1\documentclass[synpaper]{book} 2\usepackage{hyperref} 3\usepackage{makeidx} 4\usepackage{amssymb} 5\usepackage{color} 6\usepackage{alltt} 7\usepackage{graphicx} 8\usepackage{layout} 9\usepackage{appendix} 10\def\union{\cup} 11\def\intersect{\cap} 12\def\getsrandom{\stackrel{\rm R}{\gets}} 13\def\cross{\times} 14\def\cat{\hspace{0.5em} \| \hspace{0.5em}} 15\def\catn{$\|$} 16\def\divides{\hspace{0.3em} | \hspace{0.3em}} 17\def\nequiv{\not\equiv} 18\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}} 19\def\lcm{{\rm lcm}} 20\def\gcd{{\rm gcd}} 21\def\log{{\rm log}} 22\def\ord{{\rm ord}} 23\def\abs{{\mathit abs}} 24\def\rep{{\mathit rep}} 25\def\mod{{\mathit\ mod\ }} 26\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})} 27\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} 28\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil} 29\def\Or{{\rm\ or\ }} 30\def\And{{\rm\ and\ }} 31\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}} 32\def\implies{\Rightarrow} 33\def\undefined{{\rm ``undefined"}} 34\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}} 35\let\oldphi\phi 36\def\phi{\varphi} 37\def\Pr{{\rm Pr}} 38\newcommand{\str}[1]{{\mathbf{#1}}} 39\def\F{{\mathbb F}} 40\def\N{{\mathbb N}} 41\def\Z{{\mathbb Z}} 42\def\R{{\mathbb R}} 43\def\C{{\mathbb C}} 44\def\Q{{\mathbb Q}} 45\definecolor{DGray}{gray}{0.5} 46\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}} 47\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}} 48\def\gap{\vspace{0.5ex}} 49\makeindex 50\begin{document} 51\frontmatter 52\pagestyle{empty} 53\title{LibTomMath User Manual \\ v1.2.0} 54\author{LibTom Projects \\ www.libtom.net} 55\maketitle 56This text, the library and the accompanying textbook are all hereby placed in the public domain. This book has been 57formatted for B5 [176x250] paper using the \LaTeX{} {\em book} macro package. 58 59\vspace{10cm} 60 61\begin{flushright}Open Source. Open Academia. Open Minds. 62 63\mbox{ } 64LibTom Projects 65 66\& originally 67 68Tom St Denis, 69 70Ontario, Canada 71\end{flushright} 72 73\tableofcontents 74\listoffigures 75\mainmatter 76\pagestyle{headings} 77\chapter{Introduction} 78\section{What is LibTomMath?} 79LibTomMath is a library of source code which provides a series of efficient and carefully written functions for manipulating 80large integer numbers. It was written in portable ISO C source code so that it will build on any platform with a conforming 81C compiler. 82 83In a nutshell the library was written from scratch with verbose comments to help instruct computer science students how 84to implement ``bignum'' math. However, the resulting code has proven to be very useful. It has been used by numerous 85universities, commercial and open source software developers. It has been used on a variety of platforms ranging from 86Linux and Windows based x86 to ARM based Gameboys and PPC based MacOS machines. 87 88\section{License} 89As of the v0.25 the library source code has been placed in the public domain with every new release. As of the v0.28 90release the textbook ``Implementing Multiple Precision Arithmetic'' has been placed in the public domain with every new 91release as well. This textbook is meant to compliment the project by providing a more solid walkthrough of the development 92algorithms used in the library. 93 94Since both\footnote{Note that the MPI files under mtest/ are copyrighted by Michael Fromberger. They are not required to use LibTomMath.} are in the 95public domain everyone is entitled to do with them as they see fit. 96 97\section{Building LibTomMath} 98 99LibTomMath is meant to be very ``GCC friendly'' as it comes with a makefile well suited for GCC. However, the library will 100also 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 101developer. Please consider to commit such a makefile to the LibTomMath developers, currently residing at 102\url{http://github.com/libtom/libtommath}, if successfully done so. 103 104Intel's C-compiler (ICC) is sufficiently compatible with GCC, at least the newer versions, to replace GCC for building the static and the shared library. Editing the makefiles is not needed, just set the shell variable \texttt{CC} as shown below. 105\begin{alltt} 106CC=/home/czurnieden/intel/bin/icc make 107\end{alltt} 108 109ICC does not know all options available for GCC and LibTomMath uses two diagnostics \texttt{-Wbad-function-cast} and \texttt{-Wcast-align} that are not supported by ICC resulting in the warnings: 110\begin{alltt} 111icc: command line warning #10148: option '-Wbad-function-cast' not supported 112icc: command line warning #10148: option '-Wcast-align' not supported 113\end{alltt} 114It is possible to mute this ICC warning with the compiler flag \texttt{-diag-disable=10148}\footnote{It is not recommended to suppress warnings without a very good reason but there is no harm in doing so in this very special case.}. 115 116\subsection{Static Libraries} 117To build as a static library for GCC issue the following 118\begin{alltt} 119make 120\end{alltt} 121 122command. This will build the library and archive the object files in ``libtommath.a''. Now you link against 123that and include ``tommath.h'' within your programs. Alternatively to build with MSVC issue the following 124\begin{alltt} 125nmake -f makefile.msvc 126\end{alltt} 127 128This will build the library and archive the object files in ``tommath.lib''. This has been tested with MSVC 129version 6.00 with service pack 5. 130 131To run a program to adapt the Toom-Cook cut-off values to your architecture type 132\begin{alltt} 133make tune 134\end{alltt} 135This will take some time. 136 137\subsection{Shared Libraries} 138\subsubsection{GNU based Operating Systems} 139To build as a shared library for GCC issue the following 140\begin{alltt} 141make -f makefile.shared 142\end{alltt} 143This requires the ``libtool'' package (common on most Linux/BSD systems). It will build LibTomMath as both shared 144and static then install (by default) into /usr/lib as well as install the header files in /usr/include. The shared 145library (resource) will be called ``libtommath.la'' while the static library called ``libtommath.a''. Generally 146you use libtool to link your application against the shared object. 147 148To run a program to adapt the Toom-Cook cut-off values to your architecture type 149\begin{alltt} 150make -f makefile.shared tune 151\end{alltt} 152This will take some time. 153 154\subsubsection{Microsoft Windows based Operating Systems} 155There is limited support for making a ``DLL'' in windows via the ``makefile.cygwin\_dll'' makefile. It requires 156Cygwin to work with since it requires the auto-export/import functionality. The resulting DLL and import library 157``libtommath.dll.a'' can be used to link LibTomMath dynamically to any Windows program using Cygwin. 158\subsubsection{OpenBSD} 159OpenBSD replaced some of their GNU-tools, especially \texttt{libtool} with their own, slightly different versions. To ease the workload of LibTomMath's developer team, only a static library can be build with the included \texttt{makefile.unix}. 160 161The wrong \texttt{make} will result in errors like: 162\begin{alltt} 163*** Parse error in /home/user/GITHUB/libtommath: Need an operator in 'LIBNAME' ) 164*** Parse error: Need an operator in 'endif' (makefile.shared:8) 165*** Parse error: Need an operator in 'CROSS_COMPILE' (makefile_include.mk:16) 166*** Parse error: Need an operator in 'endif' (makefile_include.mk:18) 167*** Parse error: Missing dependency operator (makefile_include.mk:22) 168*** Parse error: Missing dependency operator (makefile_include.mk:23) 169... 170\end{alltt} 171The wrong \texttt{libtool} will build it all fine but when it comes to the final linking fails with 172\begin{alltt} 173... 174cc -I./ -Wall -Wsign-compare -Wextra -Wshadow -Wsystem-headers -Wdeclaration-afo... 175cc -I./ -Wall -Wsign-compare -Wextra -Wshadow -Wsystem-headers -Wdeclaration-afo... 176cc -I./ -Wall -Wsign-compare -Wextra -Wshadow -Wsystem-headers -Wdeclaration-afo... 177libtool --mode=link --tag=CC cc bn_error.lo bn_s_mp_invmod_fast.lo bn_fast_mp_mo 178libtool: link: cc bn_error.lo bn_s_mp_invmod_fast.lo bn_s_mp_montgomery_reduce_fast0 179bn_error.lo: file not recognized: File format not recognized 180cc: error: linker command failed with exit code 1 (use -v to see invocation) 181Error while executing cc bn_error.lo bn_s_mp_invmod_fast.lo bn_fast_mp_montgomery0 182gmake: *** [makefile.shared:64: libtommath.la] Error 1 183\end{alltt} 184 185To build a shared library with OpenBSD\footnote{Tested with OpenBSD version 6.4} the GNU versions of \texttt{make} and \texttt{libtool} are needed. 186\begin{alltt} 187$ sudo pkg_add gmake libtool 188\end{alltt} 189At this time two versions of \texttt{libtool} are installed and both are named \texttt{libtool}, unfortunately but GNU \texttt{libtool} has been placed in \texttt{/usr/local/bin/} and the native version in \texttt{/usr/bin/}. The path might be different in other versions of OpenBSD but both programms differ in the output of \texttt{libtool --version} 190\begin{alltt} 191$ /usr/local/bin/libtool --version 192libtool (GNU libtool) 2.4.2 193Written by Gordon Matzigkeit <gord@gnu.ai.mit.edu>, 1996 194 195Copyright (C) 2011 Free Software Foundation, Inc. 196This is free software; see the source for copying conditions. There is NO 197warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. 198$ libtool --version 199libtool (not (GNU libtool)) 1.5.26 200\end{alltt} 201 202The shared library should build now with 203\begin{alltt} 204LIBTOOL="/usr/local/bin/libtool" gmake -f makefile.shared 205\end{alltt} 206You might need to run a \texttt{gmake -f makefile.shared clean} first. 207 208\subsubsection{NetBSD} 209NetBSD is not as strict as OpenBSD but still needs \texttt{gmake} to build the shared library. \texttt{libtool} may also not exist in a fresh install. 210\begin{alltt} 211pkg_add gmake libtool 212\end{alltt} 213Please check with \texttt{libtool --version} that installed libtool is indeed a GNU libtool. 214Build the shared library by typing: 215\begin{alltt} 216gmake -f makefile.shared 217\end{alltt} 218 219\subsection{Testing} 220To build the library and the test harness type 221 222\begin{alltt} 223make test 224\end{alltt} 225 226This will build the library, ``test'' and ``mtest/mtest''. The ``test'' program will accept test vectors and verify the 227results. ``mtest/mtest'' will generate test vectors using the MPI library by Michael Fromberger\footnote{A copy of MPI 228is included in the package}. Simply pipe mtest into test using 229 230\begin{alltt} 231mtest/mtest | test 232\end{alltt} 233 234If you do not have a ``/dev/urandom'' style RNG source you will have to write your own PRNG and simply pipe that into 235mtest. For example, if your PRNG program is called ``myprng'' simply invoke 236 237\begin{alltt} 238myprng | mtest/mtest | test 239\end{alltt} 240 241This will output a row of numbers that are increasing. Each column is a different test (such as addition, multiplication, etc) 242that is being performed. The numbers represent how many times the test was invoked. If an error is detected the program 243will exit with a dump of the relevant numbers it was working with. 244 245\section{Build Configuration} 246LibTomMath can configured at build time in three phases we shall call ``depends'', ``tweaks'' and ``trims''. 247Each phase changes how the library is built and they are applied one after another respectively. 248 249To make the system more powerful you can tweak the build process. Classes are defined in the file 250``tommath\_superclass.h''. By default, the symbol ``LTM\_ALL'' shall be defined which simply 251instructs the system to build all of the functions. This is how LibTomMath used to be packaged. This will give you 252access to every function LibTomMath offers. 253 254However, there are cases where such a build is not optional. For instance, you want to perform RSA operations. You 255don't need the vast majority of the library to perform these operations. Aside from LTM\_ALL there is 256another pre--defined class ``SC\_RSA\_1'' which works in conjunction with the RSA from LibTomCrypt. Additional 257classes can be defined base on the need of the user. 258 259\subsection{Build Depends} 260In the file tommath\_class.h you will see a large list of C ``defines'' followed by a series of ``ifdefs'' 261which further define symbols. All of the symbols (technically they're macros $\ldots$) represent a given C source 262file. For instance, BN\_MP\_ADD\_C represents the file ``bn\_mp\_add.c''. When a define has been enabled the 263function in the respective file will be compiled and linked into the library. Accordingly when the define 264is absent the file will not be compiled and not contribute any size to the library. 