This Source Code Form is subject to the terms of the Mozilla Public License, v. 2.0. If a copy of the MPL was not distributed with this file, You can obtain one at http://mozilla.org/MPL/2.0/. About the MPI Library --------------------- The files 'mpi.h' and 'mpi.c' define a simple, arbitrary precision signed integer arithmetic package. The implementation is not the most efficient possible, but the code is small and should be fairly easily portable to just about any machine that supports an ANSI C compiler, as long as it is capable of at least 16-bit arithmetic (but also see below for more on this). This library was written with an eye to cryptographic applications; thus, some care is taken to make sure that temporary values are not left lying around in memory when they are no longer in use. This adds some overhead for zeroing buffers before they are released back into the free pool; however, it gives you the assurance that there is only one copy of your important values residing in your process's address space at a time. Obviously, it is difficult to guarantee anything, in a pre-emptive multitasking environment, but this at least helps you keep a lid on the more obvious ways your data can get spread around in memory. Using the Library ----------------- To use the MPI library in your program, you must include the header: #include "mpi.h" This header provides all the type and function declarations you'll need to use the library. Almost all the names defined by the library begin with the prefix 'mp_', so it should be easy to keep them from clashing with your program's namespace (he says, glibly, knowing full well there are always pathological cases). There are a few things you may want to configure about the library. By default, the MPI library uses an unsigned short for its digit type, and an unsigned int for its word type. The word type must be big enough to contain at least two digits, for the primitive arithmetic to work out. On my machine, a short is 2 bytes and an int is 4 bytes -- but if you have 64-bit ints, you might want to use a 4-byte digit and an 8-byte word. I have tested the library using 1-byte digits and 2-byte words, as well. Whatever you choose to do, the things you need to change are: (1) The type definitions for mp_digit and mp_word. (2) The macro DIGIT_FMT which tells mp_print() how to display a single digit. This is just a printf() format string, so you can adjust it appropriately. (3) The macros DIGIT_MAX and MP_WORD_MAX, which specify the largest value expressible in an mp_digit and an mp_word, respectively. Both the mp_digit and mp_word should be UNSIGNED integer types. The code relies on having the full positive precision of the type used for digits and words. The remaining type definitions should be left alone, for the most part. The code in the library does not make any significant assumptions about the sizes of things, but there is little if any reason to change the other parameters, so I would recommend you leave them as you found them. Conventions ----------- Most functions in the library return a value of type mp_err. This permits the library to communicate success or various kinds of failure to the calling program. The return values currently defined are: MP_OKAY - okay, operation succeeded, all's well MP_YES - okay, the answer is yes (same as MP_OKAY) MP_NO - okay, but answer is no (not MP_OKAY) MP_MEM - operation ran out of memory MP_RANGE - input parameter was out of range MP_BADARG - an invalid input parameter was provided MP_UNDEF - no output value is defined for this input The only function which currently uses MP_UNDEF is mp_invmod(). Division by zero is undefined, but the division functions will return MP_RANGE for a zero divisor. MP_BADARG usually means you passed a bogus mp_int structure to the function. MP_YES and MP_NO are not used by the library itself; they're defined so you can use them in your own extensions. If you need a readable interpretation of these error codes in your program, you may also use the mp_strerror() function. This function takes an mp_err as input, and returns a pointer to a human-readable string describing the meaning of the error. These strings are stored as constants within the library, so the caller should not attempt to modify or free the memory associated with these strings. The library represents values in signed-magnitude format. Values strictly less than zero are negative, all others are considered positive (zero is positive by fiat). You can access the 'sign' member of the mp_int structure directly, but better is to use the mp_cmp_z() function, to find out which side of zero the value lies on. Most arithmetic functions have a single-digit variant, as well as the full arbitrary-precision. An mp_digit is an unsigned value between 0 and DIGIT_MAX inclusive. The radix is available as RADIX. The number of bits in a given digit is given as DIGIT_BIT. Generally, input parameters are given before output parameters. Unless otherwise specified, any input parameter can be re-used as an output parameter, without