pqcrypto/crypto_kem/newhope1024cpa/clean/precomp.c

#include "inttypes.h"
#include "ntt.h"
#include "params.h"

/* Precomputed NTT contants generated by Pari/GP script as follows:
 *
 * For n = 512:
 *
 * brv = [0,256,128,384,64,320,192,448,32,288,160,416,96,352,224,480,16,272,144,400,80,336,208,464,48,304,176,432,112,368,240,496,8,
 *        264,136,392,72,328,200,456,40,296,168,424,104,360,232,488,24,280,152,408,88,344,216,472,56,312,184,440,120,376,248,504,4,
 *        260,132,388,68,324,196,452,36,292,164,420,100,356,228,484,20,276,148,404,84,340,212,468,52,308,180,436,116,372,244,500,12,
 *        268,140,396,76,332,204,460,44,300,172,428,108,364,236,492,28,284,156,412,92,348,220,476,60,316,188,444,124,380,252,508,2,
 *        258,130,386,66,322,194,450,34,290,162,418,98,354,226,482,18,274,146,402,82,338,210,466,50,306,178,434,114,370,242,498,10,
 *        266,138,394,74,330,202,458,42,298,170,426,106,362,234,490,26,282,154,410,90,346,218,474,58,314,186,442,122,378,250,506,6,
 *        262,134,390,70,326,198,454,38,294,166,422,102,358,230,486,22,278,150,406,86,342,214,470,54,310,182,438,118,374,246,502,14,
 *        270,142,398,78,334,206,462,46,302,174,430,110,366,238,494,30,286,158,414,94,350,222,478,62,318,190,446,126,382,254,510,1,
 *        257,129,385,65,321,193,449,33,289,161,417,97,353,225,481,17,273,145,401,81,337,209,465,49,305,177,433,113,369,241,497,9,
 *        265,137,393,73,329,201,457,41,297,169,425,105,361,233,489,25,281,153,409,89,345,217,473,57,313,185,441,121,377,249,505,5,
 *        261,133,389,69,325,197,453,37,293,165,421,101,357,229,485,21,277,149,405,85,341,213,469,53,309,181,437,117,373,245,501,13,
 *        269,141,397,77,333,205,461,45,301,173,429,109,365,237,493,29,285,157,413,93,349,221,477,61,317,189,445,125,381,253,509,3,
 *        259,131,387,67,323,195,451,35,291,163,419,99,355,227,483,19,275,147,403,83,339,211,467,51,307,179,435,115,371,243,499,11,
 *        267,139,395,75,331,203,459,43,299,171,427,107,363,235,491,27,283,155,411,91,347,219,475,59,315,187,443,123,379,251,507,7,
 *        263,135,391,71,327,199,455,39,295,167,423,103,359,231,487,23,279,151,407,87,343,215,471,55,311,183,439,119,375,247,503,15,
 *        271,143,399,79,335,207,463,47,303,175,431,111,367,239,495,31,287,159,415,95,351,223,479,63,319,191,447,127,383,255,511]
 * n = 512; q = 12289; mont=2^18
 * g = Mod(10968, q);
 * omegas_inv_bitrev_montgomery = lift(vector(n/2, i, (g^2)^(-brv[2*(i-1)+1])*mont))
 * gammas_bitrev_montgomery = lift(vector(n, i, g^(brv[i])*mont))
 * gammas_inv_montgomery = lift(vector(n, i, g^(-(i-1))/n*mont))
 *
 *
 *  For n = 1024:
 *
 * brv = [0,512,256,768,128,640,384,896,64,576,320,832,192,704,448,960,32,544,288,800,160,672,416,928,96,608,352,864,224,736,480,992, \
 *        16,528,272,784,144,656,400,912,80,592,336,848,208,720,464,976,48,560,304,816,176,688,432,944,112,624,368,880,240,752,496,1008, \
 *        8,520,264,776,136,648,392,904,72,584,328,840,200,712,456,968,40,552,296,808,168,680,424,936,104,616,360,872,232,744,488,1000, \
