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\documentclass[b5paper]{book}
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\def\getsrandom{\stackrel{\rm R}{\gets}}
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\def\cat{\hspace{0.5em} \| \hspace{0.5em}}
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\def\divides{\hspace{0.3em} | \hspace{0.3em}}
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\def\nequiv{\not\equiv}
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\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}}
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\def\abs{{\mathit abs}}
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\def\rep{{\mathit rep}}
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\def\mod{{\mathit\ mod\ }}
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\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})}
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\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
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\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil}
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\def\And{{\rm\ and\ }}
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\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}}
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\def\implies{\Rightarrow}
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\def\undefined{{\rm ``undefined"}}
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\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}}
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\newcommand{\str}[1]{{\mathbf{#1}}}
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\definecolor{DGray}{gray}{0.5}
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\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}}
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\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}}
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\def\gap{\vspace{0.5ex}}
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\title{LibTomMath User Manual \\ v0.32}
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\author{Tom St Denis \\ tomstdenis@iahu.ca}
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This text, the library and the accompanying textbook are all hereby placed in the public domain. This book has been
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formatted for B5 [176x250] paper using the \LaTeX{} {\em book} macro package.
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\begin{flushright}Open Source. Open Academia. Open Minds.
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\chapter{Introduction}
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\section{What is LibTomMath?}
75
LibTomMath is a library of source code which provides a series of efficient and carefully written functions for manipulating
76
large integer numbers. It was written in portable ISO C source code so that it will build on any platform with a conforming
79
In a nutshell the library was written from scratch with verbose comments to help instruct computer science students how
80
to implement ``bignum'' math. However, the resulting code has proven to be very useful. It has been used by numerous
81
universities, commercial and open source software developers. It has been used on a variety of platforms ranging from
82
Linux and Windows based x86 to ARM based Gameboys and PPC based MacOS machines.
85
As of the v0.25 the library source code has been placed in the public domain with every new release. As of the v0.28
86
release the textbook ``Implementing Multiple Precision Arithmetic'' has been placed in the public domain with every new
87
release as well. This textbook is meant to compliment the project by providing a more solid walkthrough of the development
88
algorithms used in the library.
90
Since both\footnote{Note that the MPI files under mtest/ are copyrighted by Michael Fromberger. They are not required to use LibTomMath.} are in the
91
public domain everyone is entitled to do with them as they see fit.
93
\section{Building LibTomMath}
95
LibTomMath is meant to be very ``GCC friendly'' as it comes with a makefile well suited for GCC. However, the library will
96
also build in MSVC, Borland C out of the box. For any other ISO C compiler a makefile will have to be made by the end
99
\subsection{Static Libraries}
100
To build as a static library for GCC issue the following
105
command. This will build the library and archive the object files in ``libtommath.a''. Now you link against
106
that and include ``tommath.h'' within your programs. Alternatively to build with MSVC issue the following
108
nmake -f makefile.msvc
111
This will build the library and archive the object files in ``tommath.lib''. This has been tested with MSVC
112
version 6.00 with service pack 5.
114
\subsection{Shared Libraries}
115
To build as a shared library for GCC issue the following
117
make -f makefile.shared
119
This requires the ``libtool'' package (common on most Linux/BSD systems). It will build LibTomMath as both shared
120
and static then install (by default) into /usr/lib as well as install the header files in /usr/include. The shared
121
library (resource) will be called ``libtommath.la'' while the static library called ``libtommath.a''. Generally
122
you use libtool to link your application against the shared object.
124
There is limited support for making a ``DLL'' in windows via the ``makefile.cygwin\_dll'' makefile. It requires
125
Cygwin to work with since it requires the auto-export/import functionality. The resulting DLL and import library
126
``libtommath.dll.a'' can be used to link LibTomMath dynamically to any Windows program using Cygwin.
129
To build the library and the test harness type
135
This will build the library, ``test'' and ``mtest/mtest''. The ``test'' program will accept test vectors and verify the
136
results. ``mtest/mtest'' will generate test vectors using the MPI library by Michael Fromberger\footnote{A copy of MPI
137
is included in the package}. Simply pipe mtest into test using
143
If you do not have a ``/dev/urandom'' style RNG source you will have to write your own PRNG and simply pipe that into
144
mtest. For example, if your PRNG program is called ``myprng'' simply invoke
147
myprng | mtest/mtest | test
150
This will output a row of numbers that are increasing. Each column is a different test (such as addition, multiplication, etc)
151
that is being performed. The numbers represent how many times the test was invoked. If an error is detected the program
152
will exit with a dump of the relevent numbers it was working with.
154
\section{Build Configuration}
155
LibTomMath can configured at build time in three phases we shall call ``depends'', ``tweaks'' and ``trims''.
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Each phase changes how the library is built and they are applied one after another respectively.
158
To make the system more powerful you can tweak the build process. Classes are defined in the file
159
``tommath\_superclass.h''. By default, the symbol ``LTM\_ALL'' shall be defined which simply
160
instructs the system to build all of the functions. This is how LibTomMath used to be packaged. This will give you
161
access to every function LibTomMath offers.
163
However, there are cases where such a build is not optional. For instance, you want to perform RSA operations. You
164
don't need the vast majority of the library to perform these operations. Aside from LTM\_ALL there is
165
another pre--defined class ``SC\_RSA\_1'' which works in conjunction with the RSA from LibTomCrypt. Additional
166
classes can be defined base on the need of the user.
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\subsection{Build Depends}
169
In the file tommath\_class.h you will see a large list of C ``defines'' followed by a series of ``ifdefs''
170
which further define symbols. All of the symbols (technically they're macros $\ldots$) represent a given C source
171
file. For instance, BN\_MP\_ADD\_C represents the file ``bn\_mp\_add.c''. When a define has been enabled the
172
function in the respective file will be compiled and linked into the library. Accordingly when the define
173
is absent the file will not be compiled and not contribute any size to the library.
175
You will also note that the header tommath\_class.h is actually recursively included (it includes itself twice).
176
This is to help resolve as many dependencies as possible. In the last pass the symbol LTM\_LAST will be defined.
177
This is useful for ``trims''.
179
\subsection{Build Tweaks}
180
A tweak is an algorithm ``alternative''. For example, to provide tradeoffs (usually between size and space).
181
They can be enabled at any pass of the configuration phase.
185
\begin{tabular}{|l|l|}
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\hline \textbf{Define} & \textbf{Purpose} \\
187
\hline BN\_MP\_DIV\_SMALL & Enables a slower, smaller and equally \\
188
& functional mp\_div() function \\
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\subsection{Build Trims}
195
A trim is a manner of removing functionality from a function that is not required. For instance, to perform
196
RSA cryptography you only require exponentiation with odd moduli so even moduli support can be safely removed.
197
Build trims are meant to be defined on the last pass of the configuration which means they are to be defined
198
only if LTM\_LAST has been defined.
