// -*- mode: rust; -*-
//
// This file is part of curve25519-dalek.
// Copyright (c) 2016-2021 isis lovecruft
// Copyright (c) 2016-2019 Henry de Valence
// See LICENSE for licensing information.
//
// Authors:
// - isis agora lovecruft <isis@patternsinthevoid.net>
// - Henry de Valence <hdevalence@hdevalence.ca>
//! This module contains backend-specific constant values, such as the 64-bit limbs of curve constants.
use super::field::FieldElement51;
use super::scalar::Scalar52;
use crate::backend::serial::curve_models::AffineNielsPoint;
use crate::edwards::{EdwardsBasepointTable, EdwardsPoint};
use crate::window::{LookupTable, NafLookupTable8};
/// The value of minus one, equal to `-&FieldElement::ONE`
pub(crate) const MINUS_ONE: FieldElement51 = FieldElement51([
2251799813685228,
2251799813685247,
2251799813685247,
2251799813685247,
2251799813685247,
]);
/// Edwards `d` value, equal to `-121665/121666 mod p`.
pub(crate) const EDWARDS_D: FieldElement51 = FieldElement51([
929955233495203,
466365720129213,
1662059464998953,
2033849074728123,
1442794654840575,
]);
/// Edwards `2*d` value, equal to `2*(-121665/121666) mod p`.
pub(crate) const EDWARDS_D2: FieldElement51 = FieldElement51([
1859910466990425,
932731440258426,
1072319116312658,
1815898335770999,
633789495995903,
]);
/// One minus edwards `d` value squared, equal to `(1 - (-121665/121666) mod p) pow 2`
pub(crate) const ONE_MINUS_EDWARDS_D_SQUARED: FieldElement51 = FieldElement51([
1136626929484150,
1998550399581263,
496427632559748,
118527312129759,
45110755273534,
]);
/// Edwards `d` value minus one squared, equal to `(((-121665/121666) mod p) - 1) pow 2`
pub(crate) const EDWARDS_D_MINUS_ONE_SQUARED: FieldElement51 = FieldElement51([
1507062230895904,
1572317787530805,
683053064812840,
317374165784489,
1572899562415810,
]);
/// `= sqrt(a*d - 1)`, where `a = -1 (mod p)`, `d` are the Edwards curve parameters.
pub(crate) const SQRT_AD_MINUS_ONE: FieldElement51 = FieldElement51([
2241493124984347,
425987919032274,
2207028919301688,
1220490630685848,
974799131293748,
]);
/// `= 1/sqrt(a-d)`, where `a = -1 (mod p)`, `d` are the Edwards curve parameters.
pub(crate) const INVSQRT_A_MINUS_D: FieldElement51 = FieldElement51([
278908739862762,
821645201101625,
8113234426968,
1777959178193151,
2118520810568447,
]);
/// Precomputed value of one of the square roots of -1 (mod p)
pub(crate) const SQRT_M1: FieldElement51 = FieldElement51([
1718705420411056,
234908883556509,
2233514472574048,
2117202627021982,
765476049583133,
]);
/// `APLUS2_OVER_FOUR` is (A+2)/4. (This is used internally within the Montgomery ladder.)
pub(crate) const APLUS2_OVER_FOUR: FieldElement51 = FieldElement51([121666, 0, 0, 0, 0]);
/// `MONTGOMERY_A` is equal to 486662, which is a constant of the curve equation
/// for Curve25519 in its Montgomery form. (This is used internally within the
/// Elligator map.)
pub(crate) const MONTGOMERY_A: FieldElement51 = FieldElement51([486662, 0, 0, 0, 0]);
/// `MONTGOMERY_A_NEG` is equal to -486662. (This is used internally within the
/// Elligator map.)
