1 // -*- mode: rust; -*- 2 // 3 // This file is part of curve25519-dalek. 4 // Copyright (c) 2016-2021 isis lovecruft 5 // Copyright (c) 2016-2019 Henry de Valence 6 // See LICENSE for licensing information. 7 // 8 // Authors: 9 // - isis agora lovecruft <isis@patternsinthevoid.net> 10 // - Henry de Valence <hdevalence@hdevalence.ca> 11 12 //! This module contains backend-specific constant values, such as the 64-bit limbs of curve constants. 13 14 use backend::serial::curve_models::AffineNielsPoint; 15 use super::field::FieldElement51; 16 use super::scalar::Scalar52; 17 use edwards::{EdwardsBasepointTable, EdwardsPoint}; 18 use window::{LookupTable, NafLookupTable8}; 19 20 /// The value of minus one, equal to `-&FieldElement::one()` 21 pub(crate) const MINUS_ONE: FieldElement51 = FieldElement51([ 22 2251799813685228, 23 2251799813685247, 24 2251799813685247, 25 2251799813685247, 26 2251799813685247 27 ]); 28 29 /// Edwards `d` value, equal to `-121665/121666 mod p`. 30 pub(crate) const EDWARDS_D: FieldElement51 = FieldElement51([ 31 929955233495203, 32 466365720129213, 33 1662059464998953, 34 2033849074728123, 35 1442794654840575, 36 ]); 37 38 /// Edwards `2*d` value, equal to `2*(-121665/121666) mod p`. 39 pub(crate) const EDWARDS_D2: FieldElement51 = FieldElement51([ 40 1859910466990425, 41 932731440258426, 42 1072319116312658, 43 1815898335770999, 44 633789495995903, 45 ]); 46 47 /// One minus edwards `d` value squared, equal to `(1 - (-121665/121666) mod p) pow 2` 48 pub(crate) const ONE_MINUS_EDWARDS_D_SQUARED: FieldElement51 = FieldElement51([ 49 1136626929484150, 50 1998550399581263, 51 496427632559748, 52 118527312129759, 53 45110755273534 54 ]); 55 56 /// Edwards `d` value minus one squared, equal to `(((-121665/121666) mod p) - 1) pow 2` 57 pub(crate) const EDWARDS_D_MINUS_ONE_SQUARED: FieldElement51 = FieldElement51([ 58 1507062230895904, 59 1572317787530805, 60 683053064812840, 61 317374165784489, 62 1572899562415810 63 ]); 64 65 /// `= sqrt(a*d - 1)`, where `a = -1 (mod p)`, `d` are the Edwards curve parameters. 66 pub(crate) const SQRT_AD_MINUS_ONE: FieldElement51 = FieldElement51([ 67 2241493124984347, 68 425987919032274, 69 2207028919301688, 70 1220490630685848, 71 974799131293748, 72 ]); 73 74 /// `= 1/sqrt(a-d)`, where `a = -1 (mod p)`, `d` are the Edwards curve parameters. 75 pub(crate) const INVSQRT_A_MINUS_D: FieldElement51 = FieldElement51([ 76 278908739862762, 77 821645201101625, 78 8113234426968, 79 1777959178193151, 80 2118520810568447, 81 ]); 82 83 /// Precomputed value of one of the square roots of -1 (mod p) 84 pub(crate) const SQRT_M1: FieldElement51 = FieldElement51([ 85 1718705420411056, 86 234908883556509, 87 2233514472574048, 88 2117202627021982, 89 765476049583133, 90 ]); 91 92 /// `APLUS2_OVER_FOUR` is (A+2)/4. (This is used internally within the Montgomery ladder.) 93 pub(crate) const APLUS2_OVER_FOUR: FieldElement51 = FieldElement51([121666, 0, 0, 0, 0]); 94 95 /// `MONTGOMERY_A` is equal to 486662, which is a constant of the curve equation 96 /// for Curve25519 in its Montgomery form. (This is used internally within the 97 /// Elligator map.) 98 pub(crate) const MONTGOMERY_A: FieldElement51 = FieldElement51([486662, 0, 0, 0, 0]); 99 100 /// `MONTGOMERY_A_NEG` is equal to -486662. (This is used internally within the 101 /// Elligator map.) 102 pub(crate) const MONTGOMERY_A_NEG: FieldElement51 = FieldElement51([ 103 2251799813198567, 104 2251799813685247, 105 2251799813685247, 106 2251799813685247, 107 2251799813685247, 108 ]); 109 110 /// `L` is the order of base point, i.e. 2^252 + 27742317777372353535851937790883648493 111 pub(crate) const L: Scalar52 = Scalar52([ 112 0x0002631a5cf5d3ed, 113 0x000dea2f79cd6581, 114 0x000000000014def9, 115 0x0000000000000000, 116 0x0000100000000000, 117 ]); 118 119 /// `L` * `LFACTOR` = -1 (mod 2^52) 120 pub(crate) const LFACTOR: u64 = 0x51da312547e1b; 121 122 /// `R` = R % L where R = 2^260 123 pub(crate) const R: Scalar52 = Scalar52([ 124 0x000f48bd6721e6ed, 125 0x0003bab5ac67e45a, 126 0x000fffffeb35e51b, 127 0x000fffffffffffff, 128 0x00000fffffffffff, 129 ]); 130 131 /// `RR` = (R^2) % L where R = 2^260 132 pub(crate) const RR: Scalar52 = Scalar52([ 133 0x0009d265e952d13b, 134 0x000d63c715bea69f, 135 0x0005be65cb687604, 136 0x0003dceec73d217f, 137 0x000009411b7c309a, 138 ]); 139 140 /// The Ed25519 basepoint, as an `EdwardsPoint`. 141 /// 142 /// This is called `_POINT` to distinguish it from 143 /// `ED25519_BASEPOINT_TABLE`, which should be used for scalar 144 /// multiplication (it's much faster). 