200
\subsubsection{Moduli Related}
203
\begin{tabular}{|l|l|}
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\hline \textbf{Restriction} & \textbf{Undefine} \\
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\hline Exponentiation with odd moduli only & BN\_S\_MP\_EXPTMOD\_C \\
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& BN\_MP\_REDUCE\_C \\
207
& BN\_MP\_REDUCE\_SETUP\_C \\
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& BN\_S\_MP\_MUL\_HIGH\_DIGS\_C \\
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& BN\_FAST\_S\_MP\_MUL\_HIGH\_DIGS\_C \\
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\hline Exponentiation with random odd moduli & (The above plus the following) \\
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& BN\_MP\_REDUCE\_2K\_C \\
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& BN\_MP\_REDUCE\_2K\_SETUP\_C \\
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& BN\_MP\_REDUCE\_IS\_2K\_C \\
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& BN\_MP\_DR\_IS\_MODULUS\_C \\
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& BN\_MP\_DR\_REDUCE\_C \\
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& BN\_MP\_DR\_SETUP\_C \\
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\hline Modular inverse odd moduli only & BN\_MP\_INVMOD\_SLOW\_C \\
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\hline Modular inverse (both, smaller/slower) & BN\_FAST\_MP\_INVMOD\_C \\
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\subsubsection{Operand Size Related}
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\begin{tabular}{|l|l|}
228
\hline \textbf{Restriction} & \textbf{Undefine} \\
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\hline Moduli $\le 2560$ bits & BN\_MP\_MONTGOMERY\_REDUCE\_C \\
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& BN\_S\_MP\_MUL\_DIGS\_C \\
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& BN\_S\_MP\_MUL\_HIGH\_DIGS\_C \\
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& BN\_S\_MP\_SQR\_C \\
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\hline Polynomial Schmolynomial & BN\_MP\_KARATSUBA\_MUL\_C \\
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& BN\_MP\_KARATSUBA\_SQR\_C \\
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& BN\_MP\_TOOM\_MUL\_C \\
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& BN\_MP\_TOOM\_SQR\_C \\
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\section{Purpose of LibTomMath}
245
Unlike GNU MP (GMP) Library, LIP, OpenSSL or various other commercial kits (Miracl), LibTomMath was not written with
246
bleeding edge performance in mind. First and foremost LibTomMath was written to be entirely open. Not only is the
247
source code public domain (unlike various other GPL/etc licensed code), not only is the code freely downloadable but the
248
source code is also accessible for computer science students attempting to learn ``BigNum'' or multiple precision
249
arithmetic techniques.
251
LibTomMath was written to be an instructive collection of source code. This is why there are many comments, only one
252
function per source file and often I use a ``middle-road'' approach where I don't cut corners for an extra 2\% speed
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Source code alone cannot really teach how the algorithms work which is why I also wrote a textbook that accompanies
256
the library (beat that!).
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So you may be thinking ``should I use LibTomMath?'' and the answer is a definite maybe. Let me tabulate what I think
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are the pros and cons of LibTomMath by comparing it to the math routines from GnuPG\footnote{GnuPG v1.2.3 versus LibTomMath v0.28}.
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\newpage\begin{figure}[here]
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\begin{tabular}{|l|c|c|l|}
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\hline \textbf{Criteria} & \textbf{Pro} & \textbf{Con} & \textbf{Notes} \\
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\hline Few lines of code per file & X & & GnuPG $ = 300.9$, LibTomMath $ = 76.04$ \\
267
\hline Commented function prototypes & X && GnuPG function names are cryptic. \\
268
\hline Speed && X & LibTomMath is slower. \\
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\hline Totally free & X & & GPL has unfavourable restrictions.\\
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\hline Large function base & X & & GnuPG is barebones. \\
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\hline Four modular reduction algorithms & X & & Faster modular exponentiation. \\
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\hline Portable & X & & GnuPG requires configuration to build. \\
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\caption{LibTomMath Valuation}
280
It may seem odd to compare LibTomMath to GnuPG since the math in GnuPG is only a small portion of the entire application.
281
However, LibTomMath was written with cryptography in mind. It provides essentially all of the functions a cryptosystem
282
would require when working with large integers.
284
So it may feel tempting to just rip the math code out of GnuPG (or GnuMP where it was taken from originally) in your
285
own application but I think there are reasons not to. While LibTomMath is slower than libraries such as GnuMP it is
286
not normally significantly slower. On x86 machines the difference is normally a factor of two when performing modular
289
Essentially the only time you wouldn't use LibTomMath is when blazing speed is the primary concern.
291
\chapter{Getting Started with LibTomMath}
292
\section{Building Programs}
293
In order to use LibTomMath you must include ``tommath.h'' and link against the appropriate library file (typically
294
libtommath.a). There is no library initialization required and the entire library is thread safe.
296
\section{Return Codes}
297
There are three possible return codes a function may return.
299
\index{MP\_OKAY}\index{MP\_YES}\index{MP\_NO}\index{MP\_VAL}\index{MP\_MEM}
300
\begin{figure}[here!]
303
\begin{tabular}{|l|l|}
304
\hline \textbf{Code} & \textbf{Meaning} \\
305
\hline MP\_OKAY & The function succeeded. \\
306
\hline MP\_VAL & The function input was invalid. \\
307
\hline MP\_MEM & Heap memory exhausted. \\
309
\hline MP\_YES & Response is yes. \\
310
\hline MP\_NO & Response is no. \\
315
\caption{Return Codes}
318
The last two codes listed are not actually ``return'ed'' by a function. They are placed in an integer (the caller must
319
provide the address of an integer it can store to) which the caller can access. To convert one of the three return codes
320
to a string use the following function.
322
\index{mp\_error\_to\_string}
324
char *mp_error_to_string(int code);
327
This will return a pointer to a string which describes the given error code. It will not work for the return codes
331
The basic ``multiple precision integer'' type is known as the ``mp\_int'' within LibTomMath. This data type is used to
332
organize all of the data required to manipulate the integer it represents. Within LibTomMath it has been prototyped
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int used, alloc, sign;
343
Where ``mp\_digit'' is a data type that represents individual digits of the integer. By default, an mp\_digit is the
344
ISO C ``unsigned long'' data type and each digit is $28-$bits long. The mp\_digit type can be configured to suit other
345
platforms by defining the appropriate macros.
347
All LTM functions that use the mp\_int type will expect a pointer to mp\_int structure. You must allocate memory to
348
hold the structure itself by yourself (whether off stack or heap it doesn't matter). The very first thing that must be
349
done to use an mp\_int is that it must be initialized.
351
\section{Function Organization}
353
The arithmetic functions of the library are all organized to have the same style prototype. That is source operands
354
are passed on the left and the destination is on the right. For instance,
357
mp_add(&a, &b, &c); /* c = a + b */
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mp_mul(&a, &a, &c); /* c = a * a */
359
mp_div(&a, &b, &c, &d); /* c = [a/b], d = a mod b */
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Another feature of the way the functions have been implemented is that source operands can be destination operands as well.
366
mp_add(&a, &b, &b); /* b = a + b */
367
mp_div(&a, &b, &a, &c); /* a = [a/b], c = a mod b */
370
This allows operands to be re-used which can make programming simpler.
372
\section{Initialization}
373
\subsection{Single Initialization}
374
A single mp\_int can be initialized with the ``mp\_init'' function.
378
int mp_init (mp_int * a);
381
This function expects a pointer to an mp\_int structure and will initialize the members of the structure so the mp\_int
382
represents the default integer which is zero. If the functions returns MP\_OKAY then the mp\_int is ready to be used
383
by the other LibTomMath functions.
385
\begin{small} \begin{alltt}
391
if ((result = mp_init(&number)) != MP_OKAY) \{
392
printf("Error initializing the number. \%s",
393
mp_error_to_string(result));
401
\end{alltt} \end{small}
403
\subsection{Single Free}
404
When you are finished with an mp\_int it is ideal to return the heap it used back to the system. The following function
405
provides this functionality.
409
void mp_clear (mp_int * a);
412
The function expects a pointer to a previously initialized mp\_int structure and frees the heap it uses. It sets the
413
pointer\footnote{The ``dp'' member.} within the mp\_int to \textbf{NULL} which is used to prevent double free situations.
414
Is is legal to call mp\_clear() twice on the same mp\_int in a row.
