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/* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
* All rights reserved.
*
* This package is an SSL implementation written
* by Eric Young (eay@cryptsoft.com).
* The implementation was written so as to conform with Netscapes SSL.
*
* This library is free for commercial and non-commercial use as long as
* the following conditions are aheared to. The following conditions
* apply to all code found in this distribution, be it the RC4, RSA,
* lhash, DES, etc., code; not just the SSL code. The SSL documentation
* included with this distribution is covered by the same copyright terms
* except that the holder is Tim Hudson (tjh@cryptsoft.com).
*
* Copyright remains Eric Young's, and as such any Copyright notices in
* the code are not to be removed.
* If this package is used in a product, Eric Young should be given attribution
* as the author of the parts of the library used.
* This can be in the form of a textual message at program startup or
* in documentation (online or textual) provided with the package.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* "This product includes cryptographic software written by
* Eric Young (eay@cryptsoft.com)"
* The word 'cryptographic' can be left out if the rouines from the library
* being used are not cryptographic related :-).
* 4. If you include any Windows specific code (or a derivative thereof) from
* the apps directory (application code) you must include an acknowledgement:
* "This product includes software written by Tim Hudson (tjh@cryptsoft.com)"
*
* THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*
* The licence and distribution terms for any publically available version or
* derivative of this code cannot be changed. i.e. this code cannot simply be
* copied and put under another distribution licence
* [including the GNU Public Licence.]
*/
/* ====================================================================
* Copyright (c) 1998-2001 The OpenSSL Project. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
*
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in
* the documentation and/or other materials provided with the
* distribution.
*
* 3. All advertising materials mentioning features or use of this
* software must display the following acknowledgment:
* "This product includes software developed by the OpenSSL Project
* for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
*
* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
* endorse or promote products derived from this software without
* prior written permission. For written permission, please contact
* openssl-core@openssl.org.
*
* 5. Products derived from this software may not be called "OpenSSL"
* nor may "OpenSSL" appear in their names without prior written
* permission of the OpenSSL Project.
*
* 6. Redistributions of any form whatsoever must retain the following
* acknowledgment:
* "This product includes software developed by the OpenSSL Project
* for use in the OpenSSL Toolkit (http://www.openssl.org/)"
*
* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
* EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
* ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
* STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
* OF THE POSSIBILITY OF SUCH DAMAGE.
* ====================================================================
*
* This product includes cryptographic software written by Eric Young
* (eay@cryptsoft.com). This product includes software written by Tim
* Hudson (tjh@cryptsoft.com). */
#include <openssl/bn.h>
#include <openssl/err.h>
#include <openssl/mem.h>
#include "internal.h"
#include "../../internal.h"
// The quick sieve algorithm approach to weeding out primes is Philip
// Zimmermann's, as implemented in PGP. I have had a read of his comments and
// implemented my own version.
// kPrimes contains the first 2048 primes.
static const uint16_t kPrimes[] = {
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,
};
// BN_prime_checks_for_size returns the number of Miller-Rabin iterations
// necessary for a 'bits'-bit prime.
//
//
// This table is generated using the algorithm of FIPS PUB 186-4
// Digital Signature Standard (DSS), section F.1, page 117.
// (https://doi.org/10.6028/NIST.FIPS.186-4)
// The following magma script was used to generate the output:
// securitybits:=125;
// k:=1024;
// for t:=1 to 65 do
// for M:=3 to Floor(2*Sqrt(k-1)-1) do
// S:=0;
// // Sum over m
// for m:=3 to M do
// s:=0;
// // Sum over j
// for j:=2 to m do
// s+:=(RealField(32)!2)^-(j+(k-1)/j);
// end for;
// S+:=2^(m-(m-1)*t)*s;
// end for;
// A:=2^(k-2-M*t);
// B:=8*(Pi(RealField(32))^2-6)/3*2^(k-2)*S;
// pkt:=2.00743*Log(2)*k*2^-k*(A+B);
// seclevel:=Floor(-Log(2,pkt));
// if seclevel ge securitybits then
// printf "k: %5o, security: %o bits (t: %o, M: %o)\n",k,seclevel,t,M;
// break;
// end if;
// end for;
// if seclevel ge securitybits then break; end if;
// end for;
//
// It can be run online at: http://magma.maths.usyd.edu.au/calc
// And will output:
// k: 1024, security: 129 bits (t: 6, M: 23)
// k is the number of bits of the prime, securitybits is the level we want to
// reach.
