comparison bn.tex @ 282:91fbc376f010 libtommath-orig libtommath-0.35

Import of libtommath 0.35 From ltm-0.35.tar.bz2 SHA1 of 3f193dbae9351e92d02530994fa18236f7fde01c
author Matt Johnston <matt@ucc.asn.au>
date Wed, 08 Mar 2006 13:16:18 +0000
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1 \documentclass[b5paper]{book}
2 \usepackage{hyperref}
3 \usepackage{makeidx}
4 \usepackage{amssymb}
5 \usepackage{color}
6 \usepackage{alltt}
7 \usepackage{graphicx}
8 \usepackage{layout}
9 \def\union{\cup}
10 \def\intersect{\cap}
11 \def\getsrandom{\stackrel{\rm R}{\gets}}
12 \def\cross{\times}
13 \def\cat{\hspace{0.5em} \| \hspace{0.5em}}
14 \def\catn{$\|$}
15 \def\divides{\hspace{0.3em} | \hspace{0.3em}}
16 \def\nequiv{\not\equiv}
17 \def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}}
18 \def\lcm{{\rm lcm}}
19 \def\gcd{{\rm gcd}}
20 \def\log{{\rm log}}
21 \def\ord{{\rm ord}}
22 \def\abs{{\mathit abs}}
23 \def\rep{{\mathit rep}}
24 \def\mod{{\mathit\ mod\ }}
25 \renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})}
26 \newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
27 \newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil}
28 \def\Or{{\rm\ or\ }}
29 \def\And{{\rm\ and\ }}
30 \def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}}
31 \def\implies{\Rightarrow}
32 \def\undefined{{\rm ``undefined"}}
33 \def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}}
34 \let\oldphi\phi
35 \def\phi{\varphi}
36 \def\Pr{{\rm Pr}}
37 \newcommand{\str}[1]{{\mathbf{#1}}}
38 \def\F{{\mathbb F}}
39 \def\N{{\mathbb N}}
40 \def\Z{{\mathbb Z}}
41 \def\R{{\mathbb R}}
42 \def\C{{\mathbb C}}
43 \def\Q{{\mathbb Q}}
44 \definecolor{DGray}{gray}{0.5}
45 \newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}}
46 \def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}}
47 \def\gap{\vspace{0.5ex}}
48 \makeindex
49 \begin{document}
50 \frontmatter
51 \pagestyle{empty}
52 \title{LibTomMath User Manual \\ v0.35}
53 \author{Tom St Denis \\ [email protected]}
54 \maketitle
55 This text, the library and the accompanying textbook are all hereby placed in the public domain. This book has been
56 formatted for B5 [176x250] paper using the \LaTeX{} {\em book} macro package.
57
58 \vspace{10cm}
59
60 \begin{flushright}Open Source. Open Academia. Open Minds.
61
62 \mbox{ }
63
64 Tom St Denis,
65
66 Ontario, Canada
67 \end{flushright}
68
69 \tableofcontents
70 \listoffigures
71 \mainmatter
72 \pagestyle{headings}
73 \chapter{Introduction}
74 \section{What is LibTomMath?}
75 LibTomMath is a library of source code which provides a series of efficient and carefully written functions for manipulating
76 large integer numbers. It was written in portable ISO C source code so that it will build on any platform with a conforming
77 C compiler.
78
79 In a nutshell the library was written from scratch with verbose comments to help instruct computer science students how
80 to implement ``bignum'' math. However, the resulting code has proven to be very useful. It has been used by numerous
81 universities, commercial and open source software developers. It has been used on a variety of platforms ranging from
82 Linux and Windows based x86 to ARM based Gameboys and PPC based MacOS machines.
83
84 \section{License}
85 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
86 release the textbook ``Implementing Multiple Precision Arithmetic'' has been placed in the public domain with every new
87 release as well. This textbook is meant to compliment the project by providing a more solid walkthrough of the development
88 algorithms used in the library.
89
90 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
91 public domain everyone is entitled to do with them as they see fit.
92
93 \section{Building LibTomMath}
94
95 LibTomMath is meant to be very ``GCC friendly'' as it comes with a makefile well suited for GCC. However, the library will
96 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
97 developer.
98
99 \subsection{Static Libraries}
100 To build as a static library for GCC issue the following
101 \begin{alltt}
102 make
103 \end{alltt}
104
105 command. This will build the library and archive the object files in ``libtommath.a''. Now you link against
106 that and include ``tommath.h'' within your programs. Alternatively to build with MSVC issue the following
107 \begin{alltt}
108 nmake -f makefile.msvc
109 \end{alltt}
110
111 This will build the library and archive the object files in ``tommath.lib''. This has been tested with MSVC
112 version 6.00 with service pack 5.
113
114 \subsection{Shared Libraries}
115 To build as a shared library for GCC issue the following
116 \begin{alltt}
117 make -f makefile.shared
118 \end{alltt}
119 This requires the ``libtool'' package (common on most Linux/BSD systems). It will build LibTomMath as both shared
120 and static then install (by default) into /usr/lib as well as install the header files in /usr/include. The shared
121 library (resource) will be called ``libtommath.la'' while the static library called ``libtommath.a''. Generally
122 you use libtool to link your application against the shared object.
123
124 There is limited support for making a ``DLL'' in windows via the ``makefile.cygwin\_dll'' makefile. It requires
125 Cygwin to work with since it requires the auto-export/import functionality. The resulting DLL and import library
126 ``libtommath.dll.a'' can be used to link LibTomMath dynamically to any Windows program using Cygwin.
127
128 \subsection{Testing}
129 To build the library and the test harness type
130
131 \begin{alltt}
132 make test
133 \end{alltt}
134
135 This will build the library, ``test'' and ``mtest/mtest''. The ``test'' program will accept test vectors and verify the
136 results. ``mtest/mtest'' will generate test vectors using the MPI library by Michael Fromberger\footnote{A copy of MPI
137 is included in the package}. Simply pipe mtest into test using
138
139 \begin{alltt}
140 mtest/mtest | test
141 \end{alltt}
142
143 If you do not have a ``/dev/urandom'' style RNG source you will have to write your own PRNG and simply pipe that into
144 mtest. For example, if your PRNG program is called ``myprng'' simply invoke
145
146 \begin{alltt}
147 myprng | mtest/mtest | test
148 \end{alltt}
149
150 This will output a row of numbers that are increasing. Each column is a different test (such as addition, multiplication, etc)
151 that is being performed. The numbers represent how many times the test was invoked. If an error is detected the program
152 will exit with a dump of the relevent numbers it was working with.
153
154 \section{Build Configuration}
155 LibTomMath can configured at build time in three phases we shall call ``depends'', ``tweaks'' and ``trims''.
156 Each phase changes how the library is built and they are applied one after another respectively.
157
158 To make the system more powerful you can tweak the build process. Classes are defined in the file
159 ``tommath\_superclass.h''. By default, the symbol ``LTM\_ALL'' shall be defined which simply
160 instructs the system to build all of the functions. This is how LibTomMath used to be packaged. This will give you
161 access to every function LibTomMath offers.
162
163 However, there are cases where such a build is not optional. For instance, you want to perform RSA operations. You
164 don't need the vast majority of the library to perform these operations. Aside from LTM\_ALL there is
165 another pre--defined class ``SC\_RSA\_1'' which works in conjunction with the RSA from LibTomCrypt. Additional
166 classes can be defined base on the need of the user.
167
168 \subsection{Build Depends}
169 In the file tommath\_class.h you will see a large list of C ``defines'' followed by a series of ``ifdefs''
170 which further define symbols. All of the symbols (technically they're macros $\ldots$) represent a given C source
171 file. For instance, BN\_MP\_ADD\_C represents the file ``bn\_mp\_add.c''. When a define has been enabled the
172 function in the respective file will be compiled and linked into the library. Accordingly when the define
173 is absent the file will not be compiled and not contribute any size to the library.
174
175 You will also note that the header tommath\_class.h is actually recursively included (it includes itself twice).
176 This is to help resolve as many dependencies as possible. In the last pass the symbol LTM\_LAST will be defined.
177 This is useful for ``trims''.
178
179 \subsection{Build Tweaks}
180 A tweak is an algorithm ``alternative''. For example, to provide tradeoffs (usually between size and space).
181 They can be enabled at any pass of the configuration phase.
182
183 \begin{small}
184 \begin{center}
185 \begin{tabular}{|l|l|}
186 \hline \textbf{Define} & \textbf{Purpose} \\
187 \hline BN\_MP\_DIV\_SMALL & Enables a slower, smaller and equally \\
188 & functional mp\_div() function \\
189 \hline
190 \end{tabular}
191 \end{center}
192 \end{small}
193
194 \subsection{Build Trims}
195 A trim is a manner of removing functionality from a function that is not required. For instance, to perform
196 RSA cryptography you only require exponentiation with odd moduli so even moduli support can be safely removed.
197 Build trims are meant to be defined on the last pass of the configuration which means they are to be defined
198 only if LTM\_LAST has been defined.
