | | 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 | $\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} | | | | | | | | | | | | | | | | | | | | | | | | 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 | \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 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 2400 2401 2402 2403 2404 2405 2406 2407 2408 2409 2410 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 | \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} |
︙ | ︙ | |||
2547 2548 2549 2550 2551 2552 2553 | \hline \end{tabular} \end{center} \end{small} \caption{Algorithm Comba Fixup} \end{figure} | | | | | | | | | 2536 2537 2538 2539 2540 2541 2542 2543 2544 2545 2546 2547 2548 2549 2550 2551 2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 2562 2563 2564 2565 2566 2567 2568 2569 2570 2571 2572 2573 2574 2575 2576 2577 2578 2579 2580 2581 2582 2583 2584 2585 | \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 $digs$ \\ |
︙ | ︙ | |||
2628 2629 2630 2631 2632 2633 2634 | \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. | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 2617 2618 2619 2620 2621 2622 2623 2624 2625 2626 2627 2628 2629 2630 2631 2632 2633 2634 2635 2636 2637 2638 2639 2640 2641 2642 2643 2644 2645 2646 2647 2648 2649 2650 2651 2652 2653 2654 2655 2656 2657 2658 2659 2660 2661 2662 2663 2664 2665 2666 2667 2668 2669 2670 2671 2672 2673 2674 2675 2676 2677 2678 2679 2680 2681 2682 2683 2684 2685 2686 2687 2688 2689 2690 2691 2692 2693 2694 2695 2696 2697 2698 2699 2700 2701 2702 2703 2704 2705 2706 2707 2708 2709 2710 2711 | \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}\\ |
︙ | ︙ | |||
2733 2734 2735 2736 2737 2738 2739 | \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 | | | | | | | | | | | | | 2722 2723 2724 2725 2726 2727 2728 2729 2730 2731 2732 2733 2734 2735 2736 2737 2738 2739 2740 2741 2742 2743 2744 2745 2746 2747 2748 2749 2750 2751 2752 2753 2754 2755 2756 2757 2758 2759 2760 2761 2762 2763 2764 2765 2766 2767 2768 2769 2770 2771 2772 2773 2774 2775 2776 2777 2778 2779 2780 2781 2782 2783 2784 2785 2786 2787 2788 2789 2790 2791 | \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 $b$ \\ |
︙ | ︙ | |||
2826 2827 2828 2829 2830 2831 2832 | 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}).\\ | | | | | | | | | | | | | | | | | 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 2855 2856 2857 2858 2859 2860 2861 2862 2863 2864 2865 2866 2867 2868 2869 2870 2871 2872 2873 2874 2875 2876 2877 2878 2879 2880 2881 2882 2883 2884 2885 2886 2887 2888 2889 2890 2891 2892 2893 2894 2895 2896 | 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 $b$ \\ |
︙ | ︙ | |||
2963 2964 2965 2966 2967 2968 2969 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_toom\_mul (continued)} \end{figure} \textbf{Algorithm mp\_toom\_mul.} | | | | | | | | | | | 2952 2953 2954 2955 2956 2957 2958 2959 2960 2961 2962 2963 2964 2965 2966 2967 2968 2969 2970 2971 2972 2973 2974 2975 2976 2977 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 | \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 $b$ \\ |
︙ | ︙ | |||
3036 3037 3038 3039 3040 3041 3042 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mul} \end{figure} \textbf{Algorithm mp\_mul.} | | | | | | | | | | | | | | | | 3025 3026 3027 3028 3029 3030 3031 3032 3033 3034 3035 3036 3037 3038 3039 3040 3041 3042 3043 3044 3045 3046 3047 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 | \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 $a$ \\ |
︙ | ︙ | |||
3127 3128 3129 3130 3131 3132 3133 | \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 | | | | | | | | | | | | 3116 3117 3118 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 3164 3165 3166 | \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}. \\ |
︙ | ︙ | |||
3201 3202 3203 3204 3205 3206 3207 | 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}) \\ | | | | | | | | | | | | | | | | | | 3190 3191 3192 3193 3194 3195 3196 3197 3198 3199 3200 3201 3202 3203 3204 3205 3206 3207 3208 3209 3210 3211 3212 3213 3214 3215 3216 3217 3218 3219 3220 3221 3222 3223 3224 3225 3226 3227 3228 3229 3230 3231 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 | 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 $a$ \\ |
︙ | ︙ | |||
3308 3309 3310 3311 3312 3313 3314 | 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 | | | | | | | | | | 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 3326 3327 3328 3329 3330 3331 3332 3333 3334 3335 3336 3337 3338 3339 3340 3341 3342 3343 3344 3345 3346 3347 3348 3349 | 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}. \\ |
︙ | ︙ | |||
3379 3380 3381 3382 3383 3384 3385 | \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 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 3368 3369 3370 3371 3372 3373 3374 3375 3376 3377 3378 3379 3380 3381 3382 3383 3384 3385 3386 3387 3388 3389 3390 3391 3392 3393 3394 3395 3396 3397 3398 3399 3400 3401 3402 3403 3404 3405 3406 3407 3408 3409 3410 3411 3412 3413 3414 3415 3416 3417 3418 3419 3420 3421 3422 3423 3424 3425 3426 3427 3428 3429 3430 3431 3432 3433 3434 3435 3436 3437 3438 3439 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 3474 3475 3476 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 3532 3533 3534 3535 3536 3537 3538 3539 3540 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 | \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 | \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 | | | | | | | | | | | | | | 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 | \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 | | | | | | | | | | 3655 3656 3657 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 | \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 | \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 | | | | 3698 3699 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 3736 3737 3738 3739 3740 3741 | \end{tabular} \end{center} \end{small} \caption{Example of Montgomery Reduction (I)} \label{fig:MONT1} \end{figure} | | | | | | 3724 3725 3726 3727 3728 3729 3730 3731 3732 3733 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 | \end{tabular} \end{center} \end{small} \caption{Example of Montgomery Reduction (II)} \label{fig:MONT2} \end{figure} | | | | | | 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 3818 3819 3820 3821 3822 | \hline \end{tabular} \end{center} \end{small} \caption{Algorithm Montgomery Reduction (modified II)} \end{figure} | | | | | | | | | | | 3805 3806 3807 3808 3809 3810 3811 3812 3813 3814 3815 3816 3817 3818 3819 3820 3821 3822 3823 3824 3825 3826 3827 3828 3829 3830 3831 3832 3833 3834 3835 3836 3837 3838 3839 3840 3841 3842 3843 3844 3845 3846 3847 3848 3849 3850 3851 3852 3853 3854 3855 3856 3857 | \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 $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ |
︙ | ︙ | |||
3902 3903 3904 3905 3906 3907 3908 | \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 | | | | | | | | | | | | | 3891 3892 3893 3894 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 3926 3927 3928 3929 3930 3931 3932 3933 3934 3935 3936 3937 3938 3939 3940 3941 | \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 $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ |
︙ | ︙ | |||
3987 3988 3989 3990 3991 3992 3993 | \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 | | | | | | | | | | | | | | | 3976 3977 3978 3979 3980 3981 3982 3983 3984 3985 3986 3987 3988 3989 3990 3991 3992 3993 3994 3995 3996 3997 3998 3999 4000 4001 4002 4003 4004 4005 4006 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 4060 4061 4062 4063 4064 4065 4066 4067 4068 4069 4070 4071 4072 4073 4074 4075 4076 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}. \\ |
︙ | ︙ | |||
