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pqcrypto/crypto_sign/falcon-1024/clean/vrfy.c

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/*
* Falcon signature verification.
*
* ==========================(LICENSE BEGIN)============================
*
* Copyright (c) 2017-2019 Falcon Project
*
* Permission is hereby granted, free of charge, to any person obtaining
* a copy of this software and associated documentation files (the
* "Software"), to deal in the Software without restriction, including
* without limitation the rights to use, copy, modify, merge, publish,
* distribute, sublicense, and/or sell copies of the Software, and to
* permit persons to whom the Software is furnished to do so, subject to
* the following conditions:
*
* The above copyright notice and this permission notice shall be
* included in all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
* IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
* CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
* TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
* SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
*
* ===========================(LICENSE END)=============================
*
* @author Thomas Pornin <thomas.pornin@nccgroup.com>
*/
#include "inner.h"
/* ===================================================================== */
/*
* Constants for NTT.
*
* n = 2^logn (2 <= n <= 1024)
* phi = X^n + 1
* q = 12289
* q0i = -1/q mod 2^16
* R = 2^16 mod q
* R2 = 2^32 mod q
*/
#define Q 12289
#define Q0I 12287
#define R 4091
#define R2 10952
/*
* Table for NTT, binary case:
* GMb[x] = R*(g^rev(x)) mod q
* where g = 7 (it is a 2048-th primitive root of 1 modulo q)
* and rev() is the bit-reversal function over 10 bits.
*/
static const uint16_t GMb[] = {
4091, 7888, 11060, 11208, 6960, 4342, 6275, 9759,
1591, 6399, 9477, 5266, 586, 5825, 7538, 9710,
1134, 6407, 1711, 965, 7099, 7674, 3743, 6442,
10414, 8100, 1885, 1688, 1364, 10329, 10164, 9180,
12210, 6240, 997, 117, 4783, 4407, 1549, 7072,
2829, 6458, 4431, 8877, 7144, 2564, 5664, 4042,
12189, 432, 10751, 1237, 7610, 1534, 3983, 7863,
2181, 6308, 8720, 6570, 4843, 1690, 14, 3872,
5569, 9368, 12163, 2019, 7543, 2315, 4673, 7340,
1553, 1156, 8401, 11389, 1020, 2967, 10772, 7045,
3316, 11236, 5285, 11578, 10637, 10086, 9493, 6180,
9277, 6130, 3323, 883, 10469, 489, 1502, 2851,
11061, 9729, 2742, 12241, 4970, 10481, 10078, 1195,
730, 1762, 3854, 2030, 5892, 10922, 9020, 5274,
9179, 3604, 3782, 10206, 3180, 3467, 4668, 2446,
7613, 9386, 834, 7703, 6836, 3403, 5351, 12276,
3580, 1739, 10820, 9787, 10209, 4070, 12250, 8525,
10401, 2749, 7338, 10574, 6040, 943, 9330, 1477,
6865, 9668, 3585, 6633, 12145, 4063, 3684, 7680,
8188, 6902, 3533, 9807, 6090, 727, 10099, 7003,
6945, 1949, 9731, 10559, 6057, 378, 7871, 8763,
8901, 9229, 8846, 4551, 9589, 11664, 7630, 8821,
5680, 4956, 6251, 8388, 10156, 8723, 2341, 3159,
1467, 5460, 8553, 7783, 2649, 2320, 9036, 6188,
737, 3698, 4699, 5753, 9046, 3687, 16, 914,
5186, 10531, 4552, 1964, 3509, 8436, 7516, 5381,
10733, 3281, 7037, 1060, 2895, 7156, 8887, 5357,