 *        24,536,280,792,152,664,408,920,88,600,344,856,216,728,472,984,56,568,312,824,184,696,440,952,120,632,376,888,248,760,504,1016, \
 *        4,516,260,772,132,644,388,900,68,580,324,836,196,708,452,964,36,548,292,804,164,676,420,932,100,612,356,868,228,740,484,996, \
 *        20,532,276,788,148,660,404,916,84,596,340,852,212,724,468,980,52,564,308,820,180,692,436,948,116,628,372,884,244,756,500,1012, \
 *        12,524,268,780,140,652,396,908,76,588,332,844,204,716,460,972,44,556,300,812,172,684,428,940,108,620,364,876,236,748,492,1004, \
 *        28,540,284,796,156,668,412,924,92,604,348,860,220,732,476,988,60,572,316,828,188,700,444,956,124,636,380,892,252,764,508,1020, \
 *        2,514,258,770,130,642,386,898,66,578,322,834,194,706,450,962,34,546,290,802,162,674,418,930,98,610,354,866,226,738,482,994, \
 *        18,530,274,786,146,658,402,914,82,594,338,850,210,722,466,978,50,562,306,818,178,690,434,946,114,626,370,882,242,754,498,1010, \
 *        10,522,266,778,138,650,394,906,74,586,330,842,202,714,458,970,42,554,298,810,170,682,426,938,106,618,362,874,234,746,490,1002, \
 *        26,538,282,794,154,666,410,922,90,602,346,858,218,730,474,986,58,570,314,826,186,698,442,954,122,634,378,890,250,762,506,1018, \
 *        6,518,262,774,134,646,390,902,70,582,326,838,198,710,454,966,38,550,294,806,166,678,422,934,102,614,358,870,230,742,486,998, \
 *        22,534,278,790,150,662,406,918,86,598,342,854,214,726,470,982,54,566,310,822,182,694,438,950,118,630,374,886,246,758,502,1014, \
 *        14,526,270,782,142,654,398,910,78,590,334,846,206,718,462,974,46,558,302,814,174,686,430,942,110,622,366,878,238,750,494,1006, \
 *        30,542,286,798,158,670,414,926,94,606,350,862,222,734,478,990,62,574,318,830,190,702,446,958,126,638,382,894,254,766,510,1022, \
 *        1,513,257,769,129,641,385,897,65,577,321,833,193,705,449,961,33,545,289,801,161,673,417,929,97,609,353,865,225,737,481,993, \
 *        17,529,273,785,145,657,401,913,81,593,337,849,209,721,465,977,49,561,305,817,177,689,433,945,113,625,369,881,241,753,497,1009, \
 *        9,521,265,777,137,649,393,905,73,585,329,841,201,713,457,969,41,553,297,809,169,681,425,937,105,617,361,873,233,745,489,1001, \
 *        25,537,281,793,153,665,409,921,89,601,345,857,217,729,473,985,57,569,313,825,185,697,441,953,121,633,377,889,249,761,505,1017, \
 *        5,517,261,773,133,645,389,901,69,581,325,837,197,709,453,965,37,549,293,805,165,677,421,933,101,613,357,869,229,741,485,997, \
 *        21,533,277,789,149,661,405,917,85,597,341,853,213,725,469,981,53,565,309,821,181,693,437,949,117,629,373,885,245,757,501,1013, \
 *        13,525,269,781,141,653,397,909,77,589,333,845,205,717,461,973,45,557,301,813,173,685,429,941,109,621,365,877,237,749,493,1005, \
 *        29,541,285,797,157,669,413,925,93,605,349,861,221,733,477,989,61,573,317,829,189,701,445,957,125,637,381,893,253,765,509,1021, \
 *        3,515,259,771,131,643,387,899,67,579,323,835,195,707,451,963,35,547,291,803,163,675,419,931,99,611,355,867,227,739,483,995, \
 *        19,531,275,787,147,659,403,915,83,595,339,851,211,723,467,979,51,563,307,819,179,691,435,947,115,627,371,883,243,755,499,1011, \