199
200 \subsubsection{Moduli Related}
201 \begin{small}
202 \begin{center}
203 \begin{tabular}{|l|l|}
204 \hline \textbf{Restriction} & \textbf{Undefine} \\
205 \hline Exponentiation with odd moduli only & BN\_S\_MP\_EXPTMOD\_C \\
206 & BN\_MP\_REDUCE\_C \\
207 & BN\_MP\_REDUCE\_SETUP\_C \\
208 & BN\_S\_MP\_MUL\_HIGH\_DIGS\_C \\
209 & BN\_FAST\_S\_MP\_MUL\_HIGH\_DIGS\_C \\
210 \hline Exponentiation with random odd moduli & (The above plus the following) \\
211 & BN\_MP\_REDUCE\_2K\_C \\
212 & BN\_MP\_REDUCE\_2K\_SETUP\_C \\
213 & BN\_MP\_REDUCE\_IS\_2K\_C \\
214 & BN\_MP\_DR\_IS\_MODULUS\_C \\
215 & BN\_MP\_DR\_REDUCE\_C \\
216 & BN\_MP\_DR\_SETUP\_C \\
217 \hline Modular inverse odd moduli only & BN\_MP\_INVMOD\_SLOW\_C \\
218 \hline Modular inverse (both, smaller/slower) & BN\_FAST\_MP\_INVMOD\_C \\
219 \hline
220 \end{tabular}
221 \end{center}
222 \end{small}
223
224 \subsubsection{Operand Size Related}
225 \begin{small}
226 \begin{center}
227 \begin{tabular}{|l|l|}
228 \hline \textbf{Restriction} & \textbf{Undefine} \\
229 \hline Moduli $\le 2560$ bits & BN\_MP\_MONTGOMERY\_REDUCE\_C \\
230 & BN\_S\_MP\_MUL\_DIGS\_C \\
231 & BN\_S\_MP\_MUL\_HIGH\_DIGS\_C \\
232 & BN\_S\_MP\_SQR\_C \\
233 \hline Polynomial Schmolynomial & BN\_MP\_KARATSUBA\_MUL\_C \\
234 & BN\_MP\_KARATSUBA\_SQR\_C \\
235 & BN\_MP\_TOOM\_MUL\_C \\
236 & BN\_MP\_TOOM\_SQR\_C \\
237
238 \hline
239 \end{tabular}
240 \end{center}
241 \end{small}
242
243
244 \section{Purpose of LibTomMath}
245 Unlike GNU MP (GMP) Library, LIP, OpenSSL or various other commercial kits (Miracl), LibTomMath was not written with
246 bleeding edge performance in mind. First and foremost LibTomMath was written to be entirely open. Not only is the
247 source code public domain (unlike various other GPL/etc licensed code), not only is the code freely downloadable but the
248 source code is also accessible for computer science students attempting to learn ``BigNum'' or multiple precision
249 arithmetic techniques.
250
251 LibTomMath was written to be an instructive collection of source code. This is why there are many comments, only one
252 function per source file and often I use a ``middle-road'' approach where I don't cut corners for an extra 2\% speed
253 increase.
254
255 Source code alone cannot really teach how the algorithms work which is why I also wrote a textbook that accompanies
256 the library (beat that!).
257
258 So you may be thinking ``should I use LibTomMath?'' and the answer is a definite maybe. Let me tabulate what I think
259 are the pros and cons of LibTomMath by comparing it to the math routines from GnuPG\footnote{GnuPG v1.2.3 versus LibTomMath v0.28}.
260
261 \newpage\begin{figure}[here]
262 \begin{small}
263 \begin{center}
264 \begin{tabular}{|l|c|c|l|}
265 \hline \textbf{Criteria} & \textbf{Pro} & \textbf{Con} & \textbf{Notes} \\
266 \hline Few lines of code per file & X & & GnuPG $ = 300.9$, LibTomMath $ = 71.97$ \\
267 \hline Commented function prototypes & X && GnuPG function names are cryptic. \\
268 \hline Speed && X & LibTomMath is slower. \\
269 \hline Totally free & X & & GPL has unfavourable restrictions.\\
270 \hline Large function base & X & & GnuPG is barebones. \\
271 \hline Five modular reduction algorithms & X & & Faster modular exponentiation for a variety of moduli. \\
272 \hline Portable & X & & GnuPG requires configuration to build. \\
273 \hline
274 \end{tabular}
275 \end{center}
276 \end{small}
277 \caption{LibTomMath Valuation}
278 \end{figure}
279
280 It may seem odd to compare LibTomMath to GnuPG since the math in GnuPG is only a small portion of the entire application.
281 However, LibTomMath was written with cryptography in mind. It provides essentially all of the functions a cryptosystem
282 would require when working with large integers.
283
284 So it may feel tempting to just rip the math code out of GnuPG (or GnuMP where it was taken from originally) in your
285 own application but I think there are reasons not to. While LibTomMath is slower than libraries such as GnuMP it is
286 not normally significantly slower. On x86 machines the difference is normally a factor of two when performing modular
287 exponentiations. It depends largely on the processor, compiler and the moduli being used.
288
289 Essentially the only time you wouldn't use LibTomMath is when blazing speed is the primary concern. However,
290 on the other side of the coin LibTomMath offers you a totally free (public domain) well structured math library
291 that is very flexible, complete and performs well in resource contrained environments. Fast RSA for example can
292 be performed with as little as 8KB of ram for data (again depending on build options).
293
294 \chapter{Getting Started with LibTomMath}
295 \section{Building Programs}
296 In order to use LibTomMath you must include ``tommath.h'' and link against the appropriate library file (typically
297 libtommath.a). There is no library initialization required and the entire library is thread safe.
298
299 \section{Return Codes}
300 There are three possible return codes a function may return.
301
302 \index{MP\_OKAY}\index{MP\_YES}\index{MP\_NO}\index{MP\_VAL}\index{MP\_MEM}
303 \begin{figure}[here!]
304 \begin{center}
305 \begin{small}
306 \begin{tabular}{|l|l|}
307 \hline \textbf{Code} & \textbf{Meaning} \\
308 \hline MP\_OKAY & The function succeeded. \\
309 \hline MP\_VAL & The function input was invalid. \\
310 \hline MP\_MEM & Heap memory exhausted. \\
311 \hline &\\
312 \hline MP\_YES & Response is yes. \\
313 \hline MP\_NO & Response is no. \\
314 \hline
315 \end{tabular}
316 \end{small}
317 \end{center}
318 \caption{Return Codes}
319 \end{figure}
320
321 The last two codes listed are not actually ``return'ed'' by a function. They are placed in an integer (the caller must
322 provide the address of an integer it can store to) which the caller can access. To convert one of the three return codes
323 to a string use the following function.
324
325 \index{mp\_error\_to\_string}
326 \begin{alltt}
327 char *mp_error_to_string(int code);
328 \end{alltt}
329
330 This will return a pointer to a string which describes the given error code. It will not work for the return codes
331 MP\_YES and MP\_NO.
332
333 \section{Data Types}
334 The basic ``multiple precision integer'' type is known as the ``mp\_int'' within LibTomMath. This data type is used to
335 organize all of the data required to manipulate the integer it represents. Within LibTomMath it has been prototyped
336 as the following.
337
338 \index{mp\_int}
339 \begin{alltt}
340 typedef struct \{
341 int used, alloc, sign;
342 mp_digit *dp;
343 \} mp_int;
344 \end{alltt}
345
346 Where ``mp\_digit'' is a data type that represents individual digits of the integer. By default, an mp\_digit is the
347 ISO C ``unsigned long'' data type and each digit is $28-$bits long. The mp\_digit type can be configured to suit other
348 platforms by defining the appropriate macros.
349
350 All LTM functions that use the mp\_int type will expect a pointer to mp\_int structure. You must allocate memory to
351 hold the structure itself by yourself (whether off stack or heap it doesn't matter). The very first thing that must be
352 done to use an mp\_int is that it must be initialized.
353
354 \section{Function Organization}
355
356 The arithmetic functions of the library are all organized to have the same style prototype. That is source operands
357 are passed on the left and the destination is on the right. For instance,
358
359 \begin{alltt}
360 mp_add(&a, &b, &c); /* c = a + b */
361 mp_mul(&a, &a, &c); /* c = a * a */
362 mp_div(&a, &b, &c, &d); /* c = [a/b], d = a mod b */
363 \end{alltt}
364
365 Another feature of the way the functions have been implemented is that source operands can be destination operands as well.
366 For instance,
367
368 \begin{alltt}
369 mp_add(&a, &b, &b); /* b = a + b */
370 mp_div(&a, &b, &a, &c); /* a = [a/b], c = a mod b */
371 \end{alltt}
372
373 This allows operands to be re-used which can make programming simpler.
374
375 \section{Initialization}
376 \subsection{Single Initialization}
377 A single mp\_int can be initialized with the ``mp\_init'' function.
378
379 \index{mp\_init}
380 \begin{alltt}
381 int mp_init (mp_int * a);
382 \end{alltt}
383
384 This function expects a pointer to an mp\_int structure and will initialize the members of the structure so the mp\_int
385 represents the default integer which is zero. If the functions returns MP\_OKAY then the mp\_int is ready to be used
386 by the other LibTomMath functions.
387
388 \begin{small} \begin{alltt}
389 int main(void)
390 \{
391 mp_int number;
392 int result;
393
394 if ((result = mp_init(&number)) != MP_OKAY) \{
395 printf("Error initializing the number. \%s",
396 mp_error_to_string(result));
397 return EXIT_FAILURE;
398 \}
399
400 /* use the number */
401
402 return EXIT_SUCCESS;
403 \}
404 \end{alltt} \end{small}
405
406 \subsection{Single Free}
407 When you are finished with an mp\_int it is ideal to return the heap it used back to the system. The following function
408 provides this functionality.
409
410 \index{mp\_clear}
411 \begin{alltt}
412 void mp_clear (mp_int * a);
413 \end{alltt}
414
415 The function expects a pointer to a previously initialized mp\_int structure and frees the heap it uses. It sets the
416 pointer\footnote{The ``dp'' member.} within the mp\_int to \textbf{NULL} which is used to prevent double free situations.
