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Replace GF1024 tables and syndrome constants with compile-time generated constexprs.

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pull/23577/head
Samuel Dobson 3 years ago
parent 63f7b69779
commit 14358a029d
1 changed files with 126 additions and 229 deletions
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src/bech32.cpp
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@ -8,6 +8,7 @@
#include <assert.h>
#include <optional>
#include <array>
namespace bech32
{
@ -32,182 +33,89 @@ const int8_t CHARSET_REV[128] = {
1, 0, 3, 16, 11, 28, 12, 14, 6, 4, 2, -1, -1, -1, -1, -1
};
// We work with the finite field GF(1024) defined as a degree 2 extension of the base field GF(32)
// The defining polynomial of the extension is x^2 + 9x + 23
// Let (e) be a primitive element of GF(1024), that is, a generator of the field.
// Every non-zero element of the field can then be represented as (e)^k for some power k.
// The array GF1024_EXP contains all these powers of (e) - GF1024_EXP[k] = (e)^k in GF(1024).
// Conversely, GF1024_LOG contains the discrete logarithms of these powers, so
// GF1024_LOG[GF1024_EXP[k]] == k
// Each element v of GF(1024) is encoded as a 10 bit integer in the following way:
// v = v1 || v0 where v0, v1 are 5-bit integers (elements of GF(32)).
//
// The element (e) is encoded as 9 || 15. Given (v), we compute (e)*(v) by multiplying in the following way:
// v0' = 27*v1 + 15*v0
// v1' = 6*v1 + 9*v0
// e*v = v1' || v0'
//
// The following sage code can be used to reproduce both _EXP and _LOG arrays
// GF1024_LOG = [-1] + [0] * 1023
// GF1024_EXP = [1] * 1024
// v = 1
// for i in range(1, 1023):
// v0 = v & 31
// v1 = v >> 5
// v0n = F.fetch_int(27)*F.fetch_int(v1) + F.fetch_int(15)*F.fetch_int(v0)
// v1n = F.fetch_int(6)*F.fetch_int(v1) + F.fetch_int(9)*F.fetch_int(v0)
// v = v1n.integer_representation() << 5 | v0n.integer_representation()
// GF1024_EXP[i] = v
// GF1024_LOG[v] = i
const int16_t GF1024_EXP[] = {
1, 303, 635, 446, 997, 640, 121, 142, 959, 420, 350, 438, 166, 39, 543,
335, 831, 691, 117, 632, 719, 97, 107, 374, 558, 797, 54, 150, 858, 877,
724, 1013, 294, 23, 354, 61, 164, 633, 992, 538, 469, 659, 174, 868, 184,
809, 766, 563, 866, 851, 257, 520, 45, 770, 535, 524, 408, 213, 436, 760,
472, 330, 933, 799, 616, 361, 15, 391, 756, 814, 58, 608, 554, 680, 993,
821, 942, 813, 843, 484, 193, 935, 321, 919, 572, 741, 423, 559, 562,
589, 296, 191, 493, 685, 891, 665, 435, 60, 395, 2, 606, 511, 853, 746,
32, 219, 284, 631, 840, 661, 837, 332, 78, 311, 670, 887, 111, 195, 505,
190, 194, 214, 709, 380, 819, 69, 261, 957, 1018, 161, 739, 588, 7, 708,
83, 328, 507, 736, 317, 899, 47, 348, 1000, 345, 882, 245, 367, 996, 943,
514, 304, 90, 804, 295, 312, 793, 387, 833, 249, 921, 660, 618, 823, 496,
722, 30, 782, 225, 892, 93, 480, 372, 112, 738, 867, 636, 890, 950, 968,
