* 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]. _________________________________________________________________ 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.