* NAME
* SYNOPSIS
* WARNING
* DESCRIPTION
* METHODS
* IMPLEMENTATION NOTE
* RUNNING TIME
* SEE ALSO
* BIBLIOGRAPHY
* AUTHOR
* LICENSE
* TODO
_________________________________________________________________
NAME
Crypt::Primes - Provable Prime Number Generator suitable for
Cryptographic Applications.
_________________________________________________________________
SYNOPSIS
# generate a random, provable 512-bit prime.
use Crypt::Primes qw( maurer );
my $prime = maurer ( Size => 512 );
# generate a random, provable 2048-bit prime and report
# progress on console.
my $another_prime = maurer (
Size => 2048,
Verbosity => 1
);
# generate a random 1024-bit prime and a group
# generator of Z*(n).
my $hash_ref = maurer (
Size => 1024,
Generator => 1,
Verbosity => 1
);
_________________________________________________________________
WARNING
The codebase is stable, but the API will change in a future release.
Be warned.
_________________________________________________________________
DESCRIPTION
This module implements Ueli Maurer's algorithm for generating large
provable primes and secure parameters for public-key cryptosystems.
The generated primes are almost uniformly distributed over the set of
primes of the specified bitsize and expected time for generation is
less than the time required for generating a pseudo-prime of the same
size with Miller-Rabin tests. Detailed description and running time
analysis of the algorithm can be found in Maurer's paper[1].
Crypt::Primes is a pure perl implementation. It uses Math::Pari for
multiple precision integer arithmetic and number theoretic functions.
Random numbers are gathered with Crypt::Random, a perl interface to
/dev/u?random devices found on modern Unix operating systems.
_________________________________________________________________
METHODS
maurer()
Generates a prime number of the specified bitsize. Takes a hash
as parameter and returns a Math::Pari object (prime number) or
a hash reference (prime number and generator) when group
generator computation is requested. Following hash keys are
understood:
Size
Bitsize of the required prime number.
Verbosity
Level of verbosity of progress reporting. Report is printed on
STDOUT. Level of 1 indicates normal, terse reporting. Level of
2 prints lots of intermediate computations, useful for
debugging.
Generator
When Generator key is set to a non-zero value, a group
generator of Z*(n) is computed. Group generators are required
key material in public-key cryptosystems like Elgamal and
Diffie-Hellman that are based on intractability of the discrete
logarithm problem. When this option is present, maurer()
returns a hash reference that contains two keys, Prime and
Generator.
trialdiv($n,$limit)
Performs trial division on $n to ensure it's not divisible by
any prime smaller than or equal to $limit. The module maintains
a lookup table of primes (from 2 to 65521) for this purpose. If
$limit is not provided, a suitable value is computed
automatically. trialdiv() is used by maurer() to weed out
composite random numbers before performing computationally
intensive modular exponentiation tests. It is, however,
documented should you need to use it directly.
_________________________________________________________________
IMPLEMENTATION NOTE
This module implements a modified FastPrime, as described in [1], to
facilitate group generator computation. (Refer to [1] and [2] for
description and pseudo-code of FastPrime). The modification involves
introduction of an additional constraint on relative size r of q.
While computing r, we ensure k * r is always greater than maxfact,
where maxfact is the bitsize of the largest number we can factor
easily. This value defaults to 140 bits. As a result, R is always
smaller than maxfact, which allows us to get a complete factorization
of 2Rq and use it to find a generator of the cyclic group Z*(2Rq).
_________________________________________________________________
RUNNING TIME
Crypt::Primes generates 512-bit primes in 7 seconds (on average), and
1024-bit primes in 37 seconds (on average), on my PII 300 Mhz
notebook. There are no computational limits by design; primes upto
8192-bits were generated to stress test the code. For detailed runtime
analysis see [1].
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SEE ALSO
largeprimes(1), Crypt::Random(3)
_________________________________________________________________
BIBLIOGRAPHY
1. Fast Generation of Prime Numbers and Secure Public-Key
Cryptographic Parameters, Ueli Maurer (1994).
2. Corrections to Fast Generation of Prime Numbers and Secure
Public-Key Cryptographic Parameters, Ueli Maurer (1996).
3. Handbook of Applied Cryptography by Menezes, Paul C. van Oorschot
and Scott Vanstone (1997).
Documents 1 & 2 can be found under docs/ of the source distribution.
_________________________________________________________________
AUTHOR
Vipul Ved Prakash,
_________________________________________________________________
LICENSE
Copyright (c) 1998-2000, Vipul Ved Prakash. All rights reserved. This
code is free software; you can redistribute it and/or modify it under
the same terms as Perl itself.
_________________________________________________________________
TODO
Maurer's algorithm generates primes of progressively larger bitsize
using a recursive construction method. The algorithm enters recursion
with a prime number and bitsize of the next prime to be generated.
(Bitsizes of the intermediate primes are computed using a probability
distribution that ensures generated primes are sufficiently random.)
This recursion can be distributed over multiple machines,
participating in a competitive computation model, to achieve close to
best running time of the algorithm. Support for this will be
implemented some day, possibly when the next version of Penguin hits
CPAN.