386, 622, 642, 551, 369, 234, 846, 382, 365, 442, 592, 343, 986, 122,
1023, 59, 847, 81, 790, 4, 437, 983, 931, 244, 64, 415, 529, 487, 944,
35, 938, 664, 156, 583, 53, 999, 222, 390, 987, 341, 388, 389, 170, 721,
879, 138, 522, 627, 765, 322, 230, 440, 14, 168, 143, 656, 991, 224, 595,
550, 94, 657, 752, 667, 1005, 451, 734, 744, 638, 292, 585, 157, 872,
590, 601, 827, 774, 930, 475, 571, 33, 500, 871, 969, 173, 21, 828, 450,
1009, 147, 960, 705, 201, 228, 998, 497, 1021, 613, 688, 772, 508, 36,
366, 715, 468, 956, 725, 730, 861, 425, 647, 701, 221, 759, 95, 958, 139,
805, 8, 835, 679, 614, 449, 128, 791, 299, 974, 617, 70, 628, 57, 273,
430, 67, 750, 405, 780, 703, 643, 776, 778, 340, 171, 1022, 276, 308,
495, 243, 644, 460, 857, 28, 336, 286, 41, 695, 448, 431, 364, 149, 43,
233, 63, 762, 902, 181, 240, 501, 584, 434, 275, 1008, 444, 443, 895,
812, 612, 927, 383, 66, 961, 1006, 690, 346, 3, 881, 900, 747, 271, 672,
162, 402, 456, 748, 971, 755, 490, 105, 808, 977, 72, 732, 182, 897, 625,
163, 189, 947, 850, 46, 115, 403, 231, 151, 629, 278, 874, 16, 934, 110,
492, 898, 256, 807, 598, 700, 498, 140, 481, 91, 523, 860, 134, 252, 771,
824, 119, 38, 816, 820, 641, 342, 757, 513, 577, 990, 463, 40, 920, 955,
17, 649, 533, 82, 103, 896, 862, 728, 259, 86, 466, 87, 253, 556, 323,
457, 963, 432, 845, 527, 745, 849, 863, 1015, 888, 488, 567, 727, 132,
674, 764, 109, 669, 6, 1003, 552, 246, 542, 96, 324, 781, 912, 248, 694,
239, 980, 210, 880, 683, 144, 177, 325, 546, 491, 326, 339, 623, 941, 92,
207, 783, 462, 263, 483, 517, 1012, 9, 620, 220, 984, 548, 512, 878, 421,
113, 973, 280, 962, 159, 310, 945, 268, 465, 806, 889, 199, 76, 873, 865,
34, 645, 227, 290, 418, 693, 926, 80, 569, 639, 11, 50, 291, 141, 206,
544, 949, 185, 518, 133, 909, 135, 467, 376, 646, 914, 678, 841, 954,
318, 242, 939, 951, 743, 1017, 976, 359, 167, 264, 100, 241, 218, 51, 12,
758, 368, 453, 309, 192, 648, 826, 553, 473, 101, 478, 673, 397, 1001,
118, 265, 331, 650, 356, 982, 652, 655, 510, 634, 145, 414, 830, 924,
526, 966, 298, 737, 18, 504, 401, 697, 360, 288, 1020, 842, 203, 698,
537, 676, 279, 581, 619, 536, 907, 876, 1019, 398, 152, 1010, 994, 68,
42, 454, 580, 836, 99, 565, 137, 379, 503, 22, 77, 582, 282, 412, 352,
611, 347, 300, 266, 570, 270, 911, 729, 44, 557, 108, 946, 637, 597, 461,
630, 615, 238, 763, 681, 718, 334, 528, 200, 459, 413, 79, 24, 229, 713,
906, 579, 384, 48, 893, 370, 923, 202, 917, 98, 794, 754, 197, 530, 662,
52, 712, 677, 56, 62, 981, 509, 267, 789, 885, 561, 316, 684, 596, 226,
13, 985, 779, 123, 720, 576, 753, 948, 406, 125, 315, 104, 519, 426, 502,
313, 566, 1016, 767, 796, 281, 749, 740, 136, 84, 908, 424, 936, 198,
355, 274, 735, 967, 5, 154, 428, 541, 785, 704, 486, 671, 600, 532, 381,
540, 574, 187, 88, 378, 216, 621, 499, 419, 922, 485, 494, 476, 255, 114,
188, 668, 297, 400, 918, 787, 158, 25, 458, 178, 564, 422, 768, 73, 1011,
717, 575, 404, 547, 196, 829, 237, 394, 301, 37, 65, 176, 106, 89, 85,
