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|
#!/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
563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661
673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809
811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937
941 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049
1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153
1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277
1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381
1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487
1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597
1601 1607 1609 1613 1619 1621 1627 1637 1657 1663 1667 1669 1693 1697 1699
1709 1721 1723 1733 1741 1747 1753 1759 1777 1783 1787 1789 1801 1811 1823
1831 1847 1861 1867 1871 1873 1877 1879 1889 1901 1907 1913 1931 1933 1949
1951 1973 1979 1987 1993 1997 1999 2003 2011 2017 2027 2029 2039 2053 2063
2069 2081 2083 2087 2089 2099 2111 2113 2129 2131 2137 2141 2143 2153 2161
2179 2203 2207 2213 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 2293
2297 2309 2311 2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393
2399 2411 2417 2423 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539
2543 2549 2551 2557 2579 2591 2593 2609 2617 2621 2633 2647 2657 2659 2663
2671 2677 2683 2687 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741 2749
2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851 2857 2861
2879 2887 2897 2903 2909 2917 2927 2939 2953 2957 2963 2969 2971 2999 3001
3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 3089 3109 3119 3121 3137
3163 3167 3169 3181 3187 3191 3203 3209 3217 3221 3229 3251 3253 3257 3259
3271 3299 3301 3307 3313 3319 3323 3329 3331 3343 3347 3359 3361 3371 3373
3389 3391 3407 3413 3433 3449 3457 3461 3463 3467 3469 3491 3499 3511 3517
3527 3529 3533 3539 3541 3547 3557 3559 3571 3581 3583 3593 3607 3613 3617
3623 3631 3637 3643 3659 3671 3673 3677 3691 3697 3701 3709 3719 3727 3733
3739 3761 3767 3769 3779 3793 3797 3803 3821 3823 3833 3847 3851 3853 3863
3877 3881 3889 3907 3911 3917 3919 3923 3929 3931 3943 3947 3967 3989 4001
4003 4007 4013 4019 4021 4027 4049 4051 4057 4073 4079 4091 4093 4099 4111
4127 4129 4133 4139 4153 4157 4159 4177 4201 4211 4217 4219 4229 4231 4241
4243 4253 4259 4261 4271 4273 4283 4289 4297 4327 4337 4339 4349 4357 4363
4373 4391 4397 4409 4421 4423 4441 4447 4451 4457 4463 4481 4483 4493 4507
4513 4517 4519 4523 4547 4549 4561 4567 4583 4591 4597 4603 4621 4637 4639
4643 4649 4651 4657 4663 4673 4679 4691 4703 4721 4723 4729 4733 4751 4759
4783 4787 4789 4793 4799 4801 4813 4817 4831 4861 4871 4877 4889 4903 4909
4919 4931 4933 4937 4943 4951 4957 4967 4969 4973 4987 4993 4999 5003 5009
5011 5021 5023 5039 5051 5059 5077 5081 5087 5099 5101 5107 5113 5119 5147
5153 5167 5171 5179 5189 5197 5209 5227 5231 5233 5237 5261 5273 5279 5281
5297 5303 5309 5323 5333 5347 5351 5381 5387 5393 5399 5407 5413 5417 5419
5431 5437 5441 5443 5449 5471 5477 5479 5483 5501 5503 5507 5519 5521 5527
5531 5557 5563 5569 5573 5581 5591 5623 5639 5641 5647 5651 5653 5657 5659
5669 5683 5689 5693 5701 5711 5717 5737 5741 5743 5749 5779 5783 5791 5801
5807 5813 5821 5827 5839 5843 5849 5851 5857 5861 5867 5869 5879 5881 5897
5903 5923 5927 5939 5953 5981 5987 6007 6011 6029 6037 6043 6047 6053 6067
6073 6079 6089 6091 6101 6113 6121 6131 6133 6143 6151 6163 6173 6197 6199
6203 6211 6217 6221 6229 6247 6257 6263 6269 6271 6277 6287 6299 6301 6311
6317 6323 6329 6337 6343 6353 6359 6361 6367 6373 6379 6389 6397 6421 6427
6449 6451 6469 6473 6481 6491 6521 6529 6547 6551 6553 6563 6569 6571 6577
6581 6599 6607 6619 6637 6653 6659 6661 6673 6679 6689 6691 6701 6703 6709
6719 6733 6737 6761 6763 6779 6781 6791 6793 6803 6823 6827 6829 6833 6841
6857 6863 6869 6871 6883 6899 6907 6911 6917 6947 6949 6959 6961 6967 6971
6977 6983 6991 6997 7001 7013 7019 7027 7039 7043 7057 7069 7079 7103 7109
7121 7127 7129 7151 7159 7177 7187 7193 7207 7211 7213 7219 7229 7237 7243
7247 7253 7283 7297 7307 7309 7321 7331 7333 7349 7351 7369 7393 7411 7417
7433 7451 7457 7459 7477 7481 7487 7489 7499 7507 7517 7523 7529 7537 7541
7547 7549 7559 7561 7573 7577 7583 7589 7591 7603 7607 7621 7639 7643 7649
7669 7673 7681 7687 7691 7699 7703 7717 7723 7727 7741 7753 7757 7759 7789
7793 7817 7823 7829 7841 7853 7867 7873 7877 7879 7883 7901 7907 7919 7927
7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081 8087