417 Is is legal to call mp\_clear() twice on the same mp\_int in a row.
418
419 \begin{small} \begin{alltt}
420 int main(void)
421 \{
422 mp_int number;
423 int result;
424
425 if ((result = mp_init(&number)) != MP_OKAY) \{
426 printf("Error initializing the number. \%s",
427 mp_error_to_string(result));
428 return EXIT_FAILURE;
429 \}
430
431 /* use the number */
432
433 /* We're done with it. */
434 mp_clear(&number);
435
436 return EXIT_SUCCESS;
437 \}
438 \end{alltt} \end{small}
439
440 \subsection{Multiple Initializations}
441 Certain algorithms require more than one large integer. In these instances it is ideal to initialize all of the mp\_int
442 variables in an ``all or nothing'' fashion. That is, they are either all initialized successfully or they are all
443 not initialized.
444
445 The mp\_init\_multi() function provides this functionality.
446
447 \index{mp\_init\_multi} \index{mp\_clear\_multi}
448 \begin{alltt}
449 int mp_init_multi(mp_int *mp, ...);
450 \end{alltt}
451
452 It accepts a \textbf{NULL} terminated list of pointers to mp\_int structures. It will attempt to initialize them all
453 at once. If the function returns MP\_OKAY then all of the mp\_int variables are ready to use, otherwise none of them
454 are available for use. A complementary mp\_clear\_multi() function allows multiple mp\_int variables to be free'd
455 from the heap at the same time.
456
457 \begin{small} \begin{alltt}
458 int main(void)
459 \{
460 mp_int num1, num2, num3;
461 int result;
462
463 if ((result = mp_init_multi(&num1,
464 &num2,
465 &num3, NULL)) != MP\_OKAY) \{
466 printf("Error initializing the numbers. \%s",
467 mp_error_to_string(result));
468 return EXIT_FAILURE;
469 \}
470
471 /* use the numbers */
472
473 /* We're done with them. */
474 mp_clear_multi(&num1, &num2, &num3, NULL);
475
476 return EXIT_SUCCESS;
477 \}
478 \end{alltt} \end{small}
479
480 \subsection{Other Initializers}
481 To initialized and make a copy of an mp\_int the mp\_init\_copy() function has been provided.
482
483 \index{mp\_init\_copy}
484 \begin{alltt}
485 int mp_init_copy (mp_int * a, mp_int * b);
486 \end{alltt}
487
488 This function will initialize $a$ and make it a copy of $b$ if all goes well.
489
490 \begin{small} \begin{alltt}
491 int main(void)
492 \{
493 mp_int num1, num2;
494 int result;
495
496 /* initialize and do work on num1 ... */
497
498 /* We want a copy of num1 in num2 now */
499 if ((result = mp_init_copy(&num2, &num1)) != MP_OKAY) \{
500 printf("Error initializing the copy. \%s",
501 mp_error_to_string(result));
502 return EXIT_FAILURE;
503 \}
504
505 /* now num2 is ready and contains a copy of num1 */
506
507 /* We're done with them. */
508 mp_clear_multi(&num1, &num2, NULL);
509
510 return EXIT_SUCCESS;
511 \}
512 \end{alltt} \end{small}
513
514 Another less common initializer is mp\_init\_size() which allows the user to initialize an mp\_int with a given
515 default number of digits. By default, all initializers allocate \textbf{MP\_PREC} digits. This function lets
516 you override this behaviour.
517
518 \index{mp\_init\_size}
519 \begin{alltt}
520 int mp_init_size (mp_int * a, int size);
521 \end{alltt}
522
523 The $size$ parameter must be greater than zero. If the function succeeds the mp\_int $a$ will be initialized
524 to have $size$ digits (which are all initially zero).
525
526 \begin{small} \begin{alltt}
527 int main(void)
528 \{
529 mp_int number;
530 int result;
531
532 /* we need a 60-digit number */
533 if ((result = mp_init_size(&number, 60)) != MP_OKAY) \{
534 printf("Error initializing the number. \%s",
535 mp_error_to_string(result));
536 return EXIT_FAILURE;
537 \}
538
539 /* use the number */
540
541 return EXIT_SUCCESS;
542 \}
543 \end{alltt} \end{small}
544
545 \section{Maintenance Functions}
546
547 \subsection{Reducing Memory Usage}
548 When an mp\_int is in a state where it won't be changed again\footnote{A Diffie-Hellman modulus for instance.} excess
549 digits can be removed to return memory to the heap with the mp\_shrink() function.
550
551 \index{mp\_shrink}
552 \begin{alltt}
553 int mp_shrink (mp_int * a);
554 \end{alltt}
555
556 This will remove excess digits of the mp\_int $a$. If the operation fails the mp\_int should be intact without the
557 excess digits being removed. Note that you can use a shrunk mp\_int in further computations, however, such operations
558 will require heap operations which can be slow. It is not ideal to shrink mp\_int variables that you will further
559 modify in the system (unless you are seriously low on memory).
560
561 \begin{small} \begin{alltt}
562 int main(void)
563 \{
564 mp_int number;
565 int result;
566
567 if ((result = mp_init(&number)) != MP_OKAY) \{
568 printf("Error initializing the number. \%s",
569 mp_error_to_string(result));
570 return EXIT_FAILURE;
571 \}
572
573 /* use the number [e.g. pre-computation] */
574
575 /* We're done with it for now. */
576 if ((result = mp_shrink(&number)) != MP_OKAY) \{
577 printf("Error shrinking the number. \%s",
578 mp_error_to_string(result));
579 return EXIT_FAILURE;
580 \}
581
582 /* use it .... */
583
584
585 /* we're done with it. */
586 mp_clear(&number);
587
588 return EXIT_SUCCESS;
589 \}
590 \end{alltt} \end{small}
591
592 \subsection{Adding additional digits}
593
594 Within the mp\_int structure are two parameters which control the limitations of the array of digits that represent
595 the integer the mp\_int is meant to equal. The \textit{used} parameter dictates how many digits are significant, that is,
596 contribute to the value of the mp\_int. The \textit{alloc} parameter dictates how many digits are currently available in
597 the array. If you need to perform an operation that requires more digits you will have to mp\_grow() the mp\_int to
598 your desired size.
599
600 \index{mp\_grow}
601 \begin{alltt}
602 int mp_grow (mp_int * a, int size);
603 \end{alltt}
604
605 This will grow the array of digits of $a$ to $size$. If the \textit{alloc} parameter is already bigger than
606 $size$ the function will not do anything.
607
608 \begin{small} \begin{alltt}
609 int main(void)
610 \{
611 mp_int number;
612 int result;
613
614 if ((result = mp_init(&number)) != MP_OKAY) \{
615 printf("Error initializing the number. \%s",
616 mp_error_to_string(result));
617 return EXIT_FAILURE;
618 \}
619
620 /* use the number */
621
622 /* We need to add 20 digits to the number */
623 if ((result = mp_grow(&number, number.alloc + 20)) != MP_OKAY) \{
624 printf("Error growing the number. \%s",
625 mp_error_to_string(result));
626 return EXIT_FAILURE;
627 \}
628
629
630 /* use the number */
631
632 /* we're done with it. */
633 mp_clear(&number);
634
635 return EXIT_SUCCESS;
636 \}
637 \end{alltt} \end{small}
638
639 \chapter{Basic Operations}
640 \section{Small Constants}
641 Setting mp\_ints to small constants is a relatively common operation. To accomodate these instances there are two
642 small constant assignment functions. The first function is used to set a single digit constant while the second sets
643 an ISO C style ``unsigned long'' constant. The reason for both functions is efficiency. Setting a single digit is quick but the
644 domain of a digit can change (it's always at least $0 \ldots 127$).
645
646 \subsection{Single Digit}
647
648 Setting a single digit can be accomplished with the following function.
649
650 \index{mp\_set}
651 \begin{alltt}
652 void mp_set (mp_int * a, mp_digit b);
653 \end{alltt}
654
655 This will zero the contents of $a$ and make it represent an integer equal to the value of $b$. Note that this
656 function has a return type of \textbf{void}. It cannot cause an error so it is safe to assume the function
657 succeeded.
658
659 \begin{small} \begin{alltt}
660 int main(void)
661 \{
662 mp_int number;
663 int result;
664
665 if ((result = mp_init(&number)) != MP_OKAY) \{
666 printf("Error initializing the number. \%s",
667 mp_error_to_string(result));
668 return EXIT_FAILURE;
669 \}
670
671 /* set the number to 5 */
672 mp_set(&number, 5);
673
674 /* we're done with it. */
675 mp_clear(&number);
676
677 return EXIT_SUCCESS;
678 \}
679 \end{alltt} \end{small}
680
681 \subsection{Long Constants}
682
683 To set a constant that is the size of an ISO C ``unsigned long'' and larger than a single digit the following function
684 can be used.
685
686 \index{mp\_set\_int}
687 \begin{alltt}
688 int mp_set_int (mp_int * a, unsigned long b);
689 \end{alltt}
690
691 This will assign the value of the 32-bit variable $b$ to the mp\_int $a$. Unlike mp\_set() this function will always
692 accept a 32-bit input regardless of the size of a single digit. However, since the value may span several digits
693 this function can fail if it runs out of heap memory.
