/* * 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 */ #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 Zf(to_ntt_monty)(uint16_t *h, unsigned logn) { mq_NTT(h, logn); mq_poly_tomonty(h, logn); } /* see inner.h */ int Zf(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 Zf(is_short)((int16_t *)tt, s2, logn); } /* see inner.h */ int Zf(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 Zf(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 Zf(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 Zf(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)-Zf(is_short)(s1, s2, logn); return (int)(r >> 31); } /* see inner.h */ int Zf(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; }