145 pub const ED25519_BASEPOINT_POINT: EdwardsPoint = EdwardsPoint { 146 X: FieldElement51([ 147 1738742601995546, 148 1146398526822698, 149 2070867633025821, 150 562264141797630, 151 587772402128613, 152 ]), 153 Y: FieldElement51([ 154 1801439850948184, 155 1351079888211148, 156 450359962737049, 157 900719925474099, 158 1801439850948198, 159 ]), 160 Z: FieldElement51([1, 0, 0, 0, 0]), 161 T: FieldElement51([ 162 1841354044333475, 163 16398895984059, 164 755974180946558, 165 900171276175154, 166 1821297809914039, 167 ]), 168 }; 169 170 /// The 8-torsion subgroup \\(\mathcal E [8]\\). 171 /// 172 /// In the case of Curve25519, it is cyclic; the \\(i\\)-th element of 173 /// the array is \\([i]P\\), where \\(P\\) is a point of order \\(8\\) 174 /// generating \\(\mathcal E[8]\\). 175 /// 176 /// Thus \\(\mathcal E[4]\\) is the points indexed by `0,2,4,6`, and 177 /// \\(\mathcal E[2]\\) is the points indexed by `0,4`. 178 pub const EIGHT_TORSION: [EdwardsPoint; 8] = EIGHT_TORSION_INNER_DOC_HIDDEN; 179 180 /// Inner item used to hide limb constants from cargo doc output. 181 #[doc(hidden)] 182 pub const EIGHT_TORSION_INNER_DOC_HIDDEN: [EdwardsPoint; 8] = [ 183 EdwardsPoint { 184 X: FieldElement51([0, 0, 0, 0, 0]), 185 Y: FieldElement51([1, 0, 0, 0, 0]), 186 Z: FieldElement51([1, 0, 0, 0, 0]), 187 T: FieldElement51([0, 0, 0, 0, 0]), 188 }, 189 EdwardsPoint { 190 X: FieldElement51([ 191 358744748052810, 192 1691584618240980, 193 977650209285361, 194 1429865912637724, 195 560044844278676, 196 ]), 197 Y: FieldElement51([ 198 84926274344903, 199 473620666599931, 200 365590438845504, 201 1028470286882429, 202 2146499180330972, 203 ]), 204 Z: FieldElement51([1, 0, 0, 0, 0]), 205 T: FieldElement51([ 206 1448326834587521, 207 1857896831960481, 208 1093722731865333, 209 1677408490711241, 210 1915505153018406, 211 ]), 212 }, 213 EdwardsPoint { 214 X: FieldElement51([ 215 533094393274173, 216 2016890930128738, 217 18285341111199, 218 134597186663265, 219 1486323764102114, 220 ]), 221 Y: FieldElement51([0, 0, 0, 0, 0]), 222 Z: FieldElement51([1, 0, 0, 0, 0]), 223 T: FieldElement51([0, 0, 0, 0, 0]), 224 }, 225 EdwardsPoint { 226 X: FieldElement51([ 227 358744748052810, 228 1691584618240980, 229 977650209285361, 230 1429865912637724, 231 560044844278676, 232 ]), 233 Y: FieldElement51([ 234 2166873539340326, 235 1778179147085316, 236 1886209374839743, 237 1223329526802818, 238 105300633354275, 239 ]), 240 Z: FieldElement51([1, 0, 0, 0, 0]), 241 T: FieldElement51([ 242 803472979097708, 243 393902981724766, 244 1158077081819914, 245 574391322974006, 246 336294660666841, 247 ]), 248 }, 249 EdwardsPoint { 250 X: FieldElement51([0, 0, 0, 0, 0]), 251 Y: FieldElement51([ 252 2251799813685228, 253 2251799813685247, 254 2251799813685247, 255 2251799813685247, 256 2251799813685247, 257 ]), 258 Z: FieldElement51([1, 0, 0, 0, 0]), 259 T: FieldElement51([0, 0, 0, 0, 0]), 260 }, 261 EdwardsPoint { 262 X: FieldElement51([ 263 1893055065632419, 264 560215195444267, 265 1274149604399886, 266 821933901047523, 267 1691754969406571, 268 ]), 269 Y: FieldElement51([ 270 2166873539340326, 271 1778179147085316, 272 1886209374839743, 273 1223329526802818, 274 105300633354275, 275 ]), 276 Z: FieldElement51([1, 0, 0, 0, 0]), 277 T: FieldElement51([ 278 1448326834587521, 279 1857896831960481, 280 1093722731865333, 281 1677408490711241, 282 1915505153018406, 283 ]), 284 }, 285 EdwardsPoint { 286 X: FieldElement51([ 287 1718705420411056, 288 234908883556509, 289 2233514472574048, 290 2117202627021982, 291 765476049583133, 292 ]), 293 Y: FieldElement51([0, 0, 0, 0, 0]), 294 Z: FieldElement51([1, 0, 0, 0, 0]), 295 T: FieldElement51([0, 0, 0, 0, 0]), 296 }, 297 EdwardsPoint { 298 X: FieldElement51([ 299 1893055065632419, 300 560215195444267, 301 1274149604399886, 302 821933901047523, 303 1691754969406571, 304 ]), 305 Y: FieldElement51([ 306 84926274344903, 307 473620666599931, 308 365590438845504, 309 1028470286882429, 310 2146499180330972, 311 ]), 312 Z: FieldElement51([1, 0, 0, 0, 0]), 313 T: FieldElement51([ 314 803472979097708, 315 393902981724766, 316 1158077081819914, 317 574391322974006, 318 336294660666841, 319 ]), 320 }, 321 ]; 322 323 /// Table containing precomputed multiples of the Ed25519 basepoint \\(B = (x, 4/5)\\). 324 pub const ED25519_BASEPOINT_TABLE: EdwardsBasepointTable = ED25519_BASEPOINT_TABLE_INNER_DOC_HIDDEN; 325 326 /// Inner constant, used to avoid filling the docs with precomputed points. 