6409, 8197, 2962, 6375, 5064, 6634, 5625, 278,
932, 10229, 8927, 7642, 351, 9298, 237, 5858,
7692, 3146, 12126, 7586, 2053, 11285, 3802, 5204,
4602, 1748, 11300, 340, 3711, 4614, 300, 10993,
5070, 10049, 11616, 12247, 7421, 10707, 5746, 5654,
3835, 5553, 1224, 8476, 9237, 3845, 250, 11209,
4225, 6326, 9680, 12254, 4136, 2778, 692, 8808,
6410, 6718, 10105, 10418, 3759, 7356, 11361, 8433,
6437, 3652, 6342, 8978, 5391, 2272, 6476, 7416,
8418, 10824, 11986, 5733, 876, 7030, 2167, 2436,
3442, 9217, 8206, 4858, 5964, 2746, 7178, 1434,
7389, 8879, 10661, 11457, 4220, 1432, 10832, 4328,
8557, 1867, 9454, 2416, 3816, 9076, 686, 5393,
2523, 4339, 6115, 619, 937, 2834, 7775, 3279,
2363, 7488, 6112, 5056, 824, 10204, 11690, 1113,
2727, 9848, 896, 2028, 5075, 2654, 10464, 7884,
12169, 5434, 3070, 6400, 9132, 11672, 12153, 4520,
1273, 9739, 11468, 9937, 10039, 9720, 2262, 9399,
11192, 315, 4511, 1158, 6061, 6751, 11865, 357,
7367, 4550, 983, 8534, 8352, 10126, 7530, 9253,
4367, 5221, 3999, 8777, 3161, 6990, 4130, 11652,
3374, 11477, 1753, 292, 8681, 2806, 10378, 12188,
5800, 11811, 3181, 1988, 1024, 9340, 2477, 10928,
4582, 6750, 3619, 5503, 5233, 2463, 8470, 7650,
7964, 6395, 1071, 1272, 3474, 11045, 3291, 11344,
8502, 9478, 9837, 1253, 1857, 6233, 4720, 11561,
6034, 9817, 3339, 1797, 2879, 6242, 5200, 2114,
7962, 9353, 11363, 5475, 6084, 9601, 4108, 7323,
10438, 9471, 1271, 408, 6911, 3079, 360, 8276,
11535, 9156, 9049, 11539, 850, 8617, 784, 7919,
8334, 12170, 1846, 10213, 12184, 7827, 11903, 5600,
9779, 1012, 721, 2784, 6676, 6552, 5348, 4424,
6816, 8405, 9959, 5150, 2356, 5552, 5267, 1333,
8801, 9661, 7308, 5788, 4910, 909, 11613, 4395,
8238, 6686, 4302, 3044, 2285, 12249, 1963, 9216,
4296, 11918, 695, 4371, 9793, 4884, 2411, 10230,
2650, 841, 3890, 10231, 7248, 8505, 11196, 6688,
4059, 6060, 3686, 4722, 11853, 5816, 7058, 6868,
11137, 7926, 4894, 12284, 4102, 3908, 3610, 6525,
7938, 7982, 11977, 6755, 537, 4562, 1623, 8227,
11453, 7544, 906, 11816, 9548, 10858, 9703, 2815,
11736, 6813, 6979, 819, 8903, 6271, 10843, 348,
7514, 8339, 6439, 694, 852, 5659, 2781, 3716,
11589, 3024, 1523, 8659, 4114, 10738, 3303, 5885,
2978, 7289, 11884, 9123, 9323, 11830, 98, 2526,
2116, 4131, 11407, 1844, 3645, 3916, 8133, 2224,
10871, 8092, 9651, 5989, 7140, 8480, 1670, 159,
10923, 4918, 128, 7312, 725, 9157, 5006, 6393,
3494, 6043, 10972, 6181, 11838, 3423, 10514, 7668,
3693, 6658, 6905, 11953, 10212, 11922, 9101, 8365,
5110, 45, 2400, 1921, 4377, 2720, 1695, 51,
2808, 650, 1896, 9997, 9971, 11980, 8098, 4833,
4135, 4257, 5838, 4765, 10985, 11532, 590, 12198,
482, 12173, 2006, 7064, 10018, 3912, 12016, 10519,
11362, 6954, 2210, 284, 5413, 6601, 3865, 10339,
11188, 6231, 517, 9564, 11281, 3863, 1210, 4604,
8160, 11447, 153, 7204, 5763, 5089, 9248, 12154,
11748, 1354, 6672, 179, 5532, 2646, 5941, 12185,
862, 3158, 477, 7279, 5678, 7914, 4254, 302,
2893, 10114, 6890, 9560, 9647, 11905, 4098, 9824,
10269, 1353, 10715, 5325, 6254, 3951, 1807, 6449,
5159, 1308, 8315, 3404, 1877, 1231, 112, 6398,
11724, 12272, 7286, 1459, 12274, 9896, 3456, 800,
1397, 10678, 103, 7420, 7976, 936, 764, 632,