4104 4105 4106 4107 4108 4109 4110 | \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. | | | | | | | | 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 | 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. | | | | | | | | | | | | | | | 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 | \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 | | | | | | | | | | 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 | \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 | | | | | | | 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 | \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 | | | 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 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_reduce\_is\_2k} \end{figure} \textbf{Algorithm mp\_reduce\_is\_2k.} | | | | 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 | & \\ $\left [ 2 \right ]$ & 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. \\ & \\ | | | | | | 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 4474 4475 4476 4477 4478 4479 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 | \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 | | | | | | | | 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 | \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 | | | | | | | | | 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 | \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 | | | | 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 | \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 | | | | | 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 | \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 | | | | | | | | | | | | | 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 | \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 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_exptmod} \end{figure} \textbf{Algorithm mp\_exptmod.} | | | | | | | 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 | \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. | | | | | | | | | | | | | | | | | | | | 4877 4878 4879 4880 4881 4882 4883 4884 4885 4886 4887 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 | \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 | \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 | | | | | | 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 | \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. | | | | | | | | | | | | | | | 5009 5010 5011 5012 5013 5014 5015 5016 5017 5018 5019 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 | \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. |
︙ | ︙ | |||
5080 5081 5082 5083 5084 5085 5086 | 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 | | | | | | 5069 5070 5071 5072 5073 5074 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}. \\ |
︙ | ︙ | |||
5184 5185 5186 5187 5188 5189 5190 | \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. | | | | | | | | | | | | | | | | | | | | | | | | 5173 5174 5175 5176 5177 5178 5179 5180 5181 5182 5183 5184 5185 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 5211 5212 5213 5214 5215 5216 5217 5218 5219 5220 5221 5222 5223 5224 5225 5226 5227 5228 5229 5230 5231 5232 5233 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 5319 5320 5321 5322 5323 5324 | \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mul\_d} \end{figure} \textbf{Algorithm mp\_mul\_d.} | | | | | | | 5307 5308 5309 5310 5311 5312 5313 5314 5315 5316 5317 5318 5319 5320 5321 5322 5323 5324 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 5366 5367 5368 5369 5370 5371 | \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 $\hat w$ | | | | | | | | | | | | 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 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 | \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 | | | | | | | 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 | \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 | | | | | 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 | \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_read\_radix} \end{figure} \textbf{Algorithm mp\_read\_radix.} | | | | 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 | \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_toradix} \end{figure} \textbf{Algorithm mp\_toradix.} | | | | 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 | \caption{Example of Algorithm mp\_toradix.} \label{fig:mpradix} \end{figure} EXAM,bn_mp_toradix.c \chapter{Number Theoretic Algorithms} | | | | 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 | \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 | | | | | 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 | \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 | | | | | | | 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 | \end{tabular} \end{center} \end{small} \caption{Algorithm Greatest Common Divisor (III)} \label{fig:gcd3} \end{figure} | | | | | | 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] |
︙ | ︙ | |||
5773 5774 5775 5776 5777 5778 5779 | \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 | | | | | | | | | | | | | | | | | 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 | \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}. \\ |
︙ | ︙ | |||
5844 5845 5846 5847 5848 5849 5850 | \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} | | | | | 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 | \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{)} |
︙ | ︙ | |||
5880 5881 5882 5883 5884 5885 5886 | 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 \\ | | | | | | | | | | | | | | | | | 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 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 5938 5939 5940 5941 5942 5943 | 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 $p$, $a \ge 0$, $p \ge 3$, $p \equiv 1 \mbox{ (mod }2\mbox{)}$ \\ |
︙ | ︙ | |||
5984 5985 5986 5987 5988 5989 5990 | \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 | | | | | | | | | | | | | | | | | | | | | 5973 5974 5975 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 6001 6002 6003 6004 6005 6006 6007 6008 6009 6010 6011 6012 6013 6014 6015 6016 6017 6018 6019 6020 6021 6022 6023 6024 6025 6026 6027 6028 6029 6030 6031 6032 6033 6034 6035 6036 6037 6038 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 | 15. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \end{figure} \textbf{Algorithm mp\_invmod.} | | | | | | | | | | | | | | | | | | | 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 6148 6149 6150 6151 6152 6153 6154 6155 6156 6157 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 | \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_prime\_is\_divisible} \end{figure} \textbf{Algorithm mp\_prime\_is\_divisible.} | | | | | | | | | | 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 | \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 | \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 | | | | | 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 | \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 | | | | 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 |
︙ | ︙ |
Changes to libtommath/tommath.tex.
︙ | ︙ | |||
62 63 64 65 66 67 68 | Greg Rose \\ QUALCOMM Australia \\ \end{tabular} %\end{small} } } \maketitle | | < < < < < < < < < < < | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 | 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 \\ |
︙ | ︙ | |||
400 401 402 403 404 405 406 | \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 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 | \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} |
︙ | ︙ | |||
609 610 611 612 613 614 615 | \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 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 | \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 |
︙ | ︙ | |||
756 757 758 759 760 761 762 | \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 | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 | \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 $a$ \\ |
︙ | ︙ | |||
833 834 835 836 837 838 839 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_clear} \end{figure} \textbf{Algorithm mp\_clear.} | | | > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 | \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 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_grow} \end{figure} \textbf{Algorithm mp\_grow.} | | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 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 1030 | \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 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_init\_size} \end{figure} \textbf{Algorithm mp\_init\_size.} | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 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 1110 1111 1112 1113 1114 | \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 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_init\_multi} \end{figure} \textbf{Algorithm mp\_init\_multi.} | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | | | | | | | 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 | \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 $a$ \\ |
