ark_ec/models/bn/
mod.rs

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
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
use crate::{
    models::{short_weierstrass::SWCurveConfig, CurveConfig},
    pairing::{MillerLoopOutput, Pairing, PairingOutput},
};
use ark_ff::{
    fields::{
        fp12_2over3over2::{Fp12, Fp12Config},
        fp2::Fp2Config,
        fp6_3over2::Fp6Config,
        Field, Fp2, PrimeField,
    },
    CyclotomicMultSubgroup,
};
use ark_std::{cfg_chunks_mut, marker::PhantomData, vec::*};
use educe::Educe;
use itertools::Itertools;
use num_traits::One;

#[cfg(feature = "parallel")]
use rayon::prelude::*;

pub enum TwistType {
    M,
    D,
}

pub trait BnConfig: 'static + Sized {
    /// The absolute value of the BN curve parameter `X`
    /// (as in `q = 36 X^4 + 36 X^3 + 24 X^2 + 6 X + 1`).
    const X: &'static [u64];

    /// Whether or not `X` is negative.
    const X_IS_NEGATIVE: bool;

    /// The absolute value of `6X + 2`.
    const ATE_LOOP_COUNT: &'static [i8];

    const TWIST_TYPE: TwistType;
    const TWIST_MUL_BY_Q_X: Fp2<Self::Fp2Config>;
    const TWIST_MUL_BY_Q_Y: Fp2<Self::Fp2Config>;
    type Fp: PrimeField + Into<<Self::Fp as PrimeField>::BigInt>;
    type Fp2Config: Fp2Config<Fp = Self::Fp>;
    type Fp6Config: Fp6Config<Fp2Config = Self::Fp2Config>;
    type Fp12Config: Fp12Config<Fp6Config = Self::Fp6Config>;
    type G1Config: SWCurveConfig<BaseField = Self::Fp>;
    type G2Config: SWCurveConfig<
        BaseField = Fp2<Self::Fp2Config>,
        ScalarField = <Self::G1Config as CurveConfig>::ScalarField,
    >;

    fn multi_miller_loop(
        a: impl IntoIterator<Item = impl Into<G1Prepared<Self>>>,
        b: impl IntoIterator<Item = impl Into<G2Prepared<Self>>>,
    ) -> MillerLoopOutput<Bn<Self>> {
        let mut pairs = a
            .into_iter()
            .zip_eq(b)
            .filter_map(|(p, q)| {
                let (p, q) = (p.into(), q.into());
                match !p.is_zero() && !q.is_zero() {
                    true => Some((p, q.ell_coeffs.into_iter())),
                    false => None,
                }
            })
            .collect::<Vec<_>>();

        let mut f = cfg_chunks_mut!(pairs, 4)
            .map(|pairs| {
                let mut f = <Bn<Self> as Pairing>::TargetField::one();
                for i in (1..Self::ATE_LOOP_COUNT.len()).rev() {
                    if i != Self::ATE_LOOP_COUNT.len() - 1 {
                        f.square_in_place();
                    }

                    for (p, coeffs) in pairs.iter_mut() {
                        Bn::<Self>::ell(&mut f, &coeffs.next().unwrap(), &p.0);
                    }

                    let bit = Self::ATE_LOOP_COUNT[i - 1];
                    if bit == 1 || bit == -1 {
                        for (p, coeffs) in pairs.iter_mut() {
                            Bn::<Self>::ell(&mut f, &coeffs.next().unwrap(), &p.0);
                        }
                    }
                }
                f
            })
            .product::<<Bn<Self> as Pairing>::TargetField>();

        if Self::X_IS_NEGATIVE {
            f.cyclotomic_inverse_in_place();
        }

        for (p, coeffs) in &mut pairs {
            Bn::<Self>::ell(&mut f, &coeffs.next().unwrap(), &p.0);
        }

        for (p, coeffs) in &mut pairs {
            Bn::<Self>::ell(&mut f, &coeffs.next().unwrap(), &p.0);
        }

        MillerLoopOutput(f)
    }

    #[allow(clippy::let_and_return)]
    fn final_exponentiation(f: MillerLoopOutput<Bn<Self>>) -> Option<PairingOutput<Bn<Self>>> {
        // Easy part: result = elt^((q^6-1)*(q^2+1)).
        // Follows, e.g., Beuchat et al page 9, by computing result as follows:
        //   elt^((q^6-1)*(q^2+1)) = (conj(elt) * elt^(-1))^(q^2+1)
        let f = f.0;

        // f1 = r.cyclotomic_inverse_in_place() = f^(p^6)
        let mut f1 = f;
        f1.cyclotomic_inverse_in_place();

        f.inverse().map(|mut f2| {
            // f2 = f^(-1);
            // r = f^(p^6 - 1)
            let mut r = f1 * &f2;