pub(crate) const MONTGOMERY_A_NEG: FieldElement51 = FieldElement51([
2251799813198567,
2251799813685247,
2251799813685247,
2251799813685247,
2251799813685247,
]);
/// `L` is the order of base point, i.e. 2^252 + 27742317777372353535851937790883648493
pub(crate) const L: Scalar52 = Scalar52([
0x0002631a5cf5d3ed,
0x000dea2f79cd6581,
0x000000000014def9,
0x0000000000000000,
0x0000100000000000,
]);
/// `L` * `LFACTOR` = -1 (mod 2^52)
pub(crate) const LFACTOR: u64 = 0x51da312547e1b;
/// `R` = R % L where R = 2^260
pub(crate) const R: Scalar52 = Scalar52([
0x000f48bd6721e6ed,
0x0003bab5ac67e45a,
0x000fffffeb35e51b,
0x000fffffffffffff,
0x00000fffffffffff,
]);
/// `RR` = (R^2) % L where R = 2^260
pub(crate) const RR: Scalar52 = Scalar52([
0x0009d265e952d13b,
0x000d63c715bea69f,
0x0005be65cb687604,
0x0003dceec73d217f,
0x000009411b7c309a,
]);
/// The Ed25519 basepoint, as an `EdwardsPoint`.
///
/// This is called `_POINT` to distinguish it from
/// `ED25519_BASEPOINT_TABLE`, which should be used for scalar
/// multiplication (it's much faster).
pub const ED25519_BASEPOINT_POINT: EdwardsPoint = EdwardsPoint {
X: FieldElement51([
1738742601995546,
1146398526822698,
2070867633025821,
562264141797630,
587772402128613,
]),
Y: FieldElement51([
1801439850948184,
1351079888211148,
450359962737049,
900719925474099,
1801439850948198,
]),
Z: FieldElement51([1, 0, 0, 0, 0]),
T: FieldElement51([
1841354044333475,
16398895984059,
755974180946558,
900171276175154,
1821297809914039,
]),
};
/// The 8-torsion subgroup \\(\mathcal E \[8\]\\).
///
/// In the case of Curve25519, it is cyclic; the \\(i\\)-th element of
/// the array is \\(\[i\]P\\), where \\(P\\) is a point of order \\(8\\)
/// generating \\(\mathcal E\[8\]\\).
///
/// Thus \\(\mathcal E\[4\]\\) is the points indexed by `0,2,4,6`, and
/// \\(\mathcal E\[2\]\\) is the points indexed by `0,4`.
pub const EIGHT_TORSION: [EdwardsPoint; 8] = EIGHT_TORSION_INNER_DOC_HIDDEN;
/// Inner item used to hide limb constants from cargo doc output.
#[doc(hidden)]
pub const EIGHT_TORSION_INNER_DOC_HIDDEN: [EdwardsPoint; 8] = [
EdwardsPoint {
X: FieldElement51([0, 0, 0, 0, 0]),
Y: FieldElement51([1, 0, 0, 0, 0]),
Z: FieldElement51([1, 0, 0, 0, 0]),
T: FieldElement51([0, 0, 0, 0, 0]),
},
EdwardsPoint {
X: FieldElement51([
358744748052810,
1691584618240980,
977650209285361,
1429865912637724,
560044844278676,
]),
Y: FieldElement51([
84926274344903,
473620666599931,
365590438845504,
1028470286882429,
2146499180330972,
]),
Z: FieldElement51([1, 0, 0, 0, 0]),
T: FieldElement51([
1448326834587521,
1857896831960481,
1093722731865333,
1677408490711241,
1915505153018406,
]),
},
EdwardsPoint {
X: FieldElement51([
533094393274173,
2016890930128738,
18285341111199,
134597186663265,
1486323764102114,
]),
Y: FieldElement51([0, 0, 0, 0, 0]),
Z: FieldElement51([1, 0, 0, 0, 0]),
T: FieldElement51([0, 0, 0, 0, 0]),
},
EdwardsPoint {
X: FieldElement51([
358744748052810,
1691584618240980,
977650209285361,
1429865912637724,
560044844278676,
]),
Y: FieldElement51([
2166873539340326,
1778179147085316,
1886209374839743,
1223329526802818,
105300633354275,
]),
Z: FieldElement51([1, 0, 0, 0, 0]),
T: FieldElement51([
803472979097708,
393902981724766,
1158077081819914,
574391322974006,
336294660666841,
]),
},
EdwardsPoint {
X: FieldElement51([0, 0, 0, 0, 0]),
Y: FieldElement51([
2251799813685228,
2251799813685247,
2251799813685247,
2251799813685247,
2251799813685247,
]),
Z: FieldElement51([1, 0, 0, 0, 0]),
T: FieldElement51([0, 0, 0, 0, 0]),
},
EdwardsPoint {
X: FieldElement51([
1893055065632419,
560215195444267,
1274149604399886,
821933901047523,
1691754969406571,
]),
Y: FieldElement51([
2166873539340326,
1778179147085316,
1886209374839743,
1223329526802818,
105300633354275,
]),
Z: FieldElement51([1, 0, 0, 0, 0]),
T: FieldElement51([
1448326834587521,
1857896831960481,
1093722731865333,
1677408490711241,
1915505153018406,
]),
},
EdwardsPoint {
X: FieldElement51([
1718705420411056,
234908883556509,
2233514472574048,
2117202627021982,
765476049583133,
]),
Y: FieldElement51([0, 0, 0, 0, 0]),
Z: FieldElement51([1, 0, 0, 0, 0]),
T: FieldElement51([0, 0, 0, 0, 0]),
},
EdwardsPoint {
X: FieldElement51([
1893055065632419,
560215195444267,
1274149604399886,
821933901047523,
1691754969406571,
]),
Y: FieldElement51([
84926274344903,
473620666599931,
365590438845504,
1028470286882429,
2146499180330972,
]),
Z: FieldElement51([1, 0, 0, 0, 0]),
T: FieldElement51([
803472979097708,
393902981724766,
1158077081819914,
574391322974006,
336294660666841,
]),
},
];
/// Table containing precomputed multiples of the Ed25519 basepoint \\(B = (x, 4/5)\\).
pub const ED25519_BASEPOINT_TABLE: EdwardsBasepointTable = ED25519_BASEPOINT_TABLE_INNER_DOC_HIDDEN;
/// Inner constant, used to avoid filling the docs with precomputed points.
#[doc(hidden)]
