1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131

// -*- mode: rust; -*-
//
// This file is part of curve25519-dalek.
// Copyright (c) 2016-2017 Isis Lovecruft, 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 backend::u64::field::FieldElement64;
use backend::u64::scalar::Scalar64;
use edwards::EdwardsPoint;

/// Edwards `d` value, equal to `-121665/121666 mod p`.
pub(crate) const EDWARDS_D: FieldElement64 = FieldElement64([929955233495203, 466365720129213, 1662059464998953, 2033849074728123, 1442794654840575]);

/// Edwards `2*d` value, equal to `2*(-121665/121666) mod p`.
pub(crate) const EDWARDS_D2: FieldElement64 = FieldElement64([1859910466990425, 932731440258426, 1072319116312658, 1815898335770999, 633789495995903]);

/// `= sqrt(a*d - 1)`, where `a = -1 (mod p)`, `d` are the Edwards curve parameters.
pub(crate) const SQRT_AD_MINUS_ONE: FieldElement64 = FieldElement64([
    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: FieldElement64 = FieldElement64([
    278908739862762, 821645201101625, 8113234426968, 1777959178193151, 2118520810568447
]);

/// Precomputed value of one of the square roots of -1 (mod p)
pub(crate) const SQRT_M1: FieldElement64 = FieldElement64([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: FieldElement64 = FieldElement64([121666, 0, 0, 0, 0]);

/// `L` is the order of base point, i.e. 2^252 + 27742317777372353535851937790883648493
pub(crate) const L: Scalar64 = Scalar64([ 0x0002631a5cf5d3ed, 0x000dea2f79cd6581, 0x000000000014def9, 0x0000000000000000, 0x0000100000000000 ]);

/// `L` * `LFACTOR` = -1 (mod 2^51)
pub(crate) const LFACTOR: u64 = 0x51da312547e1b;

/// `R` = R % L where R = 2^260
pub(crate) const R: Scalar64 = Scalar64([ 0x000f48bd6721e6ed, 0x0003bab5ac67e45a, 0x000fffffeb35e51b, 0x000fffffffffffff, 0x00000fffffffffff ]);

/// `RR` = (R^2) % L where R = 2^260
pub(crate) const RR: Scalar64 = Scalar64([ 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: FieldElement64([1738742601995546, 1146398526822698, 2070867633025821, 562264141797630, 587772402128613]),
    Y: FieldElement64([1801439850948184, 1351079888211148, 450359962737049, 900719925474099, 1801439850948198]),
    Z: FieldElement64([1, 0, 0, 0, 0]),
    T: FieldElement64([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[8]\\) 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: FieldElement64([0, 0, 0, 0, 0]),
        Y: FieldElement64([1, 0, 0, 0, 0]),
        Z: FieldElement64([1, 0, 0, 0, 0]),
        T: FieldElement64([0, 0, 0, 0, 0]),
    }
    ,
    EdwardsPoint {
        X: FieldElement64([358744748052810, 1691584618240980, 977650209285361, 1429865912637724, 560044844278676]),
        Y: FieldElement64([84926274344903, 473620666599931, 365590438845504, 1028470286882429, 2146499180330972]),
        Z: FieldElement64([1, 0, 0, 0, 0]),
        T: FieldElement64([1448326834587521, 1857896831960481, 1093722731865333, 1677408490711241, 1915505153018406]),
    }
    ,
    EdwardsPoint {
        X: FieldElement64([533094393274173, 2016890930128738, 18285341111199, 134597186663265, 1486323764102114]),
        Y: FieldElement64([0, 0, 0, 0, 0]),
        Z: FieldElement64([1, 0, 0, 0, 0]),
        T: FieldElement64([0, 0, 0, 0, 0]),
    }
    ,
    EdwardsPoint {
        X: FieldElement64([358744748052810, 1691584618240980, 977650209285361, 1429865912637724, 560044844278676]),
        Y: FieldElement64([2166873539340326, 1778179147085316, 1886209374839743, 1223329526802818, 105300633354275]),
        Z: FieldElement64([1, 0, 0, 0, 0]),
        T: FieldElement64([803472979097708, 393902981724766, 1158077081819914, 574391322974006, 336294660666841]),
    }
    ,
    EdwardsPoint {
        X: FieldElement64([0, 0, 0, 0, 0]),
        Y: FieldElement64([2251799813685228, 2251799813685247, 2251799813685247, 2251799813685247, 2251799813685247]),
        Z: FieldElement64([1, 0, 0, 0, 0]),
        T: FieldElement64([0, 0, 0, 0, 0]),
    }
    ,
    EdwardsPoint {
        X: FieldElement64([1893055065632419, 560215195444267, 1274149604399886, 821933901047523, 1691754969406571]),
        Y: FieldElement64([2166873539340326, 1778179147085316, 1886209374839743, 1223329526802818, 105300633354275]),
        Z: FieldElement64([1, 0, 0, 0, 0]),
        T: FieldElement64([1448326834587521, 1857896831960481, 1093722731865333, 1677408490711241, 1915505153018406]),
    }
    ,
    EdwardsPoint {
        X: FieldElement64([1718705420411056, 234908883556509, 2233514472574048, 2117202627021982, 765476049583133]),
        Y: FieldElement64([0, 0, 0, 0, 0]),
        Z: FieldElement64([1, 0, 0, 0, 0]),
        T: FieldElement64([0, 0, 0, 0, 0]),
    }
    ,
    EdwardsPoint {
        X: FieldElement64([1893055065632419, 560215195444267, 1274149604399886, 821933901047523, 1691754969406571]),
        Y: FieldElement64([84926274344903, 473620666599931, 365590438845504, 1028470286882429, 2146499180330972]),
        Z: FieldElement64([1, 0, 0, 0, 0]),
        T: FieldElement64([803472979097708, 393902981724766, 1158077081819914, 574391322974006, 336294660666841]),
    }
];