// prime length | RSA key size | # MR tests | security level
// -------------+--------------|------------+---------------
// (b) >= 6394 | >= 12788 | 3 | 256 bit
// (b) >= 3747 | >= 7494 | 3 | 192 bit
// (b) >= 1345 | >= 2690 | 4 | 128 bit
// (b) >= 1080 | >= 2160 | 5 | 128 bit
// (b) >= 852 | >= 1704 | 5 | 112 bit
// (b) >= 476 | >= 952 | 5 | 80 bit
// (b) >= 400 | >= 800 | 6 | 80 bit
// (b) >= 347 | >= 694 | 7 | 80 bit
// (b) >= 308 | >= 616 | 8 | 80 bit
// (b) >= 55 | >= 110 | 27 | 64 bit
// (b) >= 6 | >= 12 | 34 | 64 bit
static int BN_prime_checks_for_size(int bits) {
if (bits >= 3747) {
return 3;
}
if (bits >= 1345) {
return 4;
}
if (bits >= 476) {
return 5;
}
if (bits >= 400) {
return 6;
}
if (bits >= 347) {
return 7;
}
if (bits >= 308) {
return 8;
}
if (bits >= 55) {
return 27;
}
return 34;
}
// num_trial_division_primes returns the number of primes to try with trial
// division before using more expensive checks. For larger numbers, the value
// of excluding a candidate with trial division is larger.
static size_t num_trial_division_primes(const BIGNUM *n) {
if (n->width * BN_BITS2 > 1024) {
return OPENSSL_ARRAY_SIZE(kPrimes);
}
return OPENSSL_ARRAY_SIZE(kPrimes) / 4;
}
// BN_PRIME_CHECKS_BLINDED is the iteration count for blinding the constant-time
// primality test. See |BN_primality_test| for details. This number is selected
// so that, for a candidate N-bit RSA prime, picking |BN_PRIME_CHECKS_BLINDED|
// random N-bit numbers will have at least |BN_prime_checks_for_size(N)| values
// in range with high probability.
//
// The following Python script computes the blinding factor needed for the
// corresponding iteration count.
/*
import math
# We choose candidate RSA primes between sqrt(2)/2 * 2^N and 2^N and select
# witnesses by generating random N-bit numbers. Thus the probability of
# selecting one in range is at least sqrt(2)/2.
p = math.sqrt(2) / 2
# Target around 2^-8 probability of the blinding being insufficient given that
# key generation is a one-time, noisy operation.
epsilon = 2**-8
def choose(a, b):
r = 1
for i in xrange(b):
r *= a - i
r /= (i + 1)
return r
def failure_rate(min_uniform, iterations):
""" Returns the probability that, for |iterations| candidate witnesses, fewer
than |min_uniform| of them will be uniform. """
prob = 0.0
for i in xrange(min_uniform):
prob += (choose(iterations, i) *
p**i * (1-p)**(iterations - i))
return prob
for min_uniform in (3, 4, 5, 6, 8, 13, 19, 28):
# Find the smallest number of iterations under the target failure rate.
iterations = min_uniform
while True:
prob = failure_rate(min_uniform, iterations)
if prob < epsilon:
print min_uniform, iterations, prob
break
iterations += 1
Output:
3 9 0.00368894873911
4 11 0.00363319494662
5 13 0.00336215573898
6 15 0.00300145783158
8 19 0.00225214119331
13 27 0.00385610026955
19 38 0.0021410539126
28 52 0.00325405801769
16 iterations suffices for 400-bit primes and larger (6 uniform samples needed),
which is already well below the minimum acceptable key size for RSA.