265 266You will also note that the header tommath\_class.h is actually recursively included (it includes itself twice). 267This is to help resolve as many dependencies as possible. In the last pass the symbol LTM\_LAST will be defined. 268This is useful for ``trims''. 269 270\subsection{Build Tweaks} 271A tweak is an algorithm ``alternative''. For example, to provide tradeoffs (usually between size and space). 272They can be enabled at any pass of the configuration phase. 273 274\begin{small} 275\begin{center} 276\begin{tabular}{|l|l|} 277\hline \textbf{Define} & \textbf{Purpose} \\ 278\hline BN\_MP\_DIV\_SMALL & Enables a slower, smaller and equally \\ 279 & functional mp\_div() function \\ 280\hline 281\end{tabular} 282\end{center} 283\end{small} 284 285\subsection{Build Trims} 286A trim is a manner of removing functionality from a function that is not required. For instance, to perform 287RSA cryptography you only require exponentiation with odd moduli so even moduli support can be safely removed. 288Build trims are meant to be defined on the last pass of the configuration which means they are to be defined 289only if LTM\_LAST has been defined. 290 291\subsubsection{Moduli Related} 292\begin{small} 293\begin{center} 294\begin{tabular}{|l|l|} 295\hline \textbf{Restriction} & \textbf{Undefine} \\ 296\hline Exponentiation with odd moduli only & BN\_S\_MP\_EXPTMOD\_C \\ 297 & BN\_MP\_REDUCE\_C \\ 298 & BN\_MP\_REDUCE\_SETUP\_C \\ 299 & BN\_S\_MP\_MUL\_HIGH\_DIGS\_C \\ 300 & BN\_FAST\_S\_MP\_MUL\_HIGH\_DIGS\_C \\ 301\hline Exponentiation with random odd moduli & (The above plus the following) \\ 302 & BN\_MP\_REDUCE\_2K\_C \\ 303 & BN\_MP\_REDUCE\_2K\_SETUP\_C \\ 304 & BN\_MP\_REDUCE\_IS\_2K\_C \\ 305 & BN\_MP\_DR\_IS\_MODULUS\_C \\ 306 & BN\_MP\_DR\_REDUCE\_C \\ 307 & BN\_MP\_DR\_SETUP\_C \\ 308\hline Modular inverse odd moduli only & BN\_MP\_INVMOD\_SLOW\_C \\ 309\hline Modular inverse (both, smaller/slower) & BN\_FAST\_MP\_INVMOD\_C \\ 310\hline 311\end{tabular} 312\end{center} 313\end{small} 314 315\subsubsection{Operand Size Related} 316\begin{small} 317\begin{center} 318\begin{tabular}{|l|l|} 319\hline \textbf{Restriction} & \textbf{Undefine} \\ 320\hline Moduli $\le 2560$ bits & BN\_MP\_MONTGOMERY\_REDUCE\_C \\ 321 & BN\_S\_MP\_MUL\_DIGS\_C \\ 322 & BN\_S\_MP\_MUL\_HIGH\_DIGS\_C \\ 323 & BN\_S\_MP\_SQR\_C \\ 324\hline Polynomial Schmolynomial & BN\_MP\_KARATSUBA\_MUL\_C \\ 325 & BN\_MP\_KARATSUBA\_SQR\_C \\ 326 & BN\_MP\_TOOM\_MUL\_C \\ 327 & BN\_MP\_TOOM\_SQR\_C \\ 328 329\hline 330\end{tabular} 331\end{center} 332\end{small} 333 334 335\section{Purpose of LibTomMath} 336Unlike GNU MP (GMP) Library, LIP, OpenSSL or various other commercial kits (Miracl), LibTomMath was not written with 337bleeding edge performance in mind. First and foremost LibTomMath was written to be entirely open. Not only is the 338source code public domain (unlike various other GPL/etc licensed code), not only is the code freely downloadable but the 339source code is also accessible for computer science students attempting to learn ``BigNum'' or multiple precision 340arithmetic techniques. 341 342LibTomMath was written to be an instructive collection of source code. This is why there are many comments, only one 343function per source file and often I use a ``middle-road'' approach where I don't cut corners for an extra 2\% speed 344increase. 345 346Source code alone cannot really teach how the algorithms work which is why I also wrote a textbook that accompanies 347the library (beat that!). 348 349So you may be thinking ``should I use LibTomMath?'' and the answer is a definite maybe. Let me tabulate what I think 350are the pros and cons of LibTomMath by comparing it to the math routines from GnuPG\footnote{GnuPG v1.2.3 versus LibTomMath v0.28}. 351 352\newpage\begin{figure}[h] 353\begin{small} 354\begin{center} 355\begin{tabular}{|l|c|c|l|} 356\hline \textbf{Criteria} & \textbf{Pro} & \textbf{Con} & \textbf{Notes} \\ 357\hline Few lines of code per file & X & & GnuPG $ = 300.9$, LibTomMath $ = 71.97$ \\ 358\hline Commented function prototypes & X && GnuPG function names are cryptic. \\ 359\hline Speed && X & LibTomMath is slower. \\ 360\hline Totally free & X & & GPL has unfavourable restrictions.\\ 361\hline Large function base & X & & GnuPG is barebones. \\ 362\hline Five modular reduction algorithms & X & & Faster modular exponentiation for a variety of moduli. \\ 363\hline Portable & X & & GnuPG requires configuration to build. \\ 364\hline 365\end{tabular} 366\end{center} 367\end{small} 368\caption{LibTomMath Valuation} 369\end{figure} 370 371It may seem odd to compare LibTomMath to GnuPG since the math in GnuPG is only a small portion of the entire application. 372However, LibTomMath was written with cryptography in mind. It provides essentially all of the functions a cryptosystem 373would require when working with large integers. 374 375So it may feel tempting to just rip the math code out of GnuPG (or GnuMP where it was taken from originally) in your 376own application but I think there are reasons not to. While LibTomMath is slower than libraries such as GnuMP it is 377not normally significantly slower. On x86 machines the difference is normally a factor of two when performing modular 378exponentiations. It depends largely on the processor, compiler and the moduli being used. 379 380Essentially the only time you wouldn't use LibTomMath is when blazing speed is the primary concern. However, 381on the other side of the coin LibTomMath offers you a totally free (public domain) well structured math library 382that is very flexible, complete and performs well in resource constrained environments. Fast RSA for example can 383be performed with as little as 8KB of ram for data (again depending on build options). 384 385\chapter{Getting Started with LibTomMath} 386\section{Building Programs} 387In order to use LibTomMath you must include ``tommath.h'' and link against the appropriate library file (typically 388libtommath.a). There is no library initialization required and the entire library is thread safe. 389 390\section{Return Codes} 391There are three possible return codes a function may return. 392 393\index{MP\_OKAY}\index{MP\_YES}\index{MP\_NO}\index{MP\_VAL}\index{MP\_MEM} 394\begin{figure}[h!] 395\begin{center} 396\begin{small} 397\begin{tabular}{|l|l|} 398\hline \textbf{Code} & \textbf{Meaning} \\ 399\hline MP\_OKAY & The function succeeded. \\ 400\hline MP\_VAL & The function input was invalid. \\ 401\hline MP\_MEM & Heap memory exhausted. \\ 402\hline &\\ 403\hline MP\_YES & Response is yes. \\ 404\hline MP\_NO & Response is no. \\ 405\hline 406\end{tabular} 407\end{small} 408\end{center} 409\caption{Return Codes} 410\end{figure} 411 412The last two codes listed are not actually ``return'ed'' by a function. They are placed in an integer (the caller must 413provide the address of an integer it can store to) which the caller can access. To convert one of the three return codes 414to a string use the following function. 415 416\index{mp\_error\_to\_string} 417\begin{alltt} 418char *mp_error_to_string(int code); 419\end{alltt} 420 421This will return a pointer to a string which describes the given error code. It will not work for the return codes 422MP\_YES and MP\_NO. 423 424\section{Data Types} 425The basic ``multiple precision integer'' type is known as the ``mp\_int'' within LibTomMath. This data type is used to 426organize all of the data required to manipulate the integer it represents. Within LibTomMath it has been prototyped 427as the following. 428 429\index{mp\_int} 430\begin{alltt} 431typedef struct \{ 432 int used, alloc, sign; 433 mp_digit *dp; 434\} mp_int; 435\end{alltt} 436 437Where ``mp\_digit'' is a data type that represents individual digits of the integer. By default, an mp\_digit is the 438ISO C ``unsigned long'' data type and each digit is $28-$bits long. The mp\_digit type can be configured to suit other 439platforms by defining the appropriate macros. 440 441All LTM functions that use the mp\_int type will expect a pointer to mp\_int structure. You must allocate memory to 442hold the structure itself by yourself (whether off stack or heap it doesn't matter). The very first thing that must be 443done to use an mp\_int is that it must be initialized. 444 445\section{Function Organization} 446 447The arithmetic functions of the library are all organized to have the same style prototype. That is source operands 448are passed on the left and the destination is on the right. For instance, 449 450\begin{alltt} 451mp_add(&a, &b, &c); /* c = a + b */ 452mp_mul(&a, &a, &c); /* c = a * a */ 453mp_div(&a, &b, &c, &d); /* c = [a/b], d = a mod b */ 454\end{alltt} 455 456Another feature of the way the functions have been implemented is that source operands can be destination operands as well. 457For instance, 458 459\begin{alltt} 460mp_add(&a, &b, &b); /* b = a + b */ 461mp_div(&a, &b, &a, &c); /* a = [a/b], c = a mod b */ 462\end{alltt} 463 464This allows operands to be re-used which can make programming simpler. 465 466\section{Initialization} 467\subsection{Single Initialization} 468A single mp\_int can be initialized with the ``mp\_init'' function. 469 470\index{mp\_init} 471\begin{alltt} 472int mp_init (mp_int * a); 473\end{alltt} 474 475This function expects a pointer to an mp\_int structure and will initialize the members of the structure so the mp\_int 476represents the default integer which is zero. If the functions returns MP\_OKAY then the mp\_int is ready to be used 477by the other LibTomMath functions. 478 479\begin{small} \begin{alltt} 480int main(void) 481\{ 482 mp_int number; 483 int result; 484 485 if ((result = mp_init(&number)) != MP_OKAY) \{ 486 printf("Error initializing the number. \%s", 487 mp_error_to_string(result)); 488 return EXIT_FAILURE; 489 \} 490 491 /* use the number */ 492 493 return EXIT_SUCCESS; 494\} 495\end{alltt} \end{small} 496 497\subsection{Single Free} 498When you are finished with an mp\_int it is ideal to return the heap it used back to the system. The following function 499provides this functionality. 500 501\index{mp\_clear} 502\begin{alltt} 503void mp_clear (mp_int * a); 504\end{alltt} 505 506The function expects a pointer to a previously initialized mp\_int structure and frees the heap it uses. It sets the 507pointer\footnote{The ``dp'' member.} within the mp\_int to \textbf{NULL} which is used to prevent double free situations. 508Is is legal to call mp\_clear() twice on the same mp\_int in a row. 509 510\begin{small} \begin{alltt} 511int main(void) 512\{ 513 mp_int number; 514 int result; 515 516 if ((result = mp_init(&number)) != MP_OKAY) \{ 517 printf("Error initializing the number. \%s", 518 mp_error_to_string(result)); 519 return EXIT_FAILURE; 520 \} 521 522 /* use the number */ 523 524 /* We're done with it. */ 525 mp_clear(&number); 526 527 return EXIT_SUCCESS; 528\} 529\end{alltt} \end{small} 530 531\subsection{Multiple Initializations} 532Certain algorithms require more than one large integer. In these instances it is ideal to initialize all of the mp\_int 533variables in an ``all or nothing'' fashion. That is, they are either all initialized successfully or they are all 534not initialized. 535 536The mp\_init\_multi() function provides this functionality. 