694
695 To get the ``unsigned long'' copy of an mp\_int the following function can be used.
696
697 \index{mp\_get\_int}
698 \begin{alltt}
699 unsigned long mp_get_int (mp_int * a);
700 \end{alltt}
701
702 This will return the 32 least significant bits of the mp\_int $a$.
703
704 \begin{small} \begin{alltt}
705 int main(void)
706 \{
707 mp_int number;
708 int result;
709
710 if ((result = mp_init(&number)) != MP_OKAY) \{
711 printf("Error initializing the number. \%s",
712 mp_error_to_string(result));
713 return EXIT_FAILURE;
714 \}
715
716 /* set the number to 654321 (note this is bigger than 127) */
717 if ((result = mp_set_int(&number, 654321)) != MP_OKAY) \{
718 printf("Error setting the value of the number. \%s",
719 mp_error_to_string(result));
720 return EXIT_FAILURE;
721 \}
722
723 printf("number == \%lu", mp_get_int(&number));
724
725 /* we're done with it. */
726 mp_clear(&number);
727
728 return EXIT_SUCCESS;
729 \}
730 \end{alltt} \end{small}
731
732 This should output the following if the program succeeds.
733
734 \begin{alltt}
735 number == 654321
736 \end{alltt}
737
738 \subsection{Initialize and Setting Constants}
739 To both initialize and set small constants the following two functions are available.
740 \index{mp\_init\_set} \index{mp\_init\_set\_int}
741 \begin{alltt}
742 int mp_init_set (mp_int * a, mp_digit b);
743 int mp_init_set_int (mp_int * a, unsigned long b);
744 \end{alltt}
745
746 Both functions work like the previous counterparts except they first mp\_init $a$ before setting the values.
747
748 \begin{alltt}
749 int main(void)
750 \{
751 mp_int number1, number2;
752 int result;
753
754 /* initialize and set a single digit */
755 if ((result = mp_init_set(&number1, 100)) != MP_OKAY) \{
756 printf("Error setting number1: \%s",
757 mp_error_to_string(result));
758 return EXIT_FAILURE;
759 \}
760
761 /* initialize and set a long */
762 if ((result = mp_init_set_int(&number2, 1023)) != MP_OKAY) \{
763 printf("Error setting number2: \%s",
764 mp_error_to_string(result));
765 return EXIT_FAILURE;
766 \}
767
768 /* display */
769 printf("Number1, Number2 == \%lu, \%lu",
770 mp_get_int(&number1), mp_get_int(&number2));
771
772 /* clear */
773 mp_clear_multi(&number1, &number2, NULL);
774
775 return EXIT_SUCCESS;
776 \}
777 \end{alltt}
778
779 If this program succeeds it shall output.
780 \begin{alltt}
781 Number1, Number2 == 100, 1023
782 \end{alltt}
783
784 \section{Comparisons}
785
786 Comparisons in LibTomMath are always performed in a ``left to right'' fashion. There are three possible return codes
787 for any comparison.
788
789 \index{MP\_GT} \index{MP\_EQ} \index{MP\_LT}
790 \begin{figure}[here]
791 \begin{center}
792 \begin{tabular}{|c|c|}
793 \hline \textbf{Result Code} & \textbf{Meaning} \\
794 \hline MP\_GT & $a > b$ \\
795 \hline MP\_EQ & $a = b$ \\
796 \hline MP\_LT & $a < b$ \\
797 \hline
798 \end{tabular}
799 \end{center}
800 \caption{Comparison Codes for $a, b$}
801 \label{fig:CMP}
802 \end{figure}
803
804 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
805 $b$.
806
807 \subsection{Unsigned comparison}
808
809 An unsigned comparison considers only the digits themselves and not the associated \textit{sign} flag of the
810 mp\_int structures. This is analogous to an absolute comparison. The function mp\_cmp\_mag() will compare two
811 mp\_int variables based on their digits only.
812
813 \index{mp\_cmp\_mag}
814 \begin{alltt}
815 int mp_cmp_mag(mp_int * a, mp_int * b);
816 \end{alltt}
817 This will compare $a$ to $b$ placing $a$ to the left of $b$. This function cannot fail and will return one of the
818 three compare codes listed in figure \ref{fig:CMP}.
819
820 \begin{small} \begin{alltt}
821 int main(void)
822 \{
823 mp_int number1, number2;
824 int result;
825
826 if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{
827 printf("Error initializing the numbers. \%s",
828 mp_error_to_string(result));
829 return EXIT_FAILURE;
830 \}
831
832 /* set the number1 to 5 */
833 mp_set(&number1, 5);
834
835 /* set the number2 to -6 */
836 mp_set(&number2, 6);
837 if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{
838 printf("Error negating number2. \%s",
839 mp_error_to_string(result));
840 return EXIT_FAILURE;
841 \}
842
843 switch(mp_cmp_mag(&number1, &number2)) \{
844 case MP_GT: printf("|number1| > |number2|"); break;
845 case MP_EQ: printf("|number1| = |number2|"); break;
846 case MP_LT: printf("|number1| < |number2|"); break;
847 \}
848
849 /* we're done with it. */
850 mp_clear_multi(&number1, &number2, NULL);
851
852 return EXIT_SUCCESS;
853 \}
854 \end{alltt} \end{small}
855
856 If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes
857 successfully it should print the following.
858
859 \begin{alltt}
860 |number1| < |number2|
861 \end{alltt}
862
863 This is because $\vert -6 \vert = 6$ and obviously $5 < 6$.
864
865 \subsection{Signed comparison}
866
867 To compare two mp\_int variables based on their signed value the mp\_cmp() function is provided.
868
869 \index{mp\_cmp}
870 \begin{alltt}
871 int mp_cmp(mp_int * a, mp_int * b);
872 \end{alltt}
873
874 This will compare $a$ to the left of $b$. It will first compare the signs of the two mp\_int variables. If they
875 differ it will return immediately based on their signs. If the signs are equal then it will compare the digits
876 individually. This function will return one of the compare conditions codes listed in figure \ref{fig:CMP}.
877
878 \begin{small} \begin{alltt}
879 int main(void)
880 \{
881 mp_int number1, number2;
882 int result;
883
884 if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{
885 printf("Error initializing the numbers. \%s",
886 mp_error_to_string(result));
887 return EXIT_FAILURE;
888 \}
889
890 /* set the number1 to 5 */
891 mp_set(&number1, 5);
892
893 /* set the number2 to -6 */
894 mp_set(&number2, 6);
895 if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{
896 printf("Error negating number2. \%s",
897 mp_error_to_string(result));
898 return EXIT_FAILURE;
899 \}
900
901 switch(mp_cmp(&number1, &number2)) \{
902 case MP_GT: printf("number1 > number2"); break;
903 case MP_EQ: printf("number1 = number2"); break;
904 case MP_LT: printf("number1 < number2"); break;
905 \}
906
907 /* we're done with it. */
908 mp_clear_multi(&number1, &number2, NULL);
909
910 return EXIT_SUCCESS;
911 \}
912 \end{alltt} \end{small}
913
914 If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes
915 successfully it should print the following.
916
917 \begin{alltt}
918 number1 > number2
919 \end{alltt}
920
921 \subsection{Single Digit}
922
923 To compare a single digit against an mp\_int the following function has been provided.
924
925 \index{mp\_cmp\_d}
926 \begin{alltt}
927 int mp_cmp_d(mp_int * a, mp_digit b);
928 \end{alltt}
929
930 This will compare $a$ to the left of $b$ using a signed comparison. Note that it will always treat $b$ as
931 positive. This function is rather handy when you have to compare against small values such as $1$ (which often
932 comes up in cryptography). The function cannot fail and will return one of the tree compare condition codes
933 listed in figure \ref{fig:CMP}.
934
935
936 \begin{small} \begin{alltt}
937 int main(void)
938 \{
939 mp_int number;
940 int result;
941
942 if ((result = mp_init(&number)) != MP_OKAY) \{
943 printf("Error initializing the number. \%s",
944 mp_error_to_string(result));
945 return EXIT_FAILURE;
946 \}
947
948 /* set the number to 5 */
949 mp_set(&number, 5);
950
951 switch(mp_cmp_d(&number, 7)) \{
952 case MP_GT: printf("number > 7"); break;
953 case MP_EQ: printf("number = 7"); break;
954 case MP_LT: printf("number < 7"); break;
955 \}
956
957 /* we're done with it. */
958 mp_clear(&number);
959
960 return EXIT_SUCCESS;
961 \}
962 \end{alltt} \end{small}
963
964 If this program functions properly it will print out the following.
965
966 \begin{alltt}
967 number < 7
968 \end{alltt}
969
970 \section{Logical Operations}
971
972 Logical operations are operations that can be performed either with simple shifts or boolean operators such as
973 AND, XOR and OR directly. These operations are very quick.
974
975 \subsection{Multiplication by two}
976
977 Multiplications and divisions by any power of two can be performed with quick logical shifts either left or
978 right depending on the operation.
979
980 When multiplying or dividing by two a special case routine can be used which are as follows.
981 \index{mp\_mul\_2} \index{mp\_div\_2}
982 \begin{alltt}
983 int mp_mul_2(mp_int * a, mp_int * b);
984 int mp_div_2(mp_int * a, mp_int * b);
985 \end{alltt}
986
987 The former will assign twice $a$ to $b$ while the latter will assign half $a$ to $b$. These functions are fast
988 since the shift counts and maskes are hardcoded into the routines.