416
\begin{small} \begin{alltt}
422
if ((result = mp_init(&number)) != MP_OKAY) \{
423
printf("Error initializing the number. \%s",
424
mp_error_to_string(result));
430
/* We're done with it. */
435
\end{alltt} \end{small}
437
\subsection{Multiple Initializations}
438
Certain algorithms require more than one large integer. In these instances it is ideal to initialize all of the mp\_int
439
variables in an ``all or nothing'' fashion. That is, they are either all initialized successfully or they are all
442
The mp\_init\_multi() function provides this functionality.
444
\index{mp\_init\_multi} \index{mp\_clear\_multi}
446
int mp_init_multi(mp_int *mp, ...);
449
It accepts a \textbf{NULL} terminated list of pointers to mp\_int structures. It will attempt to initialize them all
450
at once. If the function returns MP\_OKAY then all of the mp\_int variables are ready to use, otherwise none of them
451
are available for use. A complementary mp\_clear\_multi() function allows multiple mp\_int variables to be free'd
452
from the heap at the same time.
454
\begin{small} \begin{alltt}
457
mp_int num1, num2, num3;
460
if ((result = mp_init_multi(&num1,
462
&num3, NULL)) != MP\_OKAY) \{
463
printf("Error initializing the numbers. \%s",
464
mp_error_to_string(result));
468
/* use the numbers */
470
/* We're done with them. */
471
mp_clear_multi(&num1, &num2, &num3, NULL);
475
\end{alltt} \end{small}
477
\subsection{Other Initializers}
478
To initialized and make a copy of an mp\_int the mp\_init\_copy() function has been provided.
480
\index{mp\_init\_copy}
482
int mp_init_copy (mp_int * a, mp_int * b);
485
This function will initialize $a$ and make it a copy of $b$ if all goes well.
487
\begin{small} \begin{alltt}
493
/* initialize and do work on num1 ... */
495
/* We want a copy of num1 in num2 now */
496
if ((result = mp_init_copy(&num2, &num1)) != MP_OKAY) \{
497
printf("Error initializing the copy. \%s",
498
mp_error_to_string(result));
502
/* now num2 is ready and contains a copy of num1 */
504
/* We're done with them. */
505
mp_clear_multi(&num1, &num2, NULL);
509
\end{alltt} \end{small}
511
Another less common initializer is mp\_init\_size() which allows the user to initialize an mp\_int with a given
512
default number of digits. By default, all initializers allocate \textbf{MP\_PREC} digits. This function lets
513
you override this behaviour.
515
\index{mp\_init\_size}
517
int mp_init_size (mp_int * a, int size);
520
The $size$ parameter must be greater than zero. If the function succeeds the mp\_int $a$ will be initialized
521
to have $size$ digits (which are all initially zero).
523
\begin{small} \begin{alltt}
529
/* we need a 60-digit number */
530
if ((result = mp_init_size(&number, 60)) != MP_OKAY) \{
531
printf("Error initializing the number. \%s",
532
mp_error_to_string(result));
540
\end{alltt} \end{small}
542
\section{Maintenance Functions}
544
\subsection{Reducing Memory Usage}
545
When an mp\_int is in a state where it won't be changed again\footnote{A Diffie-Hellman modulus for instance.} excess
546
digits can be removed to return memory to the heap with the mp\_shrink() function.
550
int mp_shrink (mp_int * a);
553
This will remove excess digits of the mp\_int $a$. If the operation fails the mp\_int should be intact without the
554
excess digits being removed. Note that you can use a shrunk mp\_int in further computations, however, such operations
555
will require heap operations which can be slow. It is not ideal to shrink mp\_int variables that you will further
556
modify in the system (unless you are seriously low on memory).
558
\begin{small} \begin{alltt}
564
if ((result = mp_init(&number)) != MP_OKAY) \{
565
printf("Error initializing the number. \%s",
566
mp_error_to_string(result));
570
/* use the number [e.g. pre-computation] */
572
/* We're done with it for now. */
573
if ((result = mp_shrink(&number)) != MP_OKAY) \{
574
printf("Error shrinking the number. \%s",
575
mp_error_to_string(result));
582
/* we're done with it. */
587
\end{alltt} \end{small}
589
\subsection{Adding additional digits}
591
Within the mp\_int structure are two parameters which control the limitations of the array of digits that represent
592
the integer the mp\_int is meant to equal. The \textit{used} parameter dictates how many digits are significant, that is,
593
contribute to the value of the mp\_int. The \textit{alloc} parameter dictates how many digits are currently available in
594
the array. If you need to perform an operation that requires more digits you will have to mp\_grow() the mp\_int to
599
int mp_grow (mp_int * a, int size);
602
This will grow the array of digits of $a$ to $size$. If the \textit{alloc} parameter is already bigger than
603
$size$ the function will not do anything.
605
\begin{small} \begin{alltt}
611
if ((result = mp_init(&number)) != MP_OKAY) \{
612
printf("Error initializing the number. \%s",
613
mp_error_to_string(result));
619
/* We need to add 20 digits to the number */
620
if ((result = mp_grow(&number, number.alloc + 20)) != MP_OKAY) \{
621
printf("Error growing the number. \%s",
622
mp_error_to_string(result));
629
/* we're done with it. */
634
\end{alltt} \end{small}
636
\chapter{Basic Operations}
637
\section{Small Constants}
638
Setting mp\_ints to small constants is a relatively common operation. To accomodate these instances there are two
639
small constant assignment functions. The first function is used to set a single digit constant while the second sets
640
an ISO C style ``unsigned long'' constant. The reason for both functions is efficiency. Setting a single digit is quick but the
641
domain of a digit can change (it's always at least $0 \ldots 127$).
643
\subsection{Single Digit}
645
Setting a single digit can be accomplished with the following function.
649
void mp_set (mp_int * a, mp_digit b);
652
This will zero the contents of $a$ and make it represent an integer equal to the value of $b$. Note that this
653
function has a return type of \textbf{void}. It cannot cause an error so it is safe to assume the function
656
\begin{small} \begin{alltt}
662
if ((result = mp_init(&number)) != MP_OKAY) \{
663
printf("Error initializing the number. \%s",
664
mp_error_to_string(result));
668
/* set the number to 5 */
671
/* we're done with it. */
676
\end{alltt} \end{small}
678
\subsection{Long Constants}
680
To set a constant that is the size of an ISO C ``unsigned long'' and larger than a single digit the following function
685
int mp_set_int (mp_int * a, unsigned long b);
688
This will assign the value of the 32-bit variable $b$ to the mp\_int $a$. Unlike mp\_set() this function will always
689
accept a 32-bit input regardless of the size of a single digit. However, since the value may span several digits
690
this function can fail if it runs out of heap memory.
692
To get the ``unsigned long'' copy of an mp\_int the following function can be used.
696
unsigned long mp_get_int (mp_int * a);
699
This will return the 32 least significant bits of the mp\_int $a$.
701
\begin{small} \begin{alltt}
707
if ((result = mp_init(&number)) != MP_OKAY) \{
708
printf("Error initializing the number. \%s",
709
mp_error_to_string(result));
713
/* set the number to 654321 (note this is bigger than 127) */
714
if ((result = mp_set_int(&number, 654321)) != MP_OKAY) \{
715
printf("Error setting the value of the number. \%s",
716
mp_error_to_string(result));
720
printf("number == \%lu", mp_get_int(&number));
722
/* we're done with it. */
727
\end{alltt} \end{small}
729
This should output the following if the program succeeds.
735
\subsection{Initialize and Setting Constants}
736
To both initialize and set small constants the following two functions are available.