8089 8093 8101 8111 8117 8123 8147 8161 8167 8171 8179 8191 8209 8219 8221
8231 8233 8237 8243 8263 8269 8273 8287 8291 8293 8297 8311 8317 8329 8353
8363 8369 8377 8387 8389 8419 8423 8429 8431 8443 8447 8461 8467 8501 8513
8521 8527 8537 8539 8543 8563 8573 8581 8597 8599 8609 8623 8627 8629 8641
8647 8663 8669 8677 8681 8689 8693 8699 8707 8713 8719 8731 8737 8741 8747
8753 8761 8779 8783 8803 8807 8819 8821 8831 8837 8839 8849 8861 8863 8867
8887 8893 8923 8929 8933 8941 8951 8963 8969 8971 8999 9001 9007 9011 9013
9029 9041 9043 9049 9059 9067 9091 9103 9109 9127 9133 9137 9151 9157 9161
9173 9181 9187 9199 9203 9209 9221 9227 9239 9241 9257 9277 9281 9283 9293
9311 9319 9323 9337 9341 9343 9349 9371 9377 9391 9397 9403 9413 9419 9421
9431 9433 9437 9439 9461 9463 9467 9473 9479 9491 9497 9511 9521 9533 9539
9547 9551 9587 9601 9613 9619 9623 9629 9631 9643 9649 9661 9677 9679 9689
9697 9719 9721 9733 9739 9743 9749 9767 9769 9781 9787 9791 9803 9811 9817
9829 9833 9839 9851 9857 9859 9871 9883 9887 9901 9907 9923 9929 9931 9941
9949 9967 9973 10007 10009 10037 10039 10061 10067 10069 10079 10091 10093
10099 10103 10111 10133 10139 10141 10151 10159 10163 10169 10177 10181
10193 10211 10223 10243 10247 10253 10259 10267 10271 10273 10289 10301
10303 10313 10321 10331 10333 10337 10343 10357 10369 10391 10399 10427
10429 10433 10453 10457 10459 10463 10477 10487 10499 10501 10513 10529
10531 10559 10567 10589 10597 10601 10607 10613 10627 10631 10639 10651
10657 10663 10667 10687 10691 10709 10711 10723 10729 10733 10739 10753
10771 10781 10789 10799 10831 10837 10847 10853 10859 10861 10867 10883
10889 10891 10903 10909 10937 10939 10949 10957 10973 10979 10987 10993
11003 11027 11047 11057 11059 11069 11071 11083 11087 11093 11113 11117
11119 11131 11149 11159 11161 11171 11173 11177 11197 11213 11239 11243
11251 11257 11261 11273 11279 11287 11299 11311 11317 11321 11329 11351
11353 11369 11383 11393 11399 11411 11423 11437 11443 11447 11467 11471
11483 11489 11491 11497 11503 11519 11527 11549 11551 11579 11587 11593
11597 11617 11621 11633 11657 11677 11681 11689 11699 11701 11717 11719
11731 11743 11777 11779 11783 11789 11801 11807 11813 11821 11827 11831
11833 11839 11863 11867 11887 11897 11903 11909 11923 11927 11933 11939
11941 11953 11959 11969 11971 11981 11987 12007 12011 12037 12041 12043
12049 12071 12073 12097 12101 12107 12109 12113 12119 12143 12149 12157
12161 12163 12197 12203 12211 12227 12239 12241 12251 12253 12263 12269
12277 12281 12289 12301 12323 12329 12343 12347 12373 12377 12379 12391
12401 12409 12413 12421 12433 12437 12451 12457 12473 12479 12487 12491
12497 12503 12511 12517 12527 12539 12541 12547 12553 12569 12577 12583
12589 12601 12611 12613 12619 12637 12641 12647 12653 12659 12671 12689
12697 12703 12713 12721 12739 12743 12757 12763 12781 12791 12799 12809
12821 12823 12829 12841 12853 12889 12893 12899 12907 12911 12917 12919
12923 12941 12953 12959 12967 12973 12979 12983 13001 13003 13007 13009
13033 13037 13043 13049 13063 13093 13099 13103 13109 13121 13127 13147
13151 13159 13163 13171 13177 13183 13187 13217 13219 13229 13241 13249
13259 13267 13291 13297 13309 13313 13327 13331 13337 13339 13367 13381
13397 13399 13411 13417 13421 13441 13451 13457 13463 13469 13477 13487
13499 13513 13523 13537 13553 13567 13577 13591 13597 13613 13619 13627
13633 13649 13669 13679 13681 13687 13691 13693 13697 13709 13711 13721
13723 13729 13751 13757 13759 13763 13781 13789 13799 13807 13829 13831
13841 13859 13873 13877 13879 13883 13901 13903 13907 13913 13921 13931
13933 13963 13967 13997 13999 14009 14011 14029 14033 14051 14057 14071
14081 14083 14087 14107 14143 14149 14153 14159 14173 14177 14197 14207
14221 14243 14249 14251 14281 14293 14303 14321 14323 14327 14341 14347
14369 14387 14389 14401 14407 14411 14419 14423 14431 14437 14447 14449
14461 14479 14489 14503 14519 14533 14537 14543 14549 14551 14557 14561
14563 14591 14593 14621 14627 14629 14633 14639 14653 14657 14669 14683
14699 14713 14717 14723 14731 14737 14741 14747 14753 14759 14767 14771
14779 14783 14797 14813 14821 14827 14831 14843 14851 14867 14869 14879
14887 14891 14897 14923 14929 14939 14947 14951 14957 14969 14983 15013
15017 15031 15053 15061 15073 15077 15083 15091 15101 15107 15121 15131
15137 15139 15149 15161 15173 15187 15193 15199 15217 15227 15233 15241
15259 15263 15269 15271 15277 15287 15289 15299 15307 15313 15319 15329
15331 15349 15359 15361 15373 15377 15383 15391 15401 15413 15427 15439
