/* 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, in order to maintain an error rate greater
// than the security level for an RSA prime of that many bits (calculated using
// the FIPS SP 800-57 security level and 186-4 Section F.1; original paper:
// Damgaard, Landrock, Pomerance: Average case error estimates for the strong
// probable prime test. -- Math. Comp. 61 (1993) 177-194)
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 >= 308) {
    return 8;
  }
  if (bits >= 205) {
    return 13;
  }
  if (bits >= 155) {
    return 19;
  }
  return 28;
}

// 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 *is_probably_prime, const BIGNUM *w,
                      int iterations, BN_CTX *ctx, int do_trial_division,
                      BN_GENCB *cb) {
  *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.
    *is_probably_prime = BN_is_word(w, 2);
    return 1;
  }

  // Miller-Rabin does not work for three.
  if (BN_is_word(w, 3)) {
    *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)) {
      *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 Rabin-Miller checks.
  //
  // Rabin-Miller 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.)
      *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);
  *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;
}