737
\index{mp\_init\_set} \index{mp\_init\_set\_int}
739
int mp_init_set (mp_int * a, mp_digit b);
740
int mp_init_set_int (mp_int * a, unsigned long b);
743
Both functions work like the previous counterparts except they first mp\_init $a$ before setting the values.
748
mp_int number1, number2;
751
/* initialize and set a single digit */
752
if ((result = mp_init_set(&number1, 100)) != MP_OKAY) \{
753
printf("Error setting number1: \%s",
754
mp_error_to_string(result));
758
/* initialize and set a long */
759
if ((result = mp_init_set_int(&number2, 1023)) != MP_OKAY) \{
760
printf("Error setting number2: \%s",
761
mp_error_to_string(result));
766
printf("Number1, Number2 == \%lu, \%lu",
767
mp_get_int(&number1), mp_get_int(&number2));
770
mp_clear_multi(&number1, &number2, NULL);
776
If this program succeeds it shall output.
778
Number1, Number2 == 100, 1023
781
\section{Comparisons}
783
Comparisons in LibTomMath are always performed in a ``left to right'' fashion. There are three possible return codes
786
\index{MP\_GT} \index{MP\_EQ} \index{MP\_LT}
789
\begin{tabular}{|c|c|}
790
\hline \textbf{Result Code} & \textbf{Meaning} \\
791
\hline MP\_GT & $a > b$ \\
792
\hline MP\_EQ & $a = b$ \\
793
\hline MP\_LT & $a < b$ \\
797
\caption{Comparison Codes for $a, b$}
801
In figure \ref{fig:CMP} two integers $a$ and $b$ are being compared. In this case $a$ is said to be ``to the left'' of
804
\subsection{Unsigned comparison}
806
An unsigned comparison considers only the digits themselves and not the associated \textit{sign} flag of the
807
mp\_int structures. This is analogous to an absolute comparison. The function mp\_cmp\_mag() will compare two
808
mp\_int variables based on their digits only.
812
int mp_cmp(mp_int * a, mp_int * b);
814
This will compare $a$ to $b$ placing $a$ to the left of $b$. This function cannot fail and will return one of the
815
three compare codes listed in figure \ref{fig:CMP}.
817
\begin{small} \begin{alltt}
820
mp_int number1, number2;
823
if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{
824
printf("Error initializing the numbers. \%s",
825
mp_error_to_string(result));
829
/* set the number1 to 5 */
832
/* set the number2 to -6 */
834
if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{
835
printf("Error negating number2. \%s",
836
mp_error_to_string(result));
840
switch(mp_cmp_mag(&number1, &number2)) \{
841
case MP_GT: printf("|number1| > |number2|"); break;
842
case MP_EQ: printf("|number1| = |number2|"); break;
843
case MP_LT: printf("|number1| < |number2|"); break;
846
/* we're done with it. */
847
mp_clear_multi(&number1, &number2, NULL);
851
\end{alltt} \end{small}
853
If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes
854
successfully it should print the following.
857
|number1| < |number2|
860
This is because $\vert -6 \vert = 6$ and obviously $5 < 6$.
862
\subsection{Signed comparison}
864
To compare two mp\_int variables based on their signed value the mp\_cmp() function is provided.
868
int mp_cmp(mp_int * a, mp_int * b);
871
This will compare $a$ to the left of $b$. It will first compare the signs of the two mp\_int variables. If they
872
differ it will return immediately based on their signs. If the signs are equal then it will compare the digits
873
individually. This function will return one of the compare conditions codes listed in figure \ref{fig:CMP}.
875
\begin{small} \begin{alltt}
878
mp_int number1, number2;
881
if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{
882
printf("Error initializing the numbers. \%s",
883
mp_error_to_string(result));
887
/* set the number1 to 5 */
890
/* set the number2 to -6 */
892
if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{
893
printf("Error negating number2. \%s",
894
mp_error_to_string(result));
898
switch(mp_cmp(&number1, &number2)) \{
899
case MP_GT: printf("number1 > number2"); break;
900
case MP_EQ: printf("number1 = number2"); break;
901
case MP_LT: printf("number1 < number2"); break;
904
/* we're done with it. */
905
mp_clear_multi(&number1, &number2, NULL);
909
\end{alltt} \end{small}
911
If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes
912
successfully it should print the following.
918
\subsection{Single Digit}
920
To compare a single digit against an mp\_int the following function has been provided.
924
int mp_cmp_d(mp_int * a, mp_digit b);
927
This will compare $a$ to the left of $b$ using a signed comparison. Note that it will always treat $b$ as
928
positive. This function is rather handy when you have to compare against small values such as $1$ (which often
929
comes up in cryptography). The function cannot fail and will return one of the tree compare condition codes
930
listed in figure \ref{fig:CMP}.
933
\begin{small} \begin{alltt}
939
if ((result = mp_init(&number)) != MP_OKAY) \{
940
printf("Error initializing the number. \%s",
941
mp_error_to_string(result));
945
/* set the number to 5 */
948
switch(mp_cmp_d(&number, 7)) \{
949
case MP_GT: printf("number > 7"); break;
950
case MP_EQ: printf("number = 7"); break;
951
case MP_LT: printf("number < 7"); break;
954
/* we're done with it. */
959
\end{alltt} \end{small}
961
If this program functions properly it will print out the following.
967
\section{Logical Operations}
969
Logical operations are operations that can be performed either with simple shifts or boolean operators such as
970
AND, XOR and OR directly. These operations are very quick.
972
\subsection{Multiplication by two}
974
Multiplications and divisions by any power of two can be performed with quick logical shifts either left or
975
right depending on the operation.
977
When multiplying or dividing by two a special case routine can be used which are as follows.
978
\index{mp\_mul\_2} \index{mp\_div\_2}
980
int mp_mul_2(mp_int * a, mp_int * b);
981
int mp_div_2(mp_int * a, mp_int * b);
984
The former will assign twice $a$ to $b$ while the latter will assign half $a$ to $b$. These functions are fast
985
since the shift counts and maskes are hardcoded into the routines.
987
\begin{small} \begin{alltt}
993
if ((result = mp_init(&number)) != MP_OKAY) \{
994
printf("Error initializing the number. \%s",
995
mp_error_to_string(result));
999
/* set the number to 5 */
1002
/* multiply by two */
1003
if ((result = mp\_mul\_2(&number, &number)) != MP_OKAY) \{
1004
printf("Error multiplying the number. \%s",
1005
mp_error_to_string(result));
1006
return EXIT_FAILURE;
1008
switch(mp_cmp_d(&number, 7)) \{
1009
case MP_GT: printf("2*number > 7"); break;
1010
case MP_EQ: printf("2*number = 7"); break;
1011
case MP_LT: printf("2*number < 7"); break;
1014
/* now divide by two */
1015
if ((result = mp\_div\_2(&number, &number)) != MP_OKAY) \{
1016
printf("Error dividing the number. \%s",
1017
mp_error_to_string(result));
1018
return EXIT_FAILURE;
1020
switch(mp_cmp_d(&number, 7)) \{
1021
case MP_GT: printf("2*number/2 > 7"); break;
1022
case MP_EQ: printf("2*number/2 = 7"); break;
1023
case MP_LT: printf("2*number/2 < 7"); break;
1026
/* we're done with it. */
1029
return EXIT_SUCCESS;
1031
\end{alltt} \end{small}
1033
If this program is successful it will print out the following text.
1040
Since $10 > 7$ and $5 < 7$. To multiply by a power of two the following function can be used.