15443 15451 15461 15467 15473 15493 15497 15511 15527 15541 15551 15559
15569 15581 15583 15601 15607 15619 15629 15641 15643 15647 15649 15661
15667 15671 15679 15683 15727 15731 15733 15737 15739 15749 15761 15767
15773 15787 15791 15797 15803 15809 15817 15823 15859 15877 15881 15887
15889 15901 15907 15913 15919 15923 15937 15959 15971 15973 15991 16001
16007 16033 16057 16061 16063 16067 16069 16073 16087 16091 16097 16103
16111 16127 16139 16141 16183 16187 16189 16193 16217 16223 16229 16231
16249 16253 16267 16273 16301 16319 16333 16339 16349 16361 16363 16369
16381 16411 16417 16421 16427 16433 16447 16451 16453 16477 16481 16487
16493 16519 16529 16547 16553 16561 16567 16573 16603 16607 16619 16631
16633 16649 16651 16657 16661 16673 16691 16693 16699 16703 16729 16741
16747 16759 16763 16787 16811 16823 16829 16831 16843 16871 16879 16883
16889 16901 16903 16921 16927 16931 16937 16943 16963 16979 16981 16987
16993 17011 17021 17027 17029 17033 17041 17047 17053 17077 17093 17099
17107 17117 17123 17137 17159 17167 17183 17189 17191 17203 17207 17209
17231 17239 17257 17291 17293 17299 17317 17321 17327 17333 17341 17351
17359 17377 17383 17387 17389 17393 17401 17417 17419 17431 17443 17449
17467 17471 17477 17483 17489 17491 17497 17509 17519 17539 17551 17569
17573 17579 17581 17597 17599 17609 17623 17627 17657 17659 17669 17681
17683 17707 17713 17729 17737 17747 17749 17761 17783 17789 17791 17807
17827 17837 17839 17851 17863 17881 17891 17903 17909 17911 17921 17923
17929 17939 17957 17959 17971 17977 17981 17987 17989 18013 18041 18043
18047 18049 18059 18061 18077 18089 18097 18119 18121 18127 18131 18133
18143 18149 18169 18181 18191 18199 18211 18217 18223 18229 18233 18251
18253 18257 18269 18287 18289 18301 18307 18311 18313 18329 18341 18353
18367 18371 18379 18397 18401 18413 18427 18433 18439 18443 18451 18457
18461 18481 18493 18503 18517 18521 18523 18539 18541 18553 18583 18587
18593 18617 18637 18661 18671 18679 18691 18701 18713 18719 18731 18743
18749 18757 18773 18787 18793 18797 18803 18839 18859 18869 18899 18911
18913 18917 18919 18947 18959 18973 18979 19001 19009 19013 19031 19037
19051 19069 19073 19079 19081 19087 19121 19139 19141 19157 19163 19181
19183 19207 19211 19213 19219 19231 19237 19249 19259 19267 19273 19289
19301 19309 19319 19333 19373 19379 19381 19387 19391 19403 19417 19421
19423 19427 19429 19433 19441 19447 19457 19463 19469 19471 19477 19483
19489 19501 19507 19531 19541 19543 19553 19559 19571 19577 19583 19597
19603 19609 19661 19681 19687 19697 19699 19709 19717 19727 19739 19751
19753 19759 19763 19777 19793 19801 19813 19819 19841 19843 19853 19861
19867 19889 19891 19913 19919 19927 19937 19949 19961 19963 19973 19979
19991 19993 19997 20011 20021 20023 20029 20047 20051 20063 20071 20089
20101 20107 20113 20117 20123 20129 20143 20147 20149 20161 20173 20177
20183 20201 20219 20231 20233 20249 20261 20269 20287 20297 20323 20327
20333 20341 20347 20353 20357 20359 20369 20389 20393 20399 20407 20411
20431 20441 20443 20477 20479 20483 20507 20509 20521 20533 20543 20549
20551 20563 20593 20599 20611 20627 20639 20641 20663 20681 20693 20707
20717 20719 20731 20743 20747 20749 20753 20759 20771 20773 20789 20807
20809 20849 20857 20873 20879 20887 20897 20899 20903 20921 20929 20939
20947 20959 20963 20981 20983 21001 21011 21013 21017 21019 21023 21031
21059 21061 21067 21089 21101 21107 21121 21139 21143 21149 21157 21163
21169 21179 21187 21191 21193 21211 21221 21227 21247 21269 21277 21283
21313 21317 21319 21323 21341 21347 21377 21379 21383 21391 21397 21401
21407 21419 21433 21467 21481 21487 21491 21493 21499 21503 21517 21521
21523 21529 21557 21559 21563 21569 21577 21587 21589 21599 21601 21611
21613 21617 21647 21649 21661 21673 21683 21701 21713 21727 21737 21739
21751 21757 21767 21773 21787 21799 21803 21817 21821 21839 21841 21851
21859 21863 21871 21881 21893 21911 21929 21937 21943 21961 21977 21991
21997 22003 22013 22027 22031 22037 22039 22051 22063 22067 22073 22079
22091 22093 22109 22111 22123 22129 22133 22147 22153 22157 22159 22171
22189 22193 22229 22247 22259 22271 22273 22277 22279 22283 22291 22303
22307 22343 22349 22367 22369 22381 22391 22397 22409 22433 22441 22447
22453 22469 22481 22483 22501 22511 22531 22541 22543 22549 22567 22571
22573 22613 22619 22621 22637 22639 22643 22651 22669 22679 22691 22697
22699 22709 22717 22721 22727 22739 22741 22751 22769 22777 22783 22787
22807 22811 22817 22853 22859 22861 22871 22877 22901 22907 22921 22937