537 538\index{mp\_init\_multi} \index{mp\_clear\_multi} 539\begin{alltt} 540int mp_init_multi(mp_int *mp, ...); 541\end{alltt} 542 543It accepts a \textbf{NULL} terminated list of pointers to mp\_int structures. It will attempt to initialize them all 544at once. If the function returns MP\_OKAY then all of the mp\_int variables are ready to use, otherwise none of them 545are available for use. A complementary mp\_clear\_multi() function allows multiple mp\_int variables to be free'd 546from the heap at the same time. 547 548\begin{small} \begin{alltt} 549int main(void) 550\{ 551 mp_int num1, num2, num3; 552 int result; 553 554 if ((result = mp_init_multi(&num1, 555 &num2, 556 &num3, NULL)) != MP\_OKAY) \{ 557 printf("Error initializing the numbers. \%s", 558 mp_error_to_string(result)); 559 return EXIT_FAILURE; 560 \} 561 562 /* use the numbers */ 563 564 /* We're done with them. */ 565 mp_clear_multi(&num1, &num2, &num3, NULL); 566 567 return EXIT_SUCCESS; 568\} 569\end{alltt} \end{small} 570 571\subsection{Other Initializers} 572To initialized and make a copy of an mp\_int the mp\_init\_copy() function has been provided. 573 574\index{mp\_init\_copy} 575\begin{alltt} 576int mp_init_copy (mp_int * a, mp_int * b); 577\end{alltt} 578 579This function will initialize $a$ and make it a copy of $b$ if all goes well. 580 581\begin{small} \begin{alltt} 582int main(void) 583\{ 584 mp_int num1, num2; 585 int result; 586 587 /* initialize and do work on num1 ... */ 588 589 /* We want a copy of num1 in num2 now */ 590 if ((result = mp_init_copy(&num2, &num1)) != MP_OKAY) \{ 591 printf("Error initializing the copy. \%s", 592 mp_error_to_string(result)); 593 return EXIT_FAILURE; 594 \} 595 596 /* now num2 is ready and contains a copy of num1 */ 597 598 /* We're done with them. */ 599 mp_clear_multi(&num1, &num2, NULL); 600 601 return EXIT_SUCCESS; 602\} 603\end{alltt} \end{small} 604 605Another less common initializer is mp\_init\_size() which allows the user to initialize an mp\_int with a given 606default number of digits. By default, all initializers allocate \textbf{MP\_PREC} digits. This function lets 607you override this behaviour. 608 609\index{mp\_init\_size} 610\begin{alltt} 611int mp_init_size (mp_int * a, int size); 612\end{alltt} 613 614The $size$ parameter must be greater than zero. If the function succeeds the mp\_int $a$ will be initialized 615to have $size$ digits (which are all initially zero). 616 617\begin{small} \begin{alltt} 618int main(void) 619\{ 620 mp_int number; 621 int result; 622 623 /* we need a 60-digit number */ 624 if ((result = mp_init_size(&number, 60)) != MP_OKAY) \{ 625 printf("Error initializing the number. \%s", 626 mp_error_to_string(result)); 627 return EXIT_FAILURE; 628 \} 629 630 /* use the number */ 631 632 return EXIT_SUCCESS; 633\} 634\end{alltt} \end{small} 635 636\section{Maintenance Functions} 637\subsection{Clear Leading Zeros} 638 639This is used to ensure that leading zero digits are trimed and the leading "used" digit will be non-zero. 640It also fixes the sign if there are no more leading digits. 641 642\index{mp\_clamp} 643\begin{alltt} 644void mp_clamp(mp_int *a); 645\end{alltt} 646 647\subsection{Zero Out} 648 649This function will set the ``bigint'' to zeros without changing the amount of allocated memory. 650 651\index{mp\_zero} 652\begin{alltt} 653void mp_zero(mp_int *a); 654\end{alltt} 655 656 657\subsection{Reducing Memory Usage} 658When an mp\_int is in a state where it won't be changed again\footnote{A Diffie-Hellman modulus for instance.} excess 659digits can be removed to return memory to the heap with the mp\_shrink() function. 660 661\index{mp\_shrink} 662\begin{alltt} 663int mp_shrink (mp_int * a); 664\end{alltt} 665 666This will remove excess digits of the mp\_int $a$. If the operation fails the mp\_int should be intact without the 667excess digits being removed. Note that you can use a shrunk mp\_int in further computations, however, such operations 668will require heap operations which can be slow. It is not ideal to shrink mp\_int variables that you will further 669modify in the system (unless you are seriously low on memory). 670 671\begin{small} \begin{alltt} 672int main(void) 673\{ 674 mp_int number; 675 int result; 676 677 if ((result = mp_init(&number)) != MP_OKAY) \{ 678 printf("Error initializing the number. \%s", 679 mp_error_to_string(result)); 680 return EXIT_FAILURE; 681 \} 682 683 /* use the number [e.g. pre-computation] */ 684 685 /* We're done with it for now. */ 686 if ((result = mp_shrink(&number)) != MP_OKAY) \{ 687 printf("Error shrinking the number. \%s", 688 mp_error_to_string(result)); 689 return EXIT_FAILURE; 690 \} 691 692 /* use it .... */ 693 694 695 /* we're done with it. */ 696 mp_clear(&number); 697 698 return EXIT_SUCCESS; 699\} 700\end{alltt} \end{small} 701 702\subsection{Adding additional digits} 703 704Within the mp\_int structure are two parameters which control the limitations of the array of digits that represent 705the integer the mp\_int is meant to equal. The \textit{used} parameter dictates how many digits are significant, that is, 706contribute to the value of the mp\_int. The \textit{alloc} parameter dictates how many digits are currently available in 707the array. If you need to perform an operation that requires more digits you will have to mp\_grow() the mp\_int to 708your desired size. 709 710\index{mp\_grow} 711\begin{alltt} 712int mp_grow (mp_int * a, int size); 713\end{alltt} 714 715This will grow the array of digits of $a$ to $size$. If the \textit{alloc} parameter is already bigger than 716$size$ the function will not do anything. 717 718\begin{small} \begin{alltt} 719int main(void) 720\{ 721 mp_int number; 722 int result; 723 724 if ((result = mp_init(&number)) != MP_OKAY) \{ 725 printf("Error initializing the number. \%s", 726 mp_error_to_string(result)); 727 return EXIT_FAILURE; 728 \} 729 730 /* use the number */ 731 732 /* We need to add 20 digits to the number */ 733 if ((result = mp_grow(&number, number.alloc + 20)) != MP_OKAY) \{ 734 printf("Error growing the number. \%s", 735 mp_error_to_string(result)); 736 return EXIT_FAILURE; 737 \} 738 739 740 /* use the number */ 741 742 /* we're done with it. */ 743 mp_clear(&number); 744 745 return EXIT_SUCCESS; 746\} 747\end{alltt} \end{small} 748 749\chapter{Basic Operations} 750\section{Copying} 751 752A so called ``deep copy'', where new memory is allocated and all contents of $a$ are copied verbatim into $b$ such that $b = a$ at the end. 753 754\index{mp\_copy} 755\begin{alltt} 756int mp_copy (mp_int * a, mp_int *b); 757\end{alltt} 758 759You can also just swap $a$ and $b$. It does the normal pointer changing with a temporary pointer variable, just that you do not have to. 760 761\index{mp\_exch} 762\begin{alltt} 763void mp_exch (mp_int * a, mp_int *b); 764\end{alltt} 765 766\section{Bit Counting} 767 768To get the position of the lowest bit set (LSB, the Lowest Significant Bit; the number of bits which are zero before the first zero bit ) 769 770\index{mp\_cnt\_lsb} 771\begin{alltt} 772int mp_cnt_lsb(const mp_int *a); 773\end{alltt} 774 775To get the position of the highest bit set (MSB, the Most Significant Bit; the number of bits in teh ``bignum'') 776 777\index{mp\_count\_bits} 778\begin{alltt} 779int mp_count_bits(const mp_int *a); 780\end{alltt} 781 782 783\section{Small Constants} 784Setting mp\_ints to small constants is a relatively common operation. To accommodate these instances there is a 785small constant assignment function. This function is used to set a single digit constant. 786The reason for this function is efficiency. Setting a single digit is quick but the 787domain of a digit can change (it's always at least $0 \ldots 127$). 788 789\subsection{Single Digit} 790 791Setting a single digit can be accomplished with the following function. 792 793\index{mp\_set} 794\begin{alltt} 795void mp_set (mp_int * a, mp_digit b); 796\end{alltt} 797 798This will zero the contents of $a$ and make it represent an integer equal to the value of $b$. Note that this 799function has a return type of \textbf{void}. It cannot cause an error so it is safe to assume the function 800succeeded. 801 802\begin{small} \begin{alltt} 803int main(void) 804\{ 805 mp_int number; 806 int result; 807 808 if ((result = mp_init(&number)) != MP_OKAY) \{ 809 printf("Error initializing the number. \%s", 810 mp_error_to_string(result)); 811 return EXIT_FAILURE; 812 \} 813 814 /* set the number to 5 */ 815 mp_set(&number, 5); 816 817 /* we're done with it. */ 818 mp_clear(&number); 819 820 return EXIT_SUCCESS; 821\} 822\end{alltt} \end{small} 823 824\subsection{Int32 and Int64 Constants} 825 826These functions can be used to set a constant with 32 or 64 bits. 827 828\index{mp\_set\_i32} \index{mp\_set\_u32} 829\index{mp\_set\_i64} \index{mp\_set\_u64} 830\begin{alltt} 831void mp_set_i32 (mp_int * a, int32_t b); 832void mp_set_u32 (mp_int * a, uint32_t b); 833void mp_set_i64 (mp_int * a, int64_t b); 834void mp_set_u64 (mp_int * a, uint64_t b); 835\end{alltt} 836 837These functions assign the sign and value of the input \texttt{b} to \texttt{mp\_int a}. 838The value can be obtained again by calling the following functions. 839 840\index{mp\_get\_i32} \index{mp\_get\_u32} \index{mp\_get\_mag\_u32} 841\index{mp\_get\_i64} \index{mp\_get\_u64} \index{mp\_get\_mag\_u64} 842\begin{alltt} 843int32_t mp_get_i32 (mp_int * a); 844uint32_t mp_get_u32 (mp_int * a); 845uint32_t mp_get_mag_u32 (mp_int * a); 846int64_t mp_get_i64 (mp_int * a); 847uint64_t mp_get_u64 (mp_int * a); 848uint64_t mp_get_mag_u64 (mp_int * a); 849\end{alltt} 850 851These functions return the 32 or 64 least significant bits of $a$ respectively. The unsigned functions 852return negative values in a twos complement representation. The absolute value or magnitude can be obtained using the mp\_get\_mag functions. 853 854\begin{small} \begin{alltt} 855int main(void) 856\{ 857 mp_int number; 858 int result; 859 860 if ((result = mp_init(&number)) != MP_OKAY) \{ 861 printf("Error initializing the number. \%s", 862 mp_error_to_string(result)); 863 return EXIT_FAILURE; 864 \} 865 866 /* set the number to 654321 (note this is bigger than 127) */ 867 mp_set_u32(&number, 654321); 868 869 printf("number == \%" PRIi32, mp_get_i32(&number)); 870 871 /* we're done with it. */ 872 mp_clear(&number); 873 874 return EXIT_SUCCESS; 875\} 876\end{alltt} \end{small} 877 878This should output the following if the program succeeds. 879 880\begin{alltt} 881number == 654321 882\end{alltt} 883 884\subsection{Long Constants - platform dependant} 885 886\index{mp\_set\_l} \index{mp\_set\_ul} 887\begin{alltt} 888void mp_set_l (mp_int * a, long b); 889void mp_set_ul (mp_int * a, unsigned long b); 890\end{alltt} 891 892This will assign the value of the platform-dependent sized variable $b$ to the mp\_int $a$. 893 894To retrieve the value, the following functions can be used. 895 896\index{mp\_get\_l} \index{mp\_get\_ul} \index{mp\_get\_mag\_ul} 897\begin{alltt} 898long mp_get_l (mp_int * a); 899unsigned long mp_get_ul (mp_int * a); 900unsigned long mp_get_mag_ul (mp_int * a); 901\end{alltt} 902 903This will return the least significant bits of the mp\_int $a$ that fit into a ``long''. 904 905\subsection{Long Long Constants - platform dependant} 906 907\index{mp\_set\_ll} \index{mp\_set\_ull} 908\begin{alltt} 909void mp_set_ll (mp_int * a, long long b); 910void mp_set_ull (mp_int * a, unsigned long long b); 911\end{alltt} 912 913This will assign the value of the platform-dependent sized variable $b$ to the mp\_int $a$. 914 915To retrieve the value, the following functions can be used. 916 917\index{mp\_get\_ll} 918\index{mp\_get\_ull} 919\index{mp\_get\_mag\_ull} 920\begin{alltt} 921long long mp_get_ll (mp_int * a); 922unsigned long long mp_get_ull (mp_int * a); 923unsigned long long mp_get_mag_ull (mp_int * a); 924\end{alltt} 925 926This will return the least significant bits of the mp\_int $a$ that fit into a ``long long''. 927 928\subsection{Initialize and Setting Constants} 929To both initialize and set small constants the following two functions are available. 930\index{mp\_init\_set} \index{mp\_init\_set\_int} 931\begin{alltt} 932int mp_init_set (mp_int * a, mp_digit b); 933int mp_init_i32 (mp_int * a, int32_t b); 934int mp_init_u32 (mp_int * a, uint32_t b); 935int mp_init_i64 (mp_int * a, int64_t b); 936int mp_init_u64 (mp_int * a, uint64_t b); 937int mp_init_l (mp_int * a, long b); 938int mp_init_ul (mp_int * a, unsigned long b); 939int mp_init_ll (mp_int * a, long long b); 940int mp_init_ull (mp_int * a, unsigned long long b); 941\end{alltt} 942 943Both functions work like the previous counterparts except they first mp\_init $a$ before setting the values. 