327 #[doc(hidden)] 328 pub const ED25519_BASEPOINT_TABLE_INNER_DOC_HIDDEN: EdwardsBasepointTable = 329 EdwardsBasepointTable([ 330 LookupTable([ 331 AffineNielsPoint { 332 y_plus_x: FieldElement51([ 333 3540182452943730, 334 2497478415033846, 335 2521227595762870, 336 1462984067271729, 337 2389212253076811, 338 ]), 339 y_minus_x: FieldElement51([ 340 62697248952638, 341 204681361388450, 342 631292143396476, 343 338455783676468, 344 1213667448819585, 345 ]), 346 xy2d: FieldElement51([ 347 301289933810280, 348 1259582250014073, 349 1422107436869536, 350 796239922652654, 351 1953934009299142, 352 ]), 353 }, 354 AffineNielsPoint { 355 y_plus_x: FieldElement51([ 356 3632771708514775, 357 790832306631235, 358 2067202295274102, 359 1995808275510000, 360 1566530869037010, 361 ]), 362 y_minus_x: FieldElement51([ 363 463307831301544, 364 432984605774163, 365 1610641361907204, 366 750899048855000, 367 1894842303421586, 368 ]), 369 xy2d: FieldElement51([ 370 748439484463711, 371 1033211726465151, 372 1396005112841647, 373 1611506220286469, 374 1972177495910992, 375 ]), 376 }, 377 AffineNielsPoint { 378 y_plus_x: FieldElement51([ 379 1601611775252272, 380 1720807796594148, 381 1132070835939856, 382 3512254832574799, 383 2147779492816910, 384 ]), 385 y_minus_x: FieldElement51([ 386 316559037616741, 387 2177824224946892, 388 1459442586438991, 389 1461528397712656, 390 751590696113597, 391 ]), 392 xy2d: FieldElement51([ 393 1850748884277385, 394 1200145853858453, 395 1068094770532492, 396 672251375690438, 397 1586055907191707, 398 ]), 399 }, 400 AffineNielsPoint { 401 y_plus_x: FieldElement51([ 402 934282339813791, 403 1846903124198670, 404 1172395437954843, 405 1007037127761661, 406 1830588347719256, 407 ]), 408 y_minus_x: FieldElement51([ 409 1694390458783935, 410 1735906047636159, 411 705069562067493, 412 648033061693059, 413 696214010414170, 414 ]), 415 xy2d: FieldElement51([ 416 1121406372216585, 417 192876649532226, 418 190294192191717, 419 1994165897297032, 420 2245000007398739, 421 ]), 422 }, 423 AffineNielsPoint { 424 y_plus_x: FieldElement51([ 425 769950342298400, 426 2384754244604994, 427 3095885746880802, 428 3225892188161580, 429 2977876099231263, 430 ]), 431 y_minus_x: FieldElement51([ 432 425251763115706, 433 608463272472562, 434 442562545713235, 435 837766094556764, 436 374555092627893, 437 ]), 438 xy2d: FieldElement51([ 439 1086255230780037, 440 274979815921559, 441 1960002765731872, 442 929474102396301, 443 1190409889297339, 444 ]), 445 }, 446 AffineNielsPoint { 447 y_plus_x: FieldElement51([ 448 1388594989461809, 449 316767091099457, 450 2646098655878230, 451 1230079486801004, 452 1440737038838979, 453 ]), 454 y_minus_x: FieldElement51([ 455 7380825640100, 456 146210432690483, 457 304903576448906, 458 1198869323871120, 459 997689833219095, 460 ]), 461 xy2d: FieldElement51([ 462 1181317918772081, 463 114573476638901, 464 262805072233344, 465 265712217171332, 466 294181933805782, 467 ]), 468 }, 469 AffineNielsPoint { 470 y_plus_x: FieldElement51([ 471 2916800678241215, 472 2065379846933858, 473 2622030924071124, 474 2602788184473875, 475 1233371373142984, 476 ]), 477 y_minus_x: FieldElement51([ 478 2019367628972465, 479 676711900706637, 480 110710997811333, 481 1108646842542025, 482 517791959672113, 483 ]), 484 xy2d: FieldElement51([ 485 965130719900578, 486 247011430587952, 487 526356006571389, 488 91986625355052, 489 2157223321444601, 490 ]), 491 }, 492 AffineNielsPoint { 493 y_plus_x: FieldElement51([ 494 4320419353804412, 495 4218074731744053, 496 957728544705548, 497 729906502578991, 498 2411634706750414, 499 ]), 500 y_minus_x: FieldElement51([ 501 2073601412052185, 502 31021124762708, 503 264500969797082, 504 248034690651703, 505 1030252227928288, 506 ]), 507 xy2d: FieldElement51([ 508 551790716293402, 509 1989538725166328, 510 801169423371717, 511 2052451893578887, 512 678432056995012, 513 ]), 514 }, 515 ]), 516 LookupTable([ 517 AffineNielsPoint { 518 y_plus_x: FieldElement51([ 519 1368953770187805, 520 3042147450398169, 521 2689308289352409, 522 2142576377050579, 523 1932081720066286, 524 ]), 525 y_minus_x: FieldElement51([ 526 953638594433374, 527 1092333936795051, 528 1419774766716690, 529 805677984380077, 530 859228993502513, 531 ]), 532 xy2d: FieldElement51([ 533 1200766035879111, 534 20142053207432, 535 1465634435977050, 536 1645256912097844, 537 295121984874596, 538 ]), 539 }, 540 AffineNielsPoint { 541 y_plus_x: FieldElement51([ 542 1735718747031538, 543 1248237894295956, 544 1204753118328107, 545 976066523550493, 546 2317743583219840, 547 ]), 548 y_minus_x: FieldElement51([ 549 1060098822528990, 550 1586825862073490, 551 212301317240126, 552 1975302711403555, 553 666724059764335, 554 ]), 555 xy2d: FieldElement51([ 556 1091990273418756, 557 1572899409348578, 558 80968014455247, 559 306009358661350, 560 1520450739132526, 561 ]), 562 }, 563 AffineNielsPoint { 564 y_plus_x: FieldElement51([ 565 3732317023121341, 566 1511153322193951, 567 3496143672676420, 568 2556587964178488, 569 2620936670181690, 570 ]), 571 y_minus_x: FieldElement51([ 572 2151330273626164, 573 762045184746182, 574 1688074332551515, 575 823046109005759, 576 907602769079491, 577 ]), 578 xy2d: FieldElement51([ 579 2047386910586836, 580 168470092900250, 581 1552838872594810, 582 340951180073789, 583 360819374702533, 584 ]), 585 }, 586 AffineNielsPoint { 587 y_plus_x: FieldElement51([ 588 1982622644432037, 589 2014393600336956, 590 2380709022489462, 591 3869592437614438, 592 2357094095599062, 593 ]), 594 y_minus_x: FieldElement51([ 595 980234343912898, 596 1712256739246056, 597 588935272190264, 598 204298813091998, 599 841798321043288, 600 ]), 601 xy2d: FieldElement51([ 602 197561292938973, 603 454817274782871, 604 1963754960082318, 605 2113372252160468, 606 971377527342673, 607 ]), 608 }, 609 AffineNielsPoint { 610 y_plus_x: FieldElement51([ 611 2416499262514576, 612 2254927265442919, 613 3451304785234000, 614 1766155447043651, 615 1899238924683527, 616 ]), 617 y_minus_x: FieldElement51([ 618 732262946680281, 619 1674412764227063, 620 2182456405662809, 621 1350894754474250, 622 558458873295247, 623 ]), 624 xy2d: FieldElement51([ 625 2103305098582922, 626 1960809151316468, 627 715134605001343, 628 1454892949167181, 629 40827143824949, 630 ]), 631 }, 632 AffineNielsPoint { 633 y_plus_x: FieldElement51([ 634 1239289043050193, 635 1744654158124578, 636 758702410031698, 637 4048562808759936, 638 2253402870349013, 639 ]), 640 y_minus_x: FieldElement51([ 641 2232056027107988, 642 987343914584615, 643 2115594492994461, 644 1819598072792159, 645 1119305654014850, 646 ]), 647 xy2d: FieldElement51([ 648 320153677847348, 649 939613871605645, 650 641883205761567, 651 1930009789398224, 652 329165806634126, 653 ]), 654 }, 655 AffineNielsPoint { 656 y_plus_x: FieldElement51([ 657 3232730304159378, 658 1242488692177892, 659 1251446316964684, 660 1086618677993530, 661 1961430968465772, 662 ]), 663 y_minus_x: FieldElement51([ 664 276821765317453, 665 1536835591188030, 666 1305212741412361, 667 61473904210175, 668 2051377036983058, 669 ]), 670 xy2d: FieldElement51([ 671 833449923882501, 672 1750270368490475, 673 1123347002068295, 674 185477424765687, 675 278090826653186, 676 ]), 677 }, 678 AffineNielsPoint { 679 y_plus_x: FieldElement51([ 680 794524995833413, 681 1849907304548286, 682 2305148486158393, 683 1272368559505216, 684 1147304168324779, 685 ]), 686 y_minus_x: FieldElement51([ 687 1504846112759364, 688 1203096289004681, 689 562139421471418, 690 274333017451844, 691 1284344053775441, 692 ]), 693 xy2d: FieldElement51([ 694 483048732424432, 695 2116063063343382, 696 30120189902313, 697 292451576741007, 698 1156379271702225, 699 ]), 700 }, 701 ]), 702 LookupTable([ 703 AffineNielsPoint { 704 y_plus_x: FieldElement51([ 705 3180171966714267, 706 2147692869914563, 707 1455665844462196, 708 1986737809425946, 709 2437006863943337, 710 ]), 711 y_minus_x: FieldElement51([ 712 137732961814206, 713 706670923917341, 714 1387038086865771, 715 1965643813686352, 716 1384777115696347, 717 ]), 718 xy2d: FieldElement51([ 719 481144981981577, 720 2053319313589856, 721 2065402289827512, 722 617954271490316, 723 1106602634668125, 724 ]), 725 }, 726 AffineNielsPoint { 727 y_plus_x: FieldElement51([ 728 2948097833334040, 729 3145099472726142, 730 1148636718636008, 731 2278533891034865, 732 2203955659340680, 733 ]), 734 y_minus_x: FieldElement51([ 735 657390353372855, 736 998499966885562, 737 991893336905797, 738 810470207106761, 739 343139804608786, 740 ]), 741 xy2d: FieldElement51([ 742 791736669492960, 743 934767652997115, 744 824656780392914, 745 1759463253018643, 746 361530362383518, 747 ]), 748 }, 749 AffineNielsPoint { 750 y_plus_x: FieldElement51([ 751 2022541353055578, 752 4346500076272714, 753 3802807888710933, 754 2494585331103411, 755 2947785218648809, 756 ]), 757 y_minus_x: FieldElement51([ 758 1287487199965223, 759 2215311941380308, 760 1552928390931986, 761 1664859529680196, 762 1125004975265243, 763 ]), 764 xy2d: FieldElement51([ 765 677434665154918, 766 989582503122485, 767 1817429540898386, 768 1052904935475344, 769 1143826298169798, 770 ]), 771 }, 772 