 *        11,523,267,779,139,651,395,907,75,587,331,843,203,715,459,971,43,555,299,811,171,683,427,939,107,619,363,875,235,747,491,1003, \
 *        27,539,283,795,155,667,411,923,91,603,347,859,219,731,475,987,59,571,315,827,187,699,443,955,123,635,379,891,251,763,507,1019, \
 *        7,519,263,775,135,647,391,903,71,583,327,839,199,711,455,967,39,551,295,807,167,679,423,935,103,615,359,871,231,743,487,999, \
 *        23,535,279,791,151,663,407,919,87,599,343,855,215,727,471,983,55,567,311,823,183,695,439,951,119,631,375,887,247,759,503,1015, \
 *        15,527,271,783,143,655,399,911,79,591,335,847,207,719,463,975,47,559,303,815,175,687,431,943,111,623,367,879,239,751,495,1007, \
 *        31,543,287,799,159,671,415,927,95,607,351,863,223,735,479,991,63,575,319,831,191,703,447,959,127,639,383,895,255,767,511,1023]
 *
 * n = 1024; q = 12289; mont=2^18
 * g = Mod(7, q);
 * omegas_inv_bitrev_montgomery = lift(vector(n/2, i, (g^2)^(-brv[2*(i-1)+1])*mont))
 * gammas_bitrev_montgomery = lift(vector(n, i, g^(brv[i])*mont))
 * gammas_inv_montgomery = lift(vector(n, i, g^(-(i-1))/n*mont))
*/


/************************************************************
* Name:         omegas_inv_bitrev_montgomery
*
* Description: Contains inverses of powers of nth root of unity
*              in Montgomery domain with R=2^18 in bit-reversed order
************************************************************/
const uint16_t PQCLEAN_NEWHOPE1024CPA_CLEAN_omegas_inv_bitrev_montgomery[NEWHOPE_N / 2] = {
    4075, 5315, 4324, 4916, 10120, 11767, 7210, 9027, 10316, 6715, 1278, 9945, 3514, 11248, 11271, 5925,
    147, 8500, 7840, 6833, 5537, 4749, 4467, 7500, 11099, 9606, 6171, 8471, 8429, 5445, 11239, 7753,
    9090, 12233, 5529, 5206, 10587, 1987, 11635, 3565, 5415, 8646, 6153, 6427, 7341, 6152, 10561, 400,
    8410, 1922, 2033, 8291, 1359, 6854, 11035, 973, 8579, 6093, 6950, 5446, 11821, 8301, 11907, 316,
    52, 3174, 10966, 9523, 6055, 8953, 11612, 6415, 2505, 5906, 10710, 11858, 8332, 9450, 10162, 151,
    3482, 787, 5468, 1010, 4169, 9162, 5241, 9369, 7509, 8844, 7232, 4698, 192, 1321, 10240, 4912,
    885, 6281, 10333, 7280, 8757, 11286, 58, 12048, 12147, 11184, 8812, 6608, 2844, 3438, 4212, 11314,
    8687, 6068, 421, 8209, 3600, 3263, 7665, 6077, 7507, 5886, 3029, 6695, 4213, 504, 11684, 2302,
    1962, 1594, 6328, 7183, 168, 2692, 8960, 4298, 5184, 11089, 6122, 9734, 10929, 3956, 5297, 6170,
    3762, 9370, 4016, 4077, 6523, 652, 11994, 6099, 1146, 11341, 11964, 10885, 6299, 1159, 8240, 8561,
    11177, 2078, 10331, 4322, 11367, 441, 4079, 11231, 3150, 1319, 8243, 709, 8049, 8719, 11454, 6224,
    3054, 6803, 3123, 10542, 4433, 6370, 7032, 3834, 8633, 12225, 9830, 683, 1566, 5782, 9786, 9341,
    12115, 723, 3009, 1693, 5735, 2655, 2738, 6421, 11942, 2925, 1975, 8532, 3315, 11863, 4754, 1858,
    1583, 6347, 2500, 10800, 6374, 1483, 12240, 1263, 1815, 5383, 10777, 350, 6920, 10232, 4493, 9087,
    8855, 8760, 9381, 218, 9928, 10446, 9259, 4115, 6147, 9842, 8326, 576, 10335, 10238, 10484, 9407,
