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Additional MPI utilities
------------------------
The files 'mpprime.h' and 'mpprime.c' define some useful extensions to
the MPI library for dealing with prime numbers (in particular, testing
for divisbility, and the Rabin-Miller probabilistic primality test).
The files 'mplogic.h' and 'mplogic.c' define extensions to the MPI
library for doing bitwise logical operations and shifting.
This document assumes you have read the help file for the MPI library
and understand its conventions.
Divisibility (mpprime.h)
------------
To test a number for divisibility by another number:
mpp_divis(a, b) - test if b|a
mpp_divis_d(a, d) - test if d|a
Each of these functions returns MP_YES if its initial argument is
divisible by its second, or MP_NO if it is not. Other errors may be
returned as appropriate (such as MP_RANGE if you try to test for
divisibility by zero).
Randomness (mpprime.h)
----------
To generate random data:
mpp_random(a) - fill a with random data
mpp_random_size(a, p) - fill a with p digits of random data
The mpp_random_size() function increases the precision of a to at
least p, then fills all those digits randomly. The mp_random()
function fills a to its current precision (as determined by the number
of significant digits, USED(a))
Note that these functions simply use the C library's rand() function
to fill a with random digits up to its precision. This should be
adequate for primality testing, but should not be used for
cryptographic applications where truly random values are required for
security.
You should call srand() in your driver program in order to seed the
random generator; this function doesn't call it.
Primality Testing (mpprime.h)
-----------------
mpp_divis_vector(a, v, s, w) - is a divisible by any of the s values
in v, and if so, w = which.
mpp_divis_primes(a, np) - is a divisible by any of the first np primes?
mpp_fermat(a, w) - is a pseudoprime with respect to witness w?
mpp_pprime(a, nt) - run nt iterations of Rabin-Miller on a.
The mpp_divis_vector() function tests a for divisibility by each
member of an array of digits. The array is v, the size of that array
is s. Returns MP_YES if a is divisible, and stores the index of the
offending digit in w. Returns MP_NO if a is not divisible by any of
the digits in the array.
A small table of primes is compiled into the library (typically the
first 128 primes, although you can change this by editing the file
'primes.c' before you build). The global variable prime_tab_size
contains the number of primes in the table, and the values themselves
are in the array prime_tab[], which is an array of mp_digit.
The mpp_divis_primes() function is basically just a wrapper around
mpp_divis_vector() that uses prime_tab[] as the test vector. The np
parameter is a pointer to an mp_digit -- on input, it should specify
the number of primes to be tested against. If a is divisible by any
of the primes, MP_YES is returned and np is given the prime value that
divided a (you can use this if you're factoring, for example).
Otherwise, MP_NO is returned and np is untouched.
The function mpp_fermat() performs Fermat's test, using w as a
witness. This test basically relies on the fact that if a is prime,
and w is relatively prime to a, then:
w^a = w (mod a)
That is,
w^(a - 1) = 1 (mod a)
The function returns MP_YES if the test passes, MP_NO if it fails. If
w is relatively prime to a, and the test fails, a is definitely
composite. If w is relatively prime to a and the test passes, then a
is either prime, or w is a false witness (the probability of this
happening depends on the choice of w and of a ... consult a number
theory textbook for more information about this).
Note: If (w, a) != 1, the output of this test is meaningless.
----
The function mpp_pprime() performs the Rabin-Miller probabilistic
primality test for nt rounds. If all the tests pass, MP_YES is
returned, and a is probably prime. The probability that an answer of
MP_YES is incorrect is no greater than 1 in 4^nt, and in fact is
usually much less than that (this is a pessimistic estimate). If any
test fails, MP_NO is returned, and a is definitely composite.
Bruce Schneier recommends at least 5 iterations of this test for most
cryptographic applications; Knuth suggests that 25 are reasonable.
Run it as many times as you feel are necessary.
See the programs 'makeprime.c' and 'isprime.c' for reasonable examples
of how to use these functions for primality testing.
Bitwise Logic (mplogic.c)
-------------
The four commonest logical operations are implemented as:
mpl_not(a, b) - Compute bitwise (one's) complement, b = ~a
mpl_and(a, b, c) - Compute bitwise AND, c = a & b
mpl_or(a, b, c) - Compute bitwise OR, c = a | b
mpl_xor(a, b, c) - Compute bitwise XOR, c = a ^ b
Left and right shifts are available as well. These take a number to
shift, a destination, and a shift amount. The shift amount must be a
digit value between 0 and DIGIT_BIT inclusive; if it is not, MP_RANGE
will be returned and the shift will not happen.
mpl_rsh(a, b, d) - Compute logical right shift, b = a >> d
mpl_lsh(a, b, d) - Compute logical left shift, b = a << d
Since these are logical shifts, they fill with zeroes (the library
uses a signed magnitude representation, so there are no sign bits to
extend anyway).
Command-line Utilities
----------------------
A handful of interesting command-line utilities are provided. These
are:
lap.c - Find the order of a mod m. Usage is 'lap <a> <m>'.
This uses a dumb algorithm, so don't use it for
a really big modulus.
invmod.c - Find the inverse of a mod m, if it exists. Usage
is 'invmod <a> <m>'
sieve.c - A simple bitmap-based implementation of the Sieve
of Eratosthenes. Used to generate the table of
primes in primes.c. Usage is 'sieve <nbits>'
prng.c - Uses the routines in bbs_rand.{h,c} to generate
one or more 32-bit pseudo-random integers. This
is mainly an example, not intended for use in a
cryptographic application (the system time is
the only source of entropy used)
dec2hex.c - Convert decimal to hexadecimal
hex2dec.c - Convert hexadecimal to decimal
basecvt.c - General radix conversion tool (supports 2-64)
isprime.c - Probabilistically test an integer for primality
using the Rabin-Miller pseudoprime test combined
with division by small primes.
primegen.c - Generate primes at random.
exptmod.c - Perform modular exponentiation
ptab.pl - A Perl script to munge the output of the sieve
program into a compilable C structure.
Other Files
-----------
PRIMES - Some randomly generated numbers which are prime with
extremely high probability.
README - You're reading me already.
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