AffineNielsPoint { 773 y_plus_x: FieldElement51([ 774 2619066141993637, 775 2570231002607651, 776 2947429167440602, 777 2885885471266079, 778 2276381426249673, 779 ]), 780 y_minus_x: FieldElement51([ 781 773360688841258, 782 1815381330538070, 783 363773437667376, 784 539629987070205, 785 783280434248437, 786 ]), 787 xy2d: FieldElement51([ 788 180820816194166, 789 168937968377394, 790 748416242794470, 791 1227281252254508, 792 1567587861004268, 793 ]), 794 }, 795 AffineNielsPoint { 796 y_plus_x: FieldElement51([ 797 2730575372268893, 798 2062896624554806, 799 2951191072970647, 800 2609899222113120, 801 1277310261461760, 802 ]), 803 y_minus_x: FieldElement51([ 804 1984740906540026, 805 1079164179400229, 806 1056021349262661, 807 1659958556483663, 808 1088529069025527, 809 ]), 810 xy2d: FieldElement51([ 811 580736401511151, 812 1842931091388998, 813 1177201471228238, 814 2075460256527244, 815 1301133425678027, 816 ]), 817 }, 818 AffineNielsPoint { 819 y_plus_x: FieldElement51([ 820 1515728832059163, 821 1575261009617579, 822 1510246567196186, 823 2442877836294952, 824 2368461529974388, 825 ]), 826 y_minus_x: FieldElement51([ 827 1295295738269652, 828 1714742313707026, 829 545583042462581, 830 2034411676262552, 831 1513248090013606, 832 ]), 833 xy2d: FieldElement51([ 834 230710545179830, 835 30821514358353, 836 760704303452229, 837 390668103790604, 838 573437871383156, 839 ]), 840 }, 841 AffineNielsPoint { 842 y_plus_x: FieldElement51([ 843 3421179921230875, 844 2514967047430861, 845 4274701112739695, 846 3071700566936367, 847 4275698278559832, 848 ]), 849 y_minus_x: FieldElement51([ 850 2102254323485823, 851 1570832666216754, 852 34696906544624, 853 1993213739807337, 854 70638552271463, 855 ]), 856 xy2d: FieldElement51([ 857 894132856735058, 858 548675863558441, 859 845349339503395, 860 1942269668326667, 861 1615682209874691, 862 ]), 863 }, 864 AffineNielsPoint { 865 y_plus_x: FieldElement51([ 866 3539470031223082, 867 1222355136884919, 868 1846481788678694, 869 1150426571265110, 870 1613523400722047, 871 ]), 872 y_minus_x: FieldElement51([ 873 793388516527298, 874 1315457083650035, 875 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1055061081337675, 5643 1458319101947665, 5644 1234195845213142, 5645 ]), 5646 xy2d: FieldElement51([ 5647 830430507734812, 5648 1780282976102377, 5649 1425386760709037, 5650 362399353095425, 5651 2168861579799910, 5652 ]), 5653 }, 5654 AffineNielsPoint { 5655 y_plus_x: FieldElement51([ 5656 3407562046415562, 5657 980662895504005, 5658 2053766700883521, 5659 2742766027762854, 5660 2762205690726604, 5661 ]), 5662 y_minus_x: FieldElement51([ 5663 1683750316716132, 5664 652278688286128, 5665 1221798761193539, 5666 1897360681476669, 5667 319658166027343, 5668 ]), 5669 xy2d: FieldElement51([ 5670 618808732869972, 5671 72755186759744, 5672 2060379135624181, 5673 1730731526741822, 5674 48862757828238, 5675 ]), 5676 }, 5677 AffineNielsPoint { 5678 y_plus_x: FieldElement51([ 5679 3714971784278753, 5680 3394840525452699, 5681 614590986558882, 5682 1409210575145591, 5683 1882816996436803, 5684 ]), 5685 y_minus_x: FieldElement51([ 5686 2230133264691131, 5687 563950955091024, 5688 2042915975426398, 5689 827314356293472, 5690 672028980152815, 5691 ]), 5692 xy2d: FieldElement51([ 5693 264204366029760, 5694 1654686424479449, 5695 2185050199932931, 5696 2207056159091748, 5697 506015669043634, 5698 ]), 5699 }, 5700 AffineNielsPoint { 5701 y_plus_x: FieldElement51([ 5702 1784446333136550, 5703 1973746527984364, 5704 334856327359575, 5705 3408569589569858, 5706 3275749938360725, 5707 ]), 5708 y_minus_x: FieldElement51([ 5709 2065270940578383, 5710 31477096270353, 5711 306421879113491, 5712 181958643936686, 5713 1907105536686083, 5714 ]), 5715 xy2d: FieldElement51([ 5716 1496516440779464, 5717 1748485652986458, 5718 872778352227340, 5719 818358834654919, 5720 97932669284220, 5721 ]), 5722 }, 5723 ]), 5724 LookupTable([ 5725 AffineNielsPoint { 5726 y_plus_x: FieldElement51([ 5727 2723435829455580, 5728 2924255216478824, 5729 1804995246884102, 5730 1842309243470804, 5731 3753662318666930, 5732 ]), 5733 y_minus_x: FieldElement51([ 5734 1013216974933691, 5735 538921919682598, 5736 1915776722521558, 5737 1742822441583877, 5738 1886550687916656, 5739 ]), 5740 xy2d: FieldElement51([ 5741 2094270000643336, 5742 303971879192276, 5743 40801275554748, 5744 649448917027930, 5745 1818544418535447, 5746 ]), 5747 }, 5748 AffineNielsPoint { 5749 y_plus_x: FieldElement51([ 5750 2241737709499146, 5751 549397817447461, 5752 838180519319392, 5753 