1044
int mp_mul_2d(mp_int * a, int b, mp_int * c);
1047
This will multiply $a$ by $2^b$ and store the result in ``c''. If the value of $b$ is less than or equal to
1048
zero the function will copy $a$ to ``c'' without performing any further actions.
1050
To divide by a power of two use the following.
1054
int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d);
1056
Which will divide $a$ by $2^b$, store the quotient in ``c'' and the remainder in ``d'. If $b \le 0$ then the
1057
function simply copies $a$ over to ``c'' and zeroes $d$. The variable $d$ may be passed as a \textbf{NULL}
1058
value to signal that the remainder is not desired.
1060
\subsection{Polynomial Basis Operations}
1062
Strictly speaking the organization of the integers within the mp\_int structures is what is known as a
1063
``polynomial basis''. This simply means a field element is stored by divisions of a radix. For example, if
1064
$f(x) = \sum_{i=0}^{k} y_ix^k$ for any vector $\vec y$ then the array of digits in $\vec y$ are said to be
1065
the polynomial basis representation of $z$ if $f(\beta) = z$ for a given radix $\beta$.
1067
To multiply by the polynomial $g(x) = x$ all you have todo is shift the digits of the basis left one place. The
1068
following function provides this operation.
1072
int mp_lshd (mp_int * a, int b);
1075
This will multiply $a$ in place by $x^b$ which is equivalent to shifting the digits left $b$ places and inserting zeroes
1076
in the least significant digits. Similarly to divide by a power of $x$ the following function is provided.
1080
void mp_rshd (mp_int * a, int b)
1082
This will divide $a$ in place by $x^b$ and discard the remainder. This function cannot fail as it performs the operations
1083
in place and no new digits are required to complete it.
1085
\subsection{AND, OR and XOR Operations}
1087
While AND, OR and XOR operations are not typical ``bignum functions'' they can be useful in several instances. The
1088
three functions are prototyped as follows.
1090
\index{mp\_or} \index{mp\_and} \index{mp\_xor}
1092
int mp_or (mp_int * a, mp_int * b, mp_int * c);
1093
int mp_and (mp_int * a, mp_int * b, mp_int * c);
1094
int mp_xor (mp_int * a, mp_int * b, mp_int * c);
1097
Which compute $c = a \odot b$ where $\odot$ is one of OR, AND or XOR.
1099
\section{Addition and Subtraction}
1101
To compute an addition or subtraction the following two functions can be used.
1103
\index{mp\_add} \index{mp\_sub}
1105
int mp_add (mp_int * a, mp_int * b, mp_int * c);
1106
int mp_sub (mp_int * a, mp_int * b, mp_int * c)
1109
Which perform $c = a \odot b$ where $\odot$ is one of signed addition or subtraction. The operations are fully sign
1112
\section{Sign Manipulation}
1113
\subsection{Negation}
1115
Simple integer negation can be performed with the following.
1119
int mp_neg (mp_int * a, mp_int * b);
1122
Which assigns $-a$ to $b$.
1124
\subsection{Absolute}
1125
Simple integer absolutes can be performed with the following.
1129
int mp_abs (mp_int * a, mp_int * b);
1132
Which assigns $\vert a \vert$ to $b$.
1134
\section{Integer Division and Remainder}
1135
To perform a complete and general integer division with remainder use the following function.
1139
int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d);
1142
This divides $a$ by $b$ and stores the quotient in $c$ and $d$. The signed quotient is computed such that
1143
$bc + d = a$. Note that either of $c$ or $d$ can be set to \textbf{NULL} if their value is not required. If
1144
$b$ is zero the function returns \textbf{MP\_VAL}.
1147
\chapter{Multiplication and Squaring}
1148
\section{Multiplication}
1149
A full signed integer multiplication can be performed with the following.
1152
int mp_mul (mp_int * a, mp_int * b, mp_int * c);
1154
Which assigns the full signed product $ab$ to $c$. This function actually breaks into one of four cases which are
1155
specific multiplication routines optimized for given parameters. First there are the Toom-Cook multiplications which
1156
should only be used with very large inputs. This is followed by the Karatsuba multiplications which are for moderate
1157
sized inputs. Then followed by the Comba and baseline multipliers.
1159
Fortunately for the developer you don't really need to know this unless you really want to fine tune the system. mp\_mul()
1160
will determine on its own\footnote{Some tweaking may be required.} what routine to use automatically when it is called.
1165
mp_int number1, number2;
1168
/* Initialize the numbers */
1169
if ((result = mp_init_multi(&number1,
1170
&number2, NULL)) != MP_OKAY) \{
1171
printf("Error initializing the numbers. \%s",
1172
mp_error_to_string(result));
1173
return EXIT_FAILURE;
1177
if ((result = mp_set_int(&number, 257)) != MP_OKAY) \{
1178
printf("Error setting number1. \%s",
1179
mp_error_to_string(result));
1180
return EXIT_FAILURE;
1183
if ((result = mp_set_int(&number2, 1023)) != MP_OKAY) \{
1184
printf("Error setting number2. \%s",
1185
mp_error_to_string(result));
1186
return EXIT_FAILURE;
1190
if ((result = mp_mul(&number1, &number2,
1191
&number1)) != MP_OKAY) \{
1192
printf("Error multiplying terms. \%s",
1193
mp_error_to_string(result));
1194
return EXIT_FAILURE;
1198
printf("number1 * number2 == \%lu", mp_get_int(&number1));
1200
/* free terms and return */
1201
mp_clear_multi(&number1, &number2, NULL);
1203
return EXIT_SUCCESS;
1207
If this program succeeds it shall output the following.
1210
number1 * number2 == 262911
1214
Since squaring can be performed faster than multiplication it is performed it's own function instead of just using
1219
int mp_sqr (mp_int * a, mp_int * b);
1222
Will square $a$ and store it in $b$. Like the case of multiplication there are four different squaring
1223
algorithms all which can be called from mp\_sqr(). It is ideal to use mp\_sqr over mp\_mul when squaring terms.
1225
\section{Tuning Polynomial Basis Routines}
1227
Both of the Toom-Cook and Karatsuba multiplication algorithms are faster than the traditional $O(n^2)$ approach that
1228
the Comba and baseline algorithms use. At $O(n^{1.464973})$ and $O(n^{1.584962})$ running times respectfully they require
1229
considerably less work. For example, a 10000-digit multiplication would take roughly 724,000 single precision
1230
multiplications with Toom-Cook or 100,000,000 single precision multiplications with the standard Comba (a factor
1233
So why not always use Karatsuba or Toom-Cook? The simple answer is that they have so much overhead that they're not
1234
actually faster than Comba until you hit distinct ``cutoff'' points. For Karatsuba with the default configuration,
1235
GCC 3.3.1 and an Athlon XP processor the cutoff point is roughly 110 digits (about 70 for the Intel P4). That is, at
1236
110 digits Karatsuba and Comba multiplications just about break even and for 110+ digits Karatsuba is faster.
1238
Toom-Cook has incredible overhead and is probably only useful for very large inputs. So far no known cutoff points
1239
exist and for the most part I just set the cutoff points very high to make sure they're not called.
1241
A demo program in the ``etc/'' directory of the project called ``tune.c'' can be used to find the cutoff points. This
1242
can be built with GCC as follows
1247
Where ``XXX'' is one of the following entries from the table \ref{fig:tuning}.