*/
#define BN_PRIME_CHECKS_BLINDED 16
static int probable_prime(BIGNUM *rnd, int bits);
static int probable_prime_dh(BIGNUM *rnd, int bits, const BIGNUM *add,
const BIGNUM *rem, BN_CTX *ctx);
static int probable_prime_dh_safe(BIGNUM *rnd, int bits, const BIGNUM *add,
const BIGNUM *rem, BN_CTX *ctx);
void BN_GENCB_set(BN_GENCB *callback,
int (*f)(int event, int n, struct bn_gencb_st *),
void *arg) {
callback->callback = f;
callback->arg = arg;
}
int BN_GENCB_call(BN_GENCB *callback, int event, int n) {
if (!callback) {
return 1;
}
return callback->callback(event, n, callback);
}
int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe, const BIGNUM *add,
const BIGNUM *rem, BN_GENCB *cb) {
BIGNUM *t;
int found = 0;
int i, j, c1 = 0;
BN_CTX *ctx;
int checks = BN_prime_checks_for_size(bits);
if (bits < 2) {
// There are no prime numbers this small.
OPENSSL_PUT_ERROR(BN, BN_R_BITS_TOO_SMALL);
return 0;
} else if (bits == 2 && safe) {
// The smallest safe prime (7) is three bits.
OPENSSL_PUT_ERROR(BN, BN_R_BITS_TOO_SMALL);
return 0;
}
ctx = BN_CTX_new();
if (ctx == NULL) {
goto err;
}
BN_CTX_start(ctx);
t = BN_CTX_get(ctx);
if (!t) {
goto err;
}
loop:
// make a random number and set the top and bottom bits
if (add == NULL) {
if (!probable_prime(ret, bits)) {
goto err;
}
} else {
if (safe) {
if (!probable_prime_dh_safe(ret, bits, add, rem, ctx)) {
goto err;
}
} else {
if (!probable_prime_dh(ret, bits, add, rem, ctx)) {
goto err;
}
}
}
if (!BN_GENCB_call(cb, BN_GENCB_GENERATED, c1++)) {
// aborted
goto err;
}
if (!safe) {
i = BN_is_prime_fasttest_ex(ret, checks, ctx, 0, cb);
if (i == -1) {
goto err;
} else if (i == 0) {
goto loop;
}
} else {
// for "safe prime" generation, check that (p-1)/2 is prime. Since a prime
// is odd, We just need to divide by 2
if (!BN_rshift1(t, ret)) {
goto err;
}
for (i = 0; i < checks; i++) {
j = BN_is_prime_fasttest_ex(ret, 1, ctx, 0, NULL);
if (j == -1) {
goto err;
} else if (j == 0) {
goto loop;
}
j = BN_is_prime_fasttest_ex(t, 1, ctx, 0, NULL);
if (j == -1) {
goto err;
} else if (j == 0) {
goto loop;
}
if (!BN_GENCB_call(cb, i, c1 - 1)) {
goto err;
}
// We have a safe prime test pass
}
}
// we have a prime :-)
found = 1;
err:
if (ctx != NULL) {
BN_CTX_end(ctx);
BN_CTX_free(ctx);
}
return found;
}
static int bn_trial_division(uint16_t *out, const BIGNUM *bn) {
const size_t num_primes = num_trial_division_primes(bn);
for (size_t i = 1; i < num_primes; i++) {
if (bn_mod_u16_consttime(bn, kPrimes[i]) == 0) {
*out = kPrimes[i];
return 1;
}
}
return 0;
}
int bn_odd_number_is_obviously_composite(const BIGNUM *bn) {
uint16_t prime;
return bn_trial_division(&prime, bn) && !BN_is_word(bn, prime);
}
int BN_primality_test(int *out_is_probably_prime, const BIGNUM *w,
int iterations, BN_CTX *ctx, int do_trial_division,
BN_GENCB *cb) {
*out_is_probably_prime = 0;
// To support RSA key generation, this function should treat |w| as secret if
// it is a large prime. Composite numbers are discarded, so they may return
// early.
if (BN_cmp(w, BN_value_one()) <= 0) {
return 1;
}
if (!BN_is_odd(w)) {
// The only even prime is two.
