--- /dev/null
+/* 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.
+
+#define NUMPRIMES 2048
+
+// primes contains all the primes that fit into a uint16_t.
+static const uint16_t primes[NUMPRIMES] = {
+ 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;
+}
+
+// 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;
+}
+
+// The following functions use a Barrett reduction variant to avoid leaking the
+// numerator. See http://ridiculousfish.com/blog/posts/labor-of-division-episode-i.html
+//
+// We use 32-bit numerator and 16-bit divisor for simplicity. This allows
+// computing |m| and |q| without architecture-specific code.
+
+// mod_u16 returns |n| mod |d|. |p| and |m| are the "magic numbers" for |d| (see
+// reference). For proof of correctness in Coq, see
+// https://github.com/davidben/fiat-crypto/blob/barrett/src/Arithmetic/BarrettReduction/RidiculousFish.v
+// Note the Coq version of |mod_u16| additionally includes the computation of
+// |p| and |m| from |bn_mod_u16_consttime| below.
+static uint16_t mod_u16(uint32_t n, uint16_t d, uint32_t p, uint32_t m) {
+ // Compute floor(n/d) per steps 3 through 5.
+ uint32_t q = ((uint64_t)m * n) >> 32;
+ // Note there is a typo in the reference. We right-shift by one, not two.
+ uint32_t t = ((n - q) >> 1) + q;
+ t = t >> (p - 1);
+
+ // Multiply and subtract to get the remainder.
+ n -= d * t;
+ assert(n < d);
+ return n;
+}
+
+// shift_and_add_mod_u16 returns |r| * 2^32 + |a| mod |d|. |p| and |m| are the
+// "magic numbers" for |d| (see reference).
+static uint16_t shift_and_add_mod_u16(uint16_t r, uint32_t a, uint16_t d,
+ uint32_t p, uint32_t m) {
+ // Incorporate |a| in two 16-bit chunks.
+ uint32_t t = r;
+ t <<= 16;
+ t |= a >> 16;
+ t = mod_u16(t, d, p, m);
+
+ t <<= 16;
+ t |= a & 0xffff;
+ t = mod_u16(t, d, p, m);
+ return t;
+}
+
+uint16_t bn_mod_u16_consttime(const BIGNUM *bn, uint16_t d) {
+ if (d <= 1) {
+ return 0;
+ }
+
+ // Compute the "magic numbers" for |d|. See steps 1 and 2.
+ // This computes p = ceil(log_2(d)).
+ uint32_t p = BN_num_bits_word(d - 1);
+ // This operation is not constant-time, but |p| and |d| are public values.
+ // Note that |p| is at most 16, so the computation fits in |uint64_t|.
+ assert(p <= 16);
+ uint32_t m = ((UINT64_C(1) << (32 + p)) + d - 1) / d;
+
+ uint16_t ret = 0;
+ for (int i = bn->width - 1; i >= 0; i--) {
+#if BN_BITS2 == 32
+ ret = shift_and_add_mod_u16(ret, bn->d[i], d, p, m);
+#elif BN_BITS2 == 64
+ ret = shift_and_add_mod_u16(ret, bn->d[i] >> 32, d, p, m);
+ ret = shift_and_add_mod_u16(ret, bn->d[i] & 0xffffffff, d, p, m);
+#else
+#error "Unknown BN_ULONG size"
+#endif
+ }
+ return ret;
+}
+
+static int bn_trial_division(uint16_t *out, const BIGNUM *bn) {
+ for (int i = 1; i < NUMPRIMES; i++) {
+ if (bn_mod_u16_consttime(bn, primes[i]) == 0) {
+ *out = primes[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));
+ }
+
+ // 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_for_modulus(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);
+ 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) {
+ int i;
+ uint16_t mods[NUMPRIMES];
+ BN_ULONG delta;
+ BN_ULONG maxdelta = BN_MASK2 - primes[NUMPRIMES - 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 (i = 1; i < NUMPRIMES; i++) {
+ mods[i] = bn_mod_u16_consttime(rnd, primes[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 primes[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 (i = 1; i < NUMPRIMES && primes[i] < rnd_word; i++) {
+ if ((mods[i] + delta) % primes[i] == 0) {
+ delta += 2;
+ if (delta > maxdelta) {
+ goto again;
+ }
+ goto loop;
+ }
+ }
+ } else {
+ for (i = 1; i < NUMPRIMES; i++) {
+ // check that rnd is not a prime and also
+ // that gcd(rnd-1,primes) == 1 (except for 2)
+ if (((mods[i] + delta) % primes[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 i, 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.
+
+loop:
+ for (i = 1; i < NUMPRIMES; i++) {
+ // check that rnd is a prime
+ if (bn_mod_u16_consttime(rnd, primes[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 i, 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;
+ }
+
+loop:
+ for (i = 1; i < NUMPRIMES; 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, primes[i]) == 0 ||
+ bn_mod_u16_consttime(q, primes[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;
+}
git://repos.xcallymotion.com / motion2.git / blobdiff