    6381, 11836, 8517, 418, 6860, 7515, 1293, 7552, 2767, 156, 8298, 8320, 10008, 5876, 5333, 10258,
    10115, 4372, 2847, 7875, 8232, 9018, 8925, 1689, 8236, 2645, 5042, 9984, 7094, 9509, 1484, 7394,
    3, 4437, 160, 3149, 113, 7370, 10123, 3915, 6998, 2704, 8653, 4938, 1426, 7635, 10512, 1663,
    6957, 3510, 2370, 2865, 3978, 9320, 3247, 9603, 6882, 3186, 10659, 10163, 1153, 9405, 8241, 10040,
    2178, 1544, 5559, 420, 8304, 4905, 476, 3531, 5191, 9153, 2399, 8889, 3000, 671, 243, 3016,
    3763, 10849, 12262, 9223, 10657, 7205, 11272, 7404, 7575, 8146, 10752, 242, 2678, 3704, 11744, 5019,
    3833, 3778, 11899, 773, 5101, 11222, 9888, 442, 2912, 5698, 11935, 4861, 7277, 9808, 11244, 2859,
    3780, 11414, 4976, 10682, 7201, 8005, 11287, 5011, 6267, 2987, 2437, 3646, 2566, 10102, 9867, 6250,
    5444, 2381, 11796, 8193, 4337, 11854, 1912, 1378, 404, 7644, 1065, 2143, 11121, 5277, 3248, 11082,
    2548, 8058, 8907, 11934, 1759, 8582, 3694, 7110, 12144, 6747, 8652, 3459, 2731, 8357, 6378, 7399,
    10861, 1696, 9863, 334, 7657, 6534, 11029, 4388, 11560, 3241, 10276, 9000, 9408, 3284, 10200, 7197,
    6498, 544, 2468, 339, 11267, 9, 2842, 480, 5331, 7300, 1673, 4278, 4177, 8705, 9764, 1381,
    7837, 2396, 8340, 8993, 4354, 130, 6915, 2837, 11462, 5767, 953, 8541, 9813, 118, 7222, 2197,
    3006, 9545, 563, 9314, 2625, 11340, 4821, 2639, 7266, 5828, 6561, 7698, 3328, 6512, 1351, 7311,
    6553, 8155, 1305, 722, 5146, 4043, 12288, 10810, 2545, 3621, 8747, 8785, 1646, 1212, 5860, 3195,
    7203, 10963, 3201, 3014, 955, 11499, 9970, 11119, 3135, 3712, 7443, 9542, 7484, 8736, 9995, 11227,
    1635, 9521, 1177, 8034, 140, 10436, 11563, 7678, 4320, 11289, 9198, 12208, 2963, 7393, 2366, 9238
};

/************************************************************
* Name:        gammas_bitrev_montgomery
*
* Description: Contains powers of nth root of -1 in Montgomery
*              domain with R=2^18 in bit-reversed order
************************************************************/
const uint16_t PQCLEAN_NEWHOPE1024CPA_CLEAN_gammas_bitrev_montgomery[NEWHOPE_N] = {
    4075, 6974, 7373, 7965, 3262, 5079, 522, 2169, 6364, 1018, 1041, 8775, 2344, 11011, 5574, 1973,
    4536, 1050, 6844, 3860, 3818, 6118, 2683, 1190, 4789, 7822, 7540, 6752, 5456, 4449, 3789, 12142,
    11973, 382, 3988, 468, 6843, 5339, 6196, 3710, 11316, 1254, 5435, 10930, 3998, 10256, 10367, 3879,
    11889, 1728, 6137, 4948, 5862, 6136, 3643, 6874, 8724, 654, 10302, 1702, 7083, 6760, 56, 3199,
    9987, 605, 11785, 8076, 5594, 9260, 6403, 4782, 6212, 4624, 9026, 8689, 4080, 11868, 6221, 3602,
    975, 8077, 8851, 9445, 5681, 3477, 1105, 142, 241, 12231, 1003, 3532, 5009, 1956, 6008, 11404,
    7377, 2049, 10968, 12097, 7591, 5057, 3445, 4780, 2920, 7048, 3127, 8120, 11279, 6821, 11502, 8807,
    12138, 2127, 2839, 3957, 431, 1579, 6383, 9784, 5874, 677, 3336, 6234, 2766, 1323, 9115, 12237,
    2031, 6956, 6413, 2281, 3969, 3991, 12133, 9522, 4737, 10996, 4774, 5429, 11871, 3772, 453, 5908,
    2882, 1805, 2051, 1954, 11713, 3963, 2447, 6142, 8174, 3030, 1843, 2361, 12071, 2908, 3529, 3434,