7996, 8223, 8445, 7758, 10870, 9571, 2508, 1946,
6524, 10158, 1044, 4338, 2457, 3641, 1659, 4139,
4688, 9733, 11148, 3946, 2082, 5261, 2036, 11850,
7636, 12236, 5366, 2380, 1399, 7720, 2100, 3217,
10912, 8898, 7578, 11995, 2791, 1215, 3355, 2711,
2267, 2004, 8568, 10176, 3214, 2337, 1750, 4729,
4997, 7415, 6315, 12044, 4374, 7157, 4844, 211,
8003, 10159, 9290, 11481, 1735, 2336, 5793, 9875,
8192, 986, 7527, 1401, 870, 3615, 8465, 2756,
9770, 2034, 10168, 3264, 6132, 54, 2880, 4763,
11805, 3074, 8286, 9428, 4881, 6933, 1090, 10038,
2567, 708, 893, 6465, 4962, 10024, 2090, 5718,
10743, 780, 4733, 4623, 2134, 2087, 4802, 884,
5372, 5795, 5938, 4333, 6559, 7549, 5269, 10664,
4252, 3260, 5917, 10814, 5768, 9983, 8096, 7791,
6800, 7491, 6272, 1907, 10947, 6289, 11803, 6032,
11449, 1171, 9201, 7933, 2479, 7970, 11337, 7062,
8911, 6728, 6542, 8114, 8828, 6595, 3545, 4348,
4610, 2205, 6999, 8106, 5560, 10390, 9321, 2499,
2413, 7272, 6881, 10582, 9308, 9437, 3554, 3326,
5991, 11969, 3415, 12283, 9838, 12063, 4332, 7830,
11329, 6605, 12271, 2044, 11611, 7353, 11201, 11582,
3733, 8943, 9978, 1627, 7168, 3935, 5050, 2762,
7496, 10383, 755, 1654, 12053, 4952, 10134, 4394,
6592, 7898, 7497, 8904, 12029, 3581, 10748, 5674,
10358, 4901, 7414, 8771, 710, 6764, 8462, 7193,
5371, 7274, 11084, 290, 7864, 6827, 11822, 2509,
6578, 4026, 5807, 1458, 5721, 5762, 4178, 2105,
11621, 4852, 8897, 2856, 11510, 9264, 2520, 8776,
7011, 2647, 1898, 7039, 5950, 11163, 5488, 6277,
9182, 11456, 633, 10046, 11554, 5633, 9587, 2333,
7008, 7084, 5047, 7199, 9865, 8997, 569, 6390,
10845, 9679, 8268, 11472, 4203, 1997, 2, 9331,
162, 6182, 2000, 3649, 9792, 6363, 7557, 6187,
8510, 9935, 5536, 9019, 3706, 12009, 1452, 3067,
5494, 9692, 4865, 6019, 7106, 9610, 4588, 10165,
6261, 5887, 2652, 10172, 1580, 10379, 4638, 9949
};
/*
* Table for inverse NTT, binary case:
* iGMb[x] = R*((1/g)^rev(x)) mod q
* Since g = 7, 1/g = 8778 mod 12289.
*/
static const uint16_t iGMb[] = {
4091, 4401, 1081, 1229, 2530, 6014, 7947, 5329,
2579, 4751, 6464, 11703, 7023, 2812, 5890, 10698,
3109, 2125, 1960, 10925, 10601, 10404, 4189, 1875,
5847, 8546, 4615, 5190, 11324, 10578, 5882, 11155,
8417, 12275, 10599, 7446, 5719, 3569, 5981, 10108,
4426, 8306, 10755, 4679, 11052, 1538, 11857, 100,
8247, 6625, 9725, 5145, 3412, 7858, 5831, 9460,
5217, 10740, 7882, 7506, 12172, 11292, 6049, 79,
13, 6938, 8886, 5453, 4586, 11455, 2903, 4676,
9843, 7621, 8822, 9109, 2083, 8507, 8685, 3110,
7015, 3269, 1367, 6397, 10259, 8435, 10527, 11559,
11094, 2211, 1808, 7319, 48, 9547, 2560, 1228,
9438, 10787, 11800, 1820, 11406, 8966, 6159, 3012,
6109, 2796, 2203, 1652, 711, 7004, 1053, 8973,
5244, 1517, 9322, 11269, 900, 3888, 11133, 10736,
4949, 7616, 9974, 4746, 10270, 126, 2921, 6720,
6635, 6543, 1582, 4868, 42, 673, 2240, 7219,
1296, 11989, 7675, 8578, 11949, 989, 10541, 7687,
7085, 8487, 1004, 10236, 4703, 163, 9143, 4597,
6431, 12052, 2991, 11938, 4647, 3362, 2060, 11357,
12011, 6664, 5655, 7225, 5914, 9327, 4092, 5880,
6932, 3402, 5133, 9394, 11229, 5252, 9008, 1556,
6908, 4773, 3853, 8780, 10325, 7737, 1758, 7103,
