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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);
mp_set_ulong(mp_int *mp, unsigned 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 <stdio.h>, 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 <stdio.h>.
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 <math.h> 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: <sting@linguist.dartmouth.edu>
Postal: 8000 Cummings Hall, Thayer School of Engineering
Dartmouth College, Hanover, New Hampshire, USA
9736 188B 5AFA 23D6 D6AA BE0D 5856 4525 289D 9907
Last updated: 16-Jan-2000