confusing anything. The basic numeric type defined by the library is an mp_int. Virtually all the functions in the library take a pointer to an mp_int as one of their parameters. An explanation of how to create and use these structures follows. And so, without further ado... Initialization and Cleanup -------------------------- The basic numeric type defined by the library is an 'mp_int'. However, it is not sufficient to simply declare a variable of type mp_int in your program. These variables also need to be initialized before they can be used, to allocate the internal storage they require for computation. This is done using one of the following functions: mp_init(mp_int *mp); mp_init_copy(mp_int *mp, mp_int *from); mp_init_size(mp_int *mp, mp_size p); Each of these requires a pointer to a structure of type mp_int. The basic mp_init() simply initializes the mp_int to a default size, and sets its value to zero. If you would like to initialize a copy of an existing mp_int, use mp_init_copy(), where the 'from' parameter is the mp_int you'd like to make a copy of. The third function, mp_init_size(), permits you to specify how many digits of precision should be preallocated for your mp_int. This can help the library avoid unnecessary re-allocations later on. The default precision used by mp_init() can be retrieved using: precision = mp_get_prec(); This returns the number of digits that will be allocated. You can change this value by using: mp_set_prec(unsigned int prec); Any positive value is acceptable -- if you pass zero, the default precision will be re-set to the compiled-in library default (this is specified in the header file 'mpi-config.h', and typically defaults to 8 or 16). Just as you must allocate an mp_int before you can use it, you must clean up the structure when you are done with it. This is performed using the mp_clear() function. Remember that any mp_int that you create as a local variable in a function must be mp_clear()'d before that function exits, or else the memory allocated to that mp_int will be orphaned and unrecoverable. To set an mp_int to a given value, the following functions are given: mp_set(mp_int *mp, mp_digit d); mp_set_int(mp_int *mp, long z); The mp_set() function sets the mp_int to a single digit value, while mp_set_int() sets the mp_int to a signed long integer value. To set an mp_int to zero, use: mp_zero(mp_int *mp); Copying and Moving ------------------ If you have two initialized mp_int's, and you want to copy the value of one into the other, use: mp_copy(from, to) This takes care of clearing the old value of 'to', and copies the new value into it. If 'to' is not yet initialized, use mp_init_copy() instead (see above). Note: The library tries, whenever possible, to avoid allocating ---- new memory. Thus, mp_copy() tries first to satisfy the needs of the copy by re-using the memory already allocated to 'to'. Only if this proves insufficient will mp_copy() actually allocate new memory. For this reason, if you know a priori that 'to' has enough available space to hold 'from', you don't need to check the return value of mp_copy() for memory failure. The USED() macro tells you how many digits are used by an mp_int, and the ALLOC() macro tells you how many are allocated. If you have two initialized mp_int's, and you want to exchange their values, use: mp_exch(a, b) This is better than using mp_copy() with a temporary, since it will not (ever) touch the memory allocator -- it just swaps the exact contents of the two structures. The mp_exch() function cannot fail; if you pass it an invalid structure, it just ignores it, and does nothing. Basic Arithmetic ---------------- Once you have initialized your integers, you can operate on them. The basic arithmetic functions on full mp_int values are: mp_add(a, b, c) - computes c = a + b mp_sub(a, b, c) - computes c = a - b mp_mul(a, b, c) - computes c = a * b mp_sqr(a, b) - computes b = a * a mp_div(a, b, q, r) - computes q, r such that a = bq + r mp_div_2d(a, d, q, r) - computes q = a / 2^d, r = a % 2^d mp_expt(a, b, c) - computes c = a ** b mp_2expt(a, k) - computes a = 2^k The mp_div_2d() function efficiently computes division by powers of two. Either the q or r parameter may be NULL, in which case that portion of the computation will be discarded. The algorithms used for some of the computations here are described in the following files which are included with this distribution: mul.txt Describes the multiplication algorithm div.txt Describes the division algorithm expt.txt Describes the exponentiation algorithm sqrt.txt Describes the square-root algorithm square.txt Describes the squaring algorithm There are single-digit versions of most of these routines, as well. In the following prototypes, 'd' is a single mp_digit: mp_add_d(a, d, c) - computes c = a + d mp_sub_d(a, d, c) - computes c = a - d mp_mul_d(a, d, c) - computes c = a * d mp_mul_2(a, c) - computes c = a * 2 mp_div_d(a, d, q, r) - computes q, r such that a = bq + r mp_div_2(a, c) - computes c = a / 2 mp_expt_d(a, d, c) - computes c = a ** d The mp_mul_2() and mp_div_2() functions take advantage of the internal representation of an mp_int to do multiplication by two more quickly than mp_mul_d() would. Other basic functions of an arithmetic variety include: mp_zero(a) - assign 0 to a mp_neg(a, c) - negate a: c = -a mp_abs(a, c) - absolute value: c = |a| Comparisons ----------- Several comparison functions are provided. Each of these, unless otherwise specified, returns zero if the comparands are equal, < 0 if the first is less than the second, and > 0 if the first is greater than the second: mp_cmp_z(a) - compare a <=> 0 mp_cmp_d(a, d) - compare a <=> d, d is a single digit mp_cmp(a, b) - compare a <=> b mp_cmp_mag(a, b) - compare |a| <=> |b| mp_isodd(a) - return nonzero if odd, zero otherwise mp_iseven(a) - return nonzero if even, zero otherwise Modular Arithmetic ------------------ Modular variations of the basic arithmetic functions are also supported. These are available if the MP_MODARITH parameter in mpi-config.h is turned on (it is by default). The modular arithmetic functions are: mp_mod(a, m, c) - compute c = a (mod m), 0 <= c < m mp_mod_d(a, d, c) - compute c = a (mod d), 0 <= c < d (see below) mp_addmod(a, b, m, c) - compute c = (a + b) mod m mp_submod(a, b, m, c) - compute c = (a - b) mod m mp_mulmod(a, b, m, c) - compute c = (a * b) mod m mp_sqrmod(a, m, c) - compute c = (a * a) mod m mp_exptmod(a, b, m, c) - compute c = (a ** b) mod m mp_exptmod_d(a, d, m, c)- compute c = (a ** d) mod m The mp_sqr() function squares its input argument. A call to mp_sqr(a, c) is identical in meaning to mp_mul(a, a, c); however, if the MP_SQUARE variable is set true in mpi-config.h (see below), then it will be implemented with a different algorithm, that is supposed to take advantage of the redundant computation that takes place during squaring. Unfortunately, some compilers result in worse performance on this code, so you can change the behaviour at will. There is a utility program "mulsqr.c" that lets you test which does better on your system. The mp_sqrmod() function is analogous to the mp_sqr() function; it uses the mp_sqr() function rather than mp_mul(), and then performs the modular reduction. This probably won't help much unless you are doing a lot of them. See the file 'square.txt' for a synopsis of the algorithm used. Note: The mp_mod_d() function computes a modular reduction around ---- a single digit d. The result is a single digit c. Because an inverse is defined for a (mod m) if and only if (a, m) = 1 (that is, if a and m are relatively prime), mp_invmod() may not be able to compute an inverse for the arguments. In this case, it returns the value MP_UNDEF, and does not modify c. If an inverse is defined, however, it returns MP_OKAY, and sets c to the value of the inverse (mod m). See the file 'redux.txt' for a description of the modular reduction algorithm used by mp_exptmod(). Greatest Common Divisor ----------------------- If The greates common divisor of two values can be found using one of the following functions: mp_gcd(a, b, c) - compute c = (a, b) using binary algorithm mp_lcm(a, b, c) - compute c = [a, b] = ab / (a, b) mp_xgcd(a, b, g, x, y) - compute g, x, y so that ax + by = g = (a, b) Also provided is a function to compute modular inverses, if they exist: mp_invmod(a, m, c) - compute c = a^-1 (mod m), if it exists The function mp_xgcd() computes the greatest common divisor, and also returns values of x and y satisfying Bezout's identity. This is used by mp_invmod() to find modular inverses. However, if you do not need these values, you will find that mp_gcd() is MUCH more efficient, since it doesn't need all the intermediate values that mp_xgcd() requires in order to compute x and y. The mp_gcd() (and mp_xgcd()) functions use the binary (extended) GCD algorithm due to Josef Stein. Input & Output Functions ------------------------ The following basic I/O routines are provided. These are present at all times: mp_read_radix(mp, str, r) - convert a string in radix r to an mp_int mp_read_raw(mp, s, len) - convert a string of bytes to an mp_int mp_radix_size(mp, r) - return length of buffer needed by mp_toradix() mp_raw_size(mp) - return length of buffer needed by mp_toraw() mp_toradix(mp, str, r) - convert an mp_int to a string of radix r digits mp_toraw(mp, str) - convert an mp_int to a string of bytes mp_tovalue(ch, r) - convert ch to its value when taken as a radix r digit, or -1 if invalid mp_strerror(err) - get a string describing mp_err value 'err' If you compile the MPI library with MP_IOFUNC defined, you will also have access to the following additional I/O function: mp_print(mp, ofp) - print an mp_int as text to output stream ofp Note that mp_radix_size() returns a size in bytes guaranteed to be AT LEAST big enough for the digits output by mp_toradix(). Because it uses an approximation technique to figure out how many digits will be needed, it may return a figure which is larger than necessary. Thus, the caller should not rely on the value to determine how many bytes will actually