︙ | ︙ | |||
1079 1080 1081 1082 1083 1084 1085 | \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 | | > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 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 1274 1275 1276 1277 1278 1279 1280 | \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. \\ & \\ |
︙ | ︙ | |||
1119 1120 1121 1122 1123 1124 1125 | % 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 | | | | | 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 | % 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 $b$. \\ |
︙ | ︙ | |||
1156 1157 1158 1159 1160 1161 1162 | \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 | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | | | | | | | | | | | | | > > > > > > > > > > > > > | | | 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 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 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 | \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} |
︙ | ︙ | |||
1306 1307 1308 1309 1310 1311 1312 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_zero} \end{figure} \textbf{Algorithm mp\_zero.} | | > > > > > > > > > > > > > > > > > | | 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 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 1579 1580 1581 | \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. |
︙ | ︙ | |||
1343 1344 1345 1346 1347 1348 1349 | \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, | | > > > > > > > > > > > > > > > > > > > > > > > > | | 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 | \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] |
︙ | ︙ | |||
1383 1384 1385 1386 1387 1388 1389 | \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 | | > > > > > > > > > > > > > > > > > > > > > | | | > > > > > > > > > > | | | | | 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 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 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 | \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] |
︙ | ︙ | |||
1469 1470 1471 1472 1473 1474 1475 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_set\_int} \end{figure} \textbf{Algorithm mp\_set\_int.} | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 | \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 \\ |
︙ | ︙ | |||
1537 1538 1539 1540 1541 1542 1543 | \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 | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 | \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 $b$ \\ |
︙ | ︙ | |||
1582 1583 1584 1585 1586 1587 1588 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_cmp} \end{figure} \textbf{Algorithm mp\_cmp.} | | | | > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | | | | | | | | 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2053 | \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} |
︙ | ︙ | |||
1690 1691 1692 1693 1694 1695 1696 | \end{tabular} \end{small} \end{center} \caption{Algorithm s\_mp\_add} \end{figure} \textbf{Algorithm s\_mp\_add.} | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | | | | | | 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 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 2152 2153 2154 2155 2156 2157 2158 2159 2160 2161 2162 2163 2164 2165 2166 2167 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 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 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 2261 2262 2263 2264 2265 | \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 \\ |
︙ | ︙ | |||
1795 1796 1797 1798 1799 1800 1801 | \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. | | | | | | | | | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | | | | | | 2281 2282 2283 2284 2285 2286 2287 2288 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 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 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 2382 2383 2384 2385 2386 2387 2388 2389 2390 2391 2392 2393 2394 2395 2396 2397 2398 2399 2400 2401 2402 2403 2404 2405 2406 2407 2408 2409 2410 2411 2412 2413 2414 2415 2416 2417 2418 | \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 $b$ \\ |
︙ | ︙ | |||
1875 1876 1877 1878 1879 1880 1881 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_add} \end{figure} \textbf{Algorithm mp\_add.} | | | | 2432 2433 2434 2435 2436 2437 2438 2439 2440 2441 2442 2443 2444 2445 2446 2447 | \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} \\ |
︙ | ︙ | |||
1906 1907 1908 1909 1910 1911 1912 | \end{tabular} \end{center} \end{small} \caption{Addition Guide Chart} \label{fig:AddChart} \end{figure} | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | 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 | \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 $c = a - b$. \\ |
︙ | ︙ | |||
1962 1963 1964 1965 1966 1967 1968 | \hline \end{tabular} \end{center} \caption{Algorithm mp\_sub} \end{figure} \textbf{Algorithm mp\_sub.} | | | 2553 2554 2555 2556 2557 2558 2559 2560 2561 2562 2563 2564 2565 2566 2567 | \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|} |
︙ | ︙ | |||
1989 1990 1991 1992 1993 1994 1995 | \end{tabular} \end{center} \end{small} \caption{Subtraction Guide Chart} \label{fig:SubChart} \end{figure} | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | 2580 2581 2582 2583 2584 2585 2586 2587 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 2626 2627 2628 2629 2630 2631 2632 2633 2634 2635 2636 2637 2638 2639 2640 2641 2642 2643 2644 2645 2646 2647 2648 2649 2650 2651 2652 2653 2654 2655 2656 2657 2658 2659 | \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 $a$ \\ |
︙ | ︙ | |||
2048 2049 2050 2051 2052 2053 2054 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mul\_2} \end{figure} \textbf{Algorithm mp\_mul\_2.} | | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | 2679 2680 2681 2682 2683 2684 2685 2686 2687 2688 2689 2690 2691 2692 2693 2694 2695 2696 2697 2698 2699 2700 2701 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 2732 2733 2734 2735 2736 2737 2738 2739 2740 2741 2742 2743 2744 2745 2746 2747 2748 2749 2750 2751 2752 2753 2754 2755 2756 2757 2758 2759 2760 2761 2762 2763 2764 2765 2766 2767 2768 2769 2770 2771 2772 2773 2774 2775 2776 2777 2778 2779 2780 | \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} |
︙ | ︙ | |||
2113 2114 2115 2116 2117 2118 2119 | \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. | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | 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 2855 2856 2857 2858 2859 2860 2861 2862 2863 2864 2865 2866 2867 2868 2869 2870 2871 2872 2873 2874 2875 2876 2877 2878 2879 2880 2881 2882 2883 2884 2885 2886 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 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_lshd} \end{figure} \textbf{Algorithm mp\_lshd.} | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | 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 2967 2968 2969 2970 2971 2972 2973 2974 2975 2976 2977 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 | \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 | \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 | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | 3028 3029 3030 3031 3032 3033 3034 3035 3036 3037 3038 3039 3040 3041 3042 3043 3044 3045 3046 3047 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 3108 3109 3110 3111 3112 3113 3114 3115 3116 3117 3118 3119 3120 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 | \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. | | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | 3148 3149 3150 3151 3152 3153 3154 3155 3156 3157 3158 3159 3160 3161 3162 3163 3164 3165 3166 3167 3168 3169 3170 3171 3172 3173 3174 3175 3176 3177 3178 3179 3180 3181 3182 3183 3184 3185 3186 3187 3188 3189 3190 3191 3192 3193 3194 3195 3196 3197 3198 3199 3200 3201 3202 3203 3204 3205 3206 3207 3208 3209 3210 3211 3212 3213 3214 3215 3216 3217 3218 3219 3220 3221 3222 3223 3224 3225 3226 3227 3228 3229 3230 3231 3232 3233 3234 3235 3236 3237 3238 3239 3240 3241 3242 3243 3244 3245 3246 3247 3248 3249 3250 3251 3252 3253 3254 3255 | \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} |