944 945\begin{alltt} 946int main(void) 947\{ 948 mp_int number1, number2; 949 int result; 950 951 /* initialize and set a single digit */ 952 if ((result = mp_init_set(&number1, 100)) != MP_OKAY) \{ 953 printf("Error setting number1: \%s", 954 mp_error_to_string(result)); 955 return EXIT_FAILURE; 956 \} 957 958 /* initialize and set a long */ 959 if ((result = mp_init_l(&number2, 1023)) != MP_OKAY) \{ 960 printf("Error setting number2: \%s", 961 mp_error_to_string(result)); 962 return EXIT_FAILURE; 963 \} 964 965 /* display */ 966 printf("Number1, Number2 == \%" PRIi32 ", \%" PRIi32, 967 mp_get_i32(&number1), mp_get_i32(&number2)); 968 969 /* clear */ 970 mp_clear_multi(&number1, &number2, NULL); 971 972 return EXIT_SUCCESS; 973\} 974\end{alltt} 975 976If this program succeeds it shall output. 977\begin{alltt} 978Number1, Number2 == 100, 1023 979\end{alltt} 980 981\section{Comparisons} 982 983Comparisons in LibTomMath are always performed in a ``left to right'' fashion. There are three possible return codes 984for any comparison. 985 986\index{MP\_GT} \index{MP\_EQ} \index{MP\_LT} 987\begin{figure}[h] 988\begin{center} 989\begin{tabular}{|c|c|} 990\hline \textbf{Result Code} & \textbf{Meaning} \\ 991\hline MP\_GT & $a > b$ \\ 992\hline MP\_EQ & $a = b$ \\ 993\hline MP\_LT & $a < b$ \\ 994\hline 995\end{tabular} 996\end{center} 997\caption{Comparison Codes for $a, b$} 998\label{fig:CMP} 999\end{figure} 1000 1001In figure \ref{fig:CMP} two integers $a$ and $b$ are being compared. In this case $a$ is said to be ``to the left'' of 1002$b$. 1003 1004\subsection{Unsigned comparison} 1005 1006An unsigned comparison considers only the digits themselves and not the associated \textit{sign} flag of the 1007mp\_int structures. This is analogous to an absolute comparison. The function mp\_cmp\_mag() will compare two 1008mp\_int variables based on their digits only. 1009 1010\index{mp\_cmp\_mag} 1011\begin{alltt} 1012int mp_cmp_mag(mp_int * a, mp_int * b); 1013\end{alltt} 1014This will compare $a$ to $b$ placing $a$ to the left of $b$. This function cannot fail and will return one of the 1015three compare codes listed in figure \ref{fig:CMP}. 1016 1017\begin{small} \begin{alltt} 1018int main(void) 1019\{ 1020 mp_int number1, number2; 1021 int result; 1022 1023 if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{ 1024 printf("Error initializing the numbers. \%s", 1025 mp_error_to_string(result)); 1026 return EXIT_FAILURE; 1027 \} 1028 1029 /* set the number1 to 5 */ 1030 mp_set(&number1, 5); 1031 1032 /* set the number2 to -6 */ 1033 mp_set(&number2, 6); 1034 if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{ 1035 printf("Error negating number2. \%s", 1036 mp_error_to_string(result)); 1037 return EXIT_FAILURE; 1038 \} 1039 1040 switch(mp_cmp_mag(&number1, &number2)) \{ 1041 case MP_GT: printf("|number1| > |number2|"); break; 1042 case MP_EQ: printf("|number1| = |number2|"); break; 1043 case MP_LT: printf("|number1| < |number2|"); break; 1044 \} 1045 1046 /* we're done with it. */ 1047 mp_clear_multi(&number1, &number2, NULL); 1048 1049 return EXIT_SUCCESS; 1050\} 1051\end{alltt} \end{small} 1052 1053If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes 1054successfully it should print the following. 1055 1056\begin{alltt} 1057|number1| < |number2| 1058\end{alltt} 1059 1060This is because $\vert -6 \vert = 6$ and obviously $5 < 6$. 1061 1062\subsection{Signed comparison} 1063 1064To compare two mp\_int variables based on their signed value the mp\_cmp() function is provided. 1065 1066\index{mp\_cmp} 1067\begin{alltt} 1068int mp_cmp(mp_int * a, mp_int * b); 1069\end{alltt} 1070 1071This will compare $a$ to the left of $b$. It will first compare the signs of the two mp\_int variables. If they 1072differ it will return immediately based on their signs. If the signs are equal then it will compare the digits 1073individually. This function will return one of the compare conditions codes listed in figure \ref{fig:CMP}. 1074 1075\begin{small} \begin{alltt} 1076int main(void) 1077\{ 1078 mp_int number1, number2; 1079 int result; 1080 1081 if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{ 1082 printf("Error initializing the numbers. \%s", 1083 mp_error_to_string(result)); 1084 return EXIT_FAILURE; 1085 \} 1086 1087 /* set the number1 to 5 */ 1088 mp_set(&number1, 5); 1089 1090 /* set the number2 to -6 */ 1091 mp_set(&number2, 6); 1092 if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{ 1093 printf("Error negating number2. \%s", 1094 mp_error_to_string(result)); 1095 return EXIT_FAILURE; 1096 \} 1097 1098 switch(mp_cmp(&number1, &number2)) \{ 1099 case MP_GT: printf("number1 > number2"); break; 1100 case MP_EQ: printf("number1 = number2"); break; 1101 case MP_LT: printf("number1 < number2"); break; 1102 \} 1103 1104 /* we're done with it. */ 1105 mp_clear_multi(&number1, &number2, NULL); 1106 1107 return EXIT_SUCCESS; 1108\} 1109\end{alltt} \end{small} 1110 1111If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes 1112successfully it should print the following. 1113 1114\begin{alltt} 1115number1 > number2 1116\end{alltt} 1117 1118\subsection{Single Digit} 1119 1120To compare a single digit against an mp\_int the following function has been provided. 1121 1122\index{mp\_cmp\_d} 1123\begin{alltt} 1124int mp_cmp_d(mp_int * a, mp_digit b); 1125\end{alltt} 1126 1127This will compare $a$ to the left of $b$ using a signed comparison. Note that it will always treat $b$ as 1128positive. This function is rather handy when you have to compare against small values such as $1$ (which often 1129comes up in cryptography). The function cannot fail and will return one of the tree compare condition codes 1130listed in figure \ref{fig:CMP}. 1131 1132 1133\begin{small} \begin{alltt} 1134int main(void) 1135\{ 1136 mp_int number; 1137 int result; 1138 1139 if ((result = mp_init(&number)) != MP_OKAY) \{ 1140 printf("Error initializing the number. \%s", 1141 mp_error_to_string(result)); 1142 return EXIT_FAILURE; 1143 \} 1144 1145 /* set the number to 5 */ 1146 mp_set(&number, 5); 1147 1148 switch(mp_cmp_d(&number, 7)) \{ 1149 case MP_GT: printf("number > 7"); break; 1150 case MP_EQ: printf("number = 7"); break; 1151 case MP_LT: printf("number < 7"); break; 1152 \} 1153 1154 /* we're done with it. */ 1155 mp_clear(&number); 1156 1157 return EXIT_SUCCESS; 1158\} 1159\end{alltt} \end{small} 1160 1161If this program functions properly it will print out the following. 1162 1163\begin{alltt} 1164number < 7 1165\end{alltt} 1166 1167\section{Logical Operations} 1168 1169Logical operations are operations that can be performed either with simple shifts or boolean operators such as 1170AND, XOR and OR directly. These operations are very quick. 1171 1172\subsection{Multiplication by two} 1173 1174Multiplications and divisions by any power of two can be performed with quick logical shifts either left or 1175right depending on the operation. 1176 1177When multiplying or dividing by two a special case routine can be used which are as follows. 1178\index{mp\_mul\_2} \index{mp\_div\_2} 1179\begin{alltt} 1180int mp_mul_2(mp_int * a, mp_int * b); 1181int mp_div_2(mp_int * a, mp_int * b); 1182\end{alltt} 1183 1184The former will assign twice $a$ to $b$ while the latter will assign half $a$ to $b$. These functions are fast 1185since the shift counts and maskes are hardcoded into the routines. 1186 1187\begin{small} \begin{alltt} 1188int main(void) 1189\{ 1190 mp_int number; 1191 int result; 1192 1193 if ((result = mp_init(&number)) != MP_OKAY) \{ 1194 printf("Error initializing the number. \%s", 1195 mp_error_to_string(result)); 1196 return EXIT_FAILURE; 1197 \} 1198 1199 /* set the number to 5 */ 1200 mp_set(&number, 5); 1201 1202 /* multiply by two */ 1203 if ((result = mp\_mul\_2(&number, &number)) != MP_OKAY) \{ 1204 printf("Error multiplying the number. \%s", 1205 mp_error_to_string(result)); 1206 return EXIT_FAILURE; 1207 \} 1208 switch(mp_cmp_d(&number, 7)) \{ 1209 case MP_GT: printf("2*number > 7"); break; 1210 case MP_EQ: printf("2*number = 7"); break; 1211 case MP_LT: printf("2*number < 7"); break; 1212 \} 1213 1214 /* now divide by two */ 1215 if ((result = mp\_div\_2(&number, &number)) != MP_OKAY) \{ 1216 printf("Error dividing the number. \%s", 1217 mp_error_to_string(result)); 1218 return EXIT_FAILURE; 1219 \} 1220 switch(mp_cmp_d(&number, 7)) \{ 1221 case MP_GT: printf("2*number/2 > 7"); break; 1222 case MP_EQ: printf("2*number/2 = 7"); break; 1223 case MP_LT: printf("2*number/2 < 7"); break; 1224 \} 1225 1226 /* we're done with it. */ 1227 mp_clear(&number); 1228 1229 return EXIT_SUCCESS; 1230\} 1231\end{alltt} \end{small} 1232 1233If this program is successful it will print out the following text. 1234 1235\begin{alltt} 12362*number > 7 12372*number/2 < 7 1238\end{alltt} 1239 1240Since $10 > 7$ and $5 < 7$. 1241 1242To multiply by a power of two the following function can be used. 1243 1244\index{mp\_mul\_2d} 1245\begin{alltt} 1246int mp_mul_2d(mp_int * a, int b, mp_int * c); 1247\end{alltt} 1248 1249This will multiply $a$ by $2^b$ and store the result in ``c''. If the value of $b$ is less than or equal to 1250zero the function will copy $a$ to ``c'' without performing any further actions. The multiplication itself 1251is implemented as a right-shift operation of $a$ by $b$ bits. 1252 1253To divide by a power of two use the following. 1254 1255\index{mp\_div\_2d} 1256\begin{alltt} 1257int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d); 1258\end{alltt} 1259Which will divide $a$ by $2^b$, store the quotient in ``c'' and the remainder in ``d'. If $b \le 0$ then the 1260function simply copies $a$ over to ``c'' and zeroes $d$. The variable $d$ may be passed as a \textbf{NULL} 1261value to signal that the remainder is not desired. The division itself is implemented as a left-shift 1262operation of $a$ by $b$ bits. 1263 1264It is also not very uncommon to need just the power of two $2^b$; for example the startvalue for the Newton method. 1265 1266\index{mp\_2expt} 1267\begin{alltt} 1268int mp_2expt(mp_int *a, int b); 1269\end{alltt} 1270It is faster than doing it by shifting $1$ with \texttt{mp\_mul\_2d}. 1271 1272\subsection{Polynomial Basis Operations} 1273 1274Strictly speaking the organization of the integers within the mp\_int structures is what is known as a 1275``polynomial basis''. This simply means a field element is stored by divisions of a radix. For example, if 1276$f(x) = \sum_{i=0}^{k} y_ix^k$ for any vector $\vec y$ then the array of digits in $\vec y$ are said to be 1277the polynomial basis representation of $z$ if $f(\beta) = z$ for a given radix $\beta$. 1278 1279To multiply by the polynomial $g(x) = x$ all you have todo is shift the digits of the basis left one place. The 1280following function provides this operation. 1281 1282\index{mp\_lshd} 1283\begin{alltt} 1284int mp_lshd (mp_int * a, int b); 1285\end{alltt} 1286 1287This will multiply $a$ in place by $x^b$ which is equivalent to shifting the digits left $b$ places and inserting zeroes 1288in the least significant digits. Similarly to divide by a power of $x$ the following function is provided. 1289 1290\index{mp\_rshd} 1291\begin{alltt} 1292void mp_rshd (mp_int * a, int b) 1293\end{alltt} 1294This will divide $a$ in place by $x^b$ and discard the remainder. This function cannot fail as it performs the operations 1295in place and no new digits are required to complete it. 1296 1297\subsection{AND, OR, XOR and COMPLEMENT Operations} 1298 1299While AND, OR and XOR operations compute arbitrary-precision bitwise operations. Negative numbers 1300are treated as if they are in two-complement representation, while internally they are sign-magnitude however. 1301 1302\index{mp\_or} \index{mp\_and} \index{mp\_xor} \index{mp\_complement} 1303\begin{alltt} 1304int mp_or (mp_int * a, mp_int * b, mp_int * c); 1305int mp_and (mp_int * a, mp_int * b, mp_int * c); 1306int mp_xor (mp_int * a, mp_int * b, mp_int * c); 1307int mp_complement(const mp_int *a, mp_int *b); 1308int mp_signed_rsh(mp_int * a, int b, mp_int * c, mp_int * d); 1309\end{alltt} 1310 1311The function \texttt{mp\_complement} computes a two-complement $b = \sim a$. The function \texttt{mp\_signed\_rsh} performs 1312sign extending right shift. For positive numbers it is equivalent to \texttt{mp\_div\_2d}. 1313 1314\subsection{Bit Picking} 1315\index{mp\_get\_bit} 1316\begin{alltt} 1317int mp_get_bit(mp_int *a, int b) 1318\end{alltt} 1319 1320Pick a bit: returns \texttt{MP\_YES} if the bit at position $b$ (0-index) is set, that is if it is 1 (one), \texttt{MP\_NO} 1321if the bit is 0 (zero) and \texttt{MP\_VAL} if $b < 0$. 