*out_is_probably_prime = BN_is_word(w, 2);
return 1;
}
// Miller-Rabin does not work for three.
if (BN_is_word(w, 3)) {
*out_is_probably_prime = 1;
return 1;
}
if (do_trial_division) {
// Perform additional trial division checks to discard small primes.
uint16_t prime;
if (bn_trial_division(&prime, w)) {
*out_is_probably_prime = BN_is_word(w, prime);
return 1;
}
if (!BN_GENCB_call(cb, 1, -1)) {
return 0;
}
}
if (iterations == BN_prime_checks) {
iterations = BN_prime_checks_for_size(BN_num_bits(w));
}
BN_CTX *new_ctx = NULL;
if (ctx == NULL) {
new_ctx = BN_CTX_new();
if (new_ctx == NULL) {
return 0;
}
ctx = new_ctx;
}
// See C.3.1 from FIPS 186-4.
int ret = 0;
BN_MONT_CTX *mont = NULL;
BN_CTX_start(ctx);
BIGNUM *w1 = BN_CTX_get(ctx);
if (w1 == NULL ||
!bn_usub_consttime(w1, w, BN_value_one())) {
goto err;
}
// Write w1 as m * 2^a (Steps 1 and 2).
int w_len = BN_num_bits(w);
int a = BN_count_low_zero_bits(w1);
BIGNUM *m = BN_CTX_get(ctx);
if (m == NULL ||
!bn_rshift_secret_shift(m, w1, a, ctx)) {
goto err;
}
// Montgomery setup for computations mod w. Additionally, compute 1 and w - 1
// in the Montgomery domain for later comparisons.
BIGNUM *b = BN_CTX_get(ctx);
BIGNUM *z = BN_CTX_get(ctx);
BIGNUM *one_mont = BN_CTX_get(ctx);
BIGNUM *w1_mont = BN_CTX_get(ctx);
mont = BN_MONT_CTX_new_consttime(w, ctx);
if (b == NULL || z == NULL || one_mont == NULL || w1_mont == NULL ||
mont == NULL ||
!bn_one_to_montgomery(one_mont, mont, ctx) ||
// w - 1 is -1 mod w, so we can compute it in the Montgomery domain, -R,
// with a subtraction. (|one_mont| cannot be zero.)
!bn_usub_consttime(w1_mont, w, one_mont)) {
goto err;
}
// The following loop performs in inner iteration of the Miller-Rabin
// Primality test (Step 4).
//
// The algorithm as specified in FIPS 186-4 leaks information on |w|, the RSA
// private key. Instead, we run through each iteration unconditionally,
// performing modular multiplications, masking off any effects to behave
// equivalently to the specified algorithm.
//
// We also blind the number of values of |b| we try. Steps 4.1–4.2 say to
// discard out-of-range values. To avoid leaking information on |w|, we use
// |bn_rand_secret_range| which, rather than discarding bad values, adjusts
// them to be in range. Though not uniformly selected, these adjusted values
// are still usable as Miller-Rabin checks.
//
// Miller-Rabin is already probabilistic, so we could reach the desired
// confidence levels by just suitably increasing the iteration count. However,
// to align with FIPS 186-4, we use a more pessimal analysis: we do not count
// the non-uniform values towards the iteration count. As a result, this
// function is more complex and has more timing risk than necessary.
//
// We count both total iterations and uniform ones and iterate until we've
// reached at least |BN_PRIME_CHECKS_BLINDED| and |iterations|, respectively.
// If the latter is large enough, it will be the limiting factor with high
// probability and we won't leak information.