1725686958520781, 5754 3957438894582995, 5755 ]), 5756 y_minus_x: FieldElement51([ 5757 1216074541925116, 5758 50120933933509, 5759 1565829004133810, 5760 721728156134580, 5761 349206064666188, 5762 ]), 5763 xy2d: FieldElement51([ 5764 948617110470858, 5765 346222547451945, 5766 1126511960599975, 5767 1759386906004538, 5768 493053284802266, 5769 ]), 5770 }, 5771 AffineNielsPoint { 5772 y_plus_x: FieldElement51([ 5773 1454933046815146, 5774 3126495827951610, 5775 1467170975468587, 5776 1432316382418897, 5777 2111710746366763, 5778 ]), 5779 y_minus_x: FieldElement51([ 5780 2105387117364450, 5781 1996463405126433, 5782 1303008614294500, 5783 851908115948209, 5784 1353742049788635, 5785 ]), 5786 xy2d: FieldElement51([ 5787 750300956351719, 5788 1487736556065813, 5789 15158817002104, 5790 1511998221598392, 5791 971739901354129, 5792 ]), 5793 }, 5794 AffineNielsPoint { 5795 y_plus_x: FieldElement51([ 5796 1874648163531674, 5797 2124487685930551, 5798 1810030029384882, 5799 918400043048335, 5800 2838148440985898, 5801 ]), 5802 y_minus_x: FieldElement51([ 5803 1235084464747900, 5804 1166111146432082, 5805 1745394857881591, 5806 1405516473883040, 5807 4463504151617, 5808 ]), 5809 xy2d: FieldElement51([ 5810 1663810156463827, 5811 327797390285791, 5812 1341846161759410, 5813 1964121122800605, 5814 1747470312055380, 5815 ]), 5816 }, 5817 AffineNielsPoint { 5818 y_plus_x: FieldElement51([ 5819 660005247548214, 5820 2071860029952887, 5821 3610548013635355, 5822 911703252219106, 5823 3266179736709079, 5824 ]), 5825 y_minus_x: FieldElement51([ 5826 2206641276178231, 5827 1690587809721504, 5828 1600173622825126, 5829 2156096097634421, 5830 1106822408548216, 5831 ]), 5832 xy2d: FieldElement51([ 5833 1344788193552206, 5834 1949552134239140, 5835 1735915881729557, 5836 675891104100469, 5837 1834220014427292, 5838 ]), 5839 }, 5840 AffineNielsPoint { 5841 y_plus_x: FieldElement51([ 5842 1920949492387945, 5843 2410685102072778, 5844 2322108077349280, 5845 2877838278583064, 5846 3719881539786256, 5847 ]), 5848 y_minus_x: FieldElement51([ 5849 622221042073383, 5850 1210146474039168, 5851 1742246422343683, 5852 1403839361379025, 5853 417189490895736, 5854 ]), 5855 xy2d: FieldElement51([ 5856 22727256592983, 5857 168471543384997, 5858 1324340989803650, 5859 1839310709638189, 5860 504999476432775, 5861 ]), 5862 }, 5863 AffineNielsPoint { 5864 y_plus_x: FieldElement51([ 5865 3565040332441556, 5866 1721896294296941, 5867 2304063388272514, 5868 2065069734239231, 5869 3056710287109878, 5870 ]), 5871 y_minus_x: FieldElement51([ 5872 1337466662091884, 5873 1287645354669772, 5874 2018019646776184, 5875 652181229374245, 5876 898011753211715, 5877 ]), 5878 xy2d: FieldElement51([ 5879 1969792547910734, 5880 779969968247557, 5881 2011350094423418, 5882 1823964252907487, 5883 1058949448296945, 5884 ]), 5885 }, 5886 AffineNielsPoint { 5887 y_plus_x: FieldElement51([ 5888 2459143550747250, 5889 1118176942430252, 5890 3010694408233412, 5891 806764629546265, 5892 1157700123092949, 5893 ]), 5894 y_minus_x: FieldElement51([ 5895 1273565321399022, 5896 1638509681964574, 5897 759235866488935, 5898 666015124346707, 5899 897983460943405, 5900 ]), 5901 xy2d: FieldElement51([ 5902 1717263794012298, 5903 1059601762860786, 5904 1837819172257618, 5905 1054130665797229, 5906 680893204263559, 5907 ]), 5908 }, 5909 ]), 5910 LookupTable([ 5911 AffineNielsPoint { 5912 y_plus_x: FieldElement51([ 5913 2237039662793603, 5914 2249022333361206, 5915 2058613546633703, 5916 2401253908530527, 5917 2215176649164581, 5918 ]), 5919 y_minus_x: FieldElement51([ 5920 79472182719605, 5921 1851130257050174, 5922 1825744808933107, 5923 821667333481068, 5924 781795293511946, 5925 ]), 5926 xy2d: FieldElement51([ 5927 755822026485370, 5928 152464789723500, 5929 1178207602290608, 5930 410307889503239, 5931 156581253571278, 5932 ]), 5933 }, 5934 AffineNielsPoint { 5935 y_plus_x: FieldElement51([ 5936 3669985309815545, 5937 2736319981413860, 5938 3898537095128197, 5939 3653287498355512, 5940 1349185550126960, 5941 ]), 5942 y_minus_x: FieldElement51([ 5943 1495380034400429, 5944 325049476417173, 5945 46346894893933, 5946 1553408840354856, 5947 828980101835683, 5948 ]), 5949 xy2d: FieldElement51([ 5950 1280337889310282, 5951 2070832742866672, 5952 1640940617225222, 5953 2098284908289951, 5954 450929509534434, 5955 ]), 5956 }, 5957 AffineNielsPoint { 5958 y_plus_x: FieldElement51([ 5959 2659503167684029, 5960 2378371955168899, 5961 2537839641198868, 5962 1999255076709337, 5963 2030511179441770, 5964 ]), 5965 y_minus_x: FieldElement51([ 