1322 1323\section{Addition and Subtraction} 1324 1325To compute an addition or subtraction the following two functions can be used. 1326 1327\index{mp\_add} \index{mp\_sub} 1328\begin{alltt} 1329int mp_add (mp_int * a, mp_int * b, mp_int * c); 1330int mp_sub (mp_int * a, mp_int * b, mp_int * c) 1331\end{alltt} 1332 1333Which perform $c = a \odot b$ where $\odot$ is one of signed addition or subtraction. The operations are fully sign 1334aware. 1335 1336\section{Sign Manipulation} 1337\subsection{Negation} 1338\label{sec:NEG} 1339Simple integer negation can be performed with the following. 1340 1341\index{mp\_neg} 1342\begin{alltt} 1343int mp_neg (mp_int * a, mp_int * b); 1344\end{alltt} 1345 1346Which assigns $-a$ to $b$. 1347 1348\subsection{Absolute} 1349Simple integer absolutes can be performed with the following. 1350 1351\index{mp\_abs} 1352\begin{alltt} 1353int mp_abs (mp_int * a, mp_int * b); 1354\end{alltt} 1355 1356Which assigns $\vert a \vert$ to $b$. 1357 1358\section{Integer Division and Remainder} 1359To perform a complete and general integer division with remainder use the following function. 1360 1361\index{mp\_div} 1362\begin{alltt} 1363int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d); 1364\end{alltt} 1365 1366This divides $a$ by $b$ and stores the quotient in $c$ and $d$. The signed quotient is computed such that 1367$bc + d = a$. Note that either of $c$ or $d$ can be set to \textbf{NULL} if their value is not required. If 1368$b$ is zero the function returns \textbf{MP\_VAL}. 1369 1370 1371\chapter{Multiplication and Squaring} 1372\section{Multiplication} 1373A full signed integer multiplication can be performed with the following. 1374\index{mp\_mul} 1375\begin{alltt} 1376int mp_mul (mp_int * a, mp_int * b, mp_int * c); 1377\end{alltt} 1378Which assigns the full signed product $ab$ to $c$. This function actually breaks into one of four cases which are 1379specific multiplication routines optimized for given parameters. First there are the Toom-Cook multiplications which 1380should only be used with very large inputs. This is followed by the Karatsuba multiplications which are for moderate 1381sized inputs. Then followed by the Comba and baseline multipliers. 1382 1383Fortunately for the developer you don't really need to know this unless you really want to fine tune the system. mp\_mul() 1384will determine on its own\footnote{Some tweaking may be required but \texttt{make tune} will put some reasonable values in \texttt{bncore.c}} what routine to use automatically when it is called. 1385 1386\begin{alltt} 1387int main(void) 1388\{ 1389 mp_int number1, number2; 1390 int result; 1391 1392 /* Initialize the numbers */ 1393 if ((result = mp_init_multi(&number1, 1394 &number2, NULL)) != MP_OKAY) \{ 1395 printf("Error initializing the numbers. \%s", 1396 mp_error_to_string(result)); 1397 return EXIT_FAILURE; 1398 \} 1399 1400 /* set the terms */ 1401 mp_set_i32(&number, 257); 1402 mp_set_i32(&number2, 1023); 1403 1404 /* multiply them */ 1405 if ((result = mp_mul(&number1, &number2, 1406 &number1)) != MP_OKAY) \{ 1407 printf("Error multiplying terms. \%s", 1408 mp_error_to_string(result)); 1409 return EXIT_FAILURE; 1410 \} 1411 1412 /* display */ 1413 printf("number1 * number2 == \%" PRIi32, mp_get_i32(&number1)); 1414 1415 /* free terms and return */ 1416 mp_clear_multi(&number1, &number2, NULL); 1417 1418 return EXIT_SUCCESS; 1419\} 1420\end{alltt} 1421 1422If this program succeeds it shall output the following. 1423 1424\begin{alltt} 1425number1 * number2 == 262911 1426\end{alltt} 1427 1428\section{Squaring} 1429Since squaring can be performed faster than multiplication it is performed it's own function instead of just using 1430mp\_mul(). 1431 1432\index{mp\_sqr} 1433\begin{alltt} 1434int mp_sqr (mp_int * a, mp_int * b); 1435\end{alltt} 1436 1437Will square $a$ and store it in $b$. Like the case of multiplication there are four different squaring 1438algorithms all which can be called from mp\_sqr(). It is ideal to use mp\_sqr over mp\_mul when squaring terms because 1439of the speed difference. 1440 1441\section{Tuning Polynomial Basis Routines} 1442 1443Both of the Toom-Cook and Karatsuba multiplication algorithms are faster than the traditional $O(n^2)$ approach that 1444the Comba and baseline algorithms use. At $O(n^{1.464973})$ and $O(n^{1.584962})$ running times respectively they require 1445considerably less work. For example, a 10000-digit multiplication would take roughly 724,000 single precision 1446multiplications with Toom-Cook or 100,000,000 single precision multiplications with the standard Comba (a factor 1447of 138). 1448 1449So why not always use Karatsuba or Toom-Cook? The simple answer is that they have so much overhead that they're not 1450actually faster than Comba until you hit distinct ``cutoff'' points. For Karatsuba with the default configuration, 1451GCC 3.3.1 and an Athlon XP processor the cutoff point is roughly 110 digits (about 70 for the Intel P4). That is, at 1452110 digits Karatsuba and Comba multiplications just about break even and for 110+ digits Karatsuba is faster. 1453 1454Toom-Cook has incredible overhead and is probably only useful for very large inputs. So far no known cutoff points 1455exist and for the most part I just set the cutoff points very high to make sure they're not called. 1456 1457To get reasonable values for the cut-off points for your architecture, type 1458 1459\begin{alltt} 1460make tune 1461\end{alltt} 1462 1463This will run a benchmark, computes the medians, rewrites \texttt{bncore.c}, and recompiles \texttt{bncore.c} and relinks the library. 1464 1465The benchmark itself can be fine-tuned in the file \texttt{etc/tune\_it.sh}. 1466 1467The program \texttt{etc/tune} is also able to print a list of values for printing curves with e.g.: \texttt{gnuplot}. type \texttt{./etc/tune -h} to get a list of all available options. 1468 1469\chapter{Modular Reduction} 1470 1471Modular reduction is process of taking the remainder of one quantity divided by another. Expressed 1472as (\ref{eqn:mod}) the modular reduction is equivalent to the remainder of $b$ divided by $c$. 1473 1474\begin{equation} 1475a \equiv b \mbox{ (mod }c\mbox{)} 1476\label{eqn:mod} 1477\end{equation} 1478 1479Of particular interest to cryptography are reductions where $b$ is limited to the range $0 \le b < c^2$ since particularly 1480fast reduction algorithms can be written for the limited range. 1481 1482Note that one of the four optimized reduction algorithms are automatically chosen in the modular exponentiation 1483algorithm mp\_exptmod when an appropriate modulus is detected. 1484 1485\section{Straight Division} 1486In order to effect an arbitrary modular reduction the following algorithm is provided. 1487 1488\index{mp\_mod} 1489\begin{alltt} 1490int mp_mod(mp_int *a, mp_int *b, mp_int *c); 1491\end{alltt} 1492 1493This reduces $a$ modulo $b$ and stores the result in $c$. The sign of $c$ shall agree with the sign 1494of $b$. This algorithm accepts an input $a$ of any range and is not limited by $0 \le a < b^2$. 1495 1496\section{Barrett Reduction} 1497 1498Barrett reduction is a generic optimized reduction algorithm that requires pre--computation to achieve 1499a decent speedup over straight division. First a $\mu$ value must be precomputed with the following function. 1500 1501\index{mp\_reduce\_setup} 1502\begin{alltt} 1503int mp_reduce_setup(mp_int *a, mp_int *b); 1504\end{alltt} 1505 1506Given a modulus in $b$ this produces the required $\mu$ value in $a$. For any given modulus this only has to 1507be computed once. Modular reduction can now be performed with the following. 1508 1509\index{mp\_reduce} 1510\begin{alltt} 1511int mp_reduce(mp_int *a, mp_int *b, mp_int *c); 1512\end{alltt} 1513 1514This will reduce $a$ in place modulo $b$ with the precomputed $\mu$ value in $c$. $a$ must be in the range 1515$0 \le a < b^2$. 1516 1517\begin{alltt} 1518int main(void) 1519\{ 1520 mp_int a, b, c, mu; 1521 int result; 1522 1523 /* initialize a,b to desired values, mp_init mu, 1524 * c and set c to 1...we want to compute a^3 mod b 1525 */ 1526 1527 /* get mu value */ 1528 if ((result = mp_reduce_setup(&mu, b)) != MP_OKAY) \{ 1529 printf("Error getting mu. \%s", 1530 mp_error_to_string(result)); 1531 return EXIT_FAILURE; 1532 \} 1533 1534 /* square a to get c = a^2 */ 1535 if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{ 1536 printf("Error squaring. \%s", 1537 mp_error_to_string(result)); 1538 return EXIT_FAILURE; 1539 \} 1540 1541 /* now reduce `c' modulo b */ 1542 if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{ 1543 printf("Error reducing. \%s", 1544 mp_error_to_string(result)); 1545 return EXIT_FAILURE; 1546 \} 1547 1548 /* multiply a to get c = a^3 */ 1549 if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{ 1550 printf("Error reducing. \%s", 1551 mp_error_to_string(result)); 1552 return EXIT_FAILURE; 1553 \} 1554 1555 /* now reduce `c' modulo b */ 1556 if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{ 1557 printf("Error reducing. \%s", 1558 mp_error_to_string(result)); 1559 return EXIT_FAILURE; 1560 \} 1561 1562 /* c now equals a^3 mod b */ 1563 1564 return EXIT_SUCCESS; 1565\} 1566\end{alltt} 1567 1568This program will calculate $a^3 \mbox{ mod }b$ if all the functions succeed. 1569 1570\section{Montgomery Reduction} 1571 1572Montgomery is a specialized reduction algorithm for any odd moduli. Like Barrett reduction a pre--computation 1573step is required. This is accomplished with the following. 1574 1575\index{mp\_montgomery\_setup} 1576\begin{alltt} 1577int mp_montgomery_setup(mp_int *a, mp_digit *mp); 1578\end{alltt} 1579 1580For the given odd moduli $a$ the precomputation value is placed in $mp$. The reduction is computed with the 1581following. 1582 1583\index{mp\_montgomery\_reduce} 1584\begin{alltt} 1585int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp); 1586\end{alltt} 1587This reduces $a$ in place modulo $m$ with the pre--computed value $mp$. $a$ must be in the range 1588$0 \le a < b^2$. 1589 1590Montgomery reduction is faster than Barrett reduction for moduli smaller than the ``comba'' limit. With the default 1591setup for instance, the limit is $127$ digits ($3556$--bits). Note that this function is not limited to 1592$127$ digits just that it falls back to a baseline algorithm after that point. 1593 1594An important observation is that this reduction does not return $a \mbox{ mod }m$ but $aR^{-1} \mbox{ mod }m$ 1595where $R = \beta^n$, $n$ is the n number of digits in $m$ and $\beta$ is radix used (default is $2^{28}$). 1596 1597To quickly calculate $R$ the following function was provided. 1598 1599\index{mp\_montgomery\_calc\_normalization} 1600\begin{alltt} 1601int mp_montgomery_calc_normalization(mp_int *a, mp_int *b); 1602\end{alltt} 1603Which calculates $a = R$ for the odd moduli $b$ without using multiplication or division. 1604 1605The normal modus operandi for Montgomery reductions is to normalize the integers before entering the system. For 1606example, to calculate $a^3 \mbox { mod }b$ using Montgomery reduction the value of $a$ can be normalized by 1607multiplying it by $R$. Consider the following code snippet. 1608 1609\begin{alltt} 1610int main(void) 1611\{ 1612 mp_int a, b, c, R; 1613 mp_digit mp; 1614 int result; 1615 1616 /* initialize a,b to desired values, 1617 * mp_init R, c and set c to 1.... 