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

1 files changed, 126 insertions, 229 deletions

diff --git a/src/bech32.cpp b/src/bech32.cpp
index 7585ee944c..d6e42bb0ac 100644
--- a/src/bech32.cpp
+++ b/src/bech32.cpp
@@ -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. */