989
990 \begin{small} \begin{alltt}
991 int main(void)
992 \{
993 mp_int number;
994 int result;
995
996 if ((result = mp_init(&number)) != MP_OKAY) \{
997 printf("Error initializing the number. \%s",
998 mp_error_to_string(result));
999 return EXIT_FAILURE;
1000 \}
1001
1002 /* set the number to 5 */
1003 mp_set(&number, 5);
1004
1005 /* multiply by two */
1006 if ((result = mp\_mul\_2(&number, &number)) != MP_OKAY) \{
1007 printf("Error multiplying the number. \%s",
1008 mp_error_to_string(result));
1009 return EXIT_FAILURE;
1010 \}
1011 switch(mp_cmp_d(&number, 7)) \{
1012 case MP_GT: printf("2*number > 7"); break;
1013 case MP_EQ: printf("2*number = 7"); break;
1014 case MP_LT: printf("2*number < 7"); break;
1015 \}
1016
1017 /* now divide by two */
1018 if ((result = mp\_div\_2(&number, &number)) != MP_OKAY) \{
1019 printf("Error dividing the number. \%s",
1020 mp_error_to_string(result));
1021 return EXIT_FAILURE;
1022 \}
1023 switch(mp_cmp_d(&number, 7)) \{
1024 case MP_GT: printf("2*number/2 > 7"); break;
1025 case MP_EQ: printf("2*number/2 = 7"); break;
1026 case MP_LT: printf("2*number/2 < 7"); break;
1027 \}
1028
1029 /* we're done with it. */
1030 mp_clear(&number);
1031
1032 return EXIT_SUCCESS;
1033 \}
1034 \end{alltt} \end{small}
1035
1036 If this program is successful it will print out the following text.
1037
1038 \begin{alltt}
1039 2*number > 7
1040 2*number/2 < 7
1041 \end{alltt}
1042
1043 Since $10 > 7$ and $5 < 7$. To multiply by a power of two the following function can be used.
1044
1045 \index{mp\_mul\_2d}
1046 \begin{alltt}
1047 int mp_mul_2d(mp_int * a, int b, mp_int * c);
1048 \end{alltt}
1049
1050 This will multiply $a$ by $2^b$ and store the result in ``c''. If the value of $b$ is less than or equal to
1051 zero the function will copy $a$ to ``c'' without performing any further actions.
1052
1053 To divide by a power of two use the following.
1054
1055 \index{mp\_div\_2d}
1056 \begin{alltt}
1057 int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d);
1058 \end{alltt}
1059 Which will divide $a$ by $2^b$, store the quotient in ``c'' and the remainder in ``d'. If $b \le 0$ then the
1060 function simply copies $a$ over to ``c'' and zeroes $d$. The variable $d$ may be passed as a \textbf{NULL}
1061 value to signal that the remainder is not desired.
1062
1063 \subsection{Polynomial Basis Operations}
1064
1065 Strictly speaking the organization of the integers within the mp\_int structures is what is known as a
1066 ``polynomial basis''. This simply means a field element is stored by divisions of a radix. For example, if
1067 $f(x) = \sum_{i=0}^{k} y_ix^k$ for any vector $\vec y$ then the array of digits in $\vec y$ are said to be
1068 the polynomial basis representation of $z$ if $f(\beta) = z$ for a given radix $\beta$.
1069
1070 To multiply by the polynomial $g(x) = x$ all you have todo is shift the digits of the basis left one place. The
1071 following function provides this operation.
1072
1073 \index{mp\_lshd}
1074 \begin{alltt}
1075 int mp_lshd (mp_int * a, int b);
1076 \end{alltt}
1077
1078 This will multiply $a$ in place by $x^b$ which is equivalent to shifting the digits left $b$ places and inserting zeroes
1079 in the least significant digits. Similarly to divide by a power of $x$ the following function is provided.
1080
1081 \index{mp\_rshd}
1082 \begin{alltt}
1083 void mp_rshd (mp_int * a, int b)
1084 \end{alltt}
1085 This will divide $a$ in place by $x^b$ and discard the remainder. This function cannot fail as it performs the operations
1086 in place and no new digits are required to complete it.
1087
1088 \subsection{AND, OR and XOR Operations}
1089
1090 While AND, OR and XOR operations are not typical ``bignum functions'' they can be useful in several instances. The
1091 three functions are prototyped as follows.
1092
1093 \index{mp\_or} \index{mp\_and} \index{mp\_xor}
1094 \begin{alltt}
1095 int mp_or (mp_int * a, mp_int * b, mp_int * c);
1096 int mp_and (mp_int * a, mp_int * b, mp_int * c);
1097 int mp_xor (mp_int * a, mp_int * b, mp_int * c);
1098 \end{alltt}
1099
1100 Which compute $c = a \odot b$ where $\odot$ is one of OR, AND or XOR.
1101
1102 \section{Addition and Subtraction}
1103
1104 To compute an addition or subtraction the following two functions can be used.
1105
1106 \index{mp\_add} \index{mp\_sub}
1107 \begin{alltt}
1108 int mp_add (mp_int * a, mp_int * b, mp_int * c);
1109 int mp_sub (mp_int * a, mp_int * b, mp_int * c)
1110 \end{alltt}
1111
1112 Which perform $c = a \odot b$ where $\odot$ is one of signed addition or subtraction. The operations are fully sign
1113 aware.
1114
1115 \section{Sign Manipulation}
1116 \subsection{Negation}
1117 \label{sec:NEG}
1118 Simple integer negation can be performed with the following.
1119
1120 \index{mp\_neg}
1121 \begin{alltt}
1122 int mp_neg (mp_int * a, mp_int * b);
1123 \end{alltt}
1124
1125 Which assigns $-a$ to $b$.
1126
1127 \subsection{Absolute}
1128 Simple integer absolutes can be performed with the following.
1129
1130 \index{mp\_neg}
1131 \begin{alltt}
1132 int mp_abs (mp_int * a, mp_int * b);
1133 \end{alltt}
1134
1135 Which assigns $\vert a \vert$ to $b$.
1136
1137 \section{Integer Division and Remainder}
1138 To perform a complete and general integer division with remainder use the following function.
1139
1140 \index{mp\_div}
1141 \begin{alltt}
1142 int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d);
1143 \end{alltt}
1144
1145 This divides $a$ by $b$ and stores the quotient in $c$ and $d$. The signed quotient is computed such that
1146 $bc + d = a$. Note that either of $c$ or $d$ can be set to \textbf{NULL} if their value is not required. If
1147 $b$ is zero the function returns \textbf{MP\_VAL}.
1148
1149
1150 \chapter{Multiplication and Squaring}
1151 \section{Multiplication}
1152 A full signed integer multiplication can be performed with the following.
1153 \index{mp\_mul}
1154 \begin{alltt}
1155 int mp_mul (mp_int * a, mp_int * b, mp_int * c);
1156 \end{alltt}
1157 Which assigns the full signed product $ab$ to $c$. This function actually breaks into one of four cases which are
1158 specific multiplication routines optimized for given parameters. First there are the Toom-Cook multiplications which
1159 should only be used with very large inputs. This is followed by the Karatsuba multiplications which are for moderate
1160 sized inputs. Then followed by the Comba and baseline multipliers.
1161
1162 Fortunately for the developer you don't really need to know this unless you really want to fine tune the system. mp\_mul()
1163 will determine on its own\footnote{Some tweaking may be required.} what routine to use automatically when it is called.
1164
1165 \begin{alltt}
1166 int main(void)
1167 \{
1168 mp_int number1, number2;
1169 int result;
1170
1171 /* Initialize the numbers */
1172 if ((result = mp_init_multi(&number1,
1173 &number2, NULL)) != MP_OKAY) \{
1174 printf("Error initializing the numbers. \%s",
1175 mp_error_to_string(result));
1176 return EXIT_FAILURE;
1177 \}
1178
1179 /* set the terms */
1180 if ((result = mp_set_int(&number, 257)) != MP_OKAY) \{
1181 printf("Error setting number1. \%s",
1182 mp_error_to_string(result));
1183 return EXIT_FAILURE;
1184 \}
1185
1186 if ((result = mp_set_int(&number2, 1023)) != MP_OKAY) \{
1187 printf("Error setting number2. \%s",
1188 mp_error_to_string(result));
1189 return EXIT_FAILURE;
1190 \}
1191
1192 /* multiply them */
1193 if ((result = mp_mul(&number1, &number2,
1194 &number1)) != MP_OKAY) \{
1195 printf("Error multiplying terms. \%s",
1196 mp_error_to_string(result));
1197 return EXIT_FAILURE;
1198 \}
1199
1200 /* display */
1201 printf("number1 * number2 == \%lu", mp_get_int(&number1));
1202
1203 /* free terms and return */
1204 mp_clear_multi(&number1, &number2, NULL);
1205
1206 return EXIT_SUCCESS;
1207 \}
1208 \end{alltt}
1209
1210 If this program succeeds it shall output the following.
1211
1212 \begin{alltt}
1213 number1 * number2 == 262911
1214 \end{alltt}
1215
1216 \section{Squaring}
1217 Since squaring can be performed faster than multiplication it is performed it's own function instead of just using
1218 mp\_mul().
1219
1220 \index{mp\_sqr}
1221 \begin{alltt}
1222 int mp_sqr (mp_int * a, mp_int * b);
1223 \end{alltt}
1224
1225 Will square $a$ and store it in $b$. Like the case of multiplication there are four different squaring
1226 algorithms all which can be called from mp\_sqr(). It is ideal to use mp\_sqr over mp\_mul when squaring terms because
1227 of the speed difference.
