ark_ec/models/mnt4/
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
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
use crate::{
    models::{short_weierstrass::SWCurveConfig, CurveConfig},
    pairing::{MillerLoopOutput, Pairing, PairingOutput},
};
use ark_ff::{
    fp2::{Fp2, Fp2Config},
    fp4::{Fp4, Fp4Config},
    AdditiveGroup, CyclotomicMultSubgroup, Field, PrimeField,
};
use educe::Educe;
use itertools::Itertools;
use num_traits::{One, Zero};

use ark_std::{marker::PhantomData, vec::*};

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

pub mod g1;
pub mod g2;

use self::g2::{AteAdditionCoefficients, AteDoubleCoefficients, G2ProjectiveExtended};
pub use self::{
    g1::{G1Affine, G1Prepared, G1Projective},
    g2::{G2Affine, G2Prepared, G2Projective},
};

pub type GT<P> = Fp4<P>;

pub trait MNT4Config: 'static + Sized {
    const TWIST: Fp2<Self::Fp2Config>;
    const TWIST_COEFF_A: Fp2<Self::Fp2Config>;
    const ATE_LOOP_COUNT: &'static [i8];
    const ATE_IS_LOOP_COUNT_NEG: bool;
    const FINAL_EXPONENT_LAST_CHUNK_1: <Self::Fp as PrimeField>::BigInt;
    const FINAL_EXPONENT_LAST_CHUNK_W0_IS_NEG: bool;
    const FINAL_EXPONENT_LAST_CHUNK_ABS_OF_W0: <Self::Fp as PrimeField>::BigInt;
    type Fp: PrimeField + Into<<Self::Fp as PrimeField>::BigInt>;
    type Fr: PrimeField + Into<<Self::Fr as PrimeField>::BigInt>;
    type Fp2Config: Fp2Config<Fp = Self::Fp>;
    type Fp4Config: Fp4Config<Fp2Config = Self::Fp2Config>;
    type G1Config: SWCurveConfig<BaseField = Self::Fp, ScalarField = Self::Fr>;
    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<MNT4<Self>> {
        let pairs = a
            .into_iter()
            .zip_eq(b)
            .map(|(a, b)| (a.into(), b.into()))
            .collect::<Vec<_>>();
        let result = ark_std::cfg_into_iter!(pairs)
            .map(|(a, b)| MNT4::<Self>::ate_miller_loop(&a, &b))
            .product();
        MillerLoopOutput(result)
    }

    fn final_exponentiation(f: MillerLoopOutput<MNT4<Self>>) -> Option<PairingOutput<MNT4<Self>>> {
        let value = f.0;
        let value_inv = value.inverse()?;
        let value_to_first_chunk =
            MNT4::<Self>::final_exponentiation_first_chunk(&value, &value_inv);
        let value_inv_to_first_chunk =
            MNT4::<Self>::final_exponentiation_first_chunk(&value_inv, &value);
        let result = MNT4::<Self>::final_exponentiation_last_chunk(
            &value_to_first_chunk,
            &value_inv_to_first_chunk,
        );
        Some(PairingOutput(result))
    }
}

#[derive(Educe)]
#[educe(Copy, Clone, PartialEq, Eq, Debug, Hash)]
pub struct MNT4<P: MNT4Config>(PhantomData<fn() -> P>);

impl<P: MNT4Config> MNT4<P> {
    fn doubling_for_flipped_miller_loop(
        r: &G2ProjectiveExtended<P>,
    ) -> (G2ProjectiveExtended<P>, AteDoubleCoefficients<P>) {
        let a = r.t.square();
        let b = r.x.square();
        let c = r.y.square();
        let d = c.square();
        let e = (r.x + &c).square() - &b - &d;
        let f = (b + &b + &b) + &(P::TWIST_COEFF_A * &a);
        let g = f.square();

        let d_eight = d.double().double().double();

        let x = -(e + &e + &e + &e) + &g;
        let y = -d_eight + &(f * &(e + &e - &x));
        let z = (r.y + &r.z).square() - &c - &r.z.square();
        let t = z.square();

        let r2 = G2ProjectiveExtended { x, y, z, t };
        let coeff = AteDoubleCoefficients {
            c_h: (r2.z + &r.t).square() - &r2.t - &a,
            c_4c: c + &c + &c + &c,
            c_j: (f + &r.t).square() - &g - &a,
            c_l: (f + &r.x).square() - &g - &b,
        };