be written by mp_toradix(). The string mp_toradix() creates will be NUL terminated, so the standard C library function strlen() should be able to ascertain this for you, if you need it. The mp_read_radix() and mp_toradix() functions support bases from 2 to 64 inclusive. If you require more general radix conversion facilities than this, you will need to write them yourself (that's why mp_div_d() is provided, after all). Note: mp_read_radix() will accept as digits either capital or ---- lower-case letters. However, the current implementation of mp_toradix() only outputs upper-case letters, when writing bases betwee 10 and 36. The underlying code supports using lower-case letters, but the interface stub does not have a selector for it. You can add one yourself if you think it is worthwhile -- I do not. Bases from 36 to 64 use lower- case letters as distinct from upper-case. Bases 63 and 64 use the characters '+' and '/' as digits. Note also that compiling with MP_IOFUNC defined will cause inclusion of , so if you are trying to write code which does not depend on the standard C library, you will probably want to avoid this option. This is needed because the mp_print() function takes a standard library FILE * as one of its parameters, and uses the fprintf() function. The mp_toraw() function converts the integer to a sequence of bytes, in big-endian ordering (most-significant byte first). Assuming your bytes are 8 bits wide, this corresponds to base 256. The sign is encoded as a single leading byte, whose value is 0 for zero or positive values, or 1 for negative values. The mp_read_raw() function reverses this process -- it takes a buffer of bytes, interprets the first as a sign indicator (0 = zero/positive, nonzero = negative), and the rest as a sequence of 1-byte digits in big-endian ordering. The mp_raw_size() function returns the exact number of bytes required to store the given integer in "raw" format (as described in the previous paragraph). Zero is returned in case of error; a valid integer will require at least three bytes of storage. In previous versions of the MPI library, an "external representation format" was supported. This was removed, however, because I found I was never using it, it was not as portable as I would have liked, and I decided it was a waste of space. Other Functions --------------- The files 'mpprime.h' and 'mpprime.c' define some routines which are useful for divisibility testing and probabilistic primality testing. The routines defined are: mpp_divis(a, b) - is a divisible by b? mpp_divis_d(a, d) - is a divisible by digit d? mpp_random(a) - set a to random value at current precision mpp_random_size(a, prec) - set a to random value at given precision Note: The mpp_random() and mpp_random_size() functions use the C ---- library's rand() function to generate random values. It is up to the caller to seed this generator before it is called. These functions are not suitable for generating quantities requiring cryptographic-quality randomness; they are intended primarily for use in primality testing. Note too that the MPI library does not call srand(), so your application should do this, if you ever want the sequence to change. mpp_divis_vector(a, v, s, w) - is a divisible by any of the s digits in v? If so, let w be the index of that digit mpp_divis_primes(a, np) - is a divisible by any of the first np primes? If so, set np to the prime which divided a. mpp_fermat(a, d) - test if w^a = w (mod a). If so, returns MP_YES, otherwise MP_NO. mpp_pprime(a, nt) - perform nt iterations of the Rabin- Miller probabilistic primality test on a. Returns MP_YES if all tests passed, or MP_NO if any test fails. The mpp_fermat() function works based on Fermat's little theorem, a consequence of which is that if p is a prime, and (w, p) = 1, then: w^p = w (mod p) Put another way, if w^p != w (mod p), then p is not prime. The test is expensive to compute, but it helps to quickly eliminate an enormous class of composite numbers prior to Rabin-Miller testing. Building the Library -------------------- The MPI library is designed to be as self-contained as possible. You should be able to compile it with your favourite ANSI C compiler, and link it into your program directly. If you are on a Unix system using the GNU C compiler (gcc), the following should work: % gcc -ansi -pedantic -Wall -O2 -c mpi.c The file 'mpi-config.h' defines several configurable parameters for the library, which you can adjust to suit your application. At the time of this writing, the available options are: MP_IOFUNC - Define true to include the mp_print() function, which is moderately useful for debugging. This implicitly includes . MP_MODARITH - Define true to include the modular arithmetic functions. If you don't need modular arithmetic in your application, you can set this to zero to leave out all the modular routines. MP_LOGTAB - If true, the file "logtab.h" is included, which is basically a static table of base 2 logarithms. These are used to compute how big the buffers for radix conversion need to