1228
1229 \section{Tuning Polynomial Basis Routines}
1230
1231 Both of the Toom-Cook and Karatsuba multiplication algorithms are faster than the traditional $O(n^2)$ approach that
1232 the Comba and baseline algorithms use. At $O(n^{1.464973})$ and $O(n^{1.584962})$ running times respectively they require
1233 considerably less work. For example, a 10000-digit multiplication would take roughly 724,000 single precision
1234 multiplications with Toom-Cook or 100,000,000 single precision multiplications with the standard Comba (a factor
1235 of 138).
1236
1237 So why not always use Karatsuba or Toom-Cook? The simple answer is that they have so much overhead that they're not
1238 actually faster than Comba until you hit distinct ``cutoff'' points. For Karatsuba with the default configuration,
1239 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
1240 110 digits Karatsuba and Comba multiplications just about break even and for 110+ digits Karatsuba is faster.
1241
1242 Toom-Cook has incredible overhead and is probably only useful for very large inputs. So far no known cutoff points
1243 exist and for the most part I just set the cutoff points very high to make sure they're not called.
1244
1245 A demo program in the ``etc/'' directory of the project called ``tune.c'' can be used to find the cutoff points. This
1246 can be built with GCC as follows
1247
1248 \begin{alltt}
1249 make XXX
1250 \end{alltt}
1251 Where ``XXX'' is one of the following entries from the table \ref{fig:tuning}.
1252
1253 \begin{figure}[here]
1254 \begin{center}
1255 \begin{small}
1256 \begin{tabular}{|l|l|}
1257 \hline \textbf{Value of XXX} & \textbf{Meaning} \\
1258 \hline tune & Builds portable tuning application \\
1259 \hline tune86 & Builds x86 (pentium and up) program for COFF \\
1260 \hline tune86c & Builds x86 program for Cygwin \\
1261 \hline tune86l & Builds x86 program for Linux (ELF format) \\
1262 \hline
1263 \end{tabular}
1264 \end{small}
1265 \end{center}
1266 \caption{Build Names for Tuning Programs}
1267 \label{fig:tuning}
1268 \end{figure}
1269
1270 When the program is running it will output a series of measurements for different cutoff points. It will first find
1271 good Karatsuba squaring and multiplication points. Then it proceeds to find Toom-Cook points. Note that the Toom-Cook
1272 tuning takes a very long time as the cutoff points are likely to be very high.
1273
1274 \chapter{Modular Reduction}
1275
1276 Modular reduction is process of taking the remainder of one quantity divided by another. Expressed
1277 as (\ref{eqn:mod}) the modular reduction is equivalent to the remainder of $b$ divided by $c$.
1278
1279 \begin{equation}
1280 a \equiv b \mbox{ (mod }c\mbox{)}
1281 \label{eqn:mod}
1282 \end{equation}
1283
1284 Of particular interest to cryptography are reductions where $b$ is limited to the range $0 \le b < c^2$ since particularly
1285 fast reduction algorithms can be written for the limited range.
1286
1287 Note that one of the four optimized reduction algorithms are automatically chosen in the modular exponentiation
1288 algorithm mp\_exptmod when an appropriate modulus is detected.
1289
1290 \section{Straight Division}
1291 In order to effect an arbitrary modular reduction the following algorithm is provided.
1292
1293 \index{mp\_mod}
1294 \begin{alltt}
1295 int mp_mod(mp_int *a, mp_int *b, mp_int *c);
1296 \end{alltt}
1297
1298 This reduces $a$ modulo $b$ and stores the result in $c$. The sign of $c$ shall agree with the sign
1299 of $b$. This algorithm accepts an input $a$ of any range and is not limited by $0 \le a < b^2$.
1300
1301 \section{Barrett Reduction}
1302
1303 Barrett reduction is a generic optimized reduction algorithm that requires pre--computation to achieve
1304 a decent speedup over straight division. First a $\mu$ value must be precomputed with the following function.
1305
1306 \index{mp\_reduce\_setup}
1307 \begin{alltt}
1308 int mp_reduce_setup(mp_int *a, mp_int *b);
1309 \end{alltt}
1310
1311 Given a modulus in $b$ this produces the required $\mu$ value in $a$. For any given modulus this only has to
1312 be computed once. Modular reduction can now be performed with the following.
1313
1314 \index{mp\_reduce}
1315 \begin{alltt}
1316 int mp_reduce(mp_int *a, mp_int *b, mp_int *c);
1317 \end{alltt}
1318
1319 This will reduce $a$ in place modulo $b$ with the precomputed $\mu$ value in $c$. $a$ must be in the range
1320 $0 \le a < b^2$.
1321
1322 \begin{alltt}
1323 int main(void)
1324 \{
1325 mp_int a, b, c, mu;
1326 int result;
1327
1328 /* initialize a,b to desired values, mp_init mu,
1329 * c and set c to 1...we want to compute a^3 mod b
1330 */
1331
1332 /* get mu value */
1333 if ((result = mp_reduce_setup(&mu, b)) != MP_OKAY) \{
1334 printf("Error getting mu. \%s",
1335 mp_error_to_string(result));
1336 return EXIT_FAILURE;
1337 \}
1338
1339 /* square a to get c = a^2 */
1340 if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{
1341 printf("Error squaring. \%s",
1342 mp_error_to_string(result));
1343 return EXIT_FAILURE;
1344 \}
1345
1346 /* now reduce `c' modulo b */
1347 if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{
1348 printf("Error reducing. \%s",
1349 mp_error_to_string(result));
1350 return EXIT_FAILURE;
1351 \}
1352
1353 /* multiply a to get c = a^3 */
1354 if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{
1355 printf("Error reducing. \%s",
1356 mp_error_to_string(result));
1357 return EXIT_FAILURE;
1358 \}
1359
1360 /* now reduce `c' modulo b */
1361 if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{
1362 printf("Error reducing. \%s",
1363 mp_error_to_string(result));
1364 return EXIT_FAILURE;
1365 \}
1366
1367 /* c now equals a^3 mod b */
1368
1369 return EXIT_SUCCESS;
1370 \}
1371 \end{alltt}
1372
1373 This program will calculate $a^3 \mbox{ mod }b$ if all the functions succeed.
1374
1375 \section{Montgomery Reduction}
1376
1377 Montgomery is a specialized reduction algorithm for any odd moduli. Like Barrett reduction a pre--computation
1378 step is required. This is accomplished with the following.
1379
1380 \index{mp\_montgomery\_setup}
1381 \begin{alltt}
1382 int mp_montgomery_setup(mp_int *a, mp_digit *mp);
1383 \end{alltt}
1384
1385 For the given odd moduli $a$ the precomputation value is placed in $mp$. The reduction is computed with the
1386 following.
1387
1388 \index{mp\_montgomery\_reduce}
1389 \begin{alltt}
1390 int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
1391 \end{alltt}
1392 This reduces $a$ in place modulo $m$ with the pre--computed value $mp$. $a$ must be in the range
1393 $0 \le a < b^2$.
1394
1395 Montgomery reduction is faster than Barrett reduction for moduli smaller than the ``comba'' limit. With the default
1396 setup for instance, the limit is $127$ digits ($3556$--bits). Note that this function is not limited to
1397 $127$ digits just that it falls back to a baseline algorithm after that point.
1398
1399 An important observation is that this reduction does not return $a \mbox{ mod }m$ but $aR^{-1} \mbox{ mod }m$
1400 where $R = \beta^n$, $n$ is the n number of digits in $m$ and $\beta$ is radix used (default is $2^{28}$).
1401
1402 To quickly calculate $R$ the following function was provided.
1403
1404 \index{mp\_montgomery\_calc\_normalization}
1405 \begin{alltt}
1406 int mp_montgomery_calc_normalization(mp_int *a, mp_int *b);
1407 \end{alltt}
1408 Which calculates $a = R$ for the odd moduli $b$ without using multiplication or division.
1409
1410 The normal modus operandi for Montgomery reductions is to normalize the integers before entering the system. For
1411 example, to calculate $a^3 \mbox { mod }b$ using Montgomery reduction the value of $a$ can be normalized by
1412 multiplying it by $R$. Consider the following code snippet.
1413
1414 \begin{alltt}
1415 int main(void)
1416 \{
1417 mp_int a, b, c, R;
1418 mp_digit mp;
1419 int result;
1420
1421 /* initialize a,b to desired values,
1422 * mp_init R, c and set c to 1....
1423 */
1424
1425 /* get normalization */
1426 if ((result = mp_montgomery_calc_normalization(&R, b)) != MP_OKAY) \{
1427 printf("Error getting norm. \%s",
1428 mp_error_to_string(result));
1429 return EXIT_FAILURE;
1430 \}
1431
1432 /* get mp value */
1433 if ((result = mp_montgomery_setup(&c, &mp)) != MP_OKAY) \{
1434 printf("Error setting up montgomery. \%s",
1435 mp_error_to_string(result));
1436 return EXIT_FAILURE;
1437 \}
1438
1439 /* normalize `a' so now a is equal to aR */
1440 if ((result = mp_mulmod(&a, &R, &b, &a)) != MP_OKAY) \{
1441 printf("Error computing aR. \%s",
1442 mp_error_to_string(result));
1443 return EXIT_FAILURE;
1444 \}
1445
1446 /* square a to get c = a^2R^2 */
1447 if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{
1448 printf("Error squaring. \%s",
1449 mp_error_to_string(result));
1450 return EXIT_FAILURE;
1451 \}
1452
1453 /* now reduce `c' back down to c = a^2R^2 * R^-1 == a^2R */
1454 if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
1455 printf("Error reducing. \%s",
1456 mp_error_to_string(result));
1457 return EXIT_FAILURE;
1458 \}
1459
1460 /* multiply a to get c = a^3R^2 */
1461 if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{
1462 printf("Error reducing. \%s",
1463 mp_error_to_string(result));
1464 return EXIT_FAILURE;
1465 \}
1466
1467 /* now reduce `c' back down to c = a^3R^2 * R^-1 == a^3R */
1468 if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
1469 printf("Error reducing. \%s",
1470 mp_error_to_string(result));
1471 return EXIT_FAILURE;
1472 \}
1473
1474 /* now reduce (again) `c' back down to c = a^3R * R^-1 == a^3 */
1475 if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
1476 printf("Error reducing. \%s",
1477 mp_error_to_string(result));
1478 return EXIT_FAILURE;
1479 \}
1480
1481 /* c now equals a^3 mod b */
1482
1483 return EXIT_SUCCESS;
1484 \}
1485 \end{alltt}
1486
1487 This particular example does not look too efficient but it demonstrates the point of the algorithm. By
1488 normalizing the inputs the reduced results are always of the form $aR$ for some variable $a$. This allows
1489 a single final reduction to correct for the normalization and the fast reduction used within the algorithm.