    3202, 7796, 2057, 5369, 11939, 1512, 6906, 10474, 11026, 49, 10806, 5915, 1489, 9789, 5942, 10706,
    10431, 7535, 426, 8974, 3757, 10314, 9364, 347, 5868, 9551, 9634, 6554, 10596, 9280, 11566, 174,
    2948, 2503, 6507, 10723, 11606, 2459, 64, 3656, 8455, 5257, 5919, 7856, 1747, 9166, 5486, 9235,
    6065, 835, 3570, 4240, 11580, 4046, 10970, 9139, 1058, 8210, 11848, 922, 7967, 1958, 10211, 1112,
    3728, 4049, 11130, 5990, 1404, 325, 948, 11143, 6190, 295, 11637, 5766, 8212, 8273, 2919, 8527,
    6119, 6992, 8333, 1360, 2555, 6167, 1200, 7105, 7991, 3329, 9597, 12121, 5106, 5961, 10695, 10327,
    3051, 9923, 4896, 9326, 81, 3091, 1000, 7969, 4611, 726, 1853, 12149, 4255, 11112, 2768, 10654,
    1062, 2294, 3553, 4805, 2747, 4846, 8577, 9154, 1170, 2319, 790, 11334, 9275, 9088, 1326, 5086,
    9094, 6429, 11077, 10643, 3504, 3542, 8668, 9744, 1479, 1, 8246, 7143, 11567, 10984, 4134, 5736,
    4978, 10938, 5777, 8961, 4591, 5728, 6461, 5023, 9650, 7468, 949, 9664, 2975, 11726, 2744, 9283,
    10092, 5067, 12171, 2476, 3748, 11336, 6522, 827, 9452, 5374, 12159, 7935, 3296, 3949, 9893, 4452,
    10908, 2525, 3584, 8112, 8011, 10616, 4989, 6958, 11809, 9447, 12280, 1022, 11950, 9821, 11745, 5791,
    5092, 2089, 9005, 2881, 3289, 2013, 9048, 729, 7901, 1260, 5755, 4632, 11955, 2426, 10593, 1428,
    4890, 5911, 3932, 9558, 8830, 3637, 5542, 145, 5179, 8595, 3707, 10530, 355, 3382, 4231, 9741,
    1207, 9041, 7012, 1168, 10146, 11224, 4645, 11885, 10911, 10377, 435, 7952, 4096, 493, 9908, 6845,
    6039, 2422, 2187, 9723, 8643, 9852, 9302, 6022, 7278, 1002, 4284, 5088, 1607, 7313, 875, 8509,
    9430, 1045, 2481, 5012, 7428, 354, 6591, 9377, 11847, 2401, 1067, 7188, 11516, 390, 8511, 8456,
    7270, 545, 8585, 9611, 12047, 1537, 4143, 4714, 4885, 1017, 5084, 1632, 3066, 27, 1440, 8526,
    9273, 12046, 11618, 9289, 3400, 9890, 3136, 7098, 8758, 11813, 7384, 3985, 11869, 6730, 10745, 10111,
    2249, 4048, 2884, 11136, 2126, 1630, 9103, 5407, 2686, 9042, 2969, 8311, 9424, 9919, 8779, 5332,
    10626, 1777, 4654, 10863, 7351, 3636, 9585, 5291, 8374, 2166, 4919, 12176, 9140, 12129, 7852, 12286,
    4895, 10805, 2780, 5195, 2305, 7247, 9644, 4053, 10600, 3364, 3271, 4057, 4414, 9442, 7917, 2174,
    3947, 11951, 2455, 6599, 10545, 10975, 3654, 2894, 7681, 7126, 7287, 12269, 4119, 3343, 2151, 1522,
    7174, 7350, 11041, 2442, 2148, 5959, 6492, 8330, 8945, 5598, 3624, 10397, 1325, 6565, 1945, 11260,
    10077, 2674, 3338, 3276, 11034, 506, 6505, 1392, 5478, 8778, 1178, 2776, 3408, 10347, 11124, 2575,
    9489, 12096, 6092, 10058, 4167, 6085, 923, 11251, 11912, 4578, 10669, 11914, 425, 10453, 392, 10104,
    8464, 4235, 8761, 7376, 2291, 3375, 7954, 8896, 6617, 7790, 1737, 11667, 3982, 9342, 6680, 636,
    6825, 7383, 512, 4670, 2900, 12050, 7735, 994, 1687, 11883, 7021, 146, 10485, 1403, 5189, 6094,
    2483, 2054, 3042, 10945, 3981, 10821, 11826, 8882, 8151, 180, 9600, 7684, 5219, 10880, 6780, 204,
    11232, 2600, 7584, 3121, 3017, 11053, 7814, 7043, 4251, 4739, 11063, 6771, 7073, 9261, 2360, 11925,
    1928, 11825, 8024, 3678, 3205, 3359, 11197, 5209, 8581, 3238, 8840, 1136, 9363, 1826, 3171, 4489,