pub const ED25519_BASEPOINT_TABLE_INNER_DOC_HIDDEN: EdwardsBasepointTable =
EdwardsBasepointTable([
LookupTable([
AffineNielsPoint {
y_plus_x: FieldElement51([
3540182452943730,
2497478415033846,
2521227595762870,
1462984067271729,
2389212253076811,
]),
y_minus_x: FieldElement51([
62697248952638,
204681361388450,
631292143396476,
338455783676468,
1213667448819585,
]),
xy2d: FieldElement51([
301289933810280,
1259582250014073,
1422107436869536,
796239922652654,
1953934009299142,
]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
3632771708514775,
790832306631235,
2067202295274102,
1995808275510000,
1566530869037010,
]),
y_minus_x: FieldElement51([
463307831301544,
432984605774163,
1610641361907204,
750899048855000,
1894842303421586,
]),
xy2d: FieldElement51([
748439484463711,
1033211726465151,
1396005112841647,
1611506220286469,
1972177495910992,
]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
1601611775252272,
1720807796594148,
1132070835939856,
3512254832574799,
2147779492816910,
]),
y_minus_x: FieldElement51([
316559037616741,
2177824224946892,
1459442586438991,
1461528397712656,
751590696113597,
]),
xy2d: FieldElement51([
1850748884277385,
1200145853858453,
1068094770532492,
672251375690438,
1586055907191707,
]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
934282339813791,
1846903124198670,
1172395437954843,
1007037127761661,
1830588347719256,
]),
y_minus_x: FieldElement51([
1694390458783935,
1735906047636159,
705069562067493,
648033061693059,
696214010414170,
]),
xy2d: FieldElement51([
1121406372216585,
192876649532226,
190294192191717,
1994165897297032,
2245000007398739,
]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
769950342298400,
2384754244604994,
3095885746880802,
3225892188161580,
2977876099231263,
]),
y_minus_x: FieldElement51([
425251763115706,
608463272472562,
442562545713235,
837766094556764,
374555092627893,
]),
xy2d: FieldElement51([
1086255230780037,
274979815921559,
1960002765731872,
929474102396301,
1190409889297339,
]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
1388594989461809,
316767091099457,
2646098655878230,
1230079486801004,
1440737038838979,
]),
y_minus_x: FieldElement51([
7380825640100,
146210432690483,
304903576448906,
1198869323871120,
997689833219095,
]),
xy2d: FieldElement51([
1181317918772081,
114573476638901,
262805072233344,
265712217171332,
294181933805782,
]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
2916800678241215,
2065379846933858,
2622030924071124,
2602788184473875,
1233371373142984,
]),
y_minus_x: FieldElement51([
2019367628972465,
676711900706637,
110710997811333,
1108646842542025,
517791959672113,
]),
xy2d: FieldElement51([
965130719900578,
247011430587952,
526356006571389,
91986625355052,
2157223321444601,
]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
4320419353804412,
4218074731744053,
957728544705548,
729906502578991,
2411634706750414,
]),
y_minus_x: FieldElement51([
2073601412052185,
31021124762708,
264500969797082,
248034690651703,
1030252227928288,
]),
xy2d: FieldElement51([
551790716293402,
1989538725166328,
801169423371717,
2052451893578887,
678432056995012,
]),
},
]),
LookupTable([
AffineNielsPoint {
y_plus_x: FieldElement51([
1368953770187805,
3042147450398169,
2689308289352409,
2142576377050579,
1932081720066286,
]),
y_minus_x: FieldElement51([
953638594433374,
1092333936795051,
1419774766716690,
805677984380077,
859228993502513,
]),
xy2d: FieldElement51([
1200766035879111,
20142053207432,
1465634435977050,
1645256912097844,
295121984874596,
]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
1735718747031538,
1248237894295956,
1204753118328107,
976066523550493,
2317743583219840,
]),
y_minus_x: FieldElement51([
1060098822528990,
1586825862073490,
212301317240126,
1975302711403555,
666724059764335,
]),
xy2d: FieldElement51([
1091990273418756,
1572899409348578,
80968014455247,
306009358661350,
1520450739132526,
]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
3732317023121341,
1511153322193951,
3496143672676420,
2556587964178488,
2620936670181690,
]),
y_minus_x: FieldElement51([
2151330273626164,
762045184746182,
1688074332551515,
823046109005759,
907602769079491,
]),
xy2d: FieldElement51([
2047386910586836,
168470092900250,
1552838872594810,
340951180073789,
360819374702533,
]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
1982622644432037,
2014393600336956,
2380709022489462,
3869592437614438,
2357094095599062,
]),
y_minus_x: FieldElement51([
980234343912898,
1712256739246056,
588935272190264,
204298813091998,
841798321043288,
]),
xy2d: FieldElement51([
197561292938973,
454817274782871,
1963754960082318,
2113372252160468,
971377527342673,
]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
2416499262514576,
2254927265442919,
3451304785234000,
1766155447043651,
1899238924683527,