//
// Note this blinding does not impact most calls when picking primes because
// composites are rejected early. Only the two secret primes see extra work.
crypto_word_t uniform_iterations = 0;
// Using |constant_time_lt_w| seems to prevent the compiler from optimizing
// this into two jumps.
for (int i = 1; (i <= BN_PRIME_CHECKS_BLINDED) |
constant_time_lt_w(uniform_iterations, iterations);
i++) {
int is_uniform;
if (// Step 4.1-4.2
!bn_rand_secret_range(b, &is_uniform, 2, w1) ||
// Step 4.3
!BN_mod_exp_mont_consttime(z, b, m, w, ctx, mont)) {
goto err;
}
uniform_iterations += is_uniform;
// loop_done is all ones if the loop has completed and all zeros otherwise.
crypto_word_t loop_done = 0;
// next_iteration is all ones if we should continue to the next iteration
// (|b| is not a composite witness for |w|). This is equivalent to going to
// step 4.7 in the original algorithm.
crypto_word_t next_iteration = 0;
// Step 4.4. If z = 1 or z = w-1, mask off the loop and continue to the next
// iteration (go to step 4.7).
loop_done = BN_equal_consttime(z, BN_value_one()) |
BN_equal_consttime(z, w1);
loop_done = 0 - loop_done; // Make it all zeros or all ones.
next_iteration = loop_done; // Go to step 4.7 if |loop_done|.
// Step 4.5. We use Montgomery-encoding for better performance and to avoid
// timing leaks.
if (!BN_to_montgomery(z, z, mont, ctx)) {
goto err;
}
// To avoid leaking |a|, we run the loop to |w_len| and mask off all
// iterations once |j| = |a|.
for (int j = 1; j < w_len; j++) {
loop_done |= constant_time_eq_int(j, a);
// Step 4.5.1.
if (!BN_mod_mul_montgomery(z, z, z, mont, ctx)) {
goto err;
}
// Step 4.5.2. If z = w-1 and the loop is not done, run through the next
// iteration.
crypto_word_t z_is_w1_mont = BN_equal_consttime(z, w1_mont) & ~loop_done;
z_is_w1_mont = 0 - z_is_w1_mont; // Make it all zeros or all ones.
loop_done |= z_is_w1_mont;
next_iteration |= z_is_w1_mont; // Go to step 4.7 if |z_is_w1_mont|.
// Step 4.5.3. If z = 1 and the loop is not done, w is composite and we
// may exit in variable time.
if (BN_equal_consttime(z, one_mont) & ~loop_done) {
assert(!next_iteration);
break;
}
}
if (!next_iteration) {
// Step 4.6. We did not see z = w-1 before z = 1, so w must be composite.
// (For any prime, the value of z immediately preceding 1 must be -1.
// There are no non-trivial square roots of 1 modulo a prime.)
*out_is_probably_prime = 0;
ret = 1;
goto err;
}
// Step 4.7
if (!BN_GENCB_call(cb, 1, i)) {
goto err;
}
}
assert(uniform_iterations >= (crypto_word_t)iterations);
*out_is_probably_prime = 1;
ret = 1;
err:
BN_MONT_CTX_free(mont);
BN_CTX_end(ctx);
BN_CTX_free(new_ctx);
return ret;
}
int BN_is_prime_ex(const BIGNUM *candidate, int checks, BN_CTX *ctx,
BN_GENCB *cb) {
return BN_is_prime_fasttest_ex(candidate, checks, ctx, 0, cb);
}
int BN_is_prime_fasttest_ex(const BIGNUM *a, int checks, BN_CTX *ctx,
int do_trial_division, BN_GENCB *cb) {
int is_probably_prime;
if (!BN_primality_test(&is_probably_prime, a, checks, ctx, do_trial_division,
cb)) {
return -1;
}
return is_probably_prime;
}
int BN_enhanced_miller_rabin_primality_test(
enum bn_primality_result_t *out_result, const BIGNUM *w, int iterations,
BN_CTX *ctx, BN_GENCB *cb) {
// Enhanced Miller-Rabin is only valid on odd integers greater than 3.
if (!BN_is_odd(w) || BN_cmp_word(w, 3) <= 0) {
OPENSSL_PUT_ERROR(BN, BN_R_INVALID_INPUT);
return 0;
}
if (iterations == BN_prime_checks) {
iterations = BN_prime_checks_for_size(BN_num_bits(w));
}
int ret = 0;
BN_MONT_CTX *mont = NULL;
BN_CTX_start(ctx);
BIGNUM *w1 = BN_CTX_get(ctx);
if (w1 == NULL ||
!BN_copy(w1, w) ||
!BN_sub_word(w1, 1)) {
goto err;
}
// Write w1 as m*2^a (Steps 1 and 2).