︙ | ︙ | |||
2369 2370 2371 2372 2373 2374 2375 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_div\_2d} \end{figure} \textbf{Algorithm mp\_div\_2d.} | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | 3279 3280 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 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 3363 3364 3365 3366 3367 3368 3369 3370 3371 3372 3373 3374 3375 3376 3377 3378 3379 3380 3381 3382 3383 3384 3385 3386 3387 3388 3389 3390 3391 3392 3393 3394 3395 | \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 $b$ \\ |
︙ | ︙ | |||
2423 2424 2425 2426 2427 2428 2429 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mod\_2d} \end{figure} \textbf{Algorithm mp\_mod\_2d.} | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | 3413 3414 3415 3416 3417 3418 3419 3420 3421 3422 3423 3424 3425 3426 3427 3428 3429 3430 3431 3432 3433 3434 3435 3436 3437 3438 3439 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 3474 3475 3476 3477 3478 3479 3480 3481 3482 | \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 2474 2475 2476 2477 2478 2479 | $\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} | | | | | | | | | | | | | | | | | | | | | | | | 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 3532 3533 3534 3535 3536 3537 3538 3539 3540 3541 3542 3543 3544 3545 3546 3547 3548 | $\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}. \\ |
︙ | ︙ | |||
2550 2551 2552 2553 2554 2555 2556 | \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 | | | | | | | | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | | | | | | | | | | | | | | | | 3578 3579 3580 3581 3582 3583 3584 3585 3586 3587 3588 3589 3590 3591 3592 3593 3594 3595 3596 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 3643 3644 3645 3646 3647 3648 3649 3650 3651 3652 3653 3654 3655 3656 3657 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 3704 3705 3706 3707 3708 3709 3710 3711 3712 3713 3714 3715 3716 3717 3718 3719 3720 3721 3722 3723 3724 3725 3726 3727 3728 3729 3730 3731 3732 3733 3734 3735 3736 3737 3738 3739 3740 3741 3742 3743 3744 3745 3746 3747 3748 3749 3750 3751 3752 3753 3754 3755 3756 3757 3758 3759 3760 3761 3762 3763 3764 3765 3766 3767 3768 3769 3770 3771 3772 3773 3774 3775 3776 3777 3778 3779 3780 | \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} |
︙ | ︙ | |||
2690 2691 2692 2693 2694 2695 2696 | \hline \end{tabular} \end{center} \end{small} \caption{Algorithm Comba Fixup} \end{figure} | | | | | | | | | 3789 3790 3791 3792 3793 3794 3795 3796 3797 3798 3799 3800 3801 3802 3803 3804 3805 3806 3807 3808 3809 3810 3811 3812 3813 3814 3815 3816 3817 3818 3819 3820 3821 3822 3823 3824 3825 3826 3827 3828 3829 3830 3831 3832 3833 3834 3835 3836 3837 3838 | \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 $digs$ \\ |
︙ | ︙ | |||
2771 2772 2773 2774 2775 2776 2777 | \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. | | | | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 3870 3871 3872 3873 3874 3875 3876 3877 3878 3879 3880 3881 3882 3883 3884 3885 3886 3887 3888 3889 3890 3891 3892 3893 3894 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 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 3954 3955 3956 3957 3958 3959 3960 3961 3962 3963 3964 3965 3966 3967 3968 3969 3970 3971 3972 3973 3974 3975 3976 3977 3978 3979 3980 3981 3982 3983 3984 3985 3986 3987 3988 3989 3990 3991 3992 3993 3994 3995 3996 3997 3998 3999 4000 4001 4002 4003 4004 4005 4006 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 | \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}\\ |
︙ | ︙ | |||
2881 2882 2883 2884 2885 2886 2887 | \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 | | | | | | | | | | | | | 4068 4069 4070 4071 4072 4073 4074 4075 4076 4077 4078 4079 4080 4081 4082 4083 4084 4085 4086 4087 4088 4089 4090 4091 4092 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 | \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 $b$ \\ |
︙ | ︙ | |||
2974 2975 2976 2977 2978 2979 2980 | 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}).\\ | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | 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 4191 4192 4193 4194 4195 4196 4197 4198 4199 4200 4201 4202 4203 4204 4205 4206 4207 4208 4209 4210 4211 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 4251 4252 4253 4254 4255 4256 4257 4258 4259 4260 4261 4262 4263 4264 4265 4266 4267 4268 4269 4270 4271 4272 4273 4274 4275 4276 4277 4278 4279 4280 4281 4282 4283 4284 4285 4286 4287 4288 4289 4290 4291 4292 4293 4294 4295 4296 4297 4298 4299 4300 4301 4302 4303 4304 4305 4306 4307 4308 4309 4310 4311 4312 4313 4314 4315 4316 4317 4318 4319 4320 4321 4322 4323 4324 4325 4326 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 4355 4356 4357 4358 4359 4360 4361 4362 4363 4364 4365 4366 4367 4368 4369 4370 4371 4372 4373 4374 4375 4376 4377 4378 4379 4380 4381 4382 4383 4384 4385 4386 4387 4388 4389 4390 4391 4392 4393 4394 4395 | 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 $b$ \\ |
︙ | ︙ | |||
3116 3117 3118 3119 3120 3121 3122 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_toom\_mul (continued)} \end{figure} \textbf{Algorithm mp\_toom\_mul.} | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | 4451 4452 4453 4454 4455 4456 4457 4458 4459 4460 4461 4462 4463 4464 4465 4466 4467 4468 4469 4470 4471 4472 4473 4474 4475 4476 4477 4478 4479 4480 4481 4482 4483 4484 4485 4486 4487 4488 4489 4490 4491 4492 4493 4494 4495 4496 4497 4498 4499 4500 4501 4502 4503 4504 4505 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 4548 4549 4550 4551 4552 4553 4554 4555 4556 4557 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 4591 4592 4593 4594 4595 4596 4597 4598 4599 4600 4601 4602 4603 4604 4605 4606 4607 4608 4609 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 4635 4636 4637 4638 4639 4640 4641 4642 4643 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 4673 4674 4675 4676 4677 4678 4679 4680 4681 4682 4683 4684 4685 4686 4687 4688 4689 4690 4691 4692 4693 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 4741 4742 4743 4744 4745 4746 4747 4748 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 | \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 3200 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mul} \end{figure} \textbf{Algorithm mp\_mul.} | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | | 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 4835 4836 4837 4838 4839 4840 4841 4842 4843 4844 4845 4846 4847 4848 4849 4850 4851 4852 4853 4854 4855 4856 4857 4858 4859 4860 4861 4862 4863 4864 4865 4866 4867 4868 4869 4870 4871 4872 4873 4874 4875 4876 4877 4878 4879 4880 4881 4882 4883 4884 4885 4886 4887 4888 4889 4890 4891 4892 4893 4894 4895 4896 4897 4898 4899 4900 4901 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 | \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 | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | 4939 4940 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 4994 4995 4996 4997 4998 4999 5000 5001 5002 5003 5004 5005 5006 5007 5008 5009 5010 5011 5012 5013 5014 5015 5016 5017 5018 5019 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 | \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}. \\ |
︙ | ︙ | |||
3368 3369 3370 3371 3372 3373 3374 | 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}) \\ | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | | | 5083 5084 5085 5086 5087 5088 5089 5090 5091 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 5131 5132 5133 5134 5135 5136 5137 5138 5139 5140 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 5169 5170 5171 5172 5173 5174 5175 5176 5177 5178 5179 5180 5181 5182 5183 5184 5185 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 5211 5212 5213 5214 5215 5216 5217 5218 5219 5220 5221 5222 5223 5224 5225 5226 5227 5228 5229 5230 5231 5232 5233 5234 5235 5236 5237 5238 5239 5240 5241 5242 5243 5244 5245 5246 5247 5248 5249 5250 | 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 $a$ \\ |
︙ | ︙ | |||