22943 22961 22963 22973 22993 23003 23011 23017 23021 23027 23029 23039
23041 23053 23057 23059 23063 23071 23081 23087 23099 23117 23131 23143
23159 23167 23173 23189 23197 23201 23203 23209 23227 23251 23269 23279
23291 23293 23297 23311 23321 23327 23333 23339 23357 23369 23371 23399
23417 23431 23447 23459 23473 23497 23509 23531 23537 23539 23549 23557
23561 23563 23567 23581 23593 23599 23603 23609 23623 23627 23629 23633
23663 23669 23671 23677 23687 23689 23719 23741 23743 23747 23753 23761
23767 23773 23789 23801 23813 23819 23827 23831 23833 23857 23869 23873
23879 23887 23893 23899 23909 23911 23917 23929 23957 23971 23977 23981
23993 24001 24007 24019 24023 24029 24043 24049 24061 24071 24077 24083
24091 24097 24103 24107 24109 24113 24121 24133 24137 24151 24169 24179
24181 24197 24203 24223 24229 24239 24247 24251 24281 24317 24329 24337
24359 24371 24373 24379 24391 24407 24413 24419 24421 24439 24443 24469
24473 24481 24499 24509 24517 24527 24533 24547 24551 24571 24593 24611
24623 24631 24659 24671 24677 24683 24691 24697 24709 24733 24749 24763
24767 24781 24793 24799 24809 24821 24841 24847 24851 24859 24877 24889
24907 24917 24919 24923 24943 24953 24967 24971 24977 24979 24989 25013
25031 25033 25037 25057 25073 25087 25097 25111 25117 25121 25127 25147
25153 25163 25169 25171 25183 25189 25219 25229 25237 25243 25247 25253
25261 25301 25303 25307 25309 25321 25339 25343 25349 25357 25367 25373
25391 25409 25411 25423 25439 25447 25453 25457 25463 25469 25471 25523
25537 25541 25561 25577 25579 25583 25589 25601 25603 25609 25621 25633
25639 25643 25657 25667 25673 25679 25693 25703 25717 25733 25741 25747
25759 25763 25771 25793 25799 25801 25819 25841 25847 25849 25867 25873
25889 25903 25913 25919 25931 25933 25939 25943 25951 25969 25981 25997
25999 26003 26017 26021 26029 26041 26053 26083 26099 26107 26111 26113
26119 26141 26153 26161 26171 26177 26183 26189 26203 26209 26227 26237
26249 26251 26261 26263 26267 26293 26297 26309 26317 26321 26339 26347
26357 26371 26387 26393 26399 26407 26417 26423 26431 26437 26449 26459
26479 26489 26497 26501 26513 26539 26557 26561 26573 26591 26597 26627
26633 26641 26647 26669 26681 26683 26687 26693 26699 26701 26711 26713
26717 26723 26729 26731 26737 26759 26777 26783 26801 26813 26821 26833
26839 26849 26861 26863 26879 26881 26891 26893 26903 26921 26927 26947
26951 26953 26959 26981 26987 26993 27011 27017 27031 27043 27059 27061
27067 27073 27077 27091 27103 27107 27109 27127 27143 27179 27191 27197
27211 27239 27241 27253 27259 27271 27277 27281 27283 27299 27329 27337
27361 27367 27397 27407 27409 27427 27431 27437 27449 27457 27479 27481
27487 27509 27527 27529 27539 27541 27551 27581 27583 27611 27617 27631
27647 27653 27673 27689 27691 27697 27701 27733 27737 27739 27743 27749
27751 27763 27767 27773 27779 27791 27793 27799 27803 27809 27817 27823
27827 27847 27851 27883 27893 27901 27917 27919 27941 27943 27947 27953
27961 27967 27983 27997 28001 28019 28027 28031 28051 28057 28069 28081
28087 28097 28099 28109 28111 28123 28151 28163 28181 28183 28201 28211
28219 28229 28277 28279 28283 28289 28297 28307 28309 28319 28349 28351
28387 28393 28403 28409 28411 28429 28433 28439 28447 28463 28477 28493
28499 28513 28517 28537 28541 28547 28549 28559 28571 28573 28579 28591
28597 28603 28607 28619 28621 28627 28631 28643 28649 28657 28661 28663
28669 28687 28697 28703 28711 28723 28729 28751 28753 28759 28771 28789
28793 28807 28813 28817 28837 28843 28859 28867 28871 28879 28901 28909
28921 28927 28933 28949 28961 28979 29009 29017 29021 29023 29027 29033
29059 29063 29077 29101 29123 29129 29131 29137 29147 29153 29167 29173
29179 29191 29201 29207 29209 29221 29231 29243 29251 29269 29287 29297
29303 29311 29327 29333 29339 29347 29363 29383 29387 29389 29399 29401
29411 29423 29429 29437 29443 29453 29473 29483 29501 29527 29531 29537
29567 29569 29573 29581 29587 29599 29611 29629 29633 29641 29663 29669
29671 29683 29717 29723 29741 29753 29759 29761 29789 29803 29819 29833
29837 29851 29863 29867 29873 29879 29881 29917 29921 29927 29947 29959
29983 29989 30011 30013 30029 30047 30059 30071 30089 30091 30097 30103
30109 30113 30119 30133 30137 30139 30161 30169 30181 30187 30197 30203
30211 30223 30241 30253 30259 30269 30271 30293 30307 30313 30319 30323
30341 30347 30367 30389 30391 30403 30427 30431 30449 30467 30469 30491
30493 30497 30509 30517 30529 30539 30553 30557 30559 30577 30593 30631