675, 979, 534, 803, 995, 363, 593, 120, 417, 452, 26, 699, 822, 223, 169,
416, 235, 609, 773, 211, 607, 208, 302, 852, 965, 603, 357, 761, 247,
817, 539, 250, 232, 272, 129, 568, 848, 624, 396, 710, 525, 183, 686, 10,
285, 856, 307, 811, 160, 972, 55, 441, 289, 723, 305, 373, 351, 153, 733,
409, 506, 975, 838, 573, 970, 988, 913, 471, 205, 337, 49, 594, 777, 549,
815, 277, 27, 916, 333, 353, 844, 800, 146, 751, 186, 375, 769, 358, 392,
883, 474, 788, 602, 74, 130, 329, 212, 155, 131, 102, 687, 293, 870, 742,
726, 427, 217, 834, 904, 29, 127, 869, 407, 338, 832, 470, 482, 810, 399,
439, 393, 604, 929, 682, 447, 714, 251, 455, 875, 319, 477, 464, 521,
258, 377, 937, 489, 792, 172, 314, 327, 124, 20, 531, 953, 591, 886, 320,
696, 71, 859, 578, 175, 587, 707, 663, 283, 179, 795, 989, 702, 940, 371,
692, 689, 555, 903, 410, 651, 75, 429, 818, 362, 894, 515, 31, 545, 666,
706, 952, 864, 269, 254, 349, 711, 802, 716, 784, 1007, 925, 801, 445,
148, 260, 658, 385, 287, 262, 204, 126, 586, 1004, 236, 165, 854, 411,
932, 560, 19, 215, 1002, 775, 653, 928, 901, 964, 884, 798, 839, 786,
433, 610, 116, 855, 180, 479, 910, 1014, 599, 915, 905, 306, 516, 731,
626, 978, 825, 344, 605, 654, 209
};
// As above, GF1024_EXP contains all elements of GF(1024) except 0
static_assert(std::size(GF1024_EXP) == 1023, "GF1024_EXP length should be 1023");
const int16_t GF1024_LOG[] = {
-1, 0, 99, 363, 198, 726, 462, 132, 297, 495, 825, 528, 561, 693, 231,
66, 396, 429, 594, 990, 924, 264, 627, 33, 660, 759, 792, 858, 330, 891,
165, 957, 104, 259, 518, 208, 280, 776, 416, 13, 426, 333, 618, 339, 641,
52, 388, 140, 666, 852, 529, 560, 678, 213, 26, 832, 681, 309, 70, 194,
97, 35, 682, 341, 203, 777, 358, 312, 617, 125, 307, 931, 379, 765, 875,
951, 515, 628, 112, 659, 525, 196, 432, 134, 717, 781, 438, 440, 740,
780, 151, 408, 487, 169, 239, 293, 467, 21, 672, 622, 557, 571, 881, 433,
704, 376, 779, 22, 643, 460, 398, 116, 172, 503, 751, 389, 1004, 18, 576,
415, 789, 6, 192, 696, 923, 702, 981, 892, 302, 816, 876, 880, 457, 537,
411, 539, 716, 624, 224, 295, 406, 531, 7, 233, 478, 586, 864, 268, 974,
338, 27, 392, 614, 839, 727, 879, 211, 250, 758, 507, 830, 129, 369, 384,
36, 985, 12, 555, 232, 796, 221, 321, 920, 263, 42, 934, 778, 479, 761,
939, 1006, 344, 381, 823, 44, 535, 866, 739, 752, 385, 119, 91, 566, 80,
120, 117, 771, 675, 721, 514, 656, 271, 670, 602, 980, 850, 532, 488,
803, 1022, 475, 801, 878, 57, 121, 991, 742, 888, 559, 105, 497, 291,
215, 795, 236, 167, 692, 520, 272, 661, 229, 391, 814, 340, 184, 798,
984, 773, 650, 473, 345, 558, 548, 326, 202, 145, 465, 810, 471, 158,
813, 908, 412, 441, 964, 750, 401, 50, 915, 437, 975, 126, 979, 491, 556,
577, 636, 685, 510, 963, 638, 367, 815, 310, 723, 349, 323, 857, 394,
606, 505, 713, 630, 938, 106, 826, 332, 978, 599, 834, 521, 530, 248,
883, 32, 153, 90, 754, 592, 304, 635, 775, 804, 1, 150, 836, 1013, 828,