1249
\begin{figure}[here]
1252
\begin{tabular}{|l|l|}
1253
\hline \textbf{Value of XXX} & \textbf{Meaning} \\
1254
\hline tune & Builds portable tuning application \\
1255
\hline tune86 & Builds x86 (pentium and up) program for COFF \\
1256
\hline tune86c & Builds x86 program for Cygwin \\
1257
\hline tune86l & Builds x86 program for Linux (ELF format) \\
1262
\caption{Build Names for Tuning Programs}
1266
When the program is running it will output a series of measurements for different cutoff points. It will first find
1267
good Karatsuba squaring and multiplication points. Then it proceeds to find Toom-Cook points. Note that the Toom-Cook
1268
tuning takes a very long time as the cutoff points are likely to be very high.
1270
\chapter{Modular Reduction}
1272
Modular reduction is process of taking the remainder of one quantity divided by another. Expressed
1273
as (\ref{eqn:mod}) the modular reduction is equivalent to the remainder of $b$ divided by $c$.
1276
a \equiv b \mbox{ (mod }c\mbox{)}
1280
Of particular interest to cryptography are reductions where $b$ is limited to the range $0 \le b < c^2$ since particularly
1281
fast reduction algorithms can be written for the limited range.
1283
Note that one of the four optimized reduction algorithms are automatically chosen in the modular exponentiation
1284
algorithm mp\_exptmod when an appropriate modulus is detected.
1286
\section{Straight Division}
1287
In order to effect an arbitrary modular reduction the following algorithm is provided.
1291
int mp_mod(mp_int *a, mp_int *b, mp_int *c);
1294
This reduces $a$ modulo $b$ and stores the result in $c$. The sign of $c$ shall agree with the sign
1295
of $b$. This algorithm accepts an input $a$ of any range and is not limited by $0 \le a < b^2$.
1297
\section{Barrett Reduction}
1299
Barrett reduction is a generic optimized reduction algorithm that requires pre--computation to achieve
1300
a decent speedup over straight division. First a $mu$ value must be precomputed with the following function.
1302
\index{mp\_reduce\_setup}
1304
int mp_reduce_setup(mp_int *a, mp_int *b);
1307
Given a modulus in $b$ this produces the required $mu$ value in $a$. For any given modulus this only has to
1308
be computed once. Modular reduction can now be performed with the following.
1312
int mp_reduce(mp_int *a, mp_int *b, mp_int *c);
1315
This will reduce $a$ in place modulo $b$ with the precomputed $mu$ value in $c$. $a$ must be in the range
1324
/* initialize a,b to desired values, mp_init mu,
1325
* c and set c to 1...we want to compute a^3 mod b
1329
if ((result = mp_reduce_setup(&mu, b)) != MP_OKAY) \{
1330
printf("Error getting mu. \%s",
1331
mp_error_to_string(result));
1332
return EXIT_FAILURE;
1335
/* square a to get c = a^2 */
1336
if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{
1337
printf("Error squaring. \%s",
1338
mp_error_to_string(result));
1339
return EXIT_FAILURE;
1342
/* now reduce `c' modulo b */
1343
if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{
1344
printf("Error reducing. \%s",
1345
mp_error_to_string(result));
1346
return EXIT_FAILURE;
1349
/* multiply a to get c = a^3 */
1350
if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{
1351
printf("Error reducing. \%s",
1352
mp_error_to_string(result));
1353
return EXIT_FAILURE;
1356
/* now reduce `c' modulo b */
1357
if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{
1358
printf("Error reducing. \%s",
1359
mp_error_to_string(result));
1360
return EXIT_FAILURE;
1363
/* c now equals a^3 mod b */
1365
return EXIT_SUCCESS;
1369
This program will calculate $a^3 \mbox{ mod }b$ if all the functions succeed.
1371
\section{Montgomery Reduction}
1373
Montgomery is a specialized reduction algorithm for any odd moduli. Like Barrett reduction a pre--computation
1374
step is required. This is accomplished with the following.
1376
\index{mp\_montgomery\_setup}
1378
int mp_montgomery_setup(mp_int *a, mp_digit *mp);
1381
For the given odd moduli $a$ the precomputation value is placed in $mp$. The reduction is computed with the
1384
\index{mp\_montgomery\_reduce}
1386
int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
1388
This reduces $a$ in place modulo $m$ with the pre--computed value $mp$. $a$ must be in the range
1391
Montgomery reduction is faster than Barrett reduction for moduli smaller than the ``comba'' limit. With the default
1392
setup for instance, the limit is $127$ digits ($3556$--bits). Note that this function is not limited to
1393
$127$ digits just that it falls back to a baseline algorithm after that point.
1395
An important observation is that this reduction does not return $a \mbox{ mod }m$ but $aR^{-1} \mbox{ mod }m$
1396
where $R = \beta^n$, $n$ is the n number of digits in $m$ and $\beta$ is radix used (default is $2^{28}$).
1398
To quickly calculate $R$ the following function was provided.
1400
\index{mp\_montgomery\_calc\_normalization}
1402
int mp_montgomery_calc_normalization(mp_int *a, mp_int *b);
1404
Which calculates $a = R$ for the odd moduli $b$ without using multiplication or division.
1406
The normal modus operandi for Montgomery reductions is to normalize the integers before entering the system. For
1407
example, to calculate $a^3 \mbox { mod }b$ using Montgomery reduction the value of $a$ can be normalized by
1408
multiplying it by $R$. Consider the following code snippet.
1417
/* initialize a,b to desired values,
1418
* mp_init R, c and set c to 1....
1421
/* get normalization */
1422
if ((result = mp_montgomery_calc_normalization(&R, b)) != MP_OKAY) \{
1423
printf("Error getting norm. \%s",
1424
mp_error_to_string(result));
1425
return EXIT_FAILURE;
1429
if ((result = mp_montgomery_setup(&c, &mp)) != MP_OKAY) \{
1430
printf("Error setting up montgomery. \%s",
1431
mp_error_to_string(result));
1432
return EXIT_FAILURE;
1435
/* normalize `a' so now a is equal to aR */
1436
if ((result = mp_mulmod(&a, &R, &b, &a)) != MP_OKAY) \{
1437
printf("Error computing aR. \%s",
1438
mp_error_to_string(result));
1439
return EXIT_FAILURE;
1442
/* square a to get c = a^2R^2 */
1443
if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{
1444
printf("Error squaring. \%s",
1445
mp_error_to_string(result));
1446
return EXIT_FAILURE;
1449
/* now reduce `c' back down to c = a^2R^2 * R^-1 == a^2R */
1450
if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
1451
printf("Error reducing. \%s",
1452
mp_error_to_string(result));
1453
return EXIT_FAILURE;
1456
/* multiply a to get c = a^3R^2 */
1457
if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{
1458
printf("Error reducing. \%s",
1459
mp_error_to_string(result));
1460
return EXIT_FAILURE;
1463
/* now reduce `c' back down to c = a^3R^2 * R^-1 == a^3R */
1464
if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
1465
printf("Error reducing. \%s",
1466
mp_error_to_string(result));
1467
return EXIT_FAILURE;
1470
/* now reduce (again) `c' back down to c = a^3R * R^-1 == a^3 */
1471
if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
1472
printf("Error reducing. \%s",
1473
mp_error_to_string(result));
1474
return EXIT_FAILURE;
1477
/* c now equals a^3 mod b */
1479
return EXIT_SUCCESS;
1483
This particular example does not look too efficient but it demonstrates the point of the algorithm. By
1484
normalizing the inputs the reduced results are always of the form $aR$ for some variable $a$. This allows
1485
a single final reduction to correct for the normalization and the fast reduction used within the algorithm.
1487
For more details consider examining the file \textit{bn\_mp\_exptmod\_fast.c}.