5966 1254958221100483, 5967 1153235960999843, 5968 942907704968834, 5969 637105404087392, 5970 1149293270147267, 5971 ]), 5972 xy2d: FieldElement51([ 5973 894249020470196, 5974 400291701616810, 5975 406878712230981, 5976 1599128793487393, 5977 1145868722604026, 5978 ]), 5979 }, 5980 AffineNielsPoint { 5981 y_plus_x: FieldElement51([ 5982 3749755063888563, 5983 2361916158338507, 5984 1128535642171975, 5985 1900106496009660, 5986 2381592531146157, 5987 ]), 5988 y_minus_x: FieldElement51([ 5989 452487513298665, 5990 1352120549024569, 5991 1173495883910956, 5992 1999111705922009, 5993 367328130454226, 5994 ]), 5995 xy2d: FieldElement51([ 5996 1717539401269642, 5997 1475188995688487, 5998 891921989653942, 5999 836824441505699, 6000 1885988485608364, 6001 ]), 6002 }, 6003 AffineNielsPoint { 6004 y_plus_x: FieldElement51([ 6005 3493583935107776, 6006 2439136865632830, 6007 3370281625921440, 6008 2680547565621609, 6009 2282158712612572, 6010 ]), 6011 y_minus_x: FieldElement51([ 6012 2022432361201842, 6013 1088816090685051, 6014 1977843398539868, 6015 1854834215890724, 6016 564238862029357, 6017 ]), 6018 xy2d: FieldElement51([ 6019 938868489100585, 6020 1100285072929025, 6021 1017806255688848, 6022 1957262154788833, 6023 152787950560442, 6024 ]), 6025 }, 6026 AffineNielsPoint { 6027 y_plus_x: FieldElement51([ 6028 3119119231364171, 6029 2872271776627789, 6030 2477832016990963, 6031 2593801257642876, 6032 1761675818237335, 6033 ]), 6034 y_minus_x: FieldElement51([ 6035 1295072362439987, 6036 931227904689414, 6037 1355731432641687, 6038 922235735834035, 6039 892227229410209, 6040 ]), 6041 xy2d: FieldElement51([ 6042 1680989767906154, 6043 535362787031440, 6044 2136691276706570, 6045 1942228485381244, 6046 1267350086882274, 6047 ]), 6048 }, 6049 AffineNielsPoint { 6050 y_plus_x: FieldElement51([ 6051 2617818047455756, 6052 2684460443440843, 6053 2378209521329782, 6054 1973842949591661, 6055 2897427157127624, 6056 ]), 6057 y_minus_x: FieldElement51([ 6058 535509430575217, 6059 546885533737322, 6060 1524675609547799, 6061 2138095752851703, 6062 1260738089896827, 6063 ]), 6064 xy2d: FieldElement51([ 6065 1159906385590467, 6066 2198530004321610, 6067 714559485023225, 6068 81880727882151, 6069 1484020820037082, 6070 ]), 6071 }, 6072 AffineNielsPoint { 6073 y_plus_x: FieldElement51([ 6074 1377485731340769, 6075 2046328105512000, 6076 1802058637158797, 6077 2313945950453421, 6078 1356993908853900, 6079 ]), 6080 y_minus_x: FieldElement51([ 6081 2013612215646735, 6082 1830770575920375, 6083 536135310219832, 6084 609272325580394, 6085 270684344495013, 6086 ]), 6087 xy2d: FieldElement51([ 6088 1237542585982777, 6089 2228682050256790, 6090 1385281931622824, 6091 593183794882890, 6092 493654978552689, 6093 ]), 6094 }, 6095 ]), 6096 LookupTable([ 6097 AffineNielsPoint { 6098 y_plus_x: FieldElement51([ 6099 2299141301692989, 6100 1891414891220256, 6101 983894663308928, 6102 2427961581972066, 6103 3378060928864955, 6104 ]), 6105 y_minus_x: FieldElement51([ 6106 1694030170963455, 6107 502038567066200, 6108 1691160065225467, 6109 949628319562187, 6110 275110186693066, 6111 ]), 6112 xy2d: FieldElement51([ 6113 1124515748676336, 6114 1661673816593408, 6115 1499640319059718, 6116 1584929449166988, 6117 558148594103306, 6118 ]), 6119 }, 6120 AffineNielsPoint { 6121 y_plus_x: FieldElement51([ 6122 1784525599998356, 6123 1619698033617383, 6124 2097300287550715, 6125 2510065271789004, 6126 1905684794832757, 6127 ]), 6128 y_minus_x: FieldElement51([ 6129 1288941072872766, 6130 931787902039402, 6131 190731008859042, 6132 2006859954667190, 6133 1005931482221702, 6134 ]), 6135 xy2d: FieldElement51([ 6136 1465551264822703, 6137 152905080555927, 6138 680334307368453, 6139 173227184634745, 6140 666407097159852, 6141 ]), 6142 }, 6143 AffineNielsPoint { 6144 y_plus_x: FieldElement51([ 6145 2111017076203943, 6146 3630560299479595, 6147 1248583954016455, 6148 3604089008549670, 6149 1895180776543895, 6150 ]), 6151 y_minus_x: FieldElement51([ 6152 171348223915638, 6153 662766099800389, 6154 462338943760497, 6155 466917763340314, 6156 656911292869115, 6157 ]), 6158 xy2d: FieldElement51([ 6159 488623681976577, 6160 866497561541722, 6161 1708105560937768, 6162 1673781214218839, 6163 1506146329818807, 6164 ]), 6165 }, 6166 AffineNielsPoint { 6167 y_plus_x: FieldElement51([ 6168 2412225278142205, 6169 950394373239688, 6170 2682296937026182, 6171 711676555398831, 6172 320964687779005, 6173 ]), 6174 y_minus_x: FieldElement51([ 6175 988979367990485, 6176 1359729327576302, 6177 1301834257246029, 