1Crypt::Primes(3) User Contributed Perl Documentation Crypt::Primes(3)
2
6 Crypt::Primes - Provable Prime Number Generator suitable for Crypto‐
7 graphic Applications.
8
10 $Revision: 0.49 $
11 $Date: 2001/06/11 01:04:23 $
12
14 # generate a random, provable 512-bit prime.
15 use Crypt::Primes qw(maurer);
17 my $prime = maurer (Size => 512);
18 # generate a random, provable 2048-bit prime and report
20 # progress on console.
21 my $another_prime = maurer (
23 Size => 2048,
24 Verbosity => 1
25 );
26 # generate a random 1024-bit prime and a group
28 # generator of Z*(n).
29 my $hash_ref = maurer (
31 Size => 1024,
32 Generator => 1,
33 Verbosity => 1
34 );
35
37 The codebase is stable, but the API will most definitely change in a
38 future release.
39
41 This module implements Ueli Maurer's algorithm for generating large
42 provable primes and secure parameters for public-key cryptosystems.
43 The generated primes are almost uniformly distributed over the set of
44 primes of the specified bitsize and expected time for generation is
45 less than the time required for generating a pseudo-prime of the same
46 size with Miller-Rabin tests. Detailed description and running time
47 analysis of the algorithm can be found in Maurer's paper[1].
48 Crypt::Primes is a pure perl implementation. It uses Math::Pari for
50 multiple precision integer arithmetic and number theoretic functions.
51 Random numbers are gathered with Crypt::Random, a perl interface to
52 /dev/u?random devices found on most modern Unix operating systems.
53
55 The following functions are availble for import. They must be
56 explicitely imported.
57 maurer(%params)
59 Generates a prime number of the specified bitsize. Takes a hash as
60 parameter and returns a Math::Pari object (prime number) or a hash
61 reference (prime number and generator) when group generator compu‐
62 tation is requested. Following hash keys are understood:
63 Size
65 Bitsize of the required prime number.
66 Verbosity
68 Level of verbosity of progress reporting. Report is printed on
69 STDOUT. Level of 1 indicates normal, terse reporting. Level of 2
70 prints lots of intermediate computations, useful for debugging.
71 Generator
73 When Generator key is set to a non-zero value, a group generator of
74 Z*(n) is computed. Group generators are required key material in
75 public-key cryptosystems like Elgamal and Diffie-Hellman that are
76 based on intractability of the discrete logarithm problem. When
77 this option is present, maurer() returns a hash reference that con‐
78 tains two keys, Prime and Generator.
79 Relprime
81 When set to 1, maurer() stores intermediate primes in a class
82 array, and ensures they are not used during construction of primes
83 in the future calls to maurer() with Reprime => 1. This is used by
84 rsaparams().
85 Intermediates
87 When set to 1, maurer() returns a hash reference that contains
88 (corresponding to the key 'Intermediates') a reference to an array
89 of intermediate primes generated.
90 Factors
92 When set to 1, maurer() returns a hash reference that contains
93 (corresponding to the key 'Factors') a reference to an array of
94 factors of p-1 where p is the prime generated, and also (corre‐
95 sponding to the key 'R') a divisor of p.
96 rsaparams(%params)
98 Generates two primes (p,q) and public exponent (e) of a RSA key
99 pair. The key pair generated with this method is resistant to iter‐
100 ative encryption attack. See Appendix 2 of [1] for more informa‐
101 tion.
102 rsaparams() takes the same arguments as maurer() with the exception
104 of `Generator' and `Relprime'. Size specifies the common bitsize
105 of p an q. Returns a hash reference with keys p, q and e.
106 trialdiv($n,$limit)
108 Performs trial division on $n to ensure it's not divisible by any
109 prime smaller than or equal to $limit. The module maintains a
110 lookup table of primes (from 2 to 65521) for this purpose. If
111 $limit is not provided, a suitable value is computed automatically.
112 trialdiv() is used by maurer() to weed out composite random numbers
113 before performing computationally intensive modular exponentiation
114 tests. It is, however, documented should you need to use it
115 directly.
116
118 This module implements a modified FastPrime, as described in [1], to
119 facilitate group generator computation. (Refer to [1] and [2] for
120 description and pseudo-code of FastPrime). The modification involves
121 introduction of an additional constraint on relative size r of q.
122 While computing r, we ensure k * r is always greater than maxfact,
123 where maxfact is the bitsize of the largest number we can factor eas‐
124 ily. This value defaults to 140 bits. As a result, R is always
125 smaller than maxfact, which allows us to get a complete factorization
126 of 2Rq and use it to find a generator of the cyclic group Z*(2Rq).
127
129 Crypt::Primes generates 512-bit primes in 7 seconds (on average), and
130 1024-bit primes in 37 seconds (on average), on my PII 300 Mhz notebook.
131 There are no computational limits by design; primes upto 8192-bits were
132 generated to stress test the code. For detailed runtime analysis see
133 [1].
134
136 largeprimes(1), Crypt::Random(3), Math::Pari(3)
137
139 1 Fast Generation of Prime Numbers and Secure Public-Key Cryptographic
140 Parameters, Ueli Maurer (1994).
141 2 Corrections to Fast Generation of Prime Numbers and Secure Public-Key
142 Cryptographic Parameters, Ueli Maurer (1996).
143 3 Handbook of Applied Cryptography by Menezes, Paul C. van Oorschot and
144 Scott Vanstone (1997).
145 Documents 1 & 2 can be found under docs/ of the source distribution.
146
148 Vipul Ved Prakash, <mail@vipul.net>
149
151 Copyright (c) 1998-2001, Vipul Ved Prakash. All rights reserved. This
152 code is free software; you can redistribute it and/or modify it under
153 the same terms as Perl itself.
154
156 Maurer's algorithm generates primes of progressively larger bitsize
157 using a recursive construction method. The algorithm enters recursion
158 with a prime number and bitsize of the next prime to be generated.
159 (Bitsizes of the intermediate primes are computed using a probability
160 distribution that ensures generated primes are sufficiently random.)
161 This recursion can be distributed over multiple machines, participating
162 in a competitive computation model, to achieve close to best running
163 time of the algorithm. Support for this will be implemented some day,
164 possibly when the next version of Penguin hits CPAN.
165perl v5.8.8 2007-04-18 Crypt::Primes(3)