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Perl POD Document | 2003-01-16 | 51.2 KB | 902 lines

#!/usr/bin/perl -s ## ## Crypt::Primes -- Provable Prime Number Generator ## for Cryptographic Applications. ## ## 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. ## ## $Id: Primes.pm,v 0.49 2001/06/11 01:04:23 vipul Exp vipul $ package Crypt::Primes; require Exporter; use vars qw($VERSION @EXPORT_OK); use Crypt::Random qw( makerandom makerandom_itv ); use Math::Pari qw( PARI Mod floor sqrt); *import = \&Exporter::import; @EXPORT_OK = qw( maurer trialdiv rsaparams ); ( $VERSION ) = '$Revision: 0.50 $' =~ /(\d+\.\d+)/; ## list of small primes for trial division. @PRIMES = qw( 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 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62687 62701 62723 62731 62743 62753 62761 62773 62791 62801 62819 62827 62851 62861 62869 62873 62897 62903 62921 62927 62929 62939 62969 62971 62981 62983 62987 62989 63029 63031 63059 63067 63073 63079 63097 63103 63113 63127 63131 63149 63179 63197 63199 63211 63241 63247 63277 63281 63299 63311 63313 63317 63331 63337 63347 63353 63361 63367 63377 63389 63391 63397 63409 63419 63421 63439 63443 63463 63467 63473 63487 63493 63499 63521 63527 63533 63541 63559 63577 63587 63589 63599 63601 63607 63611 63617 63629 63647 63649 63659 63667 63671 63689 63691 63697 63703 63709 63719 63727 63737 63743 63761 63773 63781 63793 63799 63803 63809 63823 63839 63841 63853 63857 63863 63901 63907 63913 63929 63949 63977 63997 64007 64013 64019 64033 64037 64063 64067 64081 64091 64109 64123 64151 64153 64157 64171 64187 64189 64217 64223 64231 64237 64271 64279 64283 64301 64303 64319 64327 64333 64373 64381 64399 64403 64433 64439 64451 64453 64483 64489 64499 64513 64553 64567 64577 64579 64591 64601 64609 64613 64621 64627 64633 64661 64663 64667 64679 64693 64709 64717 64747 64763 64781 64783 64793 64811 64817 64849 64853 64871 64877 64879 64891 64901 64919 64921 64927 64937 64951 64969 64997 65003 65011 65027 65029 65033 65053 65063 65071 65089 65099 65101 65111 65119 65123 65129 65141 65147 65167 65171 65173 65179 65183 65203 65213 65239 65257 65267 65269 65287 65293 65309 65323 65327 65353 65357 65371 65381 65393 65407 65413 65419 65423 65437 65447 65449 65479 65497 65519 65521 ); my $l; my @used; my @gens; sub rsaparams { my (%params) = @_; @used = (); my %paramsrsa = (%params, (Relprime => 1)); my $p = maurer (%paramsrsa); print "\n" if $params{Verbosity}; my $q = maurer (%paramsrsa); print "\n" if $params{Verbosity}; return { p => $p, q => $q, e => 65537 } } sub maurer { my ( %params ) = @_; my $k = $params { Size }; #-- bitsize of the required prime. my $v = $params { Verbosity }; #-- process reporting verbosity. #-- whether to find intermediate generators. my $p0 = 20; #-- generate primes of bitsize less #-- than $p0 with nextprime(). my $maxfact = 140; #-- bitsize of the biggest number we #-- can factor easily. $| = 1; if ( $k <= $p0 ) { my $spr = makerandom Size => $k; return Math::Pari::nextprime ( $spr ); } my $c_opt = 0.09; my $m = 20; my $B = floor ( $c_opt * ( $k ** 2 ) ); my $r = PARI ( 2 ** ( ( rand ) - 1 ) ); srand ( makerandom ( Size => 32, Strength => 0 ) ); if ( $k > ( 2 * $m ) ) { my $frs = 0; until ( $frs ) { $r = 2 ** ( (rand) - 1 ); my $rbits = $k-($r*$k); $frs = 1 if $rbits > $m; $frs = 0 if ($params{Generator} && ($rbits > $maxfact)); } } else { $r = 0.5; } my $q = maurer ( Size => floor ( $r * $k ) + 2, Verbosity => $v, Generator => $params{Generator}, Intermediates => $params{Intermediates} ); if ($params{Relprime}) { while ((grep /$q/, @used)) { print "<" if $v == 1; $q = maurer ( Size => floor ( $r * $k ) + 2, Verbosity => $v, Relprime => 1, Generator => $params{Generator}, Intermediates => $params{Intermediates} ); } } $q = $q->{Prime} if (ref $q eq 'HASH'); print "B = $B, r = $r, k = $k, q = $q\n" if $v == 2; push @used, $q; my $o1 = PARI ( 2 * $q ); my $o2 = PARI '2' ** PARI "@{[$k - 1]}"; my $I = floor ( $o2 / $o1 ); my $n = PARI (); while (1) { my $R = PARI ( makerandom_itv ( Lower => $I + 1, Upper => 2 * $I ) ); $n = floor( 2 * $R * $q + 1 ); print "n = $n\n" if $v == 2; print "." if $v == 1; my $a = PARI(); if ( trialdiv ( $n, $B ) ) { BASE: { print "(passes trial division)\n" if $v == 2; print "+" if $v == 1; $a = makerandom_itv( Lower => 2, Upper=> $n - 2 ); $a = makerandom_itv( Lower => 2, Upper=> $R ) if $params{Generator}; my $t1 = Mod (2, $n); my $t2 = Mod ($a, $n); my $b = $t2 ** ( $n - 1 ); if ( ($b == 1) && ( $q > $n ** (1/3.0) ) ) { print "$n is prime! a = $a, R = $R\n" if $v == 2; print "($k)" if $v == 1; my @s; if ($params{Generator}) { print "computing a generator...