1490
1491 For more details consider examining the file \textit{bn\_mp\_exptmod\_fast.c}.
1492
1493 \section{Restricted Dimminished Radix}
1494
1495 ``Dimminished Radix'' reduction refers to reduction with respect to moduli that are ameniable to simple
1496 digit shifting and small multiplications. In this case the ``restricted'' variant refers to moduli of the
1497 form $\beta^k - p$ for some $k \ge 0$ and $0 < p < \beta$ where $\beta$ is the radix (default to $2^{28}$).
1498
1499 As in the case of Montgomery reduction there is a pre--computation phase required for a given modulus.
1500
1501 \index{mp\_dr\_setup}
1502 \begin{alltt}
1503 void mp_dr_setup(mp_int *a, mp_digit *d);
1504 \end{alltt}
1505
1506 This computes the value required for the modulus $a$ and stores it in $d$. This function cannot fail
1507 and does not return any error codes. After the pre--computation a reduction can be performed with the
1508 following.
1509
1510 \index{mp\_dr\_reduce}
1511 \begin{alltt}
1512 int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp);
1513 \end{alltt}
1514
1515 This reduces $a$ in place modulo $b$ with the pre--computed value $mp$. $b$ must be of a restricted
1516 dimminished radix form and $a$ must be in the range $0 \le a < b^2$. Dimminished radix reductions are
1517 much faster than both Barrett and Montgomery reductions as they have a much lower asymtotic running time.
1518
1519 Since the moduli are restricted this algorithm is not particularly useful for something like Rabin, RSA or
1520 BBS cryptographic purposes. This reduction algorithm is useful for Diffie-Hellman and ECC where fixed
1521 primes are acceptable.
1522
1523 Note that unlike Montgomery reduction there is no normalization process. The result of this function is
1524 equal to the correct residue.
1525
1526 \section{Unrestricted Dimminshed Radix}
1527
1528 Unrestricted reductions work much like the restricted counterparts except in this case the moduli is of the
1529 form $2^k - p$ for $0 < p < \beta$. In this sense the unrestricted reductions are more flexible as they
1530 can be applied to a wider range of numbers.
1531
1532 \index{mp\_reduce\_2k\_setup}
1533 \begin{alltt}
1534 int mp_reduce_2k_setup(mp_int *a, mp_digit *d);
1535 \end{alltt}
1536
1537 This will compute the required $d$ value for the given moduli $a$.
1538
1539 \index{mp\_reduce\_2k}
1540 \begin{alltt}
1541 int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d);
1542 \end{alltt}
1543
1544 This will reduce $a$ in place modulo $n$ with the pre--computed value $d$. From my experience this routine is
1545 slower than mp\_dr\_reduce but faster for most moduli sizes than the Montgomery reduction.
1546
1547 \chapter{Exponentiation}
1548 \section{Single Digit Exponentiation}
1549 \index{mp\_expt\_d}
1550 \begin{alltt}
1551 int mp_expt_d (mp_int * a, mp_digit b, mp_int * c)
1552 \end{alltt}
1553 This computes $c = a^b$ using a simple binary left-to-right algorithm. It is faster than repeated multiplications by
1554 $a$ for all values of $b$ greater than three.
1555
1556 \section{Modular Exponentiation}
1557 \index{mp\_exptmod}
1558 \begin{alltt}
1559 int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
1560 \end{alltt}
1561 This computes $Y \equiv G^X \mbox{ (mod }P\mbox{)}$ using a variable width sliding window algorithm. This function
1562 will automatically detect the fastest modular reduction technique to use during the operation. For negative values of
1563 $X$ the operation is performed as $Y \equiv (G^{-1} \mbox{ mod }P)^{\vert X \vert} \mbox{ (mod }P\mbox{)}$ provided that
1564 $gcd(G, P) = 1$.
1565
1566 This function is actually a shell around the two internal exponentiation functions. This routine will automatically
1567 detect when Barrett, Montgomery, Restricted and Unrestricted Dimminished Radix based exponentiation can be used. Generally
1568 moduli of the a ``restricted dimminished radix'' form lead to the fastest modular exponentiations. Followed by Montgomery
1569 and the other two algorithms.
1570
1571 \section{Root Finding}
1572 \index{mp\_n\_root}
1573 \begin{alltt}
1574 int mp_n_root (mp_int * a, mp_digit b, mp_int * c)
1575 \end{alltt}
1576 This computes $c = a^{1/b}$ such that $c^b \le a$ and $(c+1)^b > a$. The implementation of this function is not
1577 ideal for values of $b$ greater than three. It will work but become very slow. So unless you are working with very small
1578 numbers (less than 1000 bits) I'd avoid $b > 3$ situations. Will return a positive root only for even roots and return
1579 a root with the sign of the input for odd roots. For example, performing $4^{1/2}$ will return $2$ whereas $(-8)^{1/3}$
1580 will return $-2$.
1581
1582 This algorithm uses the ``Newton Approximation'' method and will converge on the correct root fairly quickly. Since
1583 the algorithm requires raising $a$ to the power of $b$ it is not ideal to attempt to find roots for large
1584 values of $b$. If particularly large roots are required then a factor method could be used instead. For example,
1585 $a^{1/16}$ is equivalent to $\left (a^{1/4} \right)^{1/4}$ or simply
1586 $\left ( \left ( \left ( a^{1/2} \right )^{1/2} \right )^{1/2} \right )^{1/2}$
1587
1588 \chapter{Prime Numbers}
1589 \section{Trial Division}
1590 \index{mp\_prime\_is\_divisible}
1591 \begin{alltt}
1592 int mp_prime_is_divisible (mp_int * a, int *result)
1593 \end{alltt}
1594 This will attempt to evenly divide $a$ by a list of primes\footnote{Default is the first 256 primes.} and store the
1595 outcome in ``result''. That is if $result = 0$ then $a$ is not divisible by the primes, otherwise it is. Note that
1596 if the function does not return \textbf{MP\_OKAY} the value in ``result'' should be considered undefined\footnote{Currently
1597 the default is to set it to zero first.}.
1598
1599 \section{Fermat Test}
1600 \index{mp\_prime\_fermat}
1601 \begin{alltt}
1602 int mp_prime_fermat (mp_int * a, mp_int * b, int *result)
1603 \end{alltt}
1604 Performs a Fermat primality test to the base $b$. That is it computes $b^a \mbox{ mod }a$ and tests whether the value is
1605 equal to $b$ or not. If the values are equal then $a$ is probably prime and $result$ is set to one. Otherwise $result$
1606 is set to zero.
1607
1608 \section{Miller-Rabin Test}
1609 \index{mp\_prime\_miller\_rabin}
1610 \begin{alltt}
1611 int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result)
1612 \end{alltt}
1613 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
1614 fool (besides with Carmichael numbers). If $a$ passes the test (therefore is probably prime) $result$ is set to one.
1615 Otherwise $result$ is set to zero.
1616
1617 Note that is suggested that you use the Miller-Rabin test instead of the Fermat test since all of the failures of
1618 Miller-Rabin are a subset of the failures of the Fermat test.
1619
1620 \subsection{Required Number of Tests}
1621 Generally to ensure a number is very likely to be prime you have to perform the Miller-Rabin with at least a half-dozen
1622 or so unique bases. However, it has been proven that the probability of failure goes down as the size of the input goes up.
1623 This is why a simple function has been provided to help out.
1624
1625 \index{mp\_prime\_rabin\_miller\_trials}
1626 \begin{alltt}
1627 int mp_prime_rabin_miller_trials(int size)
1628 \end{alltt}
1629 This returns the number of trials required for a $2^{-96}$ (or lower) probability of failure for a given ``size'' expressed
1630 in bits. This comes in handy specially since larger numbers are slower to test. For example, a 512-bit number would
1631 require ten tests whereas a 1024-bit number would only require four tests.
1632
1633 You should always still perform a trial division before a Miller-Rabin test though.
1634
1635 \section{Primality Testing}
1636 \index{mp\_prime\_is\_prime}
1637 \begin{alltt}
1638 int mp_prime_is_prime (mp_int * a, int t, int *result)
1639 \end{alltt}
1640 This will perform a trial division followed by $t$ rounds of Miller-Rabin tests on $a$ and store the result in $result$.
1641 If $a$ passes all of the tests $result$ is set to one, otherwise it is set to zero. Note that $t$ is bounded by
1642 $1 \le t < PRIME\_SIZE$ where $PRIME\_SIZE$ is the number of primes in the prime number table (by default this is $256$).