30637 30643 30649 30661 30671 30677 30689 30697 30703 30707 30713 30727
30757 30763 30773 30781 30803 30809 30817 30829 30839 30841 30851 30853
30859 30869 30871 30881 30893 30911 30931 30937 30941 30949 30971 30977
30983 31013 31019 31033 31039 31051 31063 31069 31079 31081 31091 31121
31123 31139 31147 31151 31153 31159 31177 31181 31183 31189 31193 31219
31223 31231 31237 31247 31249 31253 31259 31267 31271 31277 31307 31319
31321 31327 31333 31337 31357 31379 31387 31391 31393 31397 31469 31477
31481 31489 31511 31513 31517 31531 31541 31543 31547 31567 31573 31583
31601 31607 31627 31643 31649 31657 31663 31667 31687 31699 31721 31723
31727 31729 31741 31751 31769 31771 31793 31799 31817 31847 31849 31859
31873 31883 31891 31907 31957 31963 31973 31981 31991 32003 32009 32027
32029 32051 32057 32059 32063 32069 32077 32083 32089 32099 32117 32119
32141 32143 32159 32173 32183 32189 32191 32203 32213 32233 32237 32251
32257 32261 32297 32299 32303 32309 32321 32323 32327 32341 32353 32359
32363 32369 32371 32377 32381 32401 32411 32413 32423 32429 32441 32443
32467 32479 32491 32497 32503 32507 32531 32533 32537 32561 32563 32569
32573 32579 32587 32603 32609 32611 32621 32633 32647 32653 32687 32693
32707 32713 32717 32719 32749 32771 32779 32783 32789 32797 32801 32803
32831 32833 32839 32843 32869 32887 32909 32911 32917 32933 32939 32941
32957 32969 32971 32983 32987 32993 32999 33013 33023 33029 33037 33049
33053 33071 33073 33083 33091 33107 33113 33119 33149 33151 33161 33179
33181 33191 33199 33203 33211 33223 33247 33287 33289 33301 33311 33317
33329 33331 33343 33347 33349 33353 33359 33377 33391 33403 33409 33413
33427 33457 33461 33469 33479 33487 33493 33503 33521 33529 33533 33547
33563 33569 33577 33581 33587 33589 33599 33601 33613 33617 33619 33623
33629 33637 33641 33647 33679 33703 33713 33721 33739 33749 33751 33757
33767 33769 33773 33791 33797 33809 33811 33827 33829 33851 33857 33863
33871 33889 33893 33911 33923 33931 33937 33941 33961 33967 33997 34019
34031 34033 34039 34057 34061 34123 34127 34129 34141 34147 34157 34159
34171 34183 34211 34213 34217 34231 34253 34259 34261 34267 34273 34283
34297 34301 34303 34313 34319 34327 34337 34351 34361 34367 34369 34381
34403 34421 34429 34439 34457 34469 34471 34483 34487 34499 34501 34511
34513 34519 34537 34543 34549 34583 34589 34591 34603 34607 34613 34631
34649 34651 34667 34673 34679 34687 34693 34703 34721 34729 34739 34747
34757 34759 34763 34781 34807 34819 34841 34843 34847 34849 34871 34877
34883 34897 34913 34919 34939 34949 34961 34963 34981 35023 35027 35051
35053 35059 35069 35081 35083 35089 35099 35107 35111 35117 35129 35141
35149 35153 35159 35171 35201 35221 35227 35251 35257 35267 35279 35281
35291 35311 35317 35323 35327 35339 35353 35363 35381 35393 35401 35407
35419 35423 35437 35447 35449 35461 35491 35507 35509 35521 35527 35531
35533 35537 35543 35569 35573 35591 35593 35597 35603 35617 35671 35677
35729 35731 35747 35753 35759 35771 35797 35801 35803 35809 35831 35837
35839 35851 35863 35869 35879 35897 35899 35911 35923 35933 35951 35963
35969 35977 35983 35993 35999 36007 36011 36013 36017 36037 36061 36067
36073 36083 36097 36107 36109 36131 36137 36151 36161 36187 36191 36209
36217 36229 36241 36251 36263 36269 36277 36293 36299 36307 36313 36319
36341 36343 36353 36373 36383 36389 36433 36451 36457 36467 36469 36473
36479 36493 36497 36523 36527 36529 36541 36551 36559 36563 36571 36583
36587 36599 36607 36629 36637 36643 36653 36671 36677 36683 36691 36697
36709 36713 36721 36739 36749 36761 36767 36779 36781 36787 36791 36793
36809 36821 36833 36847 36857 36871 36877 36887 36899 36901 36913 36919
36923 36929 36931 36943 36947 36973 36979 36997 37003 37013 37019 37021
37039 37049 37057 37061 37087 37097 37117 37123 37139 37159 37171 37181
37189 37199 37201 37217 37223 37243 37253 37273 37277 37307 37309 37313
37321 37337 37339 37357 37361 37363 37369 37379 37397 37409 37423 37441
37447 37463 37483 37489 37493 37501 37507 37511 37517 37529 37537 37547
37549 37561 37567 37571 37573 37579 37589 37591 37607 37619 37633 37643
37649 37657 37663 37691 37693 37699 37717 37747 37781 37783 37799 37811
37813 37831 37847 37853 37861 37871 37879 37889 37897 37907 37951 37957
37963 37967 37987 37991 37993 37997 38011 38039 38047 38053 38069 38083
38113 38119 38149 38153 38167 38177 38183 38189 38197 38201 38219 38231
38237 38239 38261 38273 38281 38287 38299 38303 38317 38321 38327 38329
38333 38351 38371 38377 38393 38431 38447 38449 38453 38459 38461 38501