1489
\section{Restricted Dimminished Radix}
1491
``Dimminished Radix'' reduction refers to reduction with respect to moduli that are ameniable to simple
1492
digit shifting and small multiplications. In this case the ``restricted'' variant refers to moduli of the
1493
form $\beta^k - p$ for some $k \ge 0$ and $0 < p < \beta$ where $\beta$ is the radix (default to $2^{28}$).
1495
As in the case of Montgomery reduction there is a pre--computation phase required for a given modulus.
1497
\index{mp\_dr\_setup}
1499
void mp_dr_setup(mp_int *a, mp_digit *d);
1502
This computes the value required for the modulus $a$ and stores it in $d$. This function cannot fail
1503
and does not return any error codes. After the pre--computation a reduction can be performed with the
1506
\index{mp\_dr\_reduce}
1508
int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp);
1511
This reduces $a$ in place modulo $b$ with the pre--computed value $mp$. $b$ must be of a restricted
1512
dimminished radix form and $a$ must be in the range $0 \le a < b^2$. Dimminished radix reductions are
1513
much faster than both Barrett and Montgomery reductions as they have a much lower asymtotic running time.
1515
Since the moduli are restricted this algorithm is not particularly useful for something like Rabin, RSA or
1516
BBS cryptographic purposes. This reduction algorithm is useful for Diffie-Hellman and ECC where fixed
1517
primes are acceptable.
1519
Note that unlike Montgomery reduction there is no normalization process. The result of this function is
1520
equal to the correct residue.
1522
\section{Unrestricted Dimminshed Radix}
1524
Unrestricted reductions work much like the restricted counterparts except in this case the moduli is of the
1525
form $2^k - p$ for $0 < p < \beta$. In this sense the unrestricted reductions are more flexible as they
1526
can be applied to a wider range of numbers.
1528
\index{mp\_reduce\_2k\_setup}
1530
int mp_reduce_2k_setup(mp_int *a, mp_digit *d);
1533
This will compute the required $d$ value for the given moduli $a$.
1535
\index{mp\_reduce\_2k}
1537
int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d);
1540
This will reduce $a$ in place modulo $n$ with the pre--computed value $d$. From my experience this routine is
1541
slower than mp\_dr\_reduce but faster for most moduli sizes than the Montgomery reduction.
1543
\chapter{Exponentiation}
1544
\section{Single Digit Exponentiation}
1547
int mp_expt_d (mp_int * a, mp_digit b, mp_int * c)
1549
This computes $c = a^b$ using a simple binary left-to-right algorithm. It is faster than repeated multiplications by
1550
$a$ for all values of $b$ greater than three.
1552
\section{Modular Exponentiation}
1555
int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
1557
This computes $Y \equiv G^X \mbox{ (mod }P\mbox{)}$ using a variable width sliding window algorithm. This function
1558
will automatically detect the fastest modular reduction technique to use during the operation. For negative values of
1559
$X$ the operation is performed as $Y \equiv (G^{-1} \mbox{ mod }P)^{\vert X \vert} \mbox{ (mod }P\mbox{)}$ provided that
1562
This function is actually a shell around the two internal exponentiation functions. This routine will automatically
1563
detect when Barrett, Montgomery, Restricted and Unrestricted Dimminished Radix based exponentiation can be used. Generally
1564
moduli of the a ``restricted dimminished radix'' form lead to the fastest modular exponentiations. Followed by Montgomery
1565
and the other two algorithms.
1567
\section{Root Finding}
1570
int mp_n_root (mp_int * a, mp_digit b, mp_int * c)
1572
This computes $c = a^{1/b}$ such that $c^b \le a$ and $(c+1)^b > a$. The implementation of this function is not
1573
ideal for values of $b$ greater than three. It will work but become very slow. So unless you are working with very small
1574
numbers (less than 1000 bits) I'd avoid $b > 3$ situations. Will return a positive root only for even roots and return
1575
a root with the sign of the input for odd roots. For example, performing $4^{1/2}$ will return $2$ whereas $(-8)^{1/3}$
1578
This algorithm uses the ``Newton Approximation'' method and will converge on the correct root fairly quickly. Since
1579
the algorithm requires raising $a$ to the power of $b$ it is not ideal to attempt to find roots for large
1580
values of $b$. If particularly large roots are required then a factor method could be used instead. For example,
1581
$a^{1/16}$ is equivalent to $\left (a^{1/4} \right)^{1/4}$.
1583
\chapter{Prime Numbers}
1584
\section{Trial Division}
1585
\index{mp\_prime\_is\_divisible}
1587
int mp_prime_is_divisible (mp_int * a, int *result)
1589
This will attempt to evenly divide $a$ by a list of primes\footnote{Default is the first 256 primes.} and store the
1590
outcome in ``result''. That is if $result = 0$ then $a$ is not divisible by the primes, otherwise it is. Note that
1591
if the function does not return \textbf{MP\_OKAY} the value in ``result'' should be considered undefined\footnote{Currently
1592
the default is to set it to zero first.}.
1594
\section{Fermat Test}
1595
\index{mp\_prime\_fermat}
1597
int mp_prime_fermat (mp_int * a, mp_int * b, int *result)
1599
Performs a Fermat primality test to the base $b$. That is it computes $b^a \mbox{ mod }a$ and tests whether the value is
1600
equal to $b$ or not. If the values are equal then $a$ is probably prime and $result$ is set to one. Otherwise $result$
1603
\section{Miller-Rabin Test}
1604
\index{mp\_prime\_miller\_rabin}
1606
int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result)
1608
Performs a Miller-Rabin test to the base $b$ of $a$. This test is much stronger than the Fermat test and is very hard to
1609
fool (besides with Carmichael numbers). If $a$ passes the test (therefore is probably prime) $result$ is set to one.
1610
Otherwise $result$ is set to zero.
1612
Note that is suggested that you use the Miller-Rabin test instead of the Fermat test since all of the failures of
1613
Miller-Rabin are a subset of the failures of the Fermat test.
1615
\subsection{Required Number of Tests}
1616
Generally to ensure a number is very likely to be prime you have to perform the Miller-Rabin with at least a half-dozen
1617
or so unique bases. However, it has been proven that the probability of failure goes down as the size of the input goes up.
1618
This is why a simple function has been provided to help out.
1620
\index{mp\_prime\_rabin\_miller\_trials}
1622
int mp_prime_rabin_miller_trials(int size)
1624
This returns the number of trials required for a $2^{-96}$ (or lower) probability of failure for a given ``size'' expressed
1625
in bits. This comes in handy specially since larger numbers are slower to test. For example, a 512-bit number would
1626
require ten tests whereas a 1024-bit number would only require four tests.
1628
You should always still perform a trial division before a Miller-Rabin test though.
1630
\section{Primality Testing}
1631
\index{mp\_prime\_is\_prime}
1633
int mp_prime_is_prime (mp_int * a, int t, int *result)
1635
This will perform a trial division followed by $t$ rounds of Miller-Rabin tests on $a$ and store the result in $result$.
1636
If $a$ passes all of the tests $result$ is set to one, otherwise it is set to zero. Note that $t$ is bounded by
1637
$1 \le t < PRIME\_SIZE$ where $PRIME\_SIZE$ is the number of primes in the prime number table (by default this is $256$).
1639
\section{Next Prime}
1640
\index{mp\_prime\_next\_prime}
1642
int mp_prime_next_prime(mp_int *a, int t, int bbs_style)
1644
This finds the next prime after $a$ that passes mp\_prime\_is\_prime() with $t$ tests. Set $bbs\_style$ to one if you
1645
want only the next prime congruent to $3 \mbox{ mod } 4$, otherwise set it to zero to find any next prime.