]),
y_minus_x: FieldElement51([
732262946680281,
1674412764227063,
2182456405662809,
1350894754474250,
558458873295247,
]),
xy2d: FieldElement51([
2103305098582922,
1960809151316468,
715134605001343,
1454892949167181,
40827143824949,
]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
1239289043050193,
1744654158124578,
758702410031698,
4048562808759936,
2253402870349013,
]),
y_minus_x: FieldElement51([
2232056027107988,
987343914584615,
2115594492994461,
1819598072792159,
1119305654014850,
]),
xy2d: FieldElement51([
320153677847348,
939613871605645,
641883205761567,
1930009789398224,
329165806634126,
]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
3232730304159378,
1242488692177892,
1251446316964684,
1086618677993530,
1961430968465772,
]),
y_minus_x: FieldElement51([
276821765317453,
1536835591188030,
1305212741412361,
61473904210175,
2051377036983058,
]),
xy2d: FieldElement51([
833449923882501,
1750270368490475,
1123347002068295,
185477424765687,
278090826653186,
]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
794524995833413,
1849907304548286,
2305148486158393,
1272368559505216,
1147304168324779,
]),
y_minus_x: FieldElement51([
1504846112759364,
1203096289004681,
562139421471418,
274333017451844,
1284344053775441,
]),
xy2d: FieldElement51([
483048732424432,
2116063063343382,
30120189902313,
292451576741007,
1156379271702225,
]),
},
]),
LookupTable([
AffineNielsPoint {
y_plus_x: FieldElement51([
3180171966714267,
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1455665844462196,
1986737809425946,
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]),
y_minus_x: FieldElement51([
137732961814206,
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1387038086865771,
1965643813686352,
1384777115696347,
]),
xy2d: FieldElement51([
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]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
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]),
y_minus_x: FieldElement51([
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]),
xy2d: FieldElement51([
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]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
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]),
y_minus_x: FieldElement51([
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]),
xy2d: FieldElement51([
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},
AffineNielsPoint {
y_plus_x: FieldElement51([
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]),
y_minus_x: FieldElement51([
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]),
xy2d: FieldElement51([
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]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
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]),
y_minus_x: FieldElement51([
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]),
xy2d: FieldElement51([
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]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
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]),
y_minus_x: FieldElement51([
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]),
xy2d: FieldElement51([
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]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
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]),
y_minus_x: FieldElement51([
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]),
xy2d: FieldElement51([
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},
AffineNielsPoint {
y_plus_x: FieldElement51([
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AffineNielsPoint {
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AffineNielsPoint {
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AffineNielsPoint {
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y_minus_x: FieldElement51([
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LookupTable([
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AffineNielsPoint {
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AffineNielsPoint {
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AffineNielsPoint {
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y_minus_x: FieldElement51([
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AffineNielsPoint {
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y_minus_x: FieldElement51([
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AffineNielsPoint {
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AffineNielsPoint {
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y_minus_x: FieldElement51([
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AffineNielsPoint {
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y_minus_x: FieldElement51([
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xy2d: FieldElement51([