38543 38557 38561 38567 38569 38593 38603 38609 38611 38629 38639 38651
38653 38669 38671 38677 38693 38699 38707 38711 38713 38723 38729 38737
38747 38749 38767 38783 38791 38803 38821 38833 38839 38851 38861 38867
38873 38891 38903 38917 38921 38923 38933 38953 38959 38971 38977 38993
39019 39023 39041 39043 39047 39079 39089 39097 39103 39107 39113 39119
39133 39139 39157 39161 39163 39181 39191 39199 39209 39217 39227 39229
39233 39239 39241 39251 39293 39301 39313 39317 39323 39341 39343 39359
39367 39371 39373 39383 39397 39409 39419 39439 39443 39451 39461 39499
39503 39509 39511 39521 39541 39551 39563 39569 39581 39607 39619 39623
39631 39659 39667 39671 39679 39703 39709 39719 39727 39733 39749 39761
39769 39779 39791 39799 39821 39827 39829 39839 39841 39847 39857 39863
39869 39877 39883 39887 39901 39929 39937 39953 39971 39979 39983 39989
40009 40013 40031 40037 40039 40063 40087 40093 40099 40111 40123 40127
40129 40151 40153 40163 40169 40177 40189 40193 40213 40231 40237 40241
40253 40277 40283 40289 40343 40351 40357 40361 40387 40423 40427 40429
40433 40459 40471 40483 40487 40493 40499 40507 40519 40529 40531 40543
40559 40577 40583 40591 40597 40609 40627 40637 40639 40693 40697 40699
40709 40739 40751 40759 40763 40771 40787 40801 40813 40819 40823 40829
40841 40847 40849 40853 40867 40879 40883 40897 40903 40927 40933 40939
40949 40961 40973 40993 41011 41017 41023 41039 41047 41051 41057 41077
41081 41113 41117 41131 41141 41143 41149 41161 41177 41179 41183 41189
41201 41203 41213 41221 41227 41231 41233 41243 41257 41263 41269 41281
41299 41333 41341 41351 41357 41381 41387 41389 41399 41411 41413 41443
41453 41467 41479 41491 41507 41513 41519 41521 41539 41543 41549 41579
41593 41597 41603 41609 41611 41617 41621 41627 41641 41647 41651 41659
41669 41681 41687 41719 41729 41737 41759 41761 41771 41777 41801 41809
41813 41843 41849 41851 41863 41879 41887 41893 41897 41903 41911 41927
41941 41947 41953 41957 41959 41969 41981 41983 41999 42013 42017 42019
42023 42043 42061 42071 42073 42083 42089 42101 42131 42139 42157 42169
42179 42181 42187 42193 42197 42209 42221 42223 42227 42239 42257 42281
42283 42293 42299 42307 42323 42331 42337 42349 42359 42373 42379 42391
42397 42403 42407 42409 42433 42437 42443 42451 42457 42461 42463 42467
42473 42487 42491 42499 42509 42533 42557 42569 42571 42577 42589 42611
42641 42643 42649 42667 42677 42683 42689 42697 42701 42703 42709 42719
42727 42737 42743 42751 42767 42773 42787 42793 42797 42821 42829 42839
42841 42853 42859 42863 42899 42901 42923 42929 42937 42943 42953 42961
42967 42979 42989 43003 43013 43019 43037 43049 43051 43063 43067 43093
43103 43117 43133 43151 43159 43177 43189 43201 43207 43223 43237 43261
43271 43283 43291 43313 43319 43321 43331 43391 43397 43399 43403 43411
43427 43441 43451 43457 43481 43487 43499 43517 43541 43543 43573 43577
43579 43591 43597 43607 43609 43613 43627 43633 43649 43651 43661 43669
43691 43711 43717 43721 43753 43759 43777 43781 43783 43787 43789 43793
43801 43853 43867 43889 43891 43913 43933 43943 43951 43961 43963 43969
43973 43987 43991 43997 44017 44021 44027 44029 44041 44053 44059 44071
44087 44089 44101 44111 44119 44123 44129 44131 44159 44171 44179 44189
44201 44203 44207 44221 44249 44257 44263 44267 44269 44273 44279 44281
44293 44351 44357 44371 44381 44383 44389 44417 44449 44453 44483 44491
44497 44501 44507 44519 44531 44533 44537 44543 44549 44563 44579 44587
44617 44621 44623 44633 44641 44647 44651 44657 44683 44687 44699 44701
44711 44729 44741 44753 44771 44773 44777 44789 44797 44809 44819 44839
44843 44851 44867 44879 44887 44893 44909 44917 44927 44939 44953 44959
44963 44971 44983 44987 45007 45013 45053 45061 45077 45083 45119 45121
45127 45131 45137 45139 45161 45179 45181 45191 45197 45233 45247 45259
45263 45281 45289 45293 45307 45317 45319 45329 45337 45341 45343 45361
45377 45389 45403 45413 45427 45433 45439 45481 45491 45497 45503 45523
45533 45541 45553 45557 45569 45587 45589 45599 45613 45631 45641 45659
45667 45673 45677 45691 45697 45707 45737 45751 45757 45763 45767 45779
45817 45821 45823 45827 45833 45841 45853 45863 45869 45887 45893 45943
45949 45953 45959 45971 45979 45989 46021 46027 46049 46051 46061 46073
46091 46093 46099 46103 46133 46141 46147 46153 46171 46181 46183 46187
46199 46219 46229 46237 46261 46271 46273 46279 46301 46307 46309 46327
46337 46349 46351 46381 46399 46411 46439 46441 46447 46451 46457 46471
46477 46489 46499 46507 46511 46523 46549 46559 46567 46573 46589 46591