\n" if $v == 2; print "# " if $v == 1; @s = Math::Pari::factor($R) =~ m:[\[\(\;](\d+)(?=,):g; push @s, ($q,2); my $p = PARI (1); for (@s) { $p *= $_ }; for (@s) { my $b = Mod ($a, $p) ** PARI($p/$_); if ($b == 1) { print "@!" if $v == 1; print "$a is not a generator\n" if $v == 2; next BASE; } } } my @factors = ( Factors => \@s, R => $R ) if $params{Factors}; my @intermediates = ( Intermediates => \@used ) if $params{Intermediates}; return { @factors, @intermediates, Prime => $n, Generator => $a } if $params{Generator}; return { @factors, @intermediates, Prime => $n } if $params{Factors} or $params{Intermediates}; return $n; } } } } } sub trialdiv { my ( $n, $limit, $v ) = @_; my $end = _trialdiv_limit ( $n, $limit ); for ( $i=0; $i <= $end; $i++ ) { return undef unless Math::Pari::gmod ( $n, $PRIMES[ $i ] ); } return 1; } sub _trialdiv_limit { my ($n, $limit) = @_; my $start = 0; my $mid; my $end = $#PRIMES; $limit = int(sqrt($n)) unless $limit; if ( $limit <= $PRIMES[ $end ] ) { while ( ( $start + 1 ) != $end ) { use integer; $mid = int (( $end + $start ) / 2); no integer; $start = $mid if $limit > $PRIMES[ $mid ]; $end = $mid if $limit <= $PRIMES[ $mid ]; } } return $end; } "True Value"; =head1 NAME Crypt::Primes - Provable Prime Number Generator suitable for Cryptographic Applications. =head1 VERSION $Revision: 0.49 $ $Date: 2001/06/11 01:04:23 $ =head1 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 ); =head1 WARNING The codebase is stable, but the API will most definitely change in a future release. =head1 DESCRIPTION This module implements Ueli Maurer's algorithm for generating large I<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 most modern Unix operating systems. =head1 FUNCTIONS The following functions are availble for import. They must be explicitely imported. =over 4 =item B<maurer(%params)> 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: =back =over 0 =item B<Size> Bitsize of the required prime number. =item B<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. =item B<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. =item B<Relprime> When set to 1, maurer() stores intermediate primes in a class array, and ensures they are not used during construction of primes in the future calls to maurer() with Reprime => 1. This is used by rsaparams(). =item B<Intermediates> When set to 1, maurer() returns a hash reference that contains (corresponding to the key 'Intermediates') a reference to an array of intermediate primes generated. =item B<Factors> When set to 1, maurer() returns a hash reference that contains (corresponding to the key 'Factors') a reference to an array of factors of p-1 where p is the prime generated, and also (corresponding to the key 'R') a divisor of p. =back =over 4 =item B<rsaparams(%params)> Generates two primes (p,q) and public exponent (e) of a RSA key pair. The key pair generated with this method is resistant to iterative encryption attack. See Appendix 2 of [1] for more information. rsaparams() takes the same arguments as maurer() with the exception of `Generator' and `Relprime'. Size specifies the common bitsize of p an q. Returns a hash reference with keys p, q and e. =item B<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. =back =head1 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). =head1 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]. =head1 SEE ALSO largeprimes(1), Crypt::Random(3), Math::Pari(3) =head1 BIBLIOGRAPHY =over 4 =item 1 Fast Generation of Prime Numbers and Secure Public-Key Cryptographic Parameters, Ueli Maurer (1994). =item 2 Corrections to Fast Generation of Prime Numbers and Secure Public-Key Cryptographic Parameters, Ueli Maurer (1996). =item 3 Handbook of Applied Cryptography by Menezes, Paul C. van Oorschot and Scott Vanstone (1997). =item Documents 1 & 2 can be found under docs/ of the source distribution. =back =head1 AUTHOR Vipul Ved Prakash, E<lt>mail@vipul.netE<gt> =cut =head1 LICENSE Copyright (c) 1998-2001, 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. =head1 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.