3480 3481 3482 3483 3484 3485 3486 | 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 | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | 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 5331 5332 5333 5334 5335 5336 5337 5338 5339 5340 5341 5342 5343 5344 5345 5346 5347 5348 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 5388 5389 5390 5391 5392 5393 5394 5395 5396 5397 5398 5399 5400 5401 5402 5403 5404 5405 5406 5407 5408 5409 5410 5411 5412 5413 5414 5415 5416 5417 5418 5419 5420 5421 5422 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 | 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}. \\ |
︙ | ︙ | |||
3556 3557 3558 3559 3560 3561 3562 | \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 | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 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 5498 5499 5500 5501 5502 5503 5504 5505 5506 5507 5508 5509 5510 5511 5512 5513 5514 5515 5516 5517 5518 5519 5520 5521 5522 5523 5524 5525 5526 5527 5528 5529 5530 5531 5532 5533 5534 5535 5536 5537 5538 5539 5540 5541 5542 5543 5544 5545 5546 5547 5548 5549 5550 5551 5552 5553 5554 5555 5556 5557 5558 5559 5560 5561 5562 5563 5564 5565 5566 5567 5568 5569 5570 5571 5572 5573 5574 5575 5576 5577 5578 5579 5580 5581 5582 5583 5584 5585 5586 5587 5588 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 5619 5620 5621 5622 5623 5624 5625 5626 5627 5628 5629 5630 5631 5632 5633 5634 5635 5636 5637 5638 5639 5640 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 5689 5690 5691 5692 5693 5694 5695 5696 5697 5698 5699 5700 5701 5702 5703 5704 5705 5706 5707 5708 5709 5710 5711 5712 | \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}. \\ |
︙ | ︙ | |||
3789 3790 3791 3792 3793 3794 3795 | \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 | | | | | | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | 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 | 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 | > > > > > > > > > > > > > > > | | | | | | | | | 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 | \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 | | | | 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 | \end{tabular} \end{center} \end{small} \caption{Example of Montgomery Reduction (I)} \label{fig:MONT1} \end{figure} | | | | | | 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 | \end{tabular} \end{center} \end{small} \caption{Example of Montgomery Reduction (II)} \label{fig:MONT2} \end{figure} | | | | | | 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 | \hline \end{tabular} \end{center} \end{small} \caption{Algorithm Montgomery Reduction (modified II)} \end{figure} | | | | | | | | | | | 6056 6057 6058 6059 6060 6061 6062 6063 6064 6065 6066 6067 6068 6069 6070 6071 6072 6073 6074 6075 6076 6077 6078 6079 6080 6081 6082 6083 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 | \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 $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ |
︙ | ︙ | |||
4093 4094 4095 4096 4097 4098 4099 | \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 | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | 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 6203 6204 6205 6206 6207 6208 6209 6210 6211 6212 6213 6214 6215 6216 6217 6218 6219 6220 6221 6222 6223 6224 6225 6226 6227 6228 6229 6230 6231 6232 6233 6234 6235 6236 6237 6238 6239 6240 6241 6242 6243 6244 6245 6246 6247 6248 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 6285 6286 6287 6288 6289 6290 6291 6292 6293 6294 6295 | \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 $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ |
︙ | ︙ | |||
4182 4183 4184 4185 4186 4187 4188 | \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 | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | 6330 6331 6332 6333 6334 6335 6336 6337 6338 6339 6340 6341 6342 6343 6344 6345 6346 6347 6348 6349 6350 6351 6352 6353 6354 6355 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 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 6461 6462 6463 6464 6465 6466 6467 6468 6469 6470 6471 6472 6473 6474 6475 6476 6477 6478 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 6514 6515 6516 6517 6518 6519 6520 6521 6522 6523 6524 6525 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 6558 6559 6560 6561 6562 6563 6564 6565 6566 6567 6568 6569 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 6610 6611 6612 6613 6614 6615 6616 6617 6618 6619 6620 6621 6622 6623 6624 6625 6626 6627 6628 6629 6630 6631 6632 6633 6634 | \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}. \\ |
︙ | ︙ | |||
4309 4310 4311 4312 4313 4314 4315 | \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. | | | | | | | | 6650 6651 6652 6653 6654 6655 6656 6657 6658 6659 6660 6661 6662 6663 6664 6665 6666 6667 6668 6669 6670 6671 6672 6673 6674 6675 6676 6677 6678 6679 6680 6681 6682 6683 6684 6685 6686 6687 6688 6689 6690 6691 6692 6693 6694 6695 6696 6697 6698 6699 6700 6701 6702 | \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} |
︙ | ︙ | |||
4370 4371 4372 4373 4374 4375 4376 | 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. | | | | | | | | | | | | | | | 6711 6712 6713 6714 6715 6716 6717 6718 6719 6720 6721 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 | 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}. \\ |
︙ | ︙ | |||
4428 4429 4430 4431 4432 4433 4434 | \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 | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | 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 6800 6801 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 6837 6838 6839 6840 6841 6842 6843 6844 6845 6846 6847 6848 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 6877 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. | > > > > > > > > > > > > > | 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}. | > > > > > > > > > > > > > > > > > > > > > > > > | 6952 6953 6954 6955 6956 6957 6958 6959 6960 6961 6962 6963 6964 6965 6966 6967 6968 6969 6970 6971 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 | \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 | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | 7013 7014 7015 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 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 7085 7086 7087 7088 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 | \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 | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 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 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_reduce\_is\_2k} \end{figure} \textbf{Algorithm mp\_reduce\_is\_2k.} | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | 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 | & \\ $\left [ 2 \right ]$ & 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. \\ & \\ | | | | | | 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 | \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 | | | | | | | | 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 | \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 | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | 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. |
︙ | ︙ | |||
4861 4862 4863 4864 4865 4866 4867 | \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 | | | | 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 | \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 $16$ & $2$ & $27$ & $24$ \\ |
︙ | ︙ | |||
4897 4898 4899 4900 4901 4902 4903 | \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 | | | | | 7521 7522 7523 7524 7525 7526 7527 7528 7529 7530 7531 7532 7533 7534 7535 7536 7537 7538 | \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 $16$ & $3$ & $24$ & $27$ \\ |
︙ | ︙ | |||
4951 4952 4953 4954 4955 4956 4957 | \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 | | | | | | | | | | | | | 7575 7576 7577 7578 7579 7580 7581 7582 7583 7584 7585 7586 7587 7588 7589 7590 7591 7592 7593 7594 7595 7596 7597 7598 7599 7600 7601 7602 7603 7604 7605 7606 7607 7608 7609 7610 7611 | \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 $c$ \\ |
︙ | ︙ | |||