            // f2 = f^(p^6 - 1)
            f2 = r;
            // r = f^((p^6 - 1)(p^2))
            r.frobenius_map_in_place(2);

            // r = f^((p^6 - 1)(p^2) + (p^6 - 1))
            // r = f^((p^6 - 1)(p^2 + 1))
            r *= &f2;

            // Hard part follows Laura Fuentes-Castaneda et al. "Faster hashing to G2"
            // by computing:
            //
            // result = elt^(q^3 * (12*z^3 + 6z^2 + 4z - 1) +
            //               q^2 * (12*z^3 + 6z^2 + 6z) +
            //               q   * (12*z^3 + 6z^2 + 4z) +
            //               1   * (12*z^3 + 12z^2 + 6z + 1))
            // which equals
            //
            // result = elt^( 2z * ( 6z^2 + 3z + 1 ) * (q^4 - q^2 + 1)/r ).

            let y0 = Bn::<Self>::exp_by_neg_x(r);
            let y1 = y0.cyclotomic_square();
            let y2 = y1.cyclotomic_square();
            let mut y3 = y2 * &y1;
            let y4 = Bn::<Self>::exp_by_neg_x(y3);
            let y5 = y4.cyclotomic_square();
            let mut y6 = Bn::<Self>::exp_by_neg_x(y5);
            y3.cyclotomic_inverse_in_place();
            y6.cyclotomic_inverse_in_place();
            let y7 = y6 * &y4;
            let mut y8 = y7 * &y3;
            let y9 = y8 * &y1;
            let y10 = y8 * &y4;
            let y11 = y10 * &r;
            let mut y12 = y9;
            y12.frobenius_map_in_place(1);
            let y13 = y12 * &y11;
            y8.frobenius_map_in_place(2);
            let y14 = y8 * &y13;
            r.cyclotomic_inverse_in_place();
            let mut y15 = r * &y9;
            y15.frobenius_map_in_place(3);
            let y16 = y15 * &y14;

            PairingOutput(y16)
        })
    }
}

pub mod g1;
pub mod g2;

pub use self::{
    g1::{G1Affine, G1Prepared, G1Projective},
    g2::{G2Affine, G2Prepared, G2Projective},
};

#[derive(Educe)]
#[educe(Copy, Clone, PartialEq, Eq, Debug, Hash)]
pub struct Bn<P: BnConfig>(PhantomData<fn() -> P>);

impl<P: BnConfig> Bn<P> {
    /// Evaluates the line function at point p.
    fn ell(f: &mut Fp12<P::Fp12Config>, coeffs: &g2::EllCoeff<P>, p: &G1Affine<P>) {
        let mut c0 = coeffs.0;
        let mut c1 = coeffs.1;
        let mut c2 = coeffs.2;

        match P::TWIST_TYPE {
            TwistType::M => {
                c2.mul_assign_by_fp(&p.y);
                c1.mul_assign_by_fp(&p.x);
                f.mul_by_014(&c0, &c1, &c2);
            },
            TwistType::D => {
                c0.mul_assign_by_fp(&p.y);
                c1.mul_assign_by_fp(&p.x);
                f.mul_by_034(&c0, &c1, &c2);
            },
        }
    }

    fn exp_by_neg_x(mut f: Fp12<P::Fp12Config>) -> Fp12<P::Fp12Config> {
        f = f.cyclotomic_exp(P::X);
        if !P::X_IS_NEGATIVE {
            f.cyclotomic_inverse_in_place();
        }
        f
    }
}

impl<P: BnConfig> Pairing for Bn<P> {
    type BaseField = <P::G1Config as CurveConfig>::BaseField;
    type ScalarField = <P::G1Config as CurveConfig>::ScalarField;
    type G1 = G1Projective<P>;
    type G1Affine = G1Affine<P>;
    type G1Prepared = G1Prepared<P>;
    type G2 = G2Projective<P>;
    type G2Affine = G2Affine<P>;
    type G2Prepared = G2Prepared<P>;
    type TargetField = Fp12<P::Fp12Config>;

    fn multi_miller_loop(
        a: impl IntoIterator<Item = impl Into<Self::G1Prepared>>,
        b: impl IntoIterator<Item = impl Into<Self::G2Prepared>>,
    ) -> MillerLoopOutput<Self> {
        P::multi_miller_loop(a, b)
    }

    fn final_exponentiation(f: MillerLoopOutput<Self>) -> Option<PairingOutput<Self>> {
        P::final_exponentiation(f)
    }
}