int a = 0;
while (!BN_is_bit_set(w1, a)) {
a++;
}
BIGNUM *m = BN_CTX_get(ctx);
if (m == NULL ||
!BN_rshift(m, w1, a)) {
goto err;
}
BIGNUM *b = BN_CTX_get(ctx);
BIGNUM *g = BN_CTX_get(ctx);
BIGNUM *z = BN_CTX_get(ctx);
BIGNUM *x = BN_CTX_get(ctx);
BIGNUM *x1 = BN_CTX_get(ctx);
if (b == NULL ||
g == NULL ||
z == NULL ||
x == NULL ||
x1 == NULL) {
goto err;
}
// Montgomery setup for computations mod w
mont = BN_MONT_CTX_new_for_modulus(w, ctx);
if (mont == NULL) {
goto err;
}
// The following loop performs in inner iteration of the Enhanced Miller-Rabin
// Primality test (Step 4).
for (int i = 1; i <= iterations; i++) {
// Step 4.1-4.2
if (!BN_rand_range_ex(b, 2, w1)) {
goto err;
}
// Step 4.3-4.4
if (!BN_gcd(g, b, w, ctx)) {
goto err;
}
if (BN_cmp_word(g, 1) > 0) {
*out_result = bn_composite;
ret = 1;
goto err;
}
// Step 4.5
if (!BN_mod_exp_mont(z, b, m, w, ctx, mont)) {
goto err;
}
// Step 4.6
if (BN_is_one(z) || BN_cmp(z, w1) == 0) {
goto loop;
}
// Step 4.7
for (int j = 1; j < a; j++) {
if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx)) {
goto err;
}
if (BN_cmp(z, w1) == 0) {
goto loop;
}
if (BN_is_one(z)) {
goto composite;
}
}
// Step 4.8-4.9
if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx)) {
goto err;
}
// Step 4.10-4.11
if (!BN_is_one(z) && !BN_copy(x, z)) {
goto err;
}
composite:
// Step 4.12-4.14
if (!BN_copy(x1, x) ||
!BN_sub_word(x1, 1) ||
!BN_gcd(g, x1, w, ctx)) {
goto err;
}
if (BN_cmp_word(g, 1) > 0) {
*out_result = bn_composite;
} else {
*out_result = bn_non_prime_power_composite;
}
ret = 1;
goto err;
loop:
// Step 4.15
if (!BN_GENCB_call(cb, 1, i)) {
goto err;
}
}
*out_result = bn_probably_prime;
ret = 1;
err:
BN_MONT_CTX_free(mont);
BN_CTX_end(ctx);
return ret;
}
static int probable_prime(BIGNUM *rnd, int bits) {
uint16_t mods[OPENSSL_ARRAY_SIZE(kPrimes)];
const size_t num_primes = num_trial_division_primes(rnd);
BN_ULONG delta;
BN_ULONG maxdelta = BN_MASK2 - kPrimes[num_primes - 1];
char is_single_word = bits <= BN_BITS2;
again:
if (!BN_rand(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD)) {
return 0;
}
// we now have a random number 'rnd' to test.
for (size_t i = 1; i < num_primes; i++) {
mods[i] = bn_mod_u16_consttime(rnd, kPrimes[i]);
}
// If bits is so small that it fits into a single word then we
// additionally don't want to exceed that many bits.
if (is_single_word) {
BN_ULONG size_limit;
if (bits == BN_BITS2) {
// Avoid undefined behavior.
size_limit = ~((BN_ULONG)0) - BN_get_word(rnd);
} else {
size_limit = (((BN_ULONG)1) << bits) - BN_get_word(rnd) - 1;
}
if (size_limit < maxdelta) {
maxdelta = size_limit;
}
}
delta = 0;
loop:
if (is_single_word) {
BN_ULONG rnd_word = BN_get_word(rnd);
// In the case that the candidate prime is a single word then
// we check that:
// 1) It's greater than kPrimes[i] because we shouldn't reject
// 3 as being a prime number because it's a multiple of
// three.