    7885, 346, 2068, 1389, 8257, 3163, 4840, 6127, 8062, 8921, 612, 4238, 10763, 8067, 125, 11749,
    10125, 5416, 2110, 716, 9839, 10584, 11475, 11873, 3448, 343, 1908, 4538, 10423, 7078, 4727, 1208,
    11572, 3589, 2982, 1373, 1721, 10753, 4103, 2429, 4209, 5412, 5993, 9011, 438, 3515, 7228, 1218,
    8347, 5232, 8682, 1327, 7508, 4924, 448, 1014, 10029, 12221, 4566, 5836, 12229, 2717, 1535, 3200,
    5588, 5845, 412, 5102, 7326, 3744, 3056, 2528, 7406, 8314, 9202, 6454, 6613, 1417, 10032, 7784,
    1518, 3765, 4176, 5063, 9828, 2275, 6636, 4267, 6463, 2065, 7725, 3495, 8328, 8755, 8144, 10533,
    5966, 12077, 9175, 9520, 5596, 6302, 8400, 579, 6781, 11014, 5734, 11113, 11164, 4860, 1131, 10844,
    9068, 8016, 9694, 3837, 567, 9348, 7000, 6627, 7699, 5082, 682, 11309, 5207, 4050, 7087, 844,
    7434, 3769, 293, 9057, 6940, 9344, 10883, 2633, 8190, 3944, 5530, 5604, 3480, 2171, 9282, 11024,
    2213, 8136, 3805, 767, 12239, 216, 11520, 6763, 10353, 7, 8566, 845, 7235, 3154, 4360, 3285,
    10268, 2832, 3572, 1282, 7559, 3229, 8360, 10583, 6105, 3120, 6643, 6203, 8536, 8348, 6919, 3536,
    9199, 10891, 11463, 5043, 1658, 5618, 8787, 5789, 4719, 751, 11379, 6389, 10783, 3065, 7806, 6586,
    2622, 5386, 510, 7628, 6921, 578, 10345, 11839, 8929, 4684, 12226, 7154, 9916, 7302, 8481, 3670,
    11066, 2334, 1590, 7878, 10734, 1802, 1891, 5103, 6151, 8820, 3418, 7846, 9951, 4693, 417, 9996,
    9652, 4510, 2946, 5461, 365, 881, 1927, 1015, 11675, 11009, 1371, 12265, 2485, 11385, 5039, 6742,
    8449, 1842, 12217, 8176, 9577, 4834, 7937, 9461, 2643, 11194, 3045, 6508, 4094, 3451, 7911, 11048,
    5406, 4665, 3020, 6616, 11345, 7519, 3669, 5287, 1790, 7014, 5410, 11038, 11249, 2035, 6125, 10407,
    4565, 7315, 5078, 10506, 2840, 2478, 9270, 4194, 9195, 4518, 7469, 1160, 6878, 2730, 10421, 10036,
    1734, 3815, 10939, 5832, 10595, 10759, 4423, 8420, 9617, 7119, 11010, 11424, 9173, 189, 10080, 10526,
    3466, 10588, 7592, 3578, 11511, 7785, 9663, 530, 12150, 8957, 2532, 3317, 9349, 10243, 1481, 9332,
    3454, 3758, 7899, 4218, 2593, 11410, 2276, 982, 6513, 1849, 8494, 9021, 4523, 7988, 8, 457,
    648, 150, 8000, 2307, 2301, 874, 5650, 170, 9462, 2873, 9855, 11498, 2535, 11169, 5808, 12268,
    9687, 1901, 7171, 11787, 3846, 1573, 6063, 3793, 466, 11259, 10608, 3821, 6320, 4649, 6263, 2929
};

/************************************************************
* Name:        gammas_inv_montgomery
*
* Description: Contains inverses of powers of nth  root of -1
*              divided by n in Montgomery domain with R=2^18
************************************************************/
const uint16_t PQCLEAN_NEWHOPE1024CPA_CLEAN_gammas_inv_montgomery[NEWHOPE_N] = {
    256, 10570, 1510, 7238, 1034, 7170, 6291, 7921, 11665, 3422, 4000, 2327, 2088, 5565, 795, 10647,
    1521, 5484, 2539, 7385, 1055, 7173, 8047, 11683, 1669, 1994, 3796, 5809, 4341, 9398, 11876, 12230,
    10525, 12037, 12253, 3506, 4012, 9351, 4847, 2448, 7372, 9831, 3160, 2207, 5582, 2553, 7387, 6322,