46601 46619 46633 46639 46643 46649 46663 46679 46681 46687 46691 46703
46723 46727 46747 46751 46757 46769 46771 46807 46811 46817 46819 46829
46831 46853 46861 46867 46877 46889 46901 46919 46933 46957 46993 46997
47017 47041 47051 47057 47059 47087 47093 47111 47119 47123 47129 47137
47143 47147 47149 47161 47189 47207 47221 47237 47251 47269 47279 47287
47293 47297 47303 47309 47317 47339 47351 47353 47363 47381 47387 47389
47407 47417 47419 47431 47441 47459 47491 47497 47501 47507 47513 47521
47527 47533 47543 47563 47569 47581 47591 47599 47609 47623 47629 47639
47653 47657 47659 47681 47699 47701 47711 47713 47717 47737 47741 47743
47777 47779 47791 47797 47807 47809 47819 47837 47843 47857 47869 47881
47903 47911 47917 47933 47939 47947 47951 47963 47969 47977 47981 48017
48023 48029 48049 48073 48079 48091 48109 48119 48121 48131 48157 48163
48179 48187 48193 48197 48221 48239 48247 48259 48271 48281 48299 48311
48313 48337 48341 48353 48371 48383 48397 48407 48409 48413 48437 48449
48463 48473 48479 48481 48487 48491 48497 48523 48527 48533 48539 48541
48563 48571 48589 48593 48611 48619 48623 48647 48649 48661 48673 48677
48679 48731 48733 48751 48757 48761 48767 48779 48781 48787 48799 48809
48817 48821 48823 48847 48857 48859 48869 48871 48883 48889 48907 48947
48953 48973 48989 48991 49003 49009 49019 49031 49033 49037 49043 49057
49069 49081 49103 49109 49117 49121 49123 49139 49157 49169 49171 49177
49193 49199 49201 49207 49211 49223 49253 49261 49277 49279 49297 49307
49331 49333 49339 49363 49367 49369 49391 49393 49409 49411 49417 49429
49433 49451 49459 49463 49477 49481 49499 49523 49529 49531 49537 49547
49549 49559 49597 49603 49613 49627 49633 49639 49663 49667 49669 49681
49697 49711 49727 49739 49741 49747 49757 49783 49787 49789 49801 49807
49811 49823 49831 49843 49853 49871 49877 49891 49919 49921 49927 49937
49939 49943 49957 49991 49993 49999 50021 50023 50033 50047 50051 50053
50069 50077 50087 50093 50101 50111 50119 50123 50129 50131 50147 50153
50159 50177 50207 50221 50227 50231 50261 50263 50273 50287 50291 50311
50321 50329 50333 50341 50359 50363 50377 50383 50387 50411 50417 50423
50441 50459 50461 50497 50503 50513 50527 50539 50543 50549 50551 50581
50587 50591 50593 50599 50627 50647 50651 50671 50683 50707 50723 50741
50753 50767 50773 50777 50789 50821 50833 50839 50849 50857 50867 50873
50891 50893 50909 50923 50929 50951 50957 50969 50971 50989 50993 51001
51031 51043 51047 51059 51061 51071 51109 51131 51133 51137 51151 51157
51169 51193 51197 51199 51203 51217 51229 51239 51241 51257 51263 51283
51287 51307 51329 51341 51343 51347 51349 51361 51383 51407 51413 51419
51421 51427 51431 51437 51439 51449 51461 51473 51479 51481 51487 51503
51511 51517 51521 51539 51551 51563 51577 51581 51593 51599 51607 51613
51631 51637 51647 51659 51673 51679 51683 51691 51713 51719 51721 51749
51767 51769 51787 51797 51803 51817 51827 51829 51839 51853 51859 51869
51871 51893 51899 51907 51913 51929 51941 51949 51971 51973 51977 51991
52009 52021 52027 52051 52057 52067 52069 52081 52103 52121 52127 52147
52153 52163 52177 52181 52183 52189 52201 52223 52237 52249 52253 52259
52267 52289 52291 52301 52313 52321 52361 52363 52369 52379 52387 52391
52433 52453 52457 52489 52501 52511 52517 52529 52541 52543 52553 52561
52567 52571 52579 52583 52609 52627 52631 52639 52667 52673 52691 52697
52709 52711 52721 52727 52733 52747 52757 52769 52783 52807 52813 52817
52837 52859 52861 52879 52883 52889 52901 52903 52919 52937 52951 52957
52963 52967 52973 52981 52999 53003 53017 53047 53051 53069 53077 53087
53089 53093 53101 53113 53117 53129 53147 53149 53161 53171 53173 53189
53197 53201 53231 53233 53239 53267 53269 53279 53281 53299 53309 53323
53327 53353 53359 53377 53381 53401 53407 53411 53419 53437 53441 53453
53479 53503 53507 53527 53549 53551 53569 53591 53593 53597 53609 53611
53617 53623 53629 53633 53639 53653 53657 53681 53693 53699 53717 53719
53731 53759 53773 53777 53783 53791 53813 53819 53831 53849 53857 53861
53881 53887 53891 53897 53899 53917 53923 53927 53939 53951 53959 53987
53993 54001 54011 54013 54037 54049 54059 54083 54091 54101 54121 54133
54139 54151 54163 54167 54181 54193 54217 54251 54269 54277 54287 54293
54311 54319 54323 54331 54347 54361 54367 54371 54377 54401 54403 54409
54413 54419 54421 54437 54443 54449 54469 54493 54497 54499 54503 54517
54521 54539 54541 54547 54559 54563 54577 54581 54583 54601 54617 54623