6178 294141160829308, 6179 29348272277475, 6180 ]), 6181 xy2d: FieldElement51([ 6182 1434382743317910, 6183 100082049942065, 6184 221102347892623, 6185 186982837860588, 6186 1305765053501834, 6187 ]), 6188 }, 6189 AffineNielsPoint { 6190 y_plus_x: FieldElement51([ 6191 2205916462268190, 6192 2751663643476068, 6193 961960554686615, 6194 2409862576442233, 6195 1841471168298304, 6196 ]), 6197 y_minus_x: FieldElement51([ 6198 1191737341426592, 6199 1847042034978363, 6200 1382213545049056, 6201 1039952395710448, 6202 788812858896859, 6203 ]), 6204 xy2d: FieldElement51([ 6205 1346965964571152, 6206 1291881610839830, 6207 2142916164336056, 6208 786821641205979, 6209 1571709146321039, 6210 ]), 6211 }, 6212 AffineNielsPoint { 6213 y_plus_x: FieldElement51([ 6214 787164375951248, 6215 2454669019058437, 6216 3608390234717387, 6217 1431233331032509, 6218 786341368775957, 6219 ]), 6220 y_minus_x: FieldElement51([ 6221 492448143532951, 6222 304105152670757, 6223 1761767168301056, 6224 233782684697790, 6225 1981295323106089, 6226 ]), 6227 xy2d: FieldElement51([ 6228 665807507761866, 6229 1343384868355425, 6230 895831046139653, 6231 439338948736892, 6232 1986828765695105, 6233 ]), 6234 }, 6235 AffineNielsPoint { 6236 y_plus_x: FieldElement51([ 6237 3007896024559801, 6238 1721699973539148, 6239 2510565115413133, 6240 1390588532210644, 6241 1212530909934781, 6242 ]), 6243 y_minus_x: FieldElement51([ 6244 852891097972275, 6245 1816988871354562, 6246 1543772755726524, 6247 1174710635522444, 6248 202129090724628, 6249 ]), 6250 xy2d: FieldElement51([ 6251 1205281565824323, 6252 22430498399418, 6253 992947814485516, 6254 1392458699738672, 6255 688441466734558, 6256 ]), 6257 }, 6258 AffineNielsPoint { 6259 y_plus_x: FieldElement51([ 6260 3302427242100220, 6261 1955849529137134, 6262 2171162376368357, 6263 2343545681983462, 6264 447733118757825, 6265 ]), 6266 y_minus_x: FieldElement51([ 6267 1287181461435438, 6268 622722465530711, 6269 880952150571872, 6270 741035693459198, 6271 311565274989772, 6272 ]), 6273 xy2d: FieldElement51([ 6274 1003649078149734, 6275 545233927396469, 6276 1849786171789880, 6277 1318943684880434, 6278 280345687170552, 6279 ]), 6280 }, 6281 ]), 6282 ]); 6283 6284 /// Odd multiples of the basepoint `[B, 3B, 5B, 7B, 9B, 11B, 13B, 15B, ..., 127B]`. 6285 pub(crate) const AFFINE_ODD_MULTIPLES_OF_BASEPOINT: NafLookupTable8<AffineNielsPoint> = 6286 NafLookupTable8([ 6287 AffineNielsPoint { 6288 y_plus_x: FieldElement51([ 6289 3540182452943730, 6290 2497478415033846, 6291 2521227595762870, 6292 1462984067271729, 6293 2389212253076811, 6294 ]), 6295 y_minus_x: FieldElement51([ 6296 62697248952638, 6297 204681361388450, 6298 631292143396476, 6299 338455783676468, 6300 1213667448819585, 6301 ]), 6302 xy2d: FieldElement51([ 6303 301289933810280, 6304 1259582250014073, 6305 1422107436869536, 6306 796239922652654, 6307 1953934009299142, 6308 ]), 6309 }, 6310 AffineNielsPoint { 6311 y_plus_x: FieldElement51([ 6312 1601611775252272, 6313 1720807796594148, 6314 1132070835939856, 6315 3512254832574799, 6316 2147779492816910, 6317 ]), 6318 y_minus_x: FieldElement51([ 6319 316559037616741, 6320 2177824224946892, 6321 1459442586438991, 6322 1461528397712656, 6323 751590696113597, 6324 ]), 6325 xy2d: FieldElement51([ 6326 1850748884277385, 6327 1200145853858453, 6328 1068094770532492, 6329 672251375690438, 6330 1586055907191707, 6331 ]), 6332 }, 6333 AffineNielsPoint { 6334 y_plus_x: FieldElement51([ 6335 769950342298400, 6336 2384754244604994, 6337 3095885746880802, 6338 3225892188161580, 6339 2977876099231263, 6340 ]), 6341 y_minus_x: FieldElement51([ 6342 425251763115706, 6343 608463272472562, 6344 442562545713235, 6345 837766094556764, 6346 374555092627893, 6347 ]), 6348 xy2d: FieldElement51([ 6349 1086255230780037, 6350 274979815921559, 6351 1960002765731872, 6352 929474102396301, 6353 1190409889297339, 6354 ]), 6355 }, 6356 AffineNielsPoint { 6357 y_plus_x: FieldElement51([ 6358 2916800678241215, 6359 2065379846933858, 6360 2622030924071124, 6361 2602788184473875, 6362 1233371373142984, 6363 ]), 6364 y_minus_x: FieldElement51([ 6365 2019367628972465, 6366 676711900706637, 6367 110710997811333, 6368 1108646842542025, 6369 517791959672113, 6370 ]), 6371 xy2d: FieldElement51([ 6372 965130719900578, 6373 247011430587952, 6374 526356006571389, 6375 91986625355052, 6376 2157223321444601, 6377 ]), 6378 }, 6379 AffineNielsPoint { 6380 y_plus_x: FieldElement51([ 6381 1802695059464988, 6382 1664899123557221, 6383 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