    9681, 1383, 10731, 1533, 219, 5298, 4268, 7632, 6357, 9686, 8406, 4712, 9451, 10128, 4958, 5975,
    11387, 8649, 11769, 6948, 11526, 12180, 1740, 10782, 6807, 2728, 7412, 4570, 4164, 4106, 11120, 12122,
    8754, 11784, 3439, 5758, 11356, 6889, 9762, 11928, 1704, 1999, 10819, 12079, 12259, 7018, 11536, 1648,
    1991, 2040, 2047, 2048, 10826, 12080, 8748, 8272, 8204, 1172, 1923, 7297, 2798, 7422, 6327, 4415,
    7653, 6360, 11442, 12168, 7005, 8023, 9924, 8440, 8228, 2931, 7441, 1063, 3663, 5790, 9605, 10150,
    1450, 8985, 11817, 10466, 10273, 12001, 3470, 7518, 1074, 1909, 7295, 9820, 4914, 702, 5367, 7789,
    8135, 9940, 1420, 3714, 11064, 12114, 12264, 1752, 5517, 9566, 11900, 1700, 3754, 5803, 829, 1874,
    7290, 2797, 10933, 5073, 7747, 8129, 6428, 6185, 11417, 1631, 233, 5300, 9535, 10140, 11982, 8734,
    8270, 2937, 10953, 8587, 8249, 2934, 9197, 4825, 5956, 4362, 9401, 1343, 3703, 529, 10609, 12049,
    6988, 6265, 895, 3639, 4031, 4087, 4095, 585, 10617, 8539, 4731, 4187, 9376, 3095, 9220, 10095,
    10220, 1460, 10742, 12068, 1724, 5513, 11321, 6884, 2739, 5658, 6075, 4379, 11159, 10372, 8504, 4726,
    9453, 3106, 7466, 11600, 10435, 8513, 9994, 8450, 9985, 3182, 10988, 8592, 2983, 9204, 4826, 2445,
    5616, 6069, 867, 3635, 5786, 11360, 5134, 2489, 10889, 12089, 1727, 7269, 2794, 9177, 1311, 5454,
    9557, 6632, 2703, 9164, 10087, 1441, 3717, 531, 3587, 2268, 324, 5313, 759, 1864, 5533, 2546,
    7386, 9833, 8427, 4715, 11207, 1601, 7251, 4547, 11183, 12131, 1733, 10781, 10318, 1474, 10744, 5046,
    4232, 11138, 10369, 6748, 964, 7160, 4534, 7670, 8118, 8182, 4680, 11202, 6867, 981, 8918, 1274,
    182, 26, 7026, 8026, 11680, 12202, 10521, 1503, 7237, 4545, 5916, 9623, 8397, 11733, 10454, 3249,
    9242, 6587, 941, 1890, 270, 10572, 6777, 9746, 6659, 6218, 6155, 6146, 878, 1881, 7291, 11575,
    12187, 1741, 7271, 8061, 11685, 6936, 4502, 9421, 4857, 4205, 7623, 1089, 10689, 1527, 8996, 10063,
    11971, 10488, 6765, 2722, 3900, 9335, 11867, 6962, 11528, 5158, 4248, 4118, 5855, 2592, 5637, 6072,
    2623, 7397, 8079, 9932, 4930, 5971, 853, 3633, 519, 8852, 11798, 3441, 11025, 1575, 225, 8810,
    11792, 12218, 3501, 9278, 3081, 9218, 4828, 7712, 8124, 11694, 12204, 3499, 4011, 573, 3593, 5780,
    7848, 9899, 10192, 1456, 208, 7052, 2763, 7417, 11593, 10434, 12024, 8740, 11782, 10461, 3250, 5731,
    7841, 9898, 1414, 202, 3540, 7528, 2831, 2160, 10842, 5060, 4234, 4116, 588, 84, 12, 7024,
    2759, 9172, 6577, 11473, 1639, 9012, 3043, 7457, 6332, 11438, 1634, 1989, 9062, 11828, 8712, 11778,
    12216, 10523, 6770, 9745, 10170, 4964, 9487, 6622, 946, 8913, 6540, 6201, 4397, 9406, 8366, 9973,
    8447, 8229, 11709, 8695, 10020, 3187, 5722, 2573, 10901, 6824, 4486, 4152, 9371, 8361, 2950, 2177,
    311, 1800, 9035, 8313, 11721, 3430, 490, 70, 10, 1757, 251, 3547, 7529, 11609, 3414, 7510,
    4584, 4166, 9373, 1339, 5458, 7802, 11648, 1664, 7260, 9815, 10180, 6721, 9738, 10169, 8475, 8233,
    9954, 1422, 8981, 1283, 5450, 11312, 1616, 3742, 11068, 10359, 4991, 713, 3613, 9294, 8350, 4704,
    672, 96, 7036, 9783, 11931, 3460, 5761, 823, 10651, 12055, 10500, 1500, 5481, 783, 3623, 11051,