54629 54631 54647 54667 54673 54679 54709 54713 54721 54727 54751 54767
54773 54779 54787 54799 54829 54833 54851 54869 54877 54881 54907 54917
54919 54941 54949 54959 54973 54979 54983 55001 55009 55021 55049 55051
55057 55061 55073 55079 55103 55109 55117 55127 55147 55163 55171 55201
55207 55213 55217 55219 55229 55243 55249 55259 55291 55313 55331 55333
55337 55339 55343 55351 55373 55381 55399 55411 55439 55441 55457 55469
55487 55501 55511 55529 55541 55547 55579 55589 55603 55609 55619 55621
55631 55633 55639 55661 55663 55667 55673 55681 55691 55697 55711 55717
55721 55733 55763 55787 55793 55799 55807 55813 55817 55819 55823 55829
55837 55843 55849 55871 55889 55897 55901 55903 55921 55927 55931 55933
55949 55967 55987 55997 56003 56009 56039 56041 56053 56081 56087 56093
56099 56101 56113 56123 56131 56149 56167 56171 56179 56197 56207 56209
56237 56239 56249 56263 56267 56269 56299 56311 56333 56359 56369 56377
56383 56393 56401 56417 56431 56437 56443 56453 56467 56473 56477 56479
56489 56501 56503 56509 56519 56527 56531 56533 56543 56569 56591 56597
56599 56611 56629 56633 56659 56663 56671 56681 56687 56701 56711 56713
56731 56737 56747 56767 56773 56779 56783 56807 56809 56813 56821 56827
56843 56857 56873 56891 56893 56897 56909 56911 56921 56923 56929 56941
56951 56957 56963 56983 56989 56993 56999 57037 57041 57047 57059 57073
57077 57089 57097 57107 57119 57131 57139 57143 57149 57163 57173 57179
57191 57193 57203 57221 57223 57241 57251 57259 57269 57271 57283 57287
57301 57329 57331 57347 57349 57367 57373 57383 57389 57397 57413 57427
57457 57467 57487 57493 57503 57527 57529 57557 57559 57571 57587 57593
57601 57637 57641 57649 57653 57667 57679 57689 57697 57709 57713 57719
57727 57731 57737 57751 57773 57781 57787 57791 57793 57803 57809 57829
57839 57847 57853 57859 57881 57899 57901 57917 57923 57943 57947 57973
57977 57991 58013 58027 58031 58043 58049 58057 58061 58067 58073 58099
58109 58111 58129 58147 58151 58153 58169 58171 58189 58193 58199 58207
58211 58217 58229 58231 58237 58243 58271 58309 58313 58321 58337 58363
58367 58369 58379 58391 58393 58403 58411 58417 58427 58439 58441 58451
58453 58477 58481 58511 58537 58543 58549 58567 58573 58579 58601 58603
58613 58631 58657 58661 58679 58687 58693 58699 58711 58727 58733 58741
58757 58763 58771 58787 58789 58831 58889 58897 58901 58907 58909 58913
58921 58937 58943 58963 58967 58979 58991 58997 59009 59011 59021 59023
59029 59051 59053 59063 59069 59077 59083 59093 59107 59113 59119 59123
59141 59149 59159 59167 59183 59197 59207 59209 59219 59221 59233 59239
59243 59263 59273 59281 59333 59341 59351 59357 59359 59369 59377 59387
59393 59399 59407 59417 59419 59441 59443 59447 59453 59467 59471 59473
59497 59509 59513 59539 59557 59561 59567 59581 59611 59617 59621 59627
59629 59651 59659 59663 59669 59671 59693 59699 59707 59723 59729 59743
59747 59753 59771 59779 59791 59797 59809 59833 59863 59879 59887 59921
59929 59951 59957 59971 59981 59999 60013 60017 60029 60037 60041 60077
60083 60089 60091 60101 60103 60107 60127 60133 60139 60149 60161 60167
60169 60209 60217 60223 60251 60257 60259 60271 60289 60293 60317 60331
60337 60343 60353 60373 60383 60397 60413 60427 60443 60449 60457 60493
60497 60509 60521 60527 60539 60589 60601 60607 60611 60617 60623 60631
60637 60647 60649 60659 60661 60679 60689 60703 60719 60727 60733 60737
60757 60761 60763 60773 60779 60793 60811 60821 60859 60869 60887 60889
60899 60901 60913 60917 60919 60923 60937 60943 60953 60961 61001 61007
61027 61031 61043 61051 61057 61091 61099 61121 61129 61141 61151 61153
61169 61211 61223 61231 61253 61261 61283 61291 61297 61331 61333 61339
61343 61357 61363 61379 61381 61403 61409 61417 61441 61463 61469 61471
61483 61487 61493 61507 61511 61519 61543 61547 61553 61559 61561 61583
61603 61609 61613 61627 61631 61637 61643 61651 61657 61667 61673 61681
61687 61703 61717 61723 61729 61751 61757 61781 61813 61819 61837 61843
61861 61871 61879 61909 61927 61933 61949 61961 61967 61979 61981 61987
61991 62003 62011 62017 62039 62047 62053 62057 62071 62081 62099 62119
62129 62131 62137 62141 62143 62171 62189 62191 62201 62207 62213 62219
62233 62273 62297 62299 62303 62311 62323 62327 62347 62351 62383 62401
62417 62423 62459 62467 62473 62477 62483 62497 62501 62507 62533 62539
62549 62563 62581 62591 62597 62603 62617 62627 62633 62639 62653 62659
62683 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.
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