5000 5001 5002 5003 5004 5005 5006 | \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_exptmod} \end{figure} \textbf{Algorithm mp\_exptmod.} | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | 7624 7625 7626 7627 7628 7629 7630 7631 7632 7633 7634 7635 7636 7637 7638 7639 7640 7641 7642 7643 7644 7645 7646 7647 7648 7649 7650 7651 7652 7653 7654 7655 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 7706 7707 7708 7709 7710 7711 7712 7713 7714 7715 7716 7717 7718 7719 7720 7721 7722 7723 7724 7725 7726 7727 7728 7729 7730 7731 7732 7733 7734 7735 7736 7737 7738 7739 7740 7741 7742 7743 7744 7745 7746 7747 7748 7749 7750 7751 7752 7753 7754 7755 7756 7757 7758 7759 7760 | \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} |
︙ | ︙ | |||
5133 5134 5135 5136 5137 5138 5139 | \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. | | | | | | | | | | | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | 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 8012 8013 8014 8015 8016 8017 8018 8019 8020 8021 8022 8023 8024 8025 8026 8027 8028 8029 8030 8031 8032 8033 8034 8035 8036 8037 8038 8039 8040 8041 8042 8043 8044 8045 8046 8047 8048 8049 8050 8051 8052 8053 8054 8055 8056 8057 8058 8059 8060 8061 8062 8063 8064 8065 8066 8067 8068 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 8123 8124 8125 8126 8127 8128 8129 8130 8131 8132 8133 8134 8135 8136 8137 8138 8139 8140 8141 8142 8143 8144 8145 8146 8147 8148 8149 8150 8151 8152 8153 8154 8155 8156 8157 8158 8159 8160 8161 8162 8163 8164 | \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 | \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 | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | 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 | \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. | | | | | | | | | | | | | | | 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 | 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 | | | | | | 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 | \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. | | | | | | | | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | 8426 8427 8428 8429 8430 8431 8432 8433 8434 8435 8436 8437 8438 8439 8440 8441 8442 8443 8444 8445 8446 8447 8448 8449 8450 8451 8452 8453 8454 8455 8456 8457 8458 8459 8460 8461 8462 8463 8464 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 8532 8533 8534 8535 8536 8537 8538 8539 8540 8541 8542 8543 8544 8545 8546 8547 8548 8549 8550 8551 8552 8553 8554 8555 8556 8557 8558 8559 8560 8561 8562 8563 8564 8565 8566 8567 8568 8569 8570 8571 8572 8573 8574 8575 8576 8577 8578 8579 8580 8581 8582 8583 8584 8585 8586 8587 8588 8589 8590 8591 8592 8593 8594 8595 8596 8597 8598 8599 8600 8601 8602 8603 8604 8605 8606 8607 8608 8609 8610 8611 8612 8613 8614 8615 8616 8617 8618 8619 8620 8621 8622 8623 8624 8625 8626 8627 8628 8629 8630 8631 8632 8633 8634 8635 8636 8637 8638 8639 8640 8641 8642 8643 8644 8645 8646 8647 8648 8649 8650 8651 8652 8653 8654 8655 8656 8657 8658 8659 8660 8661 8662 8663 8664 8665 8666 8667 8668 8669 8670 8671 8672 8673 8674 8675 8676 8677 8678 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 8707 8708 8709 8710 8711 8712 8713 8714 8715 8716 8717 8718 8719 8720 8721 8722 8723 8724 8725 8726 8727 8728 8729 8730 8731 8732 8733 8734 8735 8736 8737 8738 8739 8740 8741 8742 8743 8744 8745 8746 8747 8748 8749 8750 8751 8752 8753 8754 8755 8756 8757 8758 8759 8760 8761 8762 8763 8764 8765 8766 8767 8768 8769 8770 8771 8772 8773 8774 8775 8776 8777 8778 8779 8780 8781 8782 8783 8784 8785 8786 8787 8788 8789 | \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}. \\ |
︙ | ︙ | |||
5546 5547 5548 5549 5550 5551 5552 5553 5554 5555 5556 5557 5558 5559 | \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. | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 8803 8804 8805 8806 8807 8808 8809 8810 8811 8812 8813 8814 8815 8816 8817 8818 8819 8820 8821 8822 8823 8824 8825 8826 8827 8828 8829 8830 8831 8832 8833 8834 8835 8836 8837 8838 8839 8840 8841 8842 8843 8844 8845 8846 8847 8848 8849 8850 8851 8852 8853 8854 8855 8856 8857 8858 8859 8860 8861 8862 8863 8864 8865 8866 8867 8868 8869 8870 8871 8872 8873 8874 8875 8876 8877 8878 8879 8880 8881 8882 8883 8884 8885 8886 8887 8888 8889 8890 8891 8892 8893 8894 8895 8896 8897 8898 8899 8900 8901 8902 8903 8904 8905 8906 8907 8908 8909 | \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 | \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mul\_d} \end{figure} \textbf{Algorithm mp\_mul\_d.} | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | 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 8985 8986 8987 8988 8989 8990 8991 8992 8993 8994 8995 8996 8997 8998 8999 9000 9001 9002 9003 9004 9005 9006 9007 9008 9009 9010 9011 9012 9013 9014 9015 9016 9017 9018 9019 9020 9021 9022 9023 9024 9025 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 | \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 $\hat w$ | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | 9051 9052 9053 9054 9055 9056 9057 9058 9059 9060 9061 9062 9063 9064 9065 9066 9067 9068 9069 9070 9071 9072 9073 9074 9075 9076 9077 9078 9079 9080 9081 9082 9083 9084 9085 9086 9087 9088 9089 9090 9091 9092 9093 9094 9095 9096 9097 9098 9099 9100 9101 9102 9103 9104 9105 9106 9107 9108 9109 9110 9111 9112 9113 9114 9115 9116 9117 9118 9119 9120 9121 9122 9123 9124 9125 9126 9127 9128 9129 9130 9131 9132 9133 9134 9135 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 9164 9165 9166 9167 9168 9169 9170 9171 9172 9173 9174 9175 9176 9177 9178 9179 9180 9181 9182 9183 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 | \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 | | | | > > > > > > > > > > > | | | 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 | \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 | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | 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 9325 9326 9327 9328 9329 9330 9331 9332 9333 9334 9335 9336 9337 9338 9339 9340 9341 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 | \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_read\_radix} \end{figure} \textbf{Algorithm mp\_read\_radix.} | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | 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 9425 9426 9427 9428 9429 9430 9431 9432 9433 9434 9435 9436 9437 9438 9439 9440 9441 9442 9443 9444 9445 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 | \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_toradix} \end{figure} \textbf{Algorithm mp\_toradix.} | | | | 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 | \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} | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | 9532 9533 9534 9535 9536 9537 9538 9539 9540 9541 9542 9543 9544 9545 9546 9547 9548 9549 9550 9551 9552 9553 9554 9555 9556 9557 9558 9559 9560 9561 9562 9563 9564 9565 9566 9567 9568 9569 9570 9571 9572 9573 9574 9575 9576 9577 9578 9579 9580 9581 9582 9583 9584 9585 9586 9587 9588 9589 9590 9591 9592 9593 9594 9595 9596 9597 9598 9599 9600 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 | \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 | | | | | 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 | \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 | | | | | | | 9652 9653 9654 9655 9656 9657 9658 9659 9660 9661 9662 9663 9664 9665 9666 9667 9668 9669 9670 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 | \end{tabular} \end{center} \end{small} \caption{Algorithm Greatest Common Divisor (III)} \label{fig:gcd3} \end{figure} | | | | | | 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 | \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 | | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | | 9750 9751 9752 9753 9754 9755 9756 9757 9758 9759 9760 9761 9762 9763 9764 9765 9766 9767 9768 9769 9770 9771 9772 9773 9774 9775 9776 9777 9778 9779 9780 9781 9782 9783 9784 9785 9786 9787 9788 9789 9790 9791 9792 9793 9794 9795 9796 9797 9798 9799 9800 9801 9802 9803 9804 9805 9806 9807 9808 9809 9810 9811 9812 9813 9814 9815 9816 9817 9818 9819 9820 9821 9822 9823 9824 9825 9826 9827 9828 9829 9830 9831 9832 9833 9834 9835 9836 9837 9838 9839 9840 9841 9842 9843 9844 9845 9846 9847 9848 9849 9850 9851 9852 9853 9854 9855 9856 9857 9858 9859 9860 9861 9862 9863 9864 9865 9866 9867 9868 9869 9870 9871 9872 9873 9874 9875 9876 9877 9878 9879 9880 9881 9882 9883 9884 9885 9886 9887 9888 9889 9890 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 | 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} | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | 9913 9914 9915 9916 9917 9918 9919 9920 9921 9922 9923 9924 9925 9926 9927 9928 9929 9930 9931 9932 9933 9934 9935 9936 9937 9938 9939 9940 9941 9942 9943 9944 9945 9946 9947 9948 9949 9950 9951 9952 9953 9954 9955 9956 9957 9958 9959 9960 9961 9962 9963 9964 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974 9975 9976 9977 9978 9979 9980 9981 9982 9983 9984 | 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{)} |