11375, 12273, 8602, 3243, 6536, 7590, 8591, 11552,
6101, 3253, 9969, 9640, 4506, 3736, 6829, 10822,
9130, 9948, 3566, 2133, 3901, 6038, 7333, 6609,
3468, 4659, 625, 2700, 7738, 3443, 3060, 3388,
3526, 4418, 11911, 6232, 1730, 2558, 10340, 5344,
5286, 2190, 11562, 6199, 2482, 8756, 5387, 4101,
4609, 8605, 8226, 144, 5656, 8704, 2621, 5424,
10812, 2959, 11346, 6249, 1715, 4951, 9540, 1888,
3764, 39, 8219, 2080, 2502, 1469, 10550, 8709,
5601, 1093, 3784, 5041, 2058, 8399, 11448, 9639,
2059, 9878, 7405, 2496, 7918, 11594, 371, 7993,
3073, 10326, 40, 10004, 9245, 7987, 5603, 4051,
7894, 676, 11380, 7379, 6501, 4981, 2628, 3488,
10956, 7022, 6737, 9933, 7139, 2330, 3884, 5473,
7865, 6941, 5737, 5613, 9505, 11568, 11277, 2510,
6689, 386, 4462, 105, 2076, 10443, 119, 3955,
4370, 11505, 3672, 11439, 750, 3240, 3133, 754,
4013, 11929, 9210, 5378, 11881, 11018, 2818, 1851,
4966, 8181, 2688, 6205, 6814, 926, 2936, 4327,
10175, 7089, 6047, 9410, 10492, 8950, 2472, 6255,
728, 7569, 6056, 10432, 11036, 2452, 2811, 3787,
945, 8998, 1244, 8815, 11017, 11218, 5894, 4325,
4639, 3819, 9826, 7056, 6786, 8670, 5539, 7707,
1361, 9812, 2949, 11265, 10301, 9108, 478, 6489,
101, 1911, 9483, 3608, 11997, 10536, 812, 8915,
637, 8159, 5299, 9128, 3512, 8290, 7068, 7922,
3036, 4759, 2163, 3937, 3755, 11306, 7739, 4922,
11932, 424, 5538, 6228, 11131, 7778, 11974, 1097,
2890, 10027, 2569, 2250, 2352, 821, 2550, 11016,
7769, 136, 617, 3157, 5889, 9219, 6855, 120,
4405, 1825, 9635, 7214, 10261, 11393, 2441, 9562,
11176, 599, 2085, 11465, 7233, 6177, 4801, 9926,
9010, 4514, 9455, 11352, 11670, 6174, 7950, 9766,
6896, 11603, 3213, 8473, 9873, 2835, 10422, 3732,
7961, 1457, 10857, 8069, 832, 1628, 3410, 4900,
10855, 5111, 9543, 6325, 7431, 4083, 3072, 8847,
9853, 10122, 5259, 11413, 6556, 303, 1465, 3871,
4873, 5813, 10017, 6898, 3311, 5947, 8637, 5852,
3856, 928, 4933, 8530, 1871, 2184, 5571, 5879,
3481, 11597, 9511, 8153, 35, 2609, 5963, 8064,
1080, 12039, 8444, 3052, 3813, 11065, 6736, 8454,
2340, 7651, 1910, 10709, 2117, 9637, 6402, 6028,
2124, 7701, 2679, 5183, 6270, 7424, 2597, 6795,
9222, 10837, 280, 8583, 3270, 6753, 2354, 3779,
6102, 4732, 5926, 2497, 8640, 10289, 6107, 12127,
2958, 12287, 10292, 8086, 817, 4021, 2610, 1444,
5899, 11720, 3292, 2424, 5090, 7242, 5205, 5281,
9956, 2702, 6656, 735, 2243, 11656, 833, 3107,
6012, 6801, 1126, 6339, 5250, 10391, 9642, 5278,
3513, 9769, 3025, 779, 9433, 3392, 7437, 668,
10184, 8111, 6527, 6568, 10831, 6482, 8263, 5711,
9780, 467, 5462, 4425, 11999, 1205, 5015, 6918,
5096, 3827, 5525, 11579, 3518, 4875, 7388, 1931,
6615, 1541, 8708, 260, 3385, 4792, 4391, 5697,
7895, 2155, 7337, 236, 10635, 11534, 1906, 4793,
9527, 7239, 8354, 5121, 10662, 2311, 3346, 8556,
707, 1088, 4936, 678, 10245, 18, 5684, 960,
4459, 7957, 226, 2451, 6, 8874, 320, 6298,
8963, 8735, 2852, 2981, 1707, 5408, 5017, 9876,
9790, 2968, 1899, 6729, 4183, 5290, 10084, 7679,
7941, 8744, 5694, 3461, 4175, 5747, 5561, 3378,
5227, 952, 4319, 9810, 4356, 3088, 11118, 840,
6257, 486, 6000, 1342, 10382, 6017, 4798, 5489,
4498, 4193, 2306, 6521, 1475, 6372, 9029, 8037,
