/* bignum-random-prime.c
*
* Generation of random provable primes.
*/
/* Copyright (C) 2013 Red Hat
*
* The nettle library is free software; you can redistribute it and/or modify
* it under the terms of the GNU Lesser General Public License as published by
* the Free Software Foundation; either version 2.1 of the License, or (at your
* option) any later version.
*
* The nettle library is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
* or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
* License for more details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with the nettle library; see the file COPYING.LIB. If not, write to
* the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
* MA 02111-1301, USA.
*/
#if HAVE_CONFIG_H
#include <config.h>
#endif
#include <assert.h>
#include <stdlib.h>
#include <nettle/memxor.h>
#include <nettle/bignum.h>
#include <dsa-fips.h>
#include <nettle/macros.h>
static const uint16_t primes[] = {
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, 0
};
/* Tests if the (small) number provided is a prime.
*/
static unsigned small_prime_check(unsigned x)
{
unsigned j, t, square, prime;
for (j = 0; primes[j] > 0; j++) {
prime = primes[j];
square = prime * prime;
if (square > x)
break;
t = x / prime;
if (t * prime == x)
return 0;
}
return 1;
}
/* The seed in FIPS186-3 is used either as an integer or blob,
* but when used as an integer it must not be trunacated below
* the "nominal" seed size. This function returns the size
* that way. */
unsigned mpz_seed_sizeinbase_256_u(mpz_t s, unsigned nominal)
{
unsigned ret = nettle_mpz_sizeinbase_256_u(s);
if (ret < nominal)
return nominal;
return ret;
}
static int st_provable_prime_small(mpz_t p,
unsigned *prime_seed_length,
void *prime_seed,
unsigned *prime_gen_counter, unsigned bits,
unsigned seed_length, const void *seed,
void *progress_ctx,
nettle_progress_func * progress)
{
unsigned gen_counter = 0;
unsigned tseed_length;
uint8_t tseed[MAX_PVP_SEED_SIZE+1];
uint8_t h1[DIGEST_SIZE];
uint8_t h2[DIGEST_SIZE];
uint32_t c;
mpz_t s;
unsigned highbit;
assert(bits >= 2 && bits <= 32);
mpz_init(s);
/* seed is handled as mpz_t instead of simply using INCREMENT
* for the few (unlikely) cases where seed overflows. */
nettle_mpz_set_str_256_u(s, seed_length, seed);
retry:
tseed_length = mpz_seed_sizeinbase_256_u(s, seed_length);
if (tseed_length > sizeof(tseed)) {
goto fail;
}
/* c = Hash(seed) XOR Hash(seed+1) */
nettle_mpz_get_str_256(tseed_length, tseed, s);
hash(h1, tseed_length, tseed);
mpz_add_ui(s, s, 1);
tseed_length = mpz_seed_sizeinbase_256_u(s, seed_length);
if (tseed_length > sizeof(tseed))
goto fail;
nettle_mpz_get_str_256(tseed_length, tseed, s);
hash(h2, tseed_length, tseed);
memxor(h1, h2, DIGEST_SIZE);
/* c = 2^(bits-1) + (c mod 2^(bits-1)) */
highbit = 1L << (bits - 1);
c = READ_UINT32(&h1[DIGEST_SIZE - 4]);
c &= highbit - 1;
/* Make sure c is odd and high bit is set */
c |= highbit | 1;
gen_counter++;
/* seed++ */
mpz_add_ui(s, s, 1);
/* deterministic primality check on c */
if (small_prime_check(c) == 0) {
/* not a prime */
if (gen_counter >= 4 * bits)
goto fail; /* failed */
if (progress)
progress(progress_ctx, 'x');
goto retry;
}
/* success */
mpz_set_ui(p, c);
if (prime_seed != NULL) {
tseed_length = mpz_seed_sizeinbase_256_u(s, tseed_length);
if (*prime_seed_length < tseed_length)
goto fail;
nettle_mpz_get_str_256(tseed_length, prime_seed, s);
*prime_seed_length = tseed_length;
}
if (prime_gen_counter)
*prime_gen_counter = gen_counter;
mpz_clear(s);
return 1;
fail:
mpz_clear(s);
return 0;
}
/* The Shawe-Taylor algorithm described in FIPS 186-4.
*
* p: (output) the prime
* prime_seed_length: (output) the length of prime_seed. Initially
* must hold the maximum size of prime_seed. The size should be a
* byte more than seed_length.
* prime_seed: (output) the prime_seed (may be NULL)
* prime_gen_counter: A counter determined during the prime generation
* bits: The requested number of bits for prime
* seed_length: The length of seed. It is limited by MAX_PVP_SEED_SIZE.
* seed: The initial seed
*
* Returns non zero on success.