// 2) That it's not a multiple of a known prime. We don't
// check that rnd-1 is also coprime to all the known
// primes because there aren't many small primes where
// that's true.
for (size_t i = 1; i < num_primes && kPrimes[i] < rnd_word; i++) {
if ((mods[i] + delta) % kPrimes[i] == 0) {
delta += 2;
if (delta > maxdelta) {
goto again;
}
goto loop;
}
}
} else {
for (size_t i = 1; i < num_primes; i++) {
// check that rnd is not a prime and also
// that gcd(rnd-1,primes) == 1 (except for 2)
if (((mods[i] + delta) % kPrimes[i]) <= 1) {
delta += 2;
if (delta > maxdelta) {
goto again;
}
goto loop;
}
}
}
if (!BN_add_word(rnd, delta)) {
return 0;
}
if (BN_num_bits(rnd) != (unsigned)bits) {
goto again;
}
return 1;
}
static int probable_prime_dh(BIGNUM *rnd, int bits, const BIGNUM *add,
const BIGNUM *rem, BN_CTX *ctx) {
int ret = 0;
BIGNUM *t1;
BN_CTX_start(ctx);
if ((t1 = BN_CTX_get(ctx)) == NULL) {
goto err;
}
if (!BN_rand(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD)) {
goto err;
}
// we need ((rnd-rem) % add) == 0
if (!BN_mod(t1, rnd, add, ctx)) {
goto err;
}
if (!BN_sub(rnd, rnd, t1)) {
goto err;
}
if (rem == NULL) {
if (!BN_add_word(rnd, 1)) {
goto err;
}
} else {
if (!BN_add(rnd, rnd, rem)) {
goto err;
}
}
// we now have a random number 'rand' to test.
const size_t num_primes = num_trial_division_primes(rnd);
loop:
for (size_t i = 1; i < num_primes; i++) {
// check that rnd is a prime
if (bn_mod_u16_consttime(rnd, kPrimes[i]) <= 1) {
if (!BN_add(rnd, rnd, add)) {
goto err;
}
goto loop;
}
}
ret = 1;
err:
BN_CTX_end(ctx);
return ret;
}
static int probable_prime_dh_safe(BIGNUM *p, int bits, const BIGNUM *padd,
const BIGNUM *rem, BN_CTX *ctx) {
int ret = 0;
BIGNUM *t1, *qadd, *q;
bits--;
BN_CTX_start(ctx);
t1 = BN_CTX_get(ctx);
q = BN_CTX_get(ctx);
qadd = BN_CTX_get(ctx);
if (qadd == NULL) {
goto err;
}
if (!BN_rshift1(qadd, padd)) {
goto err;
}
if (!BN_rand(q, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD)) {
goto err;
}
// we need ((rnd-rem) % add) == 0
if (!BN_mod(t1, q, qadd, ctx)) {
goto err;
}
if (!BN_sub(q, q, t1)) {
goto err;
}
if (rem == NULL) {
if (!BN_add_word(q, 1)) {
goto err;
}
} else {
if (!BN_rshift1(t1, rem)) {
goto err;
}
if (!BN_add(q, q, t1)) {
goto err;
}
}
// we now have a random number 'rand' to test.
if (!BN_lshift1(p, q)) {
goto err;
}
if (!BN_add_word(p, 1)) {
goto err;
}
const size_t num_primes = num_trial_division_primes(p);
loop:
for (size_t i = 1; i < num_primes; i++) {
// check that p and q are prime
// check that for p and q
// gcd(p-1,primes) == 1 (except for 2)
if (bn_mod_u16_consttime(p, kPrimes[i]) == 0 ||
bn_mod_u16_consttime(q, kPrimes[i]) == 0) {
if (!BN_add(p, p, padd)) {
goto err;
}
if (!BN_add(q, q, qadd)) {
goto err;
}
goto loop;
}
}
ret = 1;
err:
BN_CTX_end(ctx);
return ret;
}