1625, 7020, 4740, 5730, 7956, 6351, 6494, 6917,
11405, 7487, 10202, 10155, 7666, 7556, 11509, 1546,
6571, 10199, 2265, 7327, 5824, 11396, 11581, 9722,
2251, 11199, 5356, 7408, 2861, 4003, 9215, 484,
7526, 9409, 12235, 6157, 9025, 2121, 10255, 2519,
9533, 3824, 8674, 11419, 10888, 4762, 11303, 4097,
2414, 6496, 9953, 10554, 808, 2999, 2130, 4286,
12078, 7445, 5132, 7915, 245, 5974, 4874, 7292,
7560, 10539, 9952, 9075, 2113, 3721, 10285, 10022,
9578, 8934, 11074, 9498, 294, 4711, 3391, 1377,
9072, 10189, 4569, 10890, 9909, 6923, 53, 4653,
439, 10253, 7028, 10207, 8343, 1141, 2556, 7601,
8150, 10630, 8648, 9832, 7951, 11245, 2131, 5765,
10343, 9781, 2718, 1419, 4531, 3844, 4066, 4293,
11657, 11525, 11353, 4313, 4869, 12186, 1611, 10892,
11489, 8833, 2393, 15, 10830, 5003, 17, 565,
5891, 12177, 11058, 10412, 8885, 3974, 10981, 7130,
5840, 10482, 8338, 6035, 6964, 1574, 10936, 2020,
2465, 8191, 384, 2642, 2729, 5399, 2175, 9396,
11987, 8035, 4375, 6611, 5010, 11812, 9131, 11427,
104, 6348, 9643, 6757, 12110, 5617, 10935, 541,
135, 3041, 7200, 6526, 5085, 12136, 842, 4129,
7685, 11079, 8426, 1008, 2725, 11772, 6058, 1101,
1950, 8424, 5688, 6876, 12005, 10079, 5335, 927,
1770, 273, 8377, 2271, 5225, 10283, 116, 11807,
91, 11699, 757, 1304, 7524, 6451, 8032, 8154,
7456, 4191, 309, 2318, 2292, 10393, 11639, 9481,
12238, 10594, 9569, 7912, 10368, 9889, 12244, 7179,
3924, 3188, 367, 2077, 336, 5384, 5631, 8596,
4621, 1775, 8866, 451, 6108, 1317, 6246, 8795,
5896, 7283, 3132, 11564, 4977, 12161, 7371, 1366,
12130, 10619, 3809, 5149, 6300, 2638, 4197, 1418,
10065, 4156, 8373, 8644, 10445, 882, 8158, 10173,
9763, 12191, 459, 2966, 3166, 405, 5000, 9311,
6404, 8986, 1551, 8175, 3630, 10766, 9265, 700,
8573, 9508, 6630, 11437, 11595, 5850, 3950, 4775,
11941, 1446, 6018, 3386, 11470, 5310, 5476, 553,
9474, 2586, 1431, 2741, 473, 11383, 4745, 836,
4062, 10666, 7727, 11752, 5534, 312, 4307, 4351,
5764, 8679, 8381, 8187, 5, 7395, 4363, 1152,
5421, 5231, 6473, 436, 7567, 8603, 6229, 8230
};
/*
* Reduce a small signed integer modulo q. The source integer MUST
* be between -q/2 and +q/2.
*/
static inline uint32_t
mq_conv_small(int x) {
/*
* If x < 0, the cast to uint32_t will set the high bit to 1.
*/
uint32_t y;
y = (uint32_t)x;
y += Q & -(y >> 31);
return y;
}
/*
* Addition modulo q. Operands must be in the 0..q-1 range.
*/
static inline uint32_t
mq_add(uint32_t x, uint32_t y) {
/*
* We compute x + y - q. If the result is negative, then the
* high bit will be set, and 'd >> 31' will be equal to 1;
* thus '-(d >> 31)' will be an all-one pattern. Otherwise,
* it will be an all-zero pattern. In other words, this
* implements a conditional addition of q.
*/
uint32_t d;
d = x + y - Q;
d += Q & -(d >> 31);
return d;
}
/*
* Subtraction modulo q. Operands must be in the 0..q-1 range.
*/
static inline uint32_t
mq_sub(uint32_t x, uint32_t y) {
/*
* As in mq_add(), we use a conditional addition to ensure the
* result is in the 0..q-1 range.
*/
uint32_t d;
d = x - y;
d += Q & -(d >> 31);
return d;
}
/*
* Division by 2 modulo q. Operand must be in the 0..q-1 range.