    8601, 8251, 8201, 11705, 10450, 5004, 4226, 7626, 2845, 2162, 3820, 7568, 9859, 3164, 452, 10598,
    1514, 5483, 6050, 6131, 4387, 7649, 8115, 6426, 918, 8909, 8295, 1185, 5436, 11310, 8638, 1234,
    5443, 11311, 5127, 2488, 2111, 10835, 5059, 7745, 2862, 3920, 560, 80, 1767, 2008, 3798, 11076,
    6849, 2734, 10924, 12094, 8750, 1250, 10712, 6797, 971, 7161, 1023, 8924, 4786, 7706, 4612, 4170,
    7618, 6355, 4419, 5898, 11376, 10403, 10264, 6733, 4473, 639, 5358, 2521, 9138, 3061, 5704, 4326,
    618, 5355, 765, 5376, 768, 7132, 4530, 9425, 3102, 9221, 6584, 11474, 10417, 10266, 12000, 6981,
    6264, 4406, 2385, 7363, 4563, 4163, 7617, 9866, 3165, 9230, 11852, 10471, 5007, 5982, 11388, 5138,
    734, 3616, 11050, 12112, 6997, 11533, 12181, 10518, 12036, 3475, 2252, 7344, 9827, 4915, 9480, 6621,
    4457, 7659, 9872, 6677, 4465, 4149, 7615, 4599, 657, 3605, 515, 10607, 6782, 4480, 640, 1847,
    3775, 5806, 2585, 5636, 9583, 1369, 10729, 8555, 10000, 11962, 5220, 7768, 8132, 8184, 9947, 1421,
    203, 29, 8782, 11788, 1684, 10774, 10317, 4985, 9490, 8378, 4708, 11206, 5112, 5997, 7879, 11659,
    12199, 8765, 10030, 4944, 5973, 6120, 6141, 6144, 7900, 11662, 1666, 238, 34, 3516, 5769, 9602,
    8394, 9977, 6692, 956, 10670, 6791, 9748, 11926, 8726, 11780, 5194, 742, 106, 8793, 10034, 3189,
    10989, 5081, 4237, 5872, 4350, 2377, 10873, 6820, 6241, 11425, 10410, 10265, 3222, 5727, 9596, 4882,
    2453, 2106, 3812, 11078, 12116, 5242, 4260, 11142, 8614, 11764, 12214, 5256, 4262, 4120, 11122, 5100,
    11262, 5120, 2487, 5622, 9581, 8391, 8221, 2930, 10952, 12098, 6995, 6266, 9673, 4893, 699, 3611,
    4027, 5842, 11368, 1624, 232, 8811, 8281, 1183, 169, 8802, 3013, 2186, 5579, 797, 3625, 4029,
    11109, 1587, 7249, 11569, 8675, 6506, 2685, 10917, 12093, 12261, 12285, 1755, 7273, 1039, 1904, 272,
    3550, 9285, 3082, 5707, 6082, 4380, 7648, 11626, 5172, 4250, 9385, 8363, 8217, 4685, 5936, 848,
    8899, 6538, 934, 1889, 3781, 9318, 10109, 10222, 6727, 961, 5404, 772, 5377, 9546, 8386, 1198,
    8949, 3034, 2189, 7335, 4559, 5918, 2601, 10905, 5069, 9502, 3113, 7467, 8089, 11689, 5181, 9518,
    8382, 2953, 3933, 4073, 4093, 7607, 8109, 2914, 5683, 4323, 11151, 1593, 10761, 6804, 972, 3650,
    2277, 5592, 4310, 7638, 9869, 4921, 703, 1856, 9043, 4803, 9464, 1352, 8971, 11815, 5199, 7765,
    6376, 4422, 7654, 2849, 407, 8836, 6529, 7955, 2892, 9191, 1313, 10721, 12065, 12257, 1751, 9028,
    8312, 2943, 2176, 3822, 546, 78, 8789, 11789, 10462, 12028, 6985, 4509, 9422, 1346, 5459, 4291,
    613, 10621, 6784, 9747, 3148, 7472, 2823, 5670, 810, 7138, 8042, 4660, 7688, 6365, 6176, 6149,
    2634, 5643, 9584, 10147, 11983, 5223, 9524, 11894, 10477, 8519, 1217, 3685, 2282, 326, 10580, 3267,
    7489, 4581, 2410, 5611, 11335, 6886, 8006, 8166, 11700, 3427, 11023, 8597, 10006, 3185, 455, 65,
    5276, 7776, 4622, 5927, 7869, 9902, 11948, 5218, 2501, 5624, 2559, 10899, 1557, 1978, 10816, 10323,
    8497, 4725, 675, 1852, 10798, 12076, 10503, 3256, 9243, 3076, 2195, 10847, 12083, 10504, 12034, 10497
};