1618 */ 1619 1620 /* get normalization */ 1621 if ((result = mp_montgomery_calc_normalization(&R, b)) != MP_OKAY) \{ 1622 printf("Error getting norm. \%s", 1623 mp_error_to_string(result)); 1624 return EXIT_FAILURE; 1625 \} 1626 1627 /* get mp value */ 1628 if ((result = mp_montgomery_setup(&c, &mp)) != MP_OKAY) \{ 1629 printf("Error setting up montgomery. \%s", 1630 mp_error_to_string(result)); 1631 return EXIT_FAILURE; 1632 \} 1633 1634 /* normalize `a' so now a is equal to aR */ 1635 if ((result = mp_mulmod(&a, &R, &b, &a)) != MP_OKAY) \{ 1636 printf("Error computing aR. \%s", 1637 mp_error_to_string(result)); 1638 return EXIT_FAILURE; 1639 \} 1640 1641 /* square a to get c = a^2R^2 */ 1642 if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{ 1643 printf("Error squaring. \%s", 1644 mp_error_to_string(result)); 1645 return EXIT_FAILURE; 1646 \} 1647 1648 /* now reduce `c' back down to c = a^2R^2 * R^-1 == a^2R */ 1649 if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{ 1650 printf("Error reducing. \%s", 1651 mp_error_to_string(result)); 1652 return EXIT_FAILURE; 1653 \} 1654 1655 /* multiply a to get c = a^3R^2 */ 1656 if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{ 1657 printf("Error reducing. \%s", 1658 mp_error_to_string(result)); 1659 return EXIT_FAILURE; 1660 \} 1661 1662 /* now reduce `c' back down to c = a^3R^2 * R^-1 == a^3R */ 1663 if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{ 1664 printf("Error reducing. \%s", 1665 mp_error_to_string(result)); 1666 return EXIT_FAILURE; 1667 \} 1668 1669 /* now reduce (again) `c' back down to c = a^3R * R^-1 == a^3 */ 1670 if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{ 1671 printf("Error reducing. \%s", 1672 mp_error_to_string(result)); 1673 return EXIT_FAILURE; 1674 \} 1675 1676 /* c now equals a^3 mod b */ 1677 1678 return EXIT_SUCCESS; 1679\} 1680\end{alltt} 1681 1682This particular example does not look too efficient but it demonstrates the point of the algorithm. By 1683normalizing the inputs the reduced results are always of the form $aR$ for some variable $a$. This allows 1684a single final reduction to correct for the normalization and the fast reduction used within the algorithm. 1685 1686For more details consider examining the file \textit{bn\_mp\_exptmod\_fast.c}. 1687 1688\section{Restricted Diminished Radix} 1689 1690``Diminished Radix'' reduction refers to reduction with respect to moduli that are amenable to simple 1691digit shifting and small multiplications. In this case the ``restricted'' variant refers to moduli of the 1692form $\beta^k - p$ for some $k \ge 0$ and $0 < p < \beta$ where $\beta$ is the radix (default to $2^{28}$). 1693 1694As in the case of Montgomery reduction there is a pre--computation phase required for a given modulus. 1695 1696\index{mp\_dr\_setup} 1697\begin{alltt} 1698void mp_dr_setup(mp_int *a, mp_digit *d); 1699\end{alltt} 1700 1701This computes the value required for the modulus $a$ and stores it in $d$. This function cannot fail 1702and does not return any error codes. After the pre--computation a reduction can be performed with the 1703following. 1704 1705\index{mp\_dr\_reduce} 1706\begin{alltt} 1707int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp); 1708\end{alltt} 1709 1710This reduces $a$ in place modulo $b$ with the pre--computed value $mp$. $b$ must be of a restricted 1711diminished radix form and $a$ must be in the range $0 \le a < b^2$. Diminished radix reductions are 1712much faster than both Barrett and Montgomery reductions as they have a much lower asymptotic running time. 1713 1714Since the moduli are restricted this algorithm is not particularly useful for something like Rabin, RSA or 1715BBS cryptographic purposes. This reduction algorithm is useful for Diffie-Hellman and ECC where fixed 1716primes are acceptable. 1717 1718Note that unlike Montgomery reduction there is no normalization process. The result of this function is 1719equal to the correct residue. 1720 1721\section{Unrestricted Diminished Radix} 1722 1723Unrestricted reductions work much like the restricted counterparts except in this case the moduli is of the 1724form $2^k - p$ for $0 < p < \beta$. In this sense the unrestricted reductions are more flexible as they 1725can be applied to a wider range of numbers. 1726 1727\index{mp\_reduce\_2k\_setup} 1728\begin{alltt} 1729int mp_reduce_2k_setup(mp_int *a, mp_digit *d); 1730\end{alltt} 1731 1732This will compute the required $d$ value for the given moduli $a$. 1733 1734\index{mp\_reduce\_2k} 1735\begin{alltt} 1736int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d); 1737\end{alltt} 1738 1739This will reduce $a$ in place modulo $n$ with the pre--computed value $d$. From my experience this routine is 1740slower than mp\_dr\_reduce but faster for most moduli sizes than the Montgomery reduction. 1741 1742\section{Combined Modular Reduction} 1743 1744Some of the combinations of an arithmetic operations followed by a modular reduction can be done in a faster way. The ones implemented are: 1745 1746Addition $d = (a + b) \mod c$ 1747\index{mp\_addmod} 1748\begin{alltt} 1749int mp_addmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d); 1750\end{alltt} 1751 1752Subtraction $d = (a - b) \mod c$ 1753\begin{alltt} 1754int mp_submod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d); 1755\end{alltt} 1756 1757Multiplication $d = (ab) \mod c$ 1758\begin{alltt} 1759int mp_mulmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d); 1760\end{alltt} 1761 1762Squaring $d = (a^2) \mod c$ 1763\begin{alltt} 1764int mp_sqrmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d); 1765\end{alltt} 1766 1767 1768 1769\chapter{Exponentiation} 1770\section{Single Digit Exponentiation} 1771\index{mp\_expt\_d} 1772\begin{alltt} 1773int mp_expt_d (mp_int * a, mp_digit b, mp_int * c) 1774\end{alltt} 1775This function computes $c = a^b$. 1776 1777\section{Modular Exponentiation} 1778\index{mp\_exptmod} 1779\begin{alltt} 1780int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) 1781\end{alltt} 1782This computes $Y \equiv G^X \mbox{ (mod }P\mbox{)}$ using a variable width sliding window algorithm. This function 1783will automatically detect the fastest modular reduction technique to use during the operation. For negative values of 1784$X$ the operation is performed as $Y \equiv (G^{-1} \mbox{ mod }P)^{\vert X \vert} \mbox{ (mod }P\mbox{)}$ provided that 1785$gcd(G, P) = 1$. 1786 1787This function is actually a shell around the two internal exponentiation functions. This routine will automatically 1788detect when Barrett, Montgomery, Restricted and Unrestricted Diminished Radix based exponentiation can be used. Generally 1789moduli of the a ``restricted diminished radix'' form lead to the fastest modular exponentiations. Followed by Montgomery 1790and the other two algorithms. 1791 1792\section{Modulus a Power of Two} 1793\index{mp\_mod\_2d} 1794\begin{alltt} 1795int mp_mod_2d(const mp_int *a, int b, mp_int *c) 1796\end{alltt} 1797It calculates $c = a \mod 2^b$. 1798 1799\section{Root Finding} 1800\index{mp\_n\_root} 1801\begin{alltt} 1802int mp_n_root (mp_int * a, mp_digit b, mp_int * c) 1803\end{alltt} 1804This computes $c = a^{1/b}$ such that $c^b \le a$ and $(c+1)^b > a$. Will return a positive root only for even roots and return 1805a root with the sign of the input for odd roots. For example, performing $4^{1/2}$ will return $2$ whereas $(-8)^{1/3}$ 1806will return $-2$. 1807 1808This algorithm uses the ``Newton Approximation'' method and will converge on the correct root fairly quickly. 1809 1810The square root $c = a^{1/2}$ (with the same conditions $c^2 \le a$ and $(c+1)^2 > a$) is implemented with a faster algorithm. 1811 1812\index{mp\_sqrt} 1813\begin{alltt} 1814int mp_sqrt (mp_int * a, mp_digit b, mp_int * c) 1815\end{alltt} 1816 1817 1818\chapter{Logarithm} 1819\section{Integer Logarithm} 1820A logarithm function for positive integer input \texttt{a, base} computing $\floor{\log_bx}$ such that $(\log_b x)^b \le x$. 1821\index{mp\_ilogb} 1822\begin{alltt} 1823int mp_ilogb(mp_int *a, mp_digit base, mp_int *c) 1824\end{alltt} 1825\subsection{Example} 1826\begin{alltt} 1827#include <stdlib.h> 1828#include <stdio.h> 1829#include <errno.h> 1830 1831#include <tommath.h> 1832 1833int main(int argc, char **argv) 1834{ 1835 mp_int x, output; 1836 mp_digit base; 1837 int e; 1838 1839 if (argc != 3) { 1840 fprintf(stderr,"Usage %s base x\textbackslash{}n", argv[0]); 1841 exit(EXIT_FAILURE); 1842 } 1843 if ((e = mp_init_multi(&x, &output, NULL)) != MP_OKAY) { 1844 fprintf(stderr,"mp_init failed: \textbackslash{}"%s\textbackslash{}"\textbackslash{}n", 1845 mp_error_to_string(e)); 1846 exit(EXIT_FAILURE); 1847 } 1848 errno = 0; 1849#ifdef MP_64BIT 1850 base = (mp_digit)strtoull(argv[1], NULL, 10); 1851#else 1852 base = (mp_digit)strtoul(argv[1], NULL, 10); 1853#endif 1854 if ((errno == ERANGE) || (base > (base & MP_MASK))) { 1855 fprintf(stderr,"strtoul(l) failed: input out of range\textbackslash{}n"); 1856 exit(EXIT_FAILURE); 1857 } 1858 if ((e = mp_read_radix(&x, argv[2], 10)) != MP_OKAY) { 1859 fprintf(stderr,"mp_read_radix failed: \textbackslash{}"%s\textbackslash{}"\textbackslash{}n", 1860 mp_error_to_string(e)); 1861 exit(EXIT_FAILURE); 1862 } 1863 if ((e = mp_ilogb(&x, base, &output)) != MP_OKAY) { 1864 fprintf(stderr,"mp_ilogb failed: \textbackslash{}"%s\textbackslash{}"\textbackslash{}n", 1865 mp_error_to_string(e)); 1866 exit(EXIT_FAILURE); 1867 } 1868 1869 if ((e = mp_fwrite(&output, 10, stdout)) != MP_OKAY) { 1870 fprintf(stderr,"mp_fwrite failed: \textbackslash{}"%s\textbackslash{}"\textbackslash{}n", 1871 mp_error_to_string(e)); 1872 exit(EXIT_FAILURE); 1873 } 1874 putchar('\textbackslash{}n'); 1875 1876 mp_clear_multi(&x, &output, NULL); 1877 exit(EXIT_SUCCESS); 1878} 1879\end{alltt} 1880 1881 1882 1883\chapter{Prime Numbers} 1884\section{Trial Division} 1885\index{mp\_prime\_is\_divisible} 1886\begin{alltt} 1887int mp_prime_is_divisible (mp_int * a, int *result) 1888\end{alltt} 1889This will attempt to evenly divide $a$ by a list of primes\footnote{Default is the first 256 primes.} and store the 1890outcome in ``result''. That is if $result = 0$ then $a$ is not divisible by the primes, otherwise it is. Note that 1891if the function does not return \textbf{MP\_OKAY} the value in ``result'' should be considered undefined\footnote{Currently 1892the default is to set it to zero first.}. 1893 1894\section{Fermat Test} 1895\index{mp\_prime\_fermat} 1896\begin{alltt} 1897int mp_prime_fermat (mp_int * a, mp_int * b, int *result) 1898\end{alltt} 1899Performs a Fermat primality test to the base $b$. That is it computes $b^a \mbox{ mod }a$ and tests whether the value is 1900equal to $b$ or not. If the values are equal then $a$ is probably prime and $result$ is set to one. Otherwise $result$ 1901is set to zero. 1902 1903\section{Miller-Rabin Test} 1904\index{mp\_prime\_miller\_rabin} 1905\begin{alltt} 1906int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result) 1907\end{alltt} 1908Performs a Miller-Rabin test to the base $b$ of $a$. This test is much stronger than the Fermat test and is very hard to 1909fool (besides with Carmichael numbers). If $a$ passes the test (therefore is probably prime) $result$ is set to one. 1910Otherwise $result$ is set to zero. 1911 1912Note that is suggested that you use the Miller-Rabin test instead of the Fermat test since all of the failures of 1913Miller-Rabin are a subset of the failures of the Fermat test. 1914 1915\subsection{Required Number of Tests} 1916Generally to ensure a number is very likely to be prime you have to perform the Miller-Rabin with at least a half-dozen 1917or so unique bases. However, it has been proven that the probability of failure goes down as the size of the input goes up. 1918This is why a simple function has been provided to help out. 1919 1920\index{mp\_prime\_rabin\_miller\_trials} 1921\begin{alltt} 1922int mp_prime_rabin_miller_trials(int size) 1923\end{alltt} 1924This returns the number of trials required for a low 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 18 tests for a probability of $2^{-160}$ whereas a 1024-bit number would only require 12 tests for a probability of $2^{-192}$. The exact values as implemented are listed in table \ref{table:millerrabinrunsimpl}. 1925 1926\begin{table}[h] 1927\begin{center} 1928\begin{tabular}{c c c} 1929\textbf{bits} & \textbf{Rounds} & \textbf{Error}\\ 1930 80 & -1 & Use deterministic algorithm for size <= 80 bits \\ 1931 81 & 37 & $2^{-96}$ \\ 1932 96 & 32 & $2^{-96}$ \\ 1933 128 & 40 & $2^{-112}$ \\ 1934 160 & 35 & $2^{-112}$ \\ 1935 256 & 27 & $2^{-128}$ \\ 1936 384 & 16 & $2^{-128}$ \\ 1937 512 & 18 & $2^{-160}$ \\ 1938 768 & 11 & $2^{-160}$ \\ 1939 896 & 10 & $2^{-160}$ \\ 1940 1024 & 12 & $2^{-192}$ \\ 1941 1536 & 8 & $2^{-192}$ \\ 1942 2048 & 6 & $2^{-192}$ \\ 1943 3072 & 4 & $2^{-192}$ \\ 1944 4096 & 5 & $2^{-256}$ \\ 1945 5120 & 4 & $2^{-256}$ \\ 1946 6144 & 4 & $2^{-256}$ \\ 1947 8192 & 3 & $2^{-256}$ \\ 1948 9216 & 3 & $2^{-256}$ \\ 1949 10240 & 2 & $2^{-256}$ 1950\end{tabular} 1951\caption{ Number of Miller-Rabin rounds as implemented } \label{table:millerrabinrunsimpl} 1952\end{center} 1953\end{table} 1954 1955You should always still perform a trial division before a Miller-Rabin test though. 