*/
static inline uint32_t
mq_rshift1(uint32_t x) {
x += Q & -(x & 1);
return (x >> 1);
}
/*
* Montgomery multiplication modulo q. If we set R = 2^16 mod q, then
* this function computes: x * y / R mod q
* Operands must be in the 0..q-1 range.
*/
static inline uint32_t
mq_montymul(uint32_t x, uint32_t y) {
uint32_t z, w;
/*
* We compute x*y + k*q with a value of k chosen so that the 16
* low bits of the result are 0. We can then shift the value.
* After the shift, result may still be larger than q, but it
* will be lower than 2*q, so a conditional subtraction works.
*/
z = x * y;
w = ((z * Q0I) & 0xFFFF) * Q;
/*
* When adding z and w, the result will have its low 16 bits
* equal to 0. Since x, y and z are lower than q, the sum will
* be no more than (2^15 - 1) * q + (q - 1)^2, which will
* fit on 29 bits.
*/
z = (z + w) >> 16;
/*
* After the shift, analysis shows that the value will be less
* than 2q. We do a subtraction then conditional subtraction to
* ensure the result is in the expected range.
*/
z -= Q;
z += Q & -(z >> 31);
return z;
}
/*
* Montgomery squaring (computes (x^2)/R).
*/
static inline uint32_t
mq_montysqr(uint32_t x) {
return mq_montymul(x, x);
}
/*
* Divide x by y modulo q = 12289.
*/
static inline uint32_t
mq_div_12289(uint32_t x, uint32_t y) {
/*
* We invert y by computing y^(q-2) mod q.
*
* We use the following addition chain for exponent e = 12287:
*
* e0 = 1
* e1 = 2 * e0 = 2
* e2 = e1 + e0 = 3
* e3 = e2 + e1 = 5
* e4 = 2 * e3 = 10
* e5 = 2 * e4 = 20
* e6 = 2 * e5 = 40
* e7 = 2 * e6 = 80
* e8 = 2 * e7 = 160
* e9 = e8 + e2 = 163
* e10 = e9 + e8 = 323
* e11 = 2 * e10 = 646
* e12 = 2 * e11 = 1292
* e13 = e12 + e9 = 1455
* e14 = 2 * e13 = 2910
* e15 = 2 * e14 = 5820
* e16 = e15 + e10 = 6143
* e17 = 2 * e16 = 12286
* e18 = e17 + e0 = 12287
*
* Additions on exponents are converted to Montgomery
* multiplications. We define all intermediate results as so
* many local variables, and let the C compiler work out which
* must be kept around.
*/
uint32_t y0, y1, y2, y3, y4, y5, y6, y7, y8, y9;
uint32_t y10, y11, y12, y13, y14, y15, y16, y17, y18;
y0 = mq_montymul(y, R2);
y1 = mq_montysqr(y0);
y2 = mq_montymul(y1, y0);
y3 = mq_montymul(y2, y1);
y4 = mq_montysqr(y3);
y5 = mq_montysqr(y4);
y6 = mq_montysqr(y5);
y7 = mq_montysqr(y6);
y8 = mq_montysqr(y7);
y9 = mq_montymul(y8, y2);
y10 = mq_montymul(y9, y8);
y11 = mq_montysqr(y10);
y12 = mq_montysqr(y11);
y13 = mq_montymul(y12, y9);
y14 = mq_montysqr(y13);
y15 = mq_montysqr(y14);
y16 = mq_montymul(y15, y10);
y17 = mq_montysqr(y16);
y18 = mq_montymul(y17, y0);
/*
* Final multiplication with x, which is not in Montgomery
* representation, computes the correct division result.
*/
return mq_montymul(y18, x);
}
/*
* Compute NTT on a ring element.
*/
static void
mq_NTT(uint16_t *a, unsigned logn) {
size_t n, t, m;
n = (size_t)1 << logn;
t = n;
for (m = 1; m < n; m <<= 1) {
size_t ht, i, j1;
ht = t >> 1;
for (i = 0, j1 = 0; i < m; i ++, j1 += t) {
size_t j, j2;
uint32_t s;
s = GMb[m + i];
j2 = j1 + ht;
for (j = j1; j < j2; j ++) {
uint32_t u, v;
u = a[j];
v = mq_montymul(a[j + ht], s);
a[j] = (uint16_t)mq_add(u, v);
a[j + ht] = (uint16_t)mq_sub(u, v);
}
}
t = ht;
}
}
/*
* Compute the inverse NTT on a ring element, binary case.