324, 565, 508, 113, 154, 708, 921, 703, 689, 138, 547, 911, 929, 82, 228,
443, 468, 480, 483, 922, 135, 877, 61, 578, 111, 860, 654, 15, 331, 851,
895, 484, 320, 218, 420, 190, 1019, 143, 362, 634, 141, 965, 10, 838,
632, 861, 34, 722, 580, 808, 869, 554, 598, 65, 954, 787, 337, 187, 281,
146, 563, 183, 668, 944, 171, 837, 23, 867, 541, 916, 741, 625, 123, 736,
186, 357, 665, 977, 179, 156, 219, 220, 216, 67, 870, 902, 774, 98, 820,
574, 613, 900, 755, 596, 370, 390, 769, 314, 701, 894, 56, 841, 949, 987,
631, 658, 587, 204, 797, 790, 522, 745, 9, 502, 763, 86, 719, 288, 706,
887, 728, 952, 311, 336, 446, 1002, 348, 96, 58, 199, 11, 901, 230, 833,
188, 352, 351, 973, 3, 906, 335, 301, 266, 244, 791, 564, 619, 909, 371,
444, 760, 657, 328, 647, 490, 425, 913, 511, 439, 540, 283, 40, 897, 849,
60, 570, 872, 257, 749, 912, 572, 1007, 170, 407, 898, 492, 79, 747, 732,
206, 454, 918, 375, 482, 399, 92, 748, 325, 163, 274, 405, 744, 260, 346,
707, 626, 595, 118, 842, 136, 279, 684, 584, 101, 500, 422, 149, 956,
1014, 493, 536, 705, 51, 914, 225, 409, 55, 822, 590, 448, 655, 205, 676,
925, 735, 431, 784, 54, 609, 604, 39, 812, 737, 729, 466, 14, 533, 958,
481, 770, 499, 855, 238, 182, 464, 569, 72, 947, 442, 642, 24, 87, 989,
688, 88, 47, 762, 623, 709, 455, 817, 526, 637, 258, 84, 845, 738, 768,
698, 423, 933, 664, 620, 607, 629, 212, 347, 249, 982, 935, 131, 89, 252,
927, 189, 788, 853, 237, 691, 646, 403, 1010, 734, 253, 874, 807, 903,
1020, 100, 802, 71, 799, 1003, 633, 355, 276, 300, 649, 64, 306, 161,
608, 496, 743, 180, 485, 819, 383, 1016, 226, 308, 393, 648, 107, 19, 37,
585, 2, 175, 645, 247, 527, 5, 419, 181, 317, 327, 519, 542, 289, 567,
430, 579, 950, 582, 994, 1021, 583, 234, 240, 976, 41, 160, 109, 677,
937, 210, 95, 959, 242, 753, 461, 114, 733, 368, 573, 458, 782, 605, 680,
544, 299, 73, 652, 905, 477, 690, 93, 824, 882, 277, 946, 361, 17, 945,
523, 472, 334, 930, 597, 603, 793, 404, 290, 942, 316, 731, 270, 960,
936, 133, 122, 821, 966, 679, 662, 907, 282, 968, 767, 653, 20, 697, 222,
164, 835, 30, 285, 886, 456, 436, 640, 286, 1015, 380, 840, 245, 724,
137, 593, 173, 130, 715, 85, 885, 551, 246, 449, 103, 366, 372, 714, 313,
865, 241, 699, 674, 374, 68, 421, 562, 292, 59, 809, 342, 651, 459, 227,
46, 711, 764, 868, 53, 413, 278, 800, 255, 993, 318, 854, 319, 695, 315,
469, 166, 489, 969, 730, 1001, 757, 873, 686, 197, 303, 919, 155, 673,
940, 712, 25, 999, 63, 863, 972, 967, 785, 152, 296, 512, 402, 377, 45,
899, 829, 354, 77, 69, 856, 417, 811, 953, 124, 418, 75, 794, 162, 414,
1018, 568, 254, 265, 772, 588, 16, 896, 157, 889, 298, 621, 110, 844,
1000, 108, 545, 601, 78, 862, 447, 185, 195, 818, 450, 387, 49, 805, 102,
986, 1005, 827, 329, 28, 932, 410, 287, 435, 451, 962, 517, 48, 174, 43,
893, 884, 261, 251, 516, 395, 910, 611, 29, 501, 223, 476, 364, 144, 871,
998, 687, 928, 115, 453, 513, 176, 94, 168, 667, 955, 353, 434, 382, 400,