1956 1957A small table, broke in two for typographical reasons, with the number of rounds of Miller-Rabin tests is shown below. The numbers have been compute with a PARI/GP script listed in appendix \ref{app:numberofmrcomp}. 1958The first column is the number of bits $b$ in the prime $p = 2^b$, the numbers in the first row represent the 1959probability that the number that all of the Miller-Rabin tests deemed a pseudoprime is actually a composite. There is a deterministic test for numbers smaller than $2^{80}$. 1960 1961\begin{table}[h] 1962\begin{center} 1963\begin{tabular}{c c c c c c c} 1964\textbf{bits} & $\mathbf{2^{-80}}$ & $\mathbf{2^{-96}}$ & $\mathbf{2^{-112}}$ & $\mathbf{2^{-128}}$ & $\mathbf{2^{-160}}$ & $\mathbf{2^{-192}}$ \\ 196580 & 31 & 39 & 47 & 55 & 71 & 87 \\ 196696 & 29 & 37 & 45 & 53 & 69 & 85 \\ 1967128 & 24 & 32 & 40 & 48 & 64 & 80 \\ 1968160 & 19 & 27 & 35 & 43 & 59 & 75 \\ 1969192 & 15 & 21 & 29 & 37 & 53 & 69 \\ 1970256 & 10 & 15 & 20 & 27 & 43 & 59 \\ 1971384 & 7 & 9 & 12 & 16 & 25 & 38 \\ 1972512 & 5 & 7 & 9 & 12 & 18 & 26 \\ 1973768 & 4 & 5 & 6 & 8 & 11 & 16 \\ 19741024 & 3 & 4 & 5 & 6 & 9 & 12 \\ 19751536 & 2 & 3 & 3 & 4 & 6 & 8 \\ 19762048 & 2 & 2 & 3 & 3 & 4 & 6 \\ 19773072 & 1 & 2 & 2 & 2 & 3 & 4 \\ 19784096 & 1 & 1 & 2 & 2 & 2 & 3 \\ 19796144 & 1 & 1 & 1 & 1 & 2 & 2 \\ 19808192 & 1 & 1 & 1 & 1 & 2 & 2 \\ 198112288 & 1 & 1 & 1 & 1 & 1 & 1 \\ 198216384 & 1 & 1 & 1 & 1 & 1 & 1 \\ 198324576 & 1 & 1 & 1 & 1 & 1 & 1 \\ 198432768 & 1 & 1 & 1 & 1 & 1 & 1 1985\end{tabular} 1986\caption{ Number of Miller-Rabin rounds. Part I } \label{table:millerrabinrunsp1} 1987\end{center} 1988\end{table} 1989\newpage 1990\begin{table}[h] 1991\begin{center} 1992\begin{tabular}{c c c c c c c c} 1993\textbf{bits} &$\mathbf{2^{-224}}$ & $\mathbf{2^{-256}}$ & $\mathbf{2^{-288}}$ & $\mathbf{2^{-320}}$ & $\mathbf{2^{-352}}$ & $\mathbf{2^{-384}}$ & $\mathbf{2^{-416}}$\\ 199480 & 103 & 119 & 135 & 151 & 167 & 183 & 199 \\ 199596 & 101 & 117 & 133 & 149 & 165 & 181 & 197 \\ 1996128 & 96 & 112 & 128 & 144 & 160 & 176 & 192 \\ 1997160 & 91 & 107 & 123 & 139 & 155 & 171 & 187 \\ 1998192 & 85 & 101 & 117 & 133 & 149 & 165 & 181 \\ 1999256 & 75 & 91 & 107 & 123 & 139 & 155 & 171 \\ 2000384 & 54 & 70 & 86 & 102 & 118 & 134 & 150 \\ 2001512 & 36 & 49 & 65 & 81 & 97 & 113 & 129 \\ 2002768 & 22 & 29 & 37 & 47 & 58 & 70 & 86 \\ 20031024 & 16 & 21 & 26 & 33 & 40 & 48 & 58 \\ 20041536 & 10 & 13 & 17 & 21 & 25 & 30 & 35 \\ 20052048 & 8 & 10 & 13 & 15 & 18 & 22 & 26 \\ 20063072 & 5 & 7 & 8 & 10 & 12 & 14 & 17 \\ 20074096 & 4 & 5 & 6 & 8 & 9 & 11 & 12 \\ 20086144 & 3 & 4 & 4 & 5 & 6 & 7 & 8 \\ 20098192 & 2 & 3 & 3 & 4 & 5 & 6 & 6 \\ 201012288 & 2 & 2 & 2 & 3 & 3 & 4 & 4 \\ 201116384 & 1 & 2 & 2 & 2 & 3 & 3 & 3 \\ 201224576 & 1 & 1 & 2 & 2 & 2 & 2 & 2 \\ 201332768 & 1 & 1 & 1 & 1 & 2 & 2 & 2 2014\end{tabular} 2015\caption{ Number of Miller-Rabin rounds. Part II } \label{table:millerrabinrunsp2} 2016\end{center} 2017\end{table} 2018 2019Determining the probability needed to pick the right column is a bit harder. Fips 186.4, for example has $2^{-80}$ for $512$ bit large numbers, $2^{-112}$ for $1024$ bits, and $2^{128}$ for $1536$ bits. It can be seen in table \ref{table:millerrabinrunsp1} that those combinations follow the diagonal from $(512,2^{-80})$ downwards and to the right to gain a lower probabilty of getting a composite declared a pseudoprime for the same amount of work or less. 2020 2021If this version of the library has the strong Lucas-Selfridge and/or the Frobenius-Underwood test implemented only one or two rounds of the Miller-Rabin test with a random base is necesssary for numbers larger than or equal to $1024$ bits. 2022 2023This function is meant for RSA. The number of rounds for DSA is $\lceil -log_2(p)/2\rceil$ with $p$ the probability which is just the half of the absolute value of $p$ if given as a power of two. E.g.: with $p = 2^{-128}$, $\lceil -log_2(p)/2\rceil = 64$. 2024 2025This function can be used to test a DSA prime directly if these rounds are followed by a Lucas test. 2026 2027See also table C.1 in FIPS 186-4. 2028 2029\section{Strong Lucas-Selfridge Test} 2030\index{mp\_prime\_strong\_lucas\_selfridge} 2031\begin{alltt} 2032int mp_prime_strong_lucas_selfridge(const mp_int *a, int *result) 2033\end{alltt} 2034Performs a strong Lucas-Selfridge test. The strong Lucas-Selfridge test together with the Rabin-Miler test with bases $2$ and $3$ resemble the BPSW test. The single internal use is a compile-time option in \texttt{mp\_prime\_is\_prime} and can be excluded 2035from the Libtommath build if not needed. 2036 2037\section{Frobenius (Underwood) Test} 2038\index{mp\_prime\_frobenius\_underwood} 2039\begin{alltt} 2040int mp_prime_frobenius_underwood(const mp_int *N, int *result) 2041\end{alltt} 2042Performs the variant of the Frobenius test as described by Paul Underwood. The single internal use is in 2043\texttt{mp\_prime\_is\_prime} for \texttt{MP\_8BIT} only but can be included at build-time for all other sizes 2044if the preprocessor macro \texttt{LTM\_USE\_FROBENIUS\_TEST} is defined. 2045 2046It returns \texttt{MP\_ITER} if the number of iterations is exhausted, assumes a composite as the input and sets \texttt{result} accordingly. This will reduce the set of available pseudoprimes by a very small amount: test with large datasets (more than $10^{10}$ numbers, both randomly chosen and sequences of odd numbers with a random start point) found only 31 (thirty-one) numbers with $a > 120$ and none at all with just an additional simple check for divisors $d < 2^8$. 2047 2048\section{Primality Testing} 2049Testing if a number is a square can be done a bit faster than just by calculating the square root. It is used by the primality testing function described below. 2050\index{mp\_is\_square} 2051\begin{alltt} 2052int mp_is_square(const mp_int *arg, int *ret); 2053\end{alltt} 2054 2055 2056\index{mp\_prime\_is\_prime} 2057\begin{alltt} 2058int mp_prime_is_prime (mp_int * a, int t, int *result) 2059\end{alltt} 2060This will perform a trial division followed by two rounds of Miller-Rabin with bases 2 and 3 and a Lucas-Selfridge test. The Lucas-Selfridge test is replaced with a Frobenius-Underwood for \texttt{MP\_8BIT}. The Frobenius-Underwood test for all other sizes is available as a compile-time option with the preprocessor macro \texttt{LTM\_USE\_FROBENIUS\_TEST}. See file 2061\texttt{bn\_mp\_prime\_is\_prime.c} for the necessary details. It shall be noted that both functions are much slower than 2062the Miller-Rabin test and if speed is an essential issue, the macro \texttt{LTM\_USE\_ONLY\_MR} switches both functions, the Frobenius-Underwood test and the Lucas-Selfridge test off and their code will not even be compiled into the library. 2063 2064If $t$ is set to a positive value $t$ additional rounds of the Miller-Rabin test with random bases will be performed to allow for Fips 186.4 (vid.~p.~126ff) compliance. The function \texttt{mp\_prime\_rabin\_miller\_trials} can be used to determine the number of rounds. It is vital that the function \texttt{mp\_rand()} has a cryptographically strong random number generator available. 2065 2066One Miller-Rabin tests with a random base will be run automatically, so by setting $t$ to a positive value this function will run $t + 1$ Miller-Rabin tests with random bases. 2067 2068If $t$ is set to a negative value the test will run the deterministic Miller-Rabin test for the primes up to $3317044064679887385961981$. That limit has to be checked by the caller. 2069 2070If $a$ passes all of the tests $result$ is set to one, otherwise it is set to zero. 2071 2072\section{Next Prime} 2073\index{mp\_prime\_next\_prime} 2074\begin{alltt} 2075int mp_prime_next_prime(mp_int *a, int t, int bbs_style) 2076\end{alltt} 2077This finds the next prime after $a$ that passes mp\_prime\_is\_prime() with $t$ tests but see the documentation for 2078mp\_prime\_is\_prime for details regarding the use of the argument $t$. Set $bbs\_style$ to one if you 2079want only the next prime congruent to $3 \mbox{ mod } 4$, otherwise set it to zero to find any next prime. 2080 2081\section{Random Primes} 2082\index{mp\_prime\_rand} 2083\begin{alltt} 2084int mp_prime_rand(mp_int *a, int t, 2085 int size, int flags); 2086\end{alltt} 2087This will generate a prime in $a$ using $t$ tests of the primality testing algorithms. 2088See the documentation for mp\_prime\_is\_prime for details regarding the use of the argument $t$. 2089The variable $size$ specifies the bit length of the prime desired. 2090The variable $flags$ specifies one of several options available 2091(see fig. \ref{fig:primeopts}) which can be OR'ed together. 2092 2093The function mp\_prime\_rand() is suitable for generating primes which must be secret (as in the case of RSA) since there 2094is no skew on the least significant bits. 2095 2096\textit{Note:} This function replaces the deprecated mp\_prime\_random and mp\_prime\_random\_ex functions. 2097 2098\begin{figure}[h] 2099\begin{center} 2100\begin{small} 2101\begin{tabular}{|r|l|} 2102\hline \textbf{Flag} & \textbf{Meaning} \\ 2103\hline LTM\_PRIME\_BBS & Make the prime congruent to $3$ modulo $4$ \\ 2104\hline LTM\_PRIME\_SAFE & Make a prime $p$ such that $(p - 1)/2$ is also prime. \\ 2105 & This option implies LTM\_PRIME\_BBS as well. \\ 2106\hline LTM\_PRIME\_2MSB\_OFF & Makes sure that the bit adjacent to the most significant bit \\ 2107 & Is forced to zero. \\ 2108\hline LTM\_PRIME\_2MSB\_ON & Makes sure that the bit adjacent to the most significant bit \\ 2109 & Is forced to one. \\ 2110\hline 2111\end{tabular} 2112\end{small} 2113\end{center} 2114\caption{Primality Generation Options} 2115\label{fig:primeopts} 2116\end{figure} 2117 2118\chapter{Random Number Generation} 2119\section{PRNG} 2120\index{mp\_rand\_digit} 2121\begin{alltt} 2122int mp_rand_digit(mp_digit *r) 2123\end{alltt} 2124This function generates a random number in \texttt{r} of the size given in \texttt{r} (that is, the variable is used for in- and output) but not more than \texttt{MP\_MASK} bits. 2125 2126\index{mp\_rand} 2127\begin{alltt} 2128int mp_rand(mp_int *a, int digits) 2129\end{alltt} 2130This function generates a random number of \texttt{digits} bits. 2131 2132The random number generated with these two functions is cryptographically secure if the source of random numbers the operating systems offers is cryptographically secure. It will use \texttt{arc4random()} if the OS is a BSD flavor, Wincrypt on Windows, or \texttt{/dev/urandom} on all operating systems that have it. 2133 2134 2135\chapter{Input and Output} 2136\section{ASCII Conversions} 2137\subsection{To ASCII} 2138\index{mp\_to\_radix} 2139\begin{alltt} 2140int mp_to_radix (mp_int *a, char *str, size_t maxlen, size_t *written, int radix); 2141\end{alltt} 2142This stores $a$ in \texttt{str} of maximum length \texttt{maxlen} as a base-\texttt{radix} string of ASCII chars and appends a \texttt{NUL} character to terminate the string. 2143 2144Valid values of \texttt{radix} line in the range $[2, 64]$. 2145 2146The exact number of characters in \texttt{str} plus the \texttt{NUL} will be put in \texttt{written} if that variable is not set to \texttt{NULL}. 2147 2148If \texttt{str} is not big enough to hold $a$, \texttt{str} will be filled with the least-significant digits 2149of length \texttt{maxlen-1}, then \texttt{str} will be \texttt{NUL} terminated and the error \texttt{MP\_VAL} is returned. 2150 2151Please be aware that this function cannot evaluate the actual size of the buffer, it relies on the correctness of \texttt{maxlen}! 2152 2153 2154\index{mp\_radix\_size} 2155\begin{alltt} 2156int mp_radix_size (mp_int * a, int radix, int *size) 2157\end{alltt} 2158This stores in ``size'' the number of characters (including space for the NUL terminator) required. Upon error this 2159function returns an error code and ``size'' will be zero. 2160 2161If \texttt{LTM\_NO\_FILE} is not defined a function to write to a file is also available. 