*/
static void
mq_iNTT(uint16_t *a, unsigned logn) {
size_t n, t, m;
uint32_t ni;
n = (size_t)1 << logn;
t = 1;
m = n;
while (m > 1) {
size_t hm, dt, i, j1;
hm = m >> 1;
dt = t << 1;
for (i = 0, j1 = 0; i < hm; i ++, j1 += dt) {
size_t j, j2;
uint32_t s;
j2 = j1 + t;
s = iGMb[hm + i];
for (j = j1; j < j2; j ++) {
uint32_t u, v, w;
u = a[j];
v = a[j + t];
a[j] = (uint16_t)mq_add(u, v);
w = mq_sub(u, v);
a[j + t] = (uint16_t)
mq_montymul(w, s);
}
}
t = dt;
m = hm;
}
/*
* To complete the inverse NTT, we must now divide all values by
* n (the vector size). We thus need the inverse of n, i.e. we
* need to divide 1 by 2 logn times. But we also want it in
* Montgomery representation, i.e. we also want to multiply it
* by R = 2^16. In the common case, this should be a simple right
* shift. The loop below is generic and works also in corner cases;
* its computation time is negligible.
*/
ni = R;
for (m = n; m > 1; m >>= 1) {
ni = mq_rshift1(ni);
}
for (m = 0; m < n; m ++) {
a[m] = (uint16_t)mq_montymul(a[m], ni);
}
}
/*
* Convert a polynomial (mod q) to Montgomery representation.
*/
static void
mq_poly_tomonty(uint16_t *f, unsigned logn) {
size_t u, n;
n = (size_t)1 << logn;
for (u = 0; u < n; u ++) {
f[u] = (uint16_t)mq_montymul(f[u], R2);
}
}
/*
* Multiply two polynomials together (NTT representation, and using
* a Montgomery multiplication). Result f*g is written over f.
*/
static void
mq_poly_montymul_ntt(uint16_t *f, const uint16_t *g, unsigned logn) {
size_t u, n;
n = (size_t)1 << logn;
for (u = 0; u < n; u ++) {
f[u] = (uint16_t)mq_montymul(f[u], g[u]);
}
}
/*
* Subtract polynomial g from polynomial f.
*/
static void
mq_poly_sub(uint16_t *f, const uint16_t *g, unsigned logn) {
size_t u, n;
n = (size_t)1 << logn;
for (u = 0; u < n; u ++) {
f[u] = (uint16_t)mq_sub(f[u], g[u]);
}
}
/* ===================================================================== */
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_to_ntt_monty(uint16_t *h, unsigned logn) {
mq_NTT(h, logn);
mq_poly_tomonty(h, logn);
}
/* see inner.h */
int
PQCLEAN_FALCON1024_CLEAN_verify_raw(const uint16_t *c0, const int16_t *s2,
const uint16_t *h, unsigned logn, uint8_t *tmp) {
size_t u, n;
uint16_t *tt;
n = (size_t)1 << logn;
tt = (uint16_t *)tmp;
/*
* Reduce s2 elements modulo q ([0..q-1] range).
*/
for (u = 0; u < n; u ++) {
uint32_t w;
w = (uint32_t)s2[u];
w += Q & -(w >> 31);
tt[u] = (uint16_t)w;
}
/*
* Compute -s1 = s2*h - c0 mod phi mod q (in tt[]).
*/
mq_NTT(tt, logn);
mq_poly_montymul_ntt(tt, h, logn);
mq_iNTT(tt, logn);
mq_poly_sub(tt, c0, logn);
/*
* Normalize -s1 elements into the [-q/2..q/2] range.
*/
for (u = 0; u < n; u ++) {
int32_t w;
w = (int32_t)tt[u];
w -= (int32_t)(Q & -(((Q >> 1) - (uint32_t)w) >> 31));
((int16_t *)tt)[u] = (int16_t)w;
}
/*
* Signature is valid if and only if the aggregate (-s1,s2) vector
* is short enough.