be. If you set this false, the library includes and uses log(). This typically forces you to link against math libraries. MP_ARGCHK - Set to 0, 1, or 2. This defines how the argument checking macro, ARGCHK(), gets expanded. If this is set to zero, ARGCHK() expands to nothing; no argument checks are performed. If this is 1, the ARGCHK() macro expands to code that returns MP_BADARG or similar at runtime. If it is 2, ARGCHK() expands to an assert() call that aborts the program on a bad input. MP_DEBUG - Turns on debugging output. This is probably not at all useful unless you are debugging the library. It tends to spit out a LOT of output. MP_DEFPREC - The default precision of a newly-created mp_int, in digits. The precision can be changed at runtime by the mp_set_prec() function, but this is its initial value. MP_SQUARE - If this is set to a nonzero value, the mp_sqr() function will use an alternate algorithm that takes advantage of the redundant inner product computation when both multiplicands are identical. Unfortunately, with some compilers this is actually SLOWER than just calling mp_mul() with the same argument twice. So if you set MP_SQUARE to zero, mp_sqr() will be expan- ded into a call to mp_mul(). This applies to all the uses of mp_sqr(), including mp_sqrmod() and the internal calls to s_mp_sqr() inside mpi.c The program 'mulsqr' (mulsqr.c) can be used to test which works best for your configuration. Set up the CC and CFLAGS variables in the Makefile, then type: make mulsqr Invoke it with arguments similar to the following: mulsqr 25000 1024 That is, 25000 products computed on 1024-bit values. The output will compare the two timings, and recommend a setting for MP_SQUARE. It is off by default. If you would like to use the mp_print() function (see above), be sure to define MP_IOFUNC in mpi-config.h. Many of the test drivers in the 'tests' subdirectory expect this to be defined (although the test driver 'mpi-test' doesn't need it) The Makefile which comes with the library should take care of building the library for you, if you have set the CC and CFLAGS variables at the top of the file appropriately. By default, they are set up to use the GNU C compiler: CC=gcc CFLAGS=-ansi -pedantic -Wall -O2 If all goes well, the library should compile without warnings using this combination. You should, of course, make whatever adjustments you find necessary. The MPI library distribution comes with several additional programs which are intended to demonstrate the use of the library, and provide a framework for testing it. There are a handful of test driver programs, in the files named 'mptest-X.c', where X is a digit. Also, there are some simple command-line utilities (in the 'utils' directory) for manipulating large numbers. These include: basecvt.c A radix-conversion program, supporting bases from 2 to 64 inclusive. bbsrand.c A BBS (quadratic residue) pseudo-random number generator. The file 'bbsrand.c' is just the driver for the program; the real code lives in the files 'bbs_rand.h' and 'bbs_rand.c' dec2hex.c Converts decimal to hexadecimal gcd.c Computes the greatest common divisor of two values. If invoked as 'xgcd', also computes constants x and y such that (a, b) = ax + by, in accordance with Bezout's identity. hex2dec.c Converts hexadecimal to decimal invmod.c Computes modular inverses isprime.c Performs the Rabin-Miller probabilistic primality test on a number. Values which fail this test are definitely composite, and those which pass are very likely to be prime (although there are no guarantees) lap.c Computes the order (least annihilating power) of a value v modulo m. Very dumb algorithm. primegen.c Generates large (probable) primes. prng.c A pseudo-random number generator based on the BBS generator code in 'bbs_rand.c' sieve.c Implements the Sieve of Eratosthenes, using a big bitmap, to generate a list of prime numbers. fact.c Computes the factorial of an arbitrary precision integer (iterative). exptmod.c Computes arbitrary precision modular exponentiation from the command line (exptmod a b m -> a^b (mod m)) Most of these can be built from the Makefile that comes with the library. Try 'make tools', if your environment supports it. Acknowledgements: ---------------- The algorithms used in this library were drawn primarily from Volume 2 of Donald Knuth's magnum opus, _The Art of Computer Programming_, "Semi-Numerical Methods". Barrett's algorithm for modular reduction came from Menezes, Oorschot, and Vanstone's _Handbook of Applied Cryptography_, Chapter 14. Thanks are due to Tom St. Denis, for finding an obnoxious sign-related bug in mp_read_raw() that made things break on platforms which use signed chars. About the Author ---------------- This software was written by Michael J. Fromberger. You can contact the author as follows: E-mail: Postal: 8000 Cummings Hall, Thayer School of Engineering Dartmouth College, Hanover, New Hampshire, USA PGP key: http://linguist.dartmouth.edu/~sting/keys/mjf.html 9736 188B 5AFA 23D6 D6AA BE0D 5856 4525 289D 9907 Last updated: 16-Jan-2000