139, 365, 996, 343, 948, 890, 1012, 663, 610, 718, 538, 1008, 639, 470,
848, 543, 1011, 859, 671, 756, 83, 427, 159, 746, 669, 589, 971, 524,
356, 995, 904, 256, 201, 988, 62, 397, 81, 720, 917, 209, 549, 943, 486,
76, 148, 207, 509, 644, 386, 700, 534, 177, 550, 961, 926, 546, 428, 284,
127, 294, 8, 269, 359, 506, 445, 997, 806, 591, 725, 178, 262, 846, 373,
831, 504, 305, 843, 553, 378, 1017, 783, 474, 683, 581, 200, 498, 694,
191, 217, 847, 941, 424, 235, 38, 74, 616, 786, 147, 4, 273, 214, 142,
575, 992, 463, 983, 243, 360, 970, 350, 267, 615, 766, 494, 31, 1009,
452, 710, 552, 128, 612, 600, 275, 322, 193
};
static_assert(std::size(GF1024_LOG) == 1024, "GF1024_EXP length should be 1024");
/** We work with the finite field GF(1024) defined as a degree 2 extension of the base field GF(32)
* The defining polynomial of the extension is x^2 + 9x + 23.
* Let (e) be a root of this defining polynomial. Then (e) is a primitive element of GF(1024),
* that is, a generator of the field. Every non-zero element of the field can then be represented
* as (e)^k for some power k.
* The array GF1024_EXP contains all these powers of (e) - GF1024_EXP[k] = (e)^k in GF(1024).
* Conversely, GF1024_LOG contains the discrete logarithms of these powers, so
* GF1024_LOG[GF1024_EXP[k]] == k.
* The following function generates the two tables GF1024_EXP and GF1024_LOG as constexprs. */
constexpr std::pair<std::array<int16_t, 1023>, std::array<int16_t, 1024>> GenerateGFTables()
{
// Build table for GF(32).
// We use these tables to perform arithmetic in GF(32) below, when constructing the
// tables for GF(1024).
std::array<int8_t, 31> GF32_EXP{};
std::array<int8_t, 32> GF32_LOG{};
// fmod encodes the defining polynomial of GF(32) over GF(2), x^5 + x^3 + 1.
// Because coefficients in GF(2) are binary digits, the coefficients are packed as 101001.
const int fmod = 41;
// Elements of GF(32) are encoded as vectors of length 5 over GF(2), that is,
// 5 binary digits. Each element (b_4, b_3, b_2, b_1, b_0) encodes a polynomial
// b_4*x^4 + b_3*x^3 + b_2*x^2 + b_1*x^1 + b_0 (modulo fmod).
// For example, 00001 = 1 is the multiplicative identity.
GF32_EXP[0] = 1;
GF32_LOG[0] = -1;
GF32_LOG[1] = 0;
int v = 1;
for (int i = 1; i < 31; ++i) {
// Multiplication by x is the same as shifting left by 1, as
// every coefficient of the polynomial is moved up one place.
v = v << 1;
// If the polynomial now has an x^5 term, we subtract fmod from it
// to remain working modulo fmod. Subtraction is the same as XOR in characteristic
// 2 fields.
if (v & 32) v ^= fmod;
GF32_EXP[i] = v;
GF32_LOG[v] = i;
}
// Build table for GF(1024)
std::array<int16_t, 1023> GF1024_EXP{};
std::array<int16_t, 1024> GF1024_LOG{};
GF1024_EXP[0] = 1;
GF1024_LOG[0] = -1;
GF1024_LOG[1] = 0;
// Each element v of GF(1024) is encoded as a 10 bit integer in the following way:
// v = v1 || v0 where v0, v1 are 5-bit integers (elements of GF(32)).