2162\index{mp\_fwrite} 2163\begin{alltt} 2164int mp_fwrite(const mp_int *a, int radix, FILE *stream); 2165\end{alltt} 2166 2167 2168\subsection{From ASCII} 2169\index{mp\_read\_radix} 2170\begin{alltt} 2171int mp_read_radix (mp_int * a, char *str, int radix); 2172\end{alltt} 2173This will read the base-``radix'' NUL terminated string from ``str'' into $a$. It will stop reading when it reads a 2174character it does not recognize (which happens to include th NUL char... imagine that...). A single leading $-$ sign 2175can be used to denote a negative number. 2176 2177If \texttt{LTM\_NO\_FILE} is not defined a function to read from a file is also available. 2178\index{mp\_fread} 2179\begin{alltt} 2180int mp_fread(mp_int *a, int radix, FILE *stream); 2181\end{alltt} 2182 2183 2184\section{Binary Conversions} 2185 2186Converting an mp\_int to and from binary is another keen idea. 2187 2188\index{mp\_ubin\_size} 2189\begin{alltt} 2190size_t mp_ubin_size(mp_int *a); 2191\end{alltt} 2192 2193This will return the number of bytes (octets) required to store the unsigned copy of the integer $a$. 2194 2195\index{mp\_to\_ubin} 2196\begin{alltt} 2197int mp_to_unsigned_bin(mp_int *a, unsigned char *b, size_t maxlen, size_t *len); 2198\end{alltt} 2199This will store $a$ into the buffer $b$ of size \texttt{maxlen} in big--endian format storing the number of bytes written in \texttt{len}. Fortunately this is exactly what DER (or is it ASN?) requires. It does not store the sign of the integer. 2200 2201\index{mp\_from\_ubin} 2202\begin{alltt} 2203int mp_from_ubin(mp_int *a, unsigned char *b, size_t size); 2204\end{alltt} 2205This will read in an unsigned big--endian array of bytes (octets) from $b$ of length \texttt{size} into $a$. The resulting big-integer $a$ will always be positive. 2206 2207For those who acknowledge the existence of negative numbers (heretic!) there are ``signed'' versions of the 2208previous functions. 2209\index{mp\_signed\_bin\_size} \index{mp\_to\_signed\_bin} \index{mp\_read\_signed\_bin} 2210\begin{alltt} 2211int mp_sbin_size(mp_int *a); 2212int mp_from_sbin(mp_int *a, unsigned char *b, size_t size); 2213int mp_to_sbin(mp_int *a, unsigned char *b, size_t maxsize, size_t *len); 2214\end{alltt} 2215They operate essentially the same as the unsigned copies except they prefix the data with zero or non--zero 2216byte depending on the sign. If the sign is zpos (e.g. not negative) the prefix is zero, otherwise the prefix 2217is non--zero. 2218 2219The two functions \texttt{mp\_unpack} (get your gifts out of the box, import binary data) and \texttt{mp\_pack} (put your gifts into the box, export binary data) implement the similarly working GMP functions as described at \url{http://gmplib.org/manual/Integer-Import-and-Export.html} with the exception that \texttt{mp\_pack} will not allocate memory if \texttt{rop} is \texttt{NULL}. 2220\index{mp\_unpack} \index{mp\_pack} 2221\begin{alltt} 2222int mp_unpack(mp_int *rop, size_t count, mp_order order, size_t size, 2223 mp_endian endian, size_t nails, const void *op, size_t maxsize); 2224int mp_pack(void *rop, size_t *countp, mp_order order, size_t size, 2225 mp_endian endian, size_t nails, const mp_int *op); 2226\end{alltt} 2227The function \texttt{mp\_pack} has the additional variable \texttt{maxsize} which must hold the size of the buffer \texttt{rop} in bytes. Use 2228\begin{alltt} 2229/* Parameters "nails" and "size" are the same as in mp_pack */ 2230size_t mp_pack_size(mp_int *a, size_t nails, size_t size); 2231\end{alltt} 2232To get the size in bytes necessary to be put in \texttt{maxsize}). 2233 2234To enhance the readability of your code, the following enums have been wrought for your convenience. 2235\begin{alltt} 2236typedef enum { 2237 MP_LSB_FIRST = -1, 2238 MP_MSB_FIRST = 1 2239} mp_order; 2240typedef enum { 2241 MP_LITTLE_ENDIAN = -1, 2242 MP_NATIVE_ENDIAN = 0, 2243 MP_BIG_ENDIAN = 1 2244} mp_endian; 2245\end{alltt} 2246 2247\chapter{Algebraic Functions} 2248\section{Extended Euclidean Algorithm} 2249\index{mp\_exteuclid} 2250\begin{alltt} 2251int mp_exteuclid(mp_int *a, mp_int *b, 2252 mp_int *U1, mp_int *U2, mp_int *U3); 2253\end{alltt} 2254 2255This finds the triple U1/U2/U3 using the Extended Euclidean algorithm such that the following equation holds. 2256 2257\begin{equation} 2258a \cdot U1 + b \cdot U2 = U3 2259\end{equation} 2260 2261Any of the U1/U2/U3 parameters can be set to \textbf{NULL} if they are not desired. 2262 2263\section{Greatest Common Divisor} 2264\index{mp\_gcd} 2265\begin{alltt} 2266int mp_gcd (mp_int * a, mp_int * b, mp_int * c) 2267\end{alltt} 2268This will compute the greatest common divisor of $a$ and $b$ and store it in $c$. 2269 2270\section{Least Common Multiple} 2271\index{mp\_lcm} 2272\begin{alltt} 2273int mp_lcm (mp_int * a, mp_int * b, mp_int * c) 2274\end{alltt} 2275This will compute the least common multiple of $a$ and $b$ and store it in $c$. 2276 2277\section{Jacobi Symbol} 2278\index{mp\_jacobi} 2279\begin{alltt} 2280int mp_jacobi (mp_int * a, mp_int * p, int *c) 2281\end{alltt} 2282This will compute the Jacobi symbol for $a$ with respect to $p$. If $p$ is prime this essentially computes the Legendre 2283symbol. The result is stored in $c$ and can take on one of three values $\lbrace -1, 0, 1 \rbrace$. If $p$ is prime 2284then the result will be $-1$ when $a$ is not a quadratic residue modulo $p$. The result will be $0$ if $a$ divides $p$ 2285and the result will be $1$ if $a$ is a quadratic residue modulo $p$. 2286 2287\section{Kronecker Symbol} 2288\index{mp\_kronecker} 2289\begin{alltt} 2290int mp_kronecker (mp_int * a, mp_int * p, int *c) 2291\end{alltt} 2292Extension of the Jacoby symbol to all $\lbrace a, p \rbrace \in \mathbb{Z}$ . 2293 2294 2295\section{Modular square root} 2296\index{mp\_sqrtmod\_prime} 2297\begin{alltt} 2298int mp_sqrtmod_prime(mp_int *n, mp_int *p, mp_int *r) 2299\end{alltt} 2300 2301This will solve the modular equatioon $r^2 = n \mod p$ where $p$ is a prime number greater than 2 (odd prime). 2302The result is returned in the third argument $r$, the function returns \textbf{MP\_OKAY} on success, 2303other return values indicate failure. 2304 2305The implementation is split for two different cases: 2306 23071. 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 2308$r = n^{(p+1)/4} \mod p$ 2309 23102. otherwise we use \href{https://en.wikipedia.org/wiki/Tonelli-Shanks_algorithm}{Tonelli-Shanks algorithm} 2311 2312The function does not check the primality of parameter $p$ thus it is up to the caller to assure that this parameter 2313is a prime number. When $p$ is a composite the function behaviour is undefined, it may even return a false-positive 2314\textbf{MP\_OKAY}. 2315 2316\section{Modular Inverse} 2317\index{mp\_invmod} 2318\begin{alltt} 2319int mp_invmod (mp_int * a, mp_int * b, mp_int * c) 2320\end{alltt} 2321Computes the multiplicative inverse of $a$ modulo $b$ and stores the result in $c$ such that $ac \equiv 1 \mbox{ (mod }b\mbox{)}$. 2322 2323\section{Single Digit Functions} 2324 2325For those using small numbers (\textit{snicker snicker}) there are several ``helper'' functions 2326 2327\index{mp\_add\_d} \index{mp\_sub\_d} \index{mp\_mul\_d} \index{mp\_div\_d} \index{mp\_mod\_d} 2328\begin{alltt} 2329int mp_add_d(mp_int *a, mp_digit b, mp_int *c); 2330int mp_sub_d(mp_int *a, mp_digit b, mp_int *c); 2331int mp_mul_d(mp_int *a, mp_digit b, mp_int *c); 2332int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d); 2333int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c); 2334\end{alltt} 2335 2336These work like the full mp\_int capable variants except the second parameter $b$ is a mp\_digit. These 2337functions fairly handy if you have to work with relatively small numbers since you will not have to allocate 2338an entire mp\_int to store a number like $1$ or $2$. 2339 2340The functions \texttt{mp\_incr} and \texttt{mp\_decr} mimic the postfix operators \texttt{++} and \texttt{--} respectively, to increment the input by one. They call the full single-digit functions if the addition would carry. Both functions need to be included in a minimized library because they call each other in case of a negative input, These functions change the inputs! 2341\begin{alltt} 2342int mp_incr(mp_int *a); 2343int mp_decr(mp_int *a); 2344\end{alltt} 2345 2346 2347The division by three can be made faster by replacing the division with a multiplication by the multiplicative inverse of three. 2348 2349\index{mp\_div\_3} 2350\begin{alltt} 2351int mp_div_3(const mp_int *a, mp_int *c, mp_digit *d); 2352\end{alltt} 2353 2354\chapter{Little Helpers} 2355It is never wrong to have some useful little shortcuts at hand. 2356\section{Function Macros} 2357To make this overview simpler the macros are given as function prototypes. The return of logic macros is \texttt{MP\_NO} or \texttt{MP\_YES} respectively. 2358 2359\index{mp\_iseven} 2360\begin{alltt} 2361int mp_iseven(mp_int *a) 2362\end{alltt} 2363Checks if $a = 0 mod 2$ 2364 2365\index{mp\_isodd} 2366\begin{alltt} 2367int mp_isodd(mp_int *a) 2368\end{alltt} 2369Checks if $a = 1 mod 2$ 2370 2371\index{mp\_isneg} 2372\begin{alltt} 2373int mp_isneg(mp_int *a) 2374\end{alltt} 2375Checks if $a < 0$ 2376 2377 2378\index{mp\_iszero} 2379\begin{alltt} 2380int mp_iszero(mp_int *a) 2381\end{alltt} 2382Checks if $a = 0$. It does not check if the amount of memory allocated for $a$ is also minimal. 2383 2384 2385Other macros which are either shortcuts to normal functions or just other names for them do have their place in a programmer's life, too! 2386 2387\subsection{Renamings} 2388\index{mp\_mag\_size} 2389\begin{alltt} 2390#define mp_mag_size(mp) mp_unsigned_bin_size(mp) 2391\end{alltt} 2392 2393 2394\index{mp\_raw\_size} 2395\begin{alltt} 2396#define mp_raw_size(mp) mp_signed_bin_size(mp) 2397\end{alltt} 2398 2399 2400\index{mp\_read\_mag} 2401\begin{alltt} 2402#define mp_read_mag(mp, str, len) mp_read_unsigned_bin((mp), (str), (len)) 2403\end{alltt} 2404 2405 2406\index{mp\_read\_raw} 2407\begin{alltt} 2408 #define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len)) 2409\end{alltt} 2410 2411 2412\index{mp\_tomag} 2413\begin{alltt} 2414#define mp_tomag(mp, str) mp_to_unsigned_bin((mp), (str)) 2415\end{alltt} 2416 2417 2418\index{mp\_toraw} 2419\begin{alltt} 2420#define mp_toraw(mp, str) mp_to_signed_bin((mp), (str)) 2421\end{alltt} 2422 2423 2424 2425\subsection{Shortcuts} 2426 2427\index{mp\_to\_binary} 2428\begin{alltt} 2429#define mp_to_binary(M, S, N) mp_to_radix((M), (S), (N), 2) 2430\end{alltt} 2431 2432 2433\index{mp\_to\_octal} 2434\begin{alltt} 2435#define mp_to_octal(M, S, N) mp_to_radix((M), (S), (N), 8) 2436\end{alltt} 2437 2438 2439\index{mp\_to\_decimal} 2440\begin{alltt} 2441#define mp_to_decimal(M, S, N) mp_to_radix((M), (S), (N), 10) 2442\end{alltt} 2443 2444 2445\index{mp\_to\_hex} 2446\begin{alltt} 2447#define mp_to_hex(M, S, N) mp_to_radix((M), (S), (N), 16) 2448\end{alltt} 2449 2450\begin{appendices} 2451\appendixpage 2452%\noappendicestocpagenum 2453\addappheadtotoc 2454\chapter{Computing Number of Miller-Rabin Trials}\label{app:numberofmrcomp} 2455The number of Miller-Rabin rounds in the tables \ref{millerrabinrunsimpl}, \ref{millerrabinrunsp1}, and \ref{millerrabinrunsp2} have been calculated with the formula in FIPS 186-4 appendix F.1 (page 117) implemented as a PARI/GP script. 2456\begin{alltt} 2457log2(x) = log(x)/log(2) 2458 2459fips_f1_sums(k, M, t) = { 2460 local(s = 0); 2461 s = sum(m=3,M, 2462 2^(m-t*(m-1)) * 2463 sum(j=2,m, 2464 1/ ( 2^( j + (k-1)/j ) ) 2465 ) 2466 ); 2467 return(s); 2468} 2469 2470fips_f1_2(k, t, M) = { 2471 local(common_factor, t1, t2, f1, f2, ds, res); 2472 2473 common_factor = 2.00743 * log(2) * k * 2^(-k); 2474 t1 = 2^(k - 2 - M*t); 2475 f1 = (8 * ((Pi^2) - 6))/3; 2476 f2 = 2^(k - 2); 2477 ds = t1 + f1 * f2 * fips_f1_sums(k, M, t); 2478 res = common_factor * ds; 2479 return(res); 2480} 2481 2482fips_f1_1(prime_length, ptarget)={ 2483 local(t, t_end, M, M_end, pkt); 2484 2485 t_end = ceil(-log2(ptarget)/2); 2486 M_end = floor(2 * sqrt(prime_length-1) - 1); 2487 2488 for(t = 1, t_end, 2489 for(M = 3, M_end, 2490 pkt = fips_f1_2(prime_length, t, M); 2491 if(pkt <= ptarget, 2492 return(t); 2493 ); 2494 ); 2495 ); 2496} 2497\end{alltt} 2498 2499To get the number of rounds for a $1024$ bit large prime with a probability of $2^{-160}$: 2500\begin{alltt} 2501? fips_f1_1(1024,2^(-160)) 2502%1 = 9 2503\end{alltt} 2504\end{appendices} 2505\input{bn.ind} 2506 2507\end{document} 2508