*/
int
st_provable_prime(mpz_t p,
unsigned *prime_seed_length, void *prime_seed,
unsigned *prime_gen_counter,
unsigned bits,
unsigned seed_length, const void *seed,
void *progress_ctx, nettle_progress_func * progress)
{
unsigned gen_counter;
unsigned tseed_length;
uint8_t tseed[MAX_PVP_SEED_SIZE+1];
int ret;
unsigned pseed_length, iterations;
uint8_t *pseed;
unsigned old_counter, i;
mpz_t s, tmp, r, dc0, c0, c, t, z;
uint8_t *storage = NULL;
unsigned storage_length = 0;
if (bits < 33) {
return st_provable_prime_small(p, prime_seed_length, prime_seed,
prime_gen_counter, bits,
seed_length, seed, progress_ctx,
progress);
}
mpz_init(s);
mpz_init(tmp);
mpz_init(r);
mpz_init(c);
mpz_init(z);
mpz_init(t);
mpz_init(c0);
mpz_init(dc0);
pseed_length = seed_length + 2;
pseed = gnutls_malloc(pseed_length);
if (pseed == NULL)
goto fail;
ret = st_provable_prime(c0, &pseed_length, pseed, &gen_counter,
1+div_ceil(bits, 2), seed_length, seed,
progress_ctx, progress);
if (ret == 0)
goto fail;
nettle_mpz_set_str_256_u(s, pseed_length, pseed);
/* the spec says to remove 1 here */
iterations = div_ceil(bits, DIGEST_SIZE * 8);
old_counter = gen_counter;
mpz_set_ui(tmp, 0); /* x = 0 */
if (iterations > 0) {
storage_length = iterations * DIGEST_SIZE;
storage = malloc(storage_length);
if (storage == NULL)
goto fail;
for (i = 0; i < iterations; i++) {
tseed_length = mpz_seed_sizeinbase_256_u(s, pseed_length);
if (tseed_length > sizeof(tseed))
goto fail;
nettle_mpz_get_str_256(tseed_length, tseed, s);
hash(&storage
[(iterations - i - 1) * DIGEST_SIZE],
tseed_length, tseed);
mpz_add_ui(s, s, 1);
}
nettle_mpz_set_str_256_u(tmp, storage_length, storage);
}
/* tmp = 2^(bits-1) + (tmp mod 2^(bits-1)) */
mpz_set_ui(r, 1);
mpz_mul_2exp(r, r, bits - 1); /* r = 2^(bits-1) */
mpz_fdiv_r_2exp(tmp, tmp, bits - 1);
mpz_add(tmp, tmp, r); /* tmp = x */
/* Generate candidate prime c in [2^(bits-1), 2^bits] */
/* t = u[x/2c0] */
mpz_mul_2exp(dc0, c0, 1); /* dc0 = 2*c0 */
mpz_cdiv_q(t, tmp, dc0);
retry:
/* c = t*(2c0) + 1 */
mpz_mul(c, dc0, t);
mpz_add_ui(c, c, 1);
mpz_set_ui(r, 1);
mpz_mul_2exp(r, r, bits); /* r = 2^(bits) */
/* if 2tc0+1 > 2^bits */
if (mpz_cmp(c, r) > 0) {
/* t = 2^(bits-1)/2c0 */
mpz_fdiv_q_2exp(r, r, 1); /* r = 2^(bits-1) */
mpz_cdiv_q(t, r, dc0);
/* c = t* 2c0 + 1 */
mpz_mul(c, dc0, t);
mpz_add_ui(c, c, 1);
}
gen_counter++;
mpz_set_ui(r, 0); /* a = 0 */
if (iterations > 0) {
for (i = 0; i < iterations; i++) {
tseed_length = mpz_seed_sizeinbase_256_u(s, pseed_length);
if (tseed_length > sizeof(tseed))
goto fail;
nettle_mpz_get_str_256(tseed_length, tseed, s);
hash(&storage
[(iterations - i - 1) * DIGEST_SIZE],
tseed_length, tseed);
mpz_add_ui(s, s, 1);
}
/* r = a */
nettle_mpz_set_str_256_u(r, storage_length, storage);
}
/* a = 2 + (a mod (c-3)) */
mpz_sub_ui(tmp, c, 3); /* c is too large to worry about negatives */
mpz_mod(r, r, tmp);
mpz_add_ui(r, r, 2);
/* z = a^(2t) mod c */
mpz_mul_2exp(tmp, t, 1); /* tmp = 2t */
mpz_powm(z, r, tmp, c);
mpz_sub_ui(tmp, z, 1);
mpz_gcd(r, tmp, c);
if (mpz_cmp_ui(r, 1) == 0) {
mpz_powm(tmp, z, c0, c);
if (mpz_cmp_ui(tmp, 1) == 0) {
mpz_set(p, c);
if (prime_seed != NULL) {
tseed_length = mpz_seed_sizeinbase_256_u(s, pseed_length);
if (*prime_seed_length < tseed_length)
goto fail;
nettle_mpz_get_str_256(tseed_length,
prime_seed, s);
*prime_seed_length = tseed_length;
}
if (prime_gen_counter)
*prime_gen_counter = gen_counter;
goto success;
}
}
if (progress)
progress(progress_ctx, 'x');
if (gen_counter >= (4 * bits + old_counter)) {
goto fail;
}
mpz_add_ui(t, t, 1);
goto retry;
success:
ret = 1;
goto finish;
fail:
ret = 0;
finish:
mpz_clear(c0);
mpz_clear(dc0);
mpz_clear(r);
mpz_clear(s);
mpz_clear(z);
mpz_clear(t);
mpz_clear(tmp);
mpz_clear(c);
free(pseed);
free(storage);
return ret;
}