1647
\section{Random Primes}
1648
\index{mp\_prime\_random}
1650
int mp_prime_random(mp_int *a, int t, int size, int bbs,
1651
ltm_prime_callback cb, void *dat)
1653
This will find a prime greater than $256^{size}$ which can be ``bbs\_style'' or not depending on $bbs$ and must pass
1654
$t$ rounds of tests. The ``ltm\_prime\_callback'' is a typedef for
1657
typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat);
1660
Which is a function that must read $len$ bytes (and return the amount stored) into $dst$. The $dat$ variable is simply
1661
copied from the original input. It can be used to pass RNG context data to the callback. The function
1662
mp\_prime\_random() is more suitable for generating primes which must be secret (as in the case of RSA) since there
1663
is no skew on the least significant bits.
1665
\textit{Note:} As of v0.30 of the LibTomMath library this function has been deprecated. It is still available
1666
but users are encouraged to use the new mp\_prime\_random\_ex() function instead.
1668
\subsection{Extended Generation}
1669
\index{mp\_prime\_random\_ex}
1671
int mp_prime_random_ex(mp_int *a, int t,
1672
int size, int flags,
1673
ltm_prime_callback cb, void *dat);
1675
This will generate a prime in $a$ using $t$ tests of the primality testing algorithms. The variable $size$
1676
specifies the bit length of the prime desired. The variable $flags$ specifies one of several options available
1677
(see fig. \ref{fig:primeopts}) which can be OR'ed together. The callback parameters are used as in
1678
mp\_prime\_random().
1680
\begin{figure}[here]
1683
\begin{tabular}{|r|l|}
1684
\hline \textbf{Flag} & \textbf{Meaning} \\
1685
\hline LTM\_PRIME\_BBS & Make the prime congruent to $3$ modulo $4$ \\
1686
\hline LTM\_PRIME\_SAFE & Make a prime $p$ such that $(p - 1)/2$ is also prime. \\
1687
& This option implies LTM\_PRIME\_BBS as well. \\
1688
\hline LTM\_PRIME\_2MSB\_OFF & Makes sure that the bit adjacent to the most significant bit \\
1689
& Is forced to zero. \\
1690
\hline LTM\_PRIME\_2MSB\_ON & Makes sure that the bit adjacent to the most significant bit \\
1691
& Is forced to one. \\
1696
\caption{Primality Generation Options}
1697
\label{fig:primeopts}
1700
\chapter{Input and Output}
1701
\section{ASCII Conversions}
1702
\subsection{To ASCII}
1705
int mp_toradix (mp_int * a, char *str, int radix);
1707
This still store $a$ in ``str'' as a base-``radix'' string of ASCII chars. This function appends a NUL character
1708
to terminate the string. Valid values of ``radix'' line in the range $[2, 64]$. To determine the size (exact) required
1709
by the conversion before storing any data use the following function.
1711
\index{mp\_radix\_size}
1713
int mp_radix_size (mp_int * a, int radix, int *size)
1715
This stores in ``size'' the number of characters (including space for the NUL terminator) required. Upon error this
1716
function returns an error code and ``size'' will be zero.
1718
\subsection{From ASCII}
1719
\index{mp\_read\_radix}
1721
int mp_read_radix (mp_int * a, char *str, int radix);
1723
This will read the base-``radix'' NUL terminated string from ``str'' into $a$. It will stop reading when it reads a
1724
character it does not recognize (which happens to include th NUL char... imagine that...). A single leading $-$ sign
1725
can be used to denote a negative number.
1727
\section{Binary Conversions}
1729
Converting an mp\_int to and from binary is another keen idea.
1731
\index{mp\_unsigned\_bin\_size}
1733
int mp_unsigned_bin_size(mp_int *a);
1736
This will return the number of bytes (octets) required to store the unsigned copy of the integer $a$.
1738
\index{mp\_to\_unsigned\_bin}
1740
int mp_to_unsigned_bin(mp_int *a, unsigned char *b);
1742
This will store $a$ into the buffer $b$ in big--endian format. Fortunately this is exactly what DER (or is it ASN?)
1743
requires. It does not store the sign of the integer.
1745
\index{mp\_read\_unsigned\_bin}
1747
int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c);
1749
This will read in an unsigned big--endian array of bytes (octets) from $b$ of length $c$ into $a$. The resulting
1750
integer $a$ will always be positive.
1752
For those who acknowledge the existence of negative numbers (heretic!) there are ``signed'' versions of the
1756
int mp_signed_bin_size(mp_int *a);
1757
int mp_read_signed_bin(mp_int *a, unsigned char *b, int c);
1758
int mp_to_signed_bin(mp_int *a, unsigned char *b);
1760
They operate essentially the same as the unsigned copies except they prefix the data with zero or non--zero
1761
byte depending on the sign. If the sign is zpos (e.g. not negative) the prefix is zero, otherwise the prefix
1764
\chapter{Algebraic Functions}
1765
\section{Extended Euclidean Algorithm}
1766
\index{mp\_exteuclid}
1768
int mp_exteuclid(mp_int *a, mp_int *b,
1769
mp_int *U1, mp_int *U2, mp_int *U3);
1772
This finds the triple U1/U2/U3 using the Extended Euclidean algorithm such that the following equation holds.
1775
a \cdot U1 + b \cdot U2 = U3
1778
Any of the U1/U2/U3 paramters can be set to \textbf{NULL} if they are not desired.
1780
\section{Greatest Common Divisor}
1783
int mp_gcd (mp_int * a, mp_int * b, mp_int * c)
1785
This will compute the greatest common divisor of $a$ and $b$ and store it in $c$.
1787
\section{Least Common Multiple}
1790
int mp_lcm (mp_int * a, mp_int * b, mp_int * c)
1792
This will compute the least common multiple of $a$ and $b$ and store it in $c$.
1794
\section{Jacobi Symbol}
1797
int mp_jacobi (mp_int * a, mp_int * p, int *c)
1799
This will compute the Jacobi symbol for $a$ with respect to $p$. If $p$ is prime this essentially computes the Legendre
1800
symbol. The result is stored in $c$ and can take on one of three values $\lbrace -1, 0, 1 \rbrace$. If $p$ is prime
1801
then the result will be $-1$ when $a$ is not a quadratic residue modulo $p$. The result will be $0$ if $a$ divides $p$
1802
and the result will be $1$ if $a$ is a quadratic residue modulo $p$.
1804
\section{Modular Inverse}
1807
int mp_invmod (mp_int * a, mp_int * b, mp_int * c)
1809
Computes the multiplicative inverse of $a$ modulo $b$ and stores the result in $c$ such that $ac \equiv 1 \mbox{ (mod }b\mbox{)}$.
1811
\section{Single Digit Functions}
1813
For those using small numbers (\textit{snicker snicker}) there are several ``helper'' functions
1815
\index{mp\_add\_d} \index{mp\_sub\_d} \index{mp\_mul\_d} \index{mp\_div\_d} \index{mp\_mod\_d}
1817
int mp_add_d(mp_int *a, mp_digit b, mp_int *c);
1818
int mp_sub_d(mp_int *a, mp_digit b, mp_int *c);
1819
int mp_mul_d(mp_int *a, mp_digit b, mp_int *c);
1820
int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d);
1821
int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c);
1824
These work like the full mp\_int capable variants except the second parameter $b$ is a mp\_digit. These
1825
functions fairly handy if you have to work with relatively small numbers since you will not have to allocate
1826
an entire mp\_int to store a number like $1$ or $2$.
added
removed
libtommath/bn.tex