// The element (e) is encoded as 9 || 15. Given (v), we
// compute (e)*(v) by multiplying in the following way:
//
// v0' = 27*v1 + 15*v0
// v1' = 6*v1 + 9*v0
// e*v = v1' || v0'
//
// Multiplication in GF(32) is done using the log/exp tables:
// e^x * e^y = e^(x + y) so a * b = EXP[ LOG[a] + LOG [b] ]
// for non-zero a and b.
v = 1;
for (int i = 1; i < 1023; ++i) {
int v0 = v & 31;
int v1 = v >> 5;
int v0n = (v1 ? GF32_EXP.at((GF32_LOG.at(v1) + GF32_LOG.at(27)) % 31) : 0) ^
(v0 ? GF32_EXP.at((GF32_LOG.at(v0) + GF32_LOG.at(15)) % 31) : 0);
int v1n = (v1 ? GF32_EXP.at((GF32_LOG.at(v1) + GF32_LOG.at(6)) % 31) : 0) ^
(v0 ? GF32_EXP.at((GF32_LOG.at(v0) + GF32_LOG.at(9)) % 31) : 0);
v = v1n << 5 | v0n;
GF1024_EXP[i] = v;
GF1024_LOG[v] = i;
}
return std::make_pair(GF1024_EXP, GF1024_LOG);
}
constexpr auto tables = GenerateGFTables();
constexpr const std::array<int16_t, 1023>& GF1024_EXP = tables.first;
constexpr const std::array<int16_t, 1024>& GF1024_LOG = tables.second;
/* Determine the final constant to use for the specified encoding. */
uint32_t EncodingConstant(Encoding encoding) {
@ -314,69 +222,58 @@ uint32_t PolyMod(const data& v)
* codeword, it is a multiple of G(X), so the residue is in fact just E(x) mod G(x). Note that all
* of the (e)^j are roots of G(x) by definition, so R((e)^j) = E((e)^j).
*
* Syndrome returns the three values packed into a 30-bit integer, where each 10 bits is one value.
* Let R(x) = r1*x^5 + r2*x^4 + r3*x^3 + r4*x^2 + r5*x + r6
*
* To compute R((e)^j), we are really computing:
* r1*(e)^(j*5) + r2*(e)^(j*4) + r3*(e)^(j*3) + r4*(e)^(j*2) + r5*(e)^j + r6
*
* Now note that all of the (e)^(j*i) for i in [5..0] are constants and can be precomputed.
* But even more than that, we can consider each coefficient as a bit-string.
* For example, take r5 = (b_5, b_4, b_3, b_2, b_1) written out as 5 bits. Then:
* r5*(e)^j = b_1*(e)^j + b_2*(2*(e)^j) + b_3*(4*(e)^j) + b_4*(8*(e)^j) + b_5*(16*(e)^j)
* where all the (2^i*(e)^j) are constants and can be precomputed.
*
* Then we just add each of these corresponding constants to our final value based on the
* bit values b_i. This is exactly what is done in the Syndrome function below.
*/
constexpr std::array<uint32_t, 25> GenerateSyndromeConstants() {
std::array<uint32_t, 25> SYNDROME_CONSTS{};
for (int k = 1; k < 6; ++k) {
for (int shift = 0; shift < 5; ++shift) {
int16_t b = GF1024_LOG.at(1 << shift);
int16_t c0 = GF1024_EXP.at((997*k + b) % 1023);
int16_t c1 = GF1024_EXP.at((998*k + b) % 1023);
int16_t c2 = GF1024_EXP.at((999*k + b) % 1023);
uint32_t c = c2 << 20 | c1 << 10 | c0;
int ind = 5*(k-1) + shift;
SYNDROME_CONSTS[ind] = c;
}
}
return SYNDROME_CONSTS;
}
constexpr std::array<uint32_t, 25> SYNDROME_CONSTS = GenerateSyndromeConstants();
/**
* Syndrome returns the three values s_997, s_998, and s_999 described above,
* packed into a 30-bit integer, where each group of 10 bits encodes one value.
*/
uint32_t Syndrome(const uint32_t residue) {
// Let R(x) = r1*x^5 + r2*x^4 + r3*x^3 + r4*x^2 + r5*x + r6
// low is the first 5 bits, corresponding to the r6 in the residue
// (the constant term of the polynomial).
uint32_t low = residue & 0x1f;
// Recall that XOR corresponds to addition in a characteristic 2 field.
