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precomp.nim
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precomp.nim
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import params
# NOTE: precomputed Montgomery representation of twiddle factors for the forward NTT.
const omegas_montgomery*: array[PARAM_N div 2, uint16] = [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]
# NOTE: precomputed Montgomery representation of twiddle factors for the inverse NTT.
const omegas_inv_montgomery*: array[PARAM_N div 2, uint16] = [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]
# NOTE: precomputed Montgomery representation of the NTT's psi constants in bit-reversed order.
const psis_bitrev_montgomery*: array[PARAM_N, uint16] = [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]
# NOTE: precomputed Montgomery representation of the inverse of the NTT's psi constants.
const psis_inv_montgomery*: array[PARAM_N, uint16] = [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]