︙ | ︙ | |||
6191 6192 6193 6194 6195 6196 6197 | 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 \\ | | | | | | | | | | | | | | | | | 9994 9995 9996 9997 9998 9999 10000 10001 10002 10003 10004 10005 10006 10007 10008 10009 10010 10011 10012 10013 10014 10015 10016 10017 10018 10019 10020 10021 10022 10023 10024 10025 10026 10027 10028 10029 10030 10031 10032 10033 10034 10035 10036 10037 10038 10039 10040 10041 10042 10043 10044 10045 10046 10047 10048 10049 10050 10051 10052 10053 10054 10055 10056 10057 10058 10059 10060 10061 10062 10063 10064 10065 10066 10067 10068 | 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 $p$, $a \ge 0$, $p \ge 3$, $p \equiv 1 \mbox{ (mod }2\mbox{)}$ \\ |
︙ | ︙ | |||
6295 6296 6297 6298 6299 6300 6301 | \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 | | | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | | | | | | 10098 10099 10100 10101 10102 10103 10104 10105 10106 10107 10108 10109 10110 10111 10112 10113 10114 10115 10116 10117 10118 10119 10120 10121 10122 10123 10124 10125 10126 10127 10128 10129 10130 10131 10132 10133 10134 10135 10136 10137 10138 10139 10140 10141 10142 10143 10144 10145 10146 10147 10148 10149 10150 10151 10152 10153 10154 10155 10156 10157 10158 10159 10160 10161 10162 10163 10164 10165 10166 10167 10168 10169 10170 10171 10172 10173 10174 10175 10176 10177 10178 10179 10180 10181 10182 10183 10184 10185 10186 10187 10188 10189 10190 10191 10192 10193 10194 10195 10196 10197 10198 10199 10200 10201 10202 10203 10204 10205 10206 10207 10208 10209 10210 10211 10212 10213 10214 10215 10216 10217 10218 10219 10220 10221 10222 10223 10224 10225 10226 10227 10228 10229 10230 10231 10232 10233 10234 10235 10236 10237 10238 10239 10240 10241 10242 10243 10244 10245 10246 10247 10248 10249 10250 10251 10252 10253 10254 10255 10256 10257 10258 10259 10260 10261 10262 10263 10264 10265 10266 10267 10268 10269 10270 10271 10272 10273 10274 10275 | \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 | 15. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \end{figure} \textbf{Algorithm mp\_invmod.} | | | | | | > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | | | | | | | 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 10341 10342 10343 10344 10345 10346 10347 10348 10349 10350 10351 10352 10353 10354 10355 10356 10357 10358 10359 10360 10361 10362 10363 10364 10365 10366 10367 10368 10369 10370 10371 10372 10373 10374 10375 10376 10377 10378 10379 10380 10381 10382 10383 10384 10385 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 | \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_prime\_is\_divisible} \end{figure} \textbf{Algorithm mp\_prime\_is\_divisible.} | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | | | | | 10425 10426 10427 10428 10429 10430 10431 10432 10433 10434 10435 10436 10437 10438 10439 10440 10441 10442 10443 10444 10445 10446 10447 10448 10449 10450 10451 10452 10453 10454 10455 10456 10457 10458 10459 10460 10461 10462 10463 10464 10465 10466 10467 10468 10469 10470 10471 10472 10473 10474 10475 10476 10477 10478 10479 10480 10481 10482 10483 10484 10485 10486 10487 10488 10489 10490 10491 10492 10493 10494 10495 10496 10497 10498 10499 10500 10501 10502 10503 10504 10505 10506 10507 10508 10509 10510 10511 10512 10513 10514 10515 10516 10517 10518 10519 10520 10521 10522 10523 10524 10525 10526 10527 10528 10529 10530 10531 10532 10533 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 | \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 | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | | | 10554 10555 10556 10557 10558 10559 10560 10561 10562 10563 10564 10565 10566 10567 10568 10569 10570 10571 10572 10573 10574 10575 10576 10577 10578 10579 10580 10581 10582 10583 10584 10585 10586 10587 10588 10589 10590 10591 10592 10593 10594 10595 10596 10597 10598 10599 10600 10601 10602 10603 10604 10605 10606 10607 10608 10609 10610 10611 10612 10613 10614 10615 10616 10617 10618 10619 10620 10621 10622 | \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 | \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 | | | > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 10644 10645 10646 10647 10648 10649 10650 10651 10652 10653 10654 10655 10656 10657 10658 10659 10660 10661 10662 10663 10664 10665 10666 10667 10668 10669 10670 10671 10672 10673 10674 10675 10676 10677 10678 10679 10680 10681 10682 10683 10684 10685 10686 10687 10688 10689 10690 10691 10692 10693 10694 10695 10696 10697 10698 10699 10700 10701 10702 10703 10704 10705 10706 10707 10708 10709 10710 10711 10712 10713 10714 10715 10716 10717 10718 10719 10720 10721 10722 10723 10724 10725 10726 10727 10728 10729 10730 10731 10732 10733 10734 10735 10736 10737 10738 10739 10740 10741 10742 10743 10744 10745 10746 10747 10748 10749 10750 10751 10752 | \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.
︙ | ︙ | |||
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | #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 | > > > > > | 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 | #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 |
︙ | ︙ | |||
69 70 71 72 73 74 75 76 77 78 79 80 81 82 | #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 | > | 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | #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 |
︙ | ︙ | |||
95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | #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 | > > > | 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | #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 |
︙ | ︙ | |||
310 311 312 313 314 315 316 317 318 319 320 | #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 | > > > > > > > > > > > | | | 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 | #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 |
︙ | ︙ | |||
383 384 385 386 387 388 389 | #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 | < > > > > > > > > > > > > > | 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 | #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 |
︙ | ︙ | |||
477 478 479 480 481 482 483 | #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 | | > | | | 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 | #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 |
︙ | ︙ | |||
512 513 514 515 516 517 518 | #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 | | > | 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 | #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 |
︙ | ︙ | |||
579 580 581 582 583 584 585 586 587 588 | #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 | > > > > | | 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 | #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 |
︙ | ︙ | |||
663 664 665 666 667 668 669 | #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) | | | | < | 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 | #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 |
︙ | ︙ | |||
783 784 785 786 787 788 789 790 791 792 793 794 795 796 | #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 | > > > > > > | 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 | #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 |
︙ | ︙ | |||
818 819 820 821 822 823 824 825 826 827 828 829 830 831 | #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 | > > > > > > > > > > > > > > > > > > > | 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 | #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.
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 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 | /* 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 67 68 69 70 71 72 | */ #undef BN_S_MP_MUL_DIGS_C #undef BN_S_MP_SQR_C #undef BN_MP_MONTGOMERY_REDUCE_C #endif #endif | > > > > | 66 67 68 69 70 71 72 73 74 75 76 | */ #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.