//
// To compute R((e)^j), we are really computing:
// r1*(e)^(j*5) + r2*(e)^(j*4) + r3*(e)^(j*3) + r4*(e)^(j*2) + r5*(e)^j + r6
// Now note that all of the (e)^(j*i) for i in [5..0] are constants and can be precomputed
// for efficiency. But even more than that, we can consider each coefficient as a bit-string.
// For example, take r5 = (b_5, b_4, b_3, b_2, b_1) written out as 5 bits. Then:
// r5*(e)^j = b_1*(e)^j + b_2*(2*(e)^j) + b_3*(4*(e)^j) + b_4*(8*(e)^j) + b_5*(16*(e)^j)
// where all the (2^i*(e)^j) are constants and can be precomputed. Then we just add each
// of these corresponding constants to our final value based on the bit values b_i.
// This is exactly what is done below. Note that all three values of s_j for j in (997, 998,
// 999) are computed simultaneously.
//
// We begin by setting s_j = low = r6 for all three values of j, because these are unconditional.
uint32_t result = low ^ (low << 10) ^ (low << 20);
// Then for each following bit, we add the corresponding precomputed constant if the bit is 1.
// For example, 0x31edd3c4 is 1100011110 1101110100 1111000100 when unpacked in groups of 10
// bits, corresponding exactly to a^999 || a^998 || a^997 (matching the corresponding values in
// GF1024_EXP above).
//
// The following sage code reproduces these constants:
// for k in range(1, 6):
// for b in [1,2,4,8,16]:
// c0 = GF1024_EXP[(997*k + GF1024_LOG[b]) % 1023]
// c1 = GF1024_EXP[(998*k + GF1024_LOG[b]) % 1023]
// c2 = GF1024_EXP[(999*k + GF1024_LOG[b]) % 1023]
// c = c2 << 20 | c1 << 10 | c0
// print("0x%x" % c)
return low ^ (low << 10) ^ (low << 20) ^
((residue >> 5) & 1 ? 0x31edd3c4 : 0) ^
((residue >> 6) & 1 ? 0x335f86a8 : 0) ^
((residue >> 7) & 1 ? 0x363b8870 : 0) ^
((residue >> 8) & 1 ? 0x3e6390c9 : 0) ^
((residue >> 9) & 1 ? 0x2ec72192 : 0) ^
((residue >> 10) & 1 ? 0x1046f79d : 0) ^
((residue >> 11) & 1 ? 0x208d4e33 : 0) ^
((residue >> 12) & 1 ? 0x130ebd6f : 0) ^
((residue >> 13) & 1 ? 0x2499fade : 0) ^
((residue >> 14) & 1 ? 0x1b27d4b5 : 0) ^
((residue >> 15) & 1 ? 0x04be1eb4 : 0) ^
((residue >> 16) & 1 ? 0x0968b861 : 0) ^
((residue >> 17) & 1 ? 0x1055f0c2 : 0) ^
((residue >> 18) & 1 ? 0x20ab4584 : 0) ^
((residue >> 19) & 1 ? 0x1342af08 : 0) ^
((residue >> 20) & 1 ? 0x24f1f318 : 0) ^
((residue >> 21) & 1 ? 0x1be34739 : 0) ^
((residue >> 22) & 1 ? 0x35562f7b : 0) ^
((residue >> 23) & 1 ? 0x3a3c5bff : 0) ^
((residue >> 24) & 1 ? 0x266c96f7 : 0) ^
((residue >> 25) & 1 ? 0x25c78b65 : 0) ^
((residue >> 26) & 1 ? 0x1b1f13ea : 0) ^
((residue >> 27) & 1 ? 0x34baa2f4 : 0) ^
((residue >> 28) & 1 ? 0x3b61c0e1 : 0) ^
((residue >> 29) & 1 ? 0x265325c2 : 0);
// GF1024_EXP above). In this way, we compute all three values of s_j for j in (997, 998, 999)
// simultaneously. Recall that XOR corresponds to addition in a characteristic 2 field.
for (int i = 0; i < 25; ++i) {
result ^= ((residue >> (5+i)) & 1 ? SYNDROME_CONSTS.at(i) : 0);
}
return result;
}
/** Convert to lower case. */

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