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 ***** BEGIN LICENSE BLOCK *****

 Version: MPL 1.1/GPL 2.0/LGPL 2.1

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 1.1 (the "License"); you may not use this file except in compliance with

 the License. You may obtain a copy of the License at

 http://www.mozilla.org/MPL/

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 License.

 The Original Code is the MPI Arbitrary Precision Integer Arithmetic

 library.

 The Initial Developer of the Original Code is

 Michael J. Fromberger <sting@linguist.dartmouth.edu>

 Portions created by the Initial Developer are Copyright (C) 1997-2000

 the Initial Developer. All Rights Reserved.

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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.

 The library comes with a Perl script, 'types.pl', which will scan your

 current Makefile settings, and attempt to find good definitions for

 these types.  It relies on a Unix sort of build environment, so it

 probably won't work under MacOS or Windows, but it can be convenient

 if you're porting to a new flavour of Unix.  Just run 'types.pl' at

 the command line, and it will spit out its results to the standard

 output.

 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

 mp_sqrt(a, c)           - computes c = floor(sqrt(a))

 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_cmp_int(a, z)        - compare a <=> z, z is a signed long integer

 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_NUMTH        - Define true to include number theoretic functions

                   such as mp_gcd(), mp_lcm(), and mp_invmod().

 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_MEMSET       - If true, use memset() to zero buffers.  If you run

                   into weird alignment related bugs, set this to zero

                   and an explicit loop will be used.

 MP_MEMCPY       - If true, use memcpy() to copy buffers.  If you run

                   into weird alignment bugs, set this to zero and an

                   explicit loop will be used.

 MP_CRYPTO       - If true, whenever arrays of digits are free'd, they

                   are zeroed first.  This is useful if you're using

                   the library in a cryptographic environment; however,

                   it does add overhead to each free operation.  For

                   performance, if you don't care about zeroing your

                   buffers, set this to false.

 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.

 Testing the Library

 -------------------

 Automatic test vectors are included, in the form of a program called

 'mpi-test'.  To build this program and run all the tests, simply

 invoke the shell script 'all-tests'.  If all the tests pass, you

 should see a message:

         All tests passed

 If something went wrong, you'll get:

         One or more tests failed.

 If this happens, scan back through the preceding lines, to see which

 test failed.  Any failure indicates a bug in the library, which needs

 to be fixed before it will give accurate results.  If you get any such

 thing, please let me know, and I'll try to fix it.  Please let me know

 what platform and compiler you were using, as well as which test

 failed.  If a reason for failure was given, please send me that text

 as well.

 If you're on a system where the standard Unix build tools don't work,

 you can build the 'mpi-test' program manually, and run it by hand.

 This is tedious and obnoxious, sorry.

 Further manual testing can be performed by building the manual testing

 programs, whose source is found in the 'tests' subdirectory.  Each

 test is in a source file called 'mptest-X.c'.  The Makefile contains a

 target to build all of them at once:

         make tests

 Read the comments at the top of each source file to see what the

 driver is supposed to test.  You probably don't need to do this; these

 programs were only written to help me as I was developing the library.

 The relevant files are:

 mpi-test.c              The source for the test driver

 make-test-arrays        A Perl script to generate some of the internal

                         data structures used by mpi-test.c

 test-arrays.txt         The source file for make-test-arrays

 all-tests               A Bourne shell script which runs all the

                         tests in the mpi-test suite

 Running 'make mpi-test' should build the mpi-test program.  If you

 cannot use make, here is what needs to be done:

 (1) Use 'make-test-arrays' to generate the file 'test-info.c' from

     the 'test-arrays.txt' file.  Since Perl can be found everywhere,

     this should be no trouble.  Under Unix, this looks like:

         make-test-arrays test-arrays.txt > test-info.c

 (2) Build the MPI library:

         gcc -ansi -pedantic -Wall -c mpi.c

 (3) Build the mpi-test program:

         gcc -ansi -pedantic -Wall -o mpi-test mpi.o mpi-test.c

 When you've got mpi-test, you can use 'all-tests' to run all the tests

 made available by mpi-test.  If any of them fail, there should be a

 diagnostic indicating what went wrong.  These are fairly high-level

 diagnostics, and won't really help you debug the problem; they're

 simply intended to help you isolate which function caused the problem.

 If you encounter a problem of this sort, feel free to e-mail me, and I

 will certainly attempt to help you debug it.

 Note:   Several of the tests hard-wired into 'mpi-test' operate under

 ----    the assumption that you are using at least a 16-bit mp_digit 

         type.  If that is not true, several tests might fail, because 

         of range problems with the maximum digit value.

         If you are using an 8-bit digit, you will also need to 

         modify the code for mp_read_raw(), which assumes that

         multiplication by 256 can be done with mp_mul_d(), a

         fact that fails when DIGIT_MAX is 255.  You can replace

         the call with s_mp_lshd(), which will give you the same

         effect, and without doing as much work. :)

 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

 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