︙ | ︙ | |||
162 163 164 165 166 167 168 169 170 171 172 173 174 175 | 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 */; }; F9E61D2B090A48A4002B3151 /* bn_mp_and.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D426C08F272B3004A47F5 /* bn_mp_and.c */; }; F9E61D2C090A48AC002B3151 /* bn_mp_expt_d.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */; }; F9E61D2D090A48BB002B3151 /* bn_mp_xor.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42CD08F272B3004A47F5 /* bn_mp_xor.c */; }; F9E61D2E090A48BF002B3151 /* bn_mp_or.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42A308F272B3004A47F5 /* bn_mp_or.c */; }; F9E61D2F090A48C7002B3151 /* bn_mp_shrink.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42BC08F272B3004A47F5 /* bn_mp_shrink.c */; }; F9E61D30090A48E2002B3151 /* bn_mp_to_unsigned_bin_n.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42C708F272B3004A47F5 /* bn_mp_to_unsigned_bin_n.c */; }; F9E61D31090A48F9002B3151 /* bn_mp_to_unsigned_bin.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42C608F272B3004A47F5 /* bn_mp_to_unsigned_bin.c */; }; F9E61D32090A48FA002B3151 /* bn_mp_unsigned_bin_size.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42CC08F272B3004A47F5 /* bn_mp_unsigned_bin_size.c */; }; F9F4415E0C8BAE6F00BCCD67 /* tclDTrace.d in Sources */ = {isa = PBXBuildFile; fileRef = F9F4415D0C8BAE6F00BCCD67 /* tclDTrace.d */; }; | > | 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 | 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 */; }; F9E61D2B090A48A4002B3151 /* bn_mp_and.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D426C08F272B3004A47F5 /* bn_mp_and.c */; }; F9E61D2C090A48AC002B3151 /* bn_mp_expt_d.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */; }; F9E61D2C090A48AC002B3151 /* bn_mp_expt_d_ex.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D427F08F272B3004A47F5 /* bn_mp_expt_d_ex.c */; }; F9E61D2D090A48BB002B3151 /* bn_mp_xor.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42CD08F272B3004A47F5 /* bn_mp_xor.c */; }; F9E61D2E090A48BF002B3151 /* bn_mp_or.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42A308F272B3004A47F5 /* bn_mp_or.c */; }; F9E61D2F090A48C7002B3151 /* bn_mp_shrink.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42BC08F272B3004A47F5 /* bn_mp_shrink.c */; }; F9E61D30090A48E2002B3151 /* bn_mp_to_unsigned_bin_n.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42C708F272B3004A47F5 /* bn_mp_to_unsigned_bin_n.c */; }; F9E61D31090A48F9002B3151 /* bn_mp_to_unsigned_bin.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42C608F272B3004A47F5 /* bn_mp_to_unsigned_bin.c */; }; F9E61D32090A48FA002B3151 /* bn_mp_unsigned_bin_size.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42CC08F272B3004A47F5 /* bn_mp_unsigned_bin_size.c */; }; F9F4415E0C8BAE6F00BCCD67 /* tclDTrace.d in Sources */ = {isa = PBXBuildFile; fileRef = F9F4415D0C8BAE6F00BCCD67 /* tclDTrace.d */; }; |
︙ | ︙ | |||
1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 | 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 */, | > | 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 | 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 */, |
︙ | ︙ | |||
2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 | 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 */, | > | 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085 2086 2087 | 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 macosx/Tcl.xcodeproj/project.pbxproj.
︙ | ︙ | |||
161 162 163 164 165 166 167 | F96D4AD308F272CA004A47F5 /* tclUnixTest.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D446808F272B9004A47F5 /* tclUnixTest.c */; }; 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 */; }; F9E61D2B090A48A4002B3151 /* bn_mp_and.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D426C08F272B3004A47F5 /* bn_mp_and.c */; }; | | > | 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 | F96D4AD308F272CA004A47F5 /* tclUnixTest.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D446808F272B9004A47F5 /* tclUnixTest.c */; }; 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 */; }; F9E61D2B090A48A4002B3151 /* bn_mp_and.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D426C08F272B3004A47F5 /* bn_mp_and.c */; }; F9E61D2C090A48AC002B3151 /* bn_mp_expt_d.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D427F08F272B3004A47F5 /* bn_mp_expt_d_ex.c */; }; F9E61D2C090A48AC002B3151 /* bn_mp_expt_d_ex.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */; }; F9E61D2D090A48BB002B3151 /* bn_mp_xor.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42CD08F272B3004A47F5 /* bn_mp_xor.c */; }; F9E61D2E090A48BF002B3151 /* bn_mp_or.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42A308F272B3004A47F5 /* bn_mp_or.c */; }; F9E61D2F090A48C7002B3151 /* bn_mp_shrink.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42BC08F272B3004A47F5 /* bn_mp_shrink.c */; }; F9E61D30090A48E2002B3151 /* bn_mp_to_unsigned_bin_n.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42C708F272B3004A47F5 /* bn_mp_to_unsigned_bin_n.c */; }; F9E61D31090A48F9002B3151 /* bn_mp_to_unsigned_bin.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42C608F272B3004A47F5 /* bn_mp_to_unsigned_bin.c */; }; F9E61D32090A48FA002B3151 /* bn_mp_unsigned_bin_size.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42CC08F272B3004A47F5 /* bn_mp_unsigned_bin_size.c */; }; F9F4415E0C8BAE6F00BCCD67 /* tclDTrace.d in Sources */ = {isa = PBXBuildFile; fileRef = F9F4415D0C8BAE6F00BCCD67 /* tclDTrace.d */; }; |
︙ | ︙ | |||
575 576 577 578 579 580 581 582 583 584 585 586 587 588 | F96D427608F272B3004A47F5 /* bn_mp_div.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_div.c; sourceTree = "<group>"; }; F96D427708F272B3004A47F5 /* bn_mp_div_2.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_div_2.c; sourceTree = "<group>"; }; F96D427808F272B3004A47F5 /* bn_mp_div_2d.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_div_2d.c; sourceTree = "<group>"; }; F96D427908F272B3004A47F5 /* bn_mp_div_3.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_div_3.c; sourceTree = "<group>"; }; F96D427A08F272B3004A47F5 /* bn_mp_div_d.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_div_d.c; sourceTree = "<group>"; }; F96D427E08F272B3004A47F5 /* bn_mp_exch.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_exch.c; sourceTree = "<group>"; }; F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_expt_d.c; sourceTree = "<group>"; }; F96D428708F272B3004A47F5 /* bn_mp_grow.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_grow.c; sourceTree = "<group>"; }; 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>"; }; | > | 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 | F96D427608F272B3004A47F5 /* bn_mp_div.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_div.c; sourceTree = "<group>"; }; F96D427708F272B3004A47F5 /* bn_mp_div_2.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_div_2.c; sourceTree = "<group>"; }; F96D427808F272B3004A47F5 /* bn_mp_div_2d.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_div_2d.c; sourceTree = "<group>"; }; F96D427908F272B3004A47F5 /* bn_mp_div_3.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_div_3.c; sourceTree = "<group>"; }; F96D427A08F272B3004A47F5 /* bn_mp_div_d.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_div_d.c; sourceTree = "<group>"; }; F96D427E08F272B3004A47F5 /* bn_mp_exch.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_exch.c; sourceTree = "<group>"; }; F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_expt_d.c; sourceTree = "<group>"; }; F96D427F08F272B3004A47F5 /* bn_mp_expt_d_ex.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_expt_d_ex.c; sourceTree = "<group>"; }; F96D428708F272B3004A47F5 /* bn_mp_grow.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_grow.c; sourceTree = "<group>"; }; 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>"; }; |
︙ | ︙ | |||
1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 | 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 */, | > | 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 | 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 */, |
︙ | ︙ | |||
2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 | 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 */, | > | 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085 2086 2087 2088 | 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.
︙ | ︙ | |||
316 317 318 319 320 321 322 | 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 \ | | | 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 | 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 \ |
︙ | ︙ | |||
499 500 501 502 503 504 505 506 507 508 509 510 511 512 | $(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 \ | > | 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 | $(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 \ |
︙ | ︙ | |||
1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 | $(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 | > > > | 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 | $(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) $(TOMMATH_DIR)/bn_mp_init.c |
︙ | ︙ |
Changes to win/Makefile.in.
︙ | ︙ | |||
324 325 326 327 328 329 330 331 332 333 334 335 336 337 | bn_mp_div.${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.${OBJEXT} \ | > | 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 | bn_mp_div.${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.
︙ | ︙ | |||
379 380 381 382 383 384 385 386 387 388 389 390 391 392 | $(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 \ | > | 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 | $(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 \ |
︙ | ︙ |