*/
return PQCLEAN_FALCON1024_CLEAN_is_short((int16_t *)tt, s2, logn);
}
/* see inner.h */
int
PQCLEAN_FALCON1024_CLEAN_compute_public(uint16_t *h,
const int8_t *f, const int8_t *g, unsigned logn, uint8_t *tmp) {
size_t u, n;
uint16_t *tt;
n = (size_t)1 << logn;
tt = (uint16_t *)tmp;
for (u = 0; u < n; u ++) {
tt[u] = (uint16_t)mq_conv_small(f[u]);
h[u] = (uint16_t)mq_conv_small(g[u]);
}
mq_NTT(h, logn);
mq_NTT(tt, logn);
for (u = 0; u < n; u ++) {
if (tt[u] == 0) {
return 0;
}
h[u] = (uint16_t)mq_div_12289(h[u], tt[u]);
}
mq_iNTT(h, logn);
return 1;
}
/* see inner.h */
int
PQCLEAN_FALCON1024_CLEAN_complete_private(int8_t *G,
const int8_t *f, const int8_t *g, const int8_t *F,
unsigned logn, uint8_t *tmp) {
size_t u, n;
uint16_t *t1, *t2;
n = (size_t)1 << logn;
t1 = (uint16_t *)tmp;
t2 = t1 + n;
for (u = 0; u < n; u ++) {
t1[u] = (uint16_t)mq_conv_small(g[u]);
t2[u] = (uint16_t)mq_conv_small(F[u]);
}
mq_NTT(t1, logn);
mq_NTT(t2, logn);
mq_poly_tomonty(t1, logn);
mq_poly_montymul_ntt(t1, t2, logn);
for (u = 0; u < n; u ++) {
t2[u] = (uint16_t)mq_conv_small(f[u]);
}
mq_NTT(t2, logn);
for (u = 0; u < n; u ++) {
if (t2[u] == 0) {
return 0;
}
t1[u] = (uint16_t)mq_div_12289(t1[u], t2[u]);
}
mq_iNTT(t1, logn);
for (u = 0; u < n; u ++) {
uint32_t w;
int32_t gi;
w = t1[u];
w -= (Q & ~ -((w - (Q >> 1)) >> 31));
gi = *(int32_t *)&w;
if (gi < -127 || gi > +127) {
return 0;
}
G[u] = (int8_t)gi;
}
return 1;
}
/* see inner.h */
int
PQCLEAN_FALCON1024_CLEAN_is_invertible(
const int16_t *s2, unsigned logn, uint8_t *tmp) {
size_t u, n;
uint16_t *tt;
uint32_t r;
n = (size_t)1 << logn;
tt = (uint16_t *)tmp;
for (u = 0; u < n; u ++) {
uint32_t w;
w = (uint32_t)s2[u];
w += Q & -(w >> 31);
tt[u] = (uint16_t)w;
}
mq_NTT(tt, logn);
r = 0;
for (u = 0; u < n; u ++) {
r |= (uint32_t)(tt[u] - 1);
}
return (int)(1u - (r >> 31));
}
/* see inner.h */
int
PQCLEAN_FALCON1024_CLEAN_verify_recover(uint16_t *h,
const uint16_t *c0, const int16_t *s1, const int16_t *s2,
unsigned logn, uint8_t *tmp) {
size_t u, n;
uint16_t *tt;
uint32_t r;
n = (size_t)1 << logn;
/*
* Reduce elements of s1 and s2 modulo q; then write s2 into tt[]
* and c0 - s1 into h[].
*/
tt = (uint16_t *)tmp;
for (u = 0; u < n; u ++) {
uint32_t w;
w = (uint32_t)s2[u];
w += Q & -(w >> 31);
tt[u] = (uint16_t)w;
w = (uint32_t)s1[u];
w += Q & -(w >> 31);
w = mq_sub(c0[u], w);
h[u] = (uint16_t)w;
}
/*
* Compute h = (c0 - s1) / s2. If one of the coefficients of s2
* is zero (in NTT representation) then the operation fails. We
* keep that information into a flag so that we do not deviate
* from strict constant-time processing; if all coefficients of
* s2 are non-zero, then the high bit of r will be zero.
*/
mq_NTT(tt, logn);
mq_NTT(h, logn);
r = 0;
for (u = 0; u < n; u ++) {
r |= (uint32_t)(tt[u] - 1);
h[u] = (uint16_t)mq_div_12289(h[u], tt[u]);
}
mq_iNTT(h, logn);
/*
* Signature is acceptable if and only if it is short enough,
* and s2 was invertible mod phi mod q. The caller must still
* check that the rebuilt public key matches the expected
* value (e.g. through a hash).
*/
r = ~r & (uint32_t) - PQCLEAN_FALCON1024_CLEAN_is_short(s1, s2, logn);
return (int)(r >> 31);
}
/* see inner.h */
int
PQCLEAN_FALCON1024_CLEAN_count_nttzero(const int16_t *sig, unsigned logn, uint8_t *tmp) {
uint16_t *s2;
size_t u, n;
uint32_t r;
n = (size_t)1 << logn;
s2 = (uint16_t *)tmp;
for (u = 0; u < n; u ++) {
uint32_t w;
w = (uint32_t)sig[u];
w += Q & -(w >> 31);
s2[u] = (uint16_t)w;
}
mq_NTT(s2, logn);
r = 0;
for (u = 0; u < n; u ++) {
uint32_t w;
w = (uint32_t)s2[u] - 1u;
r += (w >> 31);
}
return (int)r;
}
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