1643
1644 \section{Next Prime}
1645 \index{mp\_prime\_next\_prime}
1646 \begin{alltt}
1647 int mp_prime_next_prime(mp_int *a, int t, int bbs_style)
1648 \end{alltt}
1649 This finds the next prime after $a$ that passes mp\_prime\_is\_prime() with $t$ tests. Set $bbs\_style$ to one if you
1650 want only the next prime congruent to $3 \mbox{ mod } 4$, otherwise set it to zero to find any next prime.
1651
1652 \section{Random Primes}
1653 \index{mp\_prime\_random}
1654 \begin{alltt}
1655 int mp_prime_random(mp_int *a, int t, int size, int bbs,
1656 ltm_prime_callback cb, void *dat)
1657 \end{alltt}
1658 This will find a prime greater than $256^{size}$ which can be ``bbs\_style'' or not depending on $bbs$ and must pass
1659 $t$ rounds of tests. The ``ltm\_prime\_callback'' is a typedef for
1660
1661 \begin{alltt}
1662 typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat);
1663 \end{alltt}
1664
1665 Which is a function that must read $len$ bytes (and return the amount stored) into $dst$. The $dat$ variable is simply
1666 copied from the original input. It can be used to pass RNG context data to the callback. The function
1667 mp\_prime\_random() is more suitable for generating primes which must be secret (as in the case of RSA) since there
1668 is no skew on the least significant bits.
1669
1670 \textit{Note:} As of v0.30 of the LibTomMath library this function has been deprecated. It is still available
1671 but users are encouraged to use the new mp\_prime\_random\_ex() function instead.
1672
1673 \subsection{Extended Generation}
1674 \index{mp\_prime\_random\_ex}
1675 \begin{alltt}
1676 int mp_prime_random_ex(mp_int *a, int t,
1677 int size, int flags,
1678 ltm_prime_callback cb, void *dat);
1679 \end{alltt}
1680 This will generate a prime in $a$ using $t$ tests of the primality testing algorithms. The variable $size$
1681 specifies the bit length of the prime desired. The variable $flags$ specifies one of several options available
1682 (see fig. \ref{fig:primeopts}) which can be OR'ed together. The callback parameters are used as in
1683 mp\_prime\_random().
1684
1685 \begin{figure}[here]
1686 \begin{center}
1687 \begin{small}
1688 \begin{tabular}{|r|l|}
1689 \hline \textbf{Flag} & \textbf{Meaning} \\
1690 \hline LTM\_PRIME\_BBS & Make the prime congruent to $3$ modulo $4$ \\
1691 \hline LTM\_PRIME\_SAFE & Make a prime $p$ such that $(p - 1)/2$ is also prime. \\
1692 & This option implies LTM\_PRIME\_BBS as well. \\
1693 \hline LTM\_PRIME\_2MSB\_OFF & Makes sure that the bit adjacent to the most significant bit \\
1694 & Is forced to zero. \\
1695 \hline LTM\_PRIME\_2MSB\_ON & Makes sure that the bit adjacent to the most significant bit \\
1696 & Is forced to one. \\
1697 \hline
1698 \end{tabular}
1699 \end{small}
1700 \end{center}
1701 \caption{Primality Generation Options}
1702 \label{fig:primeopts}
1703 \end{figure}
1704
1705 \chapter{Input and Output}
1706 \section{ASCII Conversions}
1707 \subsection{To ASCII}
1708 \index{mp\_toradix}
1709 \begin{alltt}
1710 int mp_toradix (mp_int * a, char *str, int radix);
1711 \end{alltt}
1712 This still store $a$ in ``str'' as a base-``radix'' string of ASCII chars. This function appends a NUL character
1713 to terminate the string. Valid values of ``radix'' line in the range $[2, 64]$. To determine the size (exact) required
1714 by the conversion before storing any data use the following function.
1715
1716 \index{mp\_radix\_size}
1717 \begin{alltt}
1718 int mp_radix_size (mp_int * a, int radix, int *size)
1719 \end{alltt}
1720 This stores in ``size'' the number of characters (including space for the NUL terminator) required. Upon error this
1721 function returns an error code and ``size'' will be zero.
1722
1723 \subsection{From ASCII}
1724 \index{mp\_read\_radix}
1725 \begin{alltt}
1726 int mp_read_radix (mp_int * a, char *str, int radix);
1727 \end{alltt}
1728 This will read the base-``radix'' NUL terminated string from ``str'' into $a$. It will stop reading when it reads a
1729 character it does not recognize (which happens to include th NUL char... imagine that...). A single leading $-$ sign
1730 can be used to denote a negative number.
1731
1732 \section{Binary Conversions}
1733
1734 Converting an mp\_int to and from binary is another keen idea.
1735
1736 \index{mp\_unsigned\_bin\_size}
1737 \begin{alltt}
1738 int mp_unsigned_bin_size(mp_int *a);
1739 \end{alltt}
1740
1741 This will return the number of bytes (octets) required to store the unsigned copy of the integer $a$.
1742
1743 \index{mp\_to\_unsigned\_bin}
1744 \begin{alltt}
1745 int mp_to_unsigned_bin(mp_int *a, unsigned char *b);
1746 \end{alltt}
1747 This will store $a$ into the buffer $b$ in big--endian format. Fortunately this is exactly what DER (or is it ASN?)
1748 requires. It does not store the sign of the integer.
1749
1750 \index{mp\_read\_unsigned\_bin}
1751 \begin{alltt}
1752 int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c);
1753 \end{alltt}
1754 This will read in an unsigned big--endian array of bytes (octets) from $b$ of length $c$ into $a$. The resulting
1755 integer $a$ will always be positive.
1756
1757 For those who acknowledge the existence of negative numbers (heretic!) there are ``signed'' versions of the
1758 previous functions.
1759
1760 \begin{alltt}
1761 int mp_signed_bin_size(mp_int *a);
1762 int mp_read_signed_bin(mp_int *a, unsigned char *b, int c);
1763 int mp_to_signed_bin(mp_int *a, unsigned char *b);
1764 \end{alltt}
1765 They operate essentially the same as the unsigned copies except they prefix the data with zero or non--zero
1766 byte depending on the sign. If the sign is zpos (e.g. not negative) the prefix is zero, otherwise the prefix
1767 is non--zero.
1768
1769 \chapter{Algebraic Functions}
1770 \section{Extended Euclidean Algorithm}
1771 \index{mp\_exteuclid}
1772 \begin{alltt}
1773 int mp_exteuclid(mp_int *a, mp_int *b,
1774 mp_int *U1, mp_int *U2, mp_int *U3);
1775 \end{alltt}
1776
1777 This finds the triple U1/U2/U3 using the Extended Euclidean algorithm such that the following equation holds.
1778
1779 \begin{equation}
1780 a \cdot U1 + b \cdot U2 = U3
1781 \end{equation}
1782
1783 Any of the U1/U2/U3 paramters can be set to \textbf{NULL} if they are not desired.
1784
1785 \section{Greatest Common Divisor}
1786 \index{mp\_gcd}
1787 \begin{alltt}
1788 int mp_gcd (mp_int * a, mp_int * b, mp_int * c)
1789 \end{alltt}
1790 This will compute the greatest common divisor of $a$ and $b$ and store it in $c$.
1791
1792 \section{Least Common Multiple}
1793 \index{mp\_lcm}
1794 \begin{alltt}
1795 int mp_lcm (mp_int * a, mp_int * b, mp_int * c)
1796 \end{alltt}
1797 This will compute the least common multiple of $a$ and $b$ and store it in $c$.
1798
1799 \section{Jacobi Symbol}
1800 \index{mp\_jacobi}
1801 \begin{alltt}
1802 int mp_jacobi (mp_int * a, mp_int * p, int *c)
1803 \end{alltt}
1804 This will compute the Jacobi symbol for $a$ with respect to $p$. If $p$ is prime this essentially computes the Legendre
1805 symbol. The result is stored in $c$ and can take on one of three values $\lbrace -1, 0, 1 \rbrace$. If $p$ is prime
1806 then the result will be $-1$ when $a$ is not a quadratic residue modulo $p$. The result will be $0$ if $a$ divides $p$
1807 and the result will be $1$ if $a$ is a quadratic residue modulo $p$.
1808
1809 \section{Modular Inverse}
1810 \index{mp\_invmod}
1811 \begin{alltt}
1812 int mp_invmod (mp_int * a, mp_int * b, mp_int * c)
1813 \end{alltt}
1814 Computes the multiplicative inverse of $a$ modulo $b$ and stores the result in $c$ such that $ac \equiv 1 \mbox{ (mod }b\mbox{)}$.
1815
1816 \section{Single Digit Functions}
1817
1818 For those using small numbers (\textit{snicker snicker}) there are several ``helper'' functions
1819
1820 \index{mp\_add\_d} \index{mp\_sub\_d} \index{mp\_mul\_d} \index{mp\_div\_d} \index{mp\_mod\_d}
1821 \begin{alltt}
1822 int mp_add_d(mp_int *a, mp_digit b, mp_int *c);
1823 int mp_sub_d(mp_int *a, mp_digit b, mp_int *c);
1824 int mp_mul_d(mp_int *a, mp_digit b, mp_int *c);
1825 int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d);
1826 int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c);
1827 \end{alltt}
1828
1829 These work like the full mp\_int capable variants except the second parameter $b$ is a mp\_digit. These
1830 functions fairly handy if you have to work with relatively small numbers since you will not have to allocate
1831 an entire mp\_int to store a number like $1$ or $2$.
1832
1833 \input{bn.ind}
1834
1835 \end{document}