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]),
LookupTable([
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y_minus_x: FieldElement51([
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AffineNielsPoint {
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AffineNielsPoint {
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AffineNielsPoint {
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AffineNielsPoint {
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AffineNielsPoint {
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AffineNielsPoint {
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AffineNielsPoint {
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y_minus_x: FieldElement51([
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xy2d: FieldElement51([
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LookupTable([
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AffineNielsPoint {
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AffineNielsPoint {
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AffineNielsPoint {
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AffineNielsPoint {
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AffineNielsPoint {
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AffineNielsPoint {
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y_minus_x: FieldElement51([
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]),
]);
/// Odd multiples of the basepoint `[B, 3B, 5B, 7B, 9B, 11B, 13B, 15B, ..., 127B]`.
#[allow(dead_code)]
pub(crate) const AFFINE_ODD_MULTIPLES_OF_BASEPOINT: NafLookupTable8<AffineNielsPoint> =
NafLookupTable8([
AffineNielsPoint {
y_plus_x: FieldElement51([
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y_minus_x: FieldElement51([
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xy2d: FieldElement51([
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},
AffineNielsPoint {
y_plus_x: FieldElement51([
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y_minus_x: FieldElement51([
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xy2d: FieldElement51([
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},
AffineNielsPoint {
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y_minus_x: FieldElement51([
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xy2d: FieldElement51([
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},
AffineNielsPoint {
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y_minus_x: FieldElement51([
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xy2d: FieldElement51([
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},
AffineNielsPoint {
y_plus_x: FieldElement51([
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y_minus_x: FieldElement51([
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xy2d: FieldElement51([
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AffineNielsPoint {
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y_minus_x: FieldElement51([
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xy2d: FieldElement51([
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},
AffineNielsPoint {
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]),
y_minus_x: FieldElement51([
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xy2d: FieldElement51([
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]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
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y_minus_x: FieldElement51([
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xy2d: FieldElement51([
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},
AffineNielsPoint {
y_plus_x: FieldElement51([
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y_minus_x: FieldElement51([
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]),
xy2d: FieldElement51([
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]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
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]),
y_minus_x: FieldElement51([
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]),
xy2d: FieldElement51([
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]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
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]),
y_minus_x: FieldElement51([
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xy2d: FieldElement51([
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},
AffineNielsPoint {
y_plus_x: FieldElement51([
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y_minus_x: FieldElement51([
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]),
xy2d: FieldElement51([
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]),
},
AffineNielsPoint {
y_plus_x: FieldElement51([
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