1use ark_ec::{
  2    bls12,
  3    bls12::Bls12Parameters,
  4    models::CurveConfig,
  5    short_weierstrass::{Affine, SWCurveConfig},
  6    AffineCurve,
  7};
  8use ark_ff::{Field, MontFp, Zero};
  9
 10use crate::*;
 11
 12pub type G2Affine = bls12::G2Affine<crate::Parameters>;
 13pub type G2Projective = bls12::G2Projective<crate::Parameters>;
 14
 15#[derive(Clone, Default, PartialEq, Eq)]
 16pub struct Parameters;
 17
 18impl CurveConfig for Parameters {
 19    type BaseField = Fq2;
 20    type ScalarField = Fr;
 21
 22    /// COFACTOR = (x^8 - 4 x^7 + 5 x^6) - (4 x^4 + 6 x^3 - 4 x^2 - 4 x + 13) //
 23    /// 9
 24    /// = 305502333931268344200999753193121504214466019254188142667664032982267604182971884026507427359259977847832272839041616661285803823378372096355777062779109
 25    #[rustfmt::skip]
 26    const COFACTOR: &'static [u64] = &[
 27        0xcf1c38e31c7238e5,
 28        0x1616ec6e786f0c70,
 29        0x21537e293a6691ae,
 30        0xa628f1cb4d9e82ef,
 31        0xa68a205b2e5a7ddf,
 32        0xcd91de4547085aba,
 33        0x91d50792876a202,
 34        0x5d543a95414e7f1,
 35    ];
 36
 37    /// COFACTOR_INV = COFACTOR^{-1} mod r
 38    /// 26652489039290660355457965112010883481355318854675681319708643586776743290055
 39    const COFACTOR_INV: Fr =
 40        MontFp!("26652489039290660355457965112010883481355318854675681319708643586776743290055");
 41}
 42
 43impl SWCurveConfig for Parameters {
 44    /// COEFF_A = [0, 0]
 45    const COEFF_A: Fq2 = Fq2::new(g1::Parameters::COEFF_A, g1::Parameters::COEFF_A);
 46
 47    /// COEFF_B = [4, 4]
 48    const COEFF_B: Fq2 = Fq2::new(g1::Parameters::COEFF_B, g1::Parameters::COEFF_B);
 49
 50    /// AFFINE_GENERATOR_COEFFS = (G2_GENERATOR_X, G2_GENERATOR_Y)
 51    const GENERATOR: G2Affine = G2Affine::new_unchecked(G2_GENERATOR_X, G2_GENERATOR_Y);
 52
 53    #[inline(always)]
 54    fn mul_by_a(_: &Self::BaseField) -> Self::BaseField {
 55        Self::BaseField::zero()
 56    }
 57
 58    fn is_in_correct_subgroup_assuming_on_curve(point: &G2Affine) -> bool {
 59        // Algorithm from Section 4 of https://eprint.iacr.org/2021/1130.
 60        //
 61        // Checks that [p]P = [X]P
 62
 63        let mut x_times_point = point.mul_bigint(crate::Parameters::X);
 64        if crate::Parameters::X_IS_NEGATIVE {
 65            x_times_point = -x_times_point;
 66        }
 67
 68        let p_times_point = p_power_endomorphism(point);
 69
 70        x_times_point.eq(&p_times_point)
 71    }
 72}
 73
 74pub const G2_GENERATOR_X: Fq2 = Fq2::new(G2_GENERATOR_X_C0, G2_GENERATOR_X_C1);
 75pub const G2_GENERATOR_Y: Fq2 = Fq2::new(G2_GENERATOR_Y_C0, G2_GENERATOR_Y_C1);
 76
 77/// G2_GENERATOR_X_C0 =
 78/// 352701069587466618187139116011060144890029952792775240219908644239793785735715026873347600343865175952761926303160
 79pub const G2_GENERATOR_X_C0: Fq = MontFp!("352701069587466618187139116011060144890029952792775240219908644239793785735715026873347600343865175952761926303160");
 80
 81/// G2_GENERATOR_X_C1 =
 82/// 3059144344244213709971259814753781636986470325476647558659373206291635324768958432433509563104347017837885763365758
 83pub const G2_GENERATOR_X_C1: Fq = MontFp!("3059144344244213709971259814753781636986470325476647558659373206291635324768958432433509563104347017837885763365758");
 84
 85/// G2_GENERATOR_Y_C0 =
 86/// 1985150602287291935568054521177171638300868978215655730859378665066344726373823718423869104263333984641494340347905
 87pub const G2_GENERATOR_Y_C0: Fq = MontFp!("1985150602287291935568054521177171638300868978215655730859378665066344726373823718423869104263333984641494340347905");
 88
 89/// G2_GENERATOR_Y_C1 =
 90/// 927553665492332455747201965776037880757740193453592970025027978793976877002675564980949289727957565575433344219582
 91pub const G2_GENERATOR_Y_C1: Fq = MontFp!("927553665492332455747201965776037880757740193453592970025027978793976877002675564980949289727957565575433344219582");
 92
 93// psi(x,y) = (x**p * PSI_X, y**p * PSI_Y) is the Frobenius composed
 94// with the quadratic twist and its inverse
 95
 96// PSI_X = 1/(u+1)^((p-1)/3)
 97pub const P_POWER_ENDOMORPHISM_COEFF_0 : Fq2 = Fq2::new(
 98    Fq::ZERO,
 99    MontFp!(
100                "4002409555221667392624310435006688643935503118305586438271171395842971157480381377015405980053539358417135540939437"
101    )
102);
103
104// PSI_Y = 1/(u+1)^((p-1)/2)
105pub const P_POWER_ENDOMORPHISM_COEFF_1: Fq2 = Fq2::new(
106    MontFp!(
107                "2973677408986561043442465346520108879172042883009249989176415018091420807192182638567116318576472649347015917690530"),
108    MontFp!(
109                "1028732146235106349975324479215795277384839936929757896155643118032610843298655225875571310552543014690878354869257")
110);
111
112pub fn p_power_endomorphism(p: &Affine<Parameters>) -> Affine<Parameters> {
113    // The p-power endomorphism for G2 is defined as follows:
114    // 1. Note that G2 is defined on curve E': y^2 = x^3 + 4(u+1).
115    //    To map a point (x, y) in E' to (s, t) in E,
116    //    one set s = x / ((u+1) ^ (1/3)), t = y / ((u+1) ^ (1/2)),
117    //    because E: y^2 = x^3 + 4.
118    // 2. Apply the Frobenius endomorphism (s, t) => (s', t'),
119    //    another point on curve E, where s' = s^p, t' = t^p.
120    // 3. Map the point from E back to E'; that is,
121    //    one set x' = s' * ((u+1) ^ (1/3)), y' = t' * ((u+1) ^ (1/2)).
122    //
123    // To sum up, it maps
124    // (x,y) -> (x^p / ((u+1)^((p-1)/3)), y^p / ((u+1)^((p-1)/2)))
125    // as implemented in the code as follows.
126
127    let mut res = *p;
128    res.x.frobenius_map(1);
129    res.y.frobenius_map(1);
130
131    let tmp_x = res.x.clone();
132    res.x.c0 = -P_POWER_ENDOMORPHISM_COEFF_0.c1 * &tmp_x.c1;
133    res.x.c1 = P_POWER_ENDOMORPHISM_COEFF_0.c1 * &tmp_x.c0;
134    res.y *= P_POWER_ENDOMORPHISM_COEFF_1;
135
136    res
137}