        (r2, coeff)
    }

    fn mixed_addition_for_flipped_miller_loop(
        x: &Fp2<P::Fp2Config>,
        y: &Fp2<P::Fp2Config>,
        r: &G2ProjectiveExtended<P>,
    ) -> (G2ProjectiveExtended<P>, AteAdditionCoefficients<P>) {
        let a = y.square();
        let b = r.t * x;
        let d = ((r.z + y).square() - &a - &r.t) * &r.t;
        let h = b - &r.x;
        let i = h.square();
        let e = i + &i + &i + &i;
        let j = h * &e;
        let v = r.x * &e;
        let l1 = d - &(r.y + &r.y);

        let x = l1.square() - &j - &(v + &v);
        let y = l1 * &(v - &x) - &(j * &(r.y + &r.y));
        let z = (r.z + &h).square() - &r.t - &i;
        let t = z.square();

        let r2 = G2ProjectiveExtended { x, y, z, t };
        let coeff = AteAdditionCoefficients { c_l1: l1, c_rz: z };

        (r2, coeff)
    }

    pub fn ate_miller_loop(p: &G1Prepared<P>, q: &G2Prepared<P>) -> Fp4<P::Fp4Config> {
        let l1_coeff = Fp2::new(p.x, P::Fp::zero()) - &q.x_over_twist;

        let mut f = <Fp4<P::Fp4Config>>::one();

        let mut add_idx: usize = 0;

        // code below gets executed for all bits (EXCEPT the MSB itself) of
        // mnt6_param_p (skipping leading zeros) in MSB to LSB order
        let y_over_twist_neg = -q.y_over_twist;
        assert_eq!(P::ATE_LOOP_COUNT.len() - 1, q.double_coefficients.len());
        for (bit, dc) in P::ATE_LOOP_COUNT.iter().skip(1).zip(&q.double_coefficients) {
            let g_rr_at_p = Fp4::new(
                -dc.c_4c - &(dc.c_j * &p.x_twist) + &dc.c_l,
                dc.c_h * &p.y_twist,
            );

            f = f.square() * &g_rr_at_p;

            // Compute l_{R,Q}(P) if bit == 1, and l_{R,-Q}(P) if bit == -1
            let g_rq_at_p = if *bit == 1 {
                let ac = &q.addition_coefficients[add_idx];
                add_idx += 1;

                Fp4::new(
                    ac.c_rz * &p.y_twist,
                    -(q.y_over_twist * &ac.c_rz + &(l1_coeff * &ac.c_l1)),
                )
            } else if *bit == -1 {
                let ac = &q.addition_coefficients[add_idx];
                add_idx += 1;

                Fp4::new(
                    ac.c_rz * &p.y_twist,
                    -(y_over_twist_neg * &ac.c_rz + &(l1_coeff * &ac.c_l1)),
                )
            } else if *bit == 0 {
                continue;
            } else {
                unreachable!();
            };
            f *= &g_rq_at_p;
        }

        if P::ATE_IS_LOOP_COUNT_NEG {
            let ac = &q.addition_coefficients[add_idx];

            let g_rnegr_at_p = Fp4::new(
                ac.c_rz * &p.y_twist,
                -(q.y_over_twist * &ac.c_rz + &(l1_coeff * &ac.c_l1)),
            );
            f = (f * &g_rnegr_at_p).inverse().unwrap();
        }

        f
    }

    fn final_exponentiation_first_chunk(
        elt: &Fp4<P::Fp4Config>,
        elt_inv: &Fp4<P::Fp4Config>,
    ) -> Fp4<P::Fp4Config> {
        // (q^2-1)

        // elt_q2 = elt^(q^2)
        let mut elt_q2 = *elt;
        elt_q2.cyclotomic_inverse_in_place();
        // elt_q2_over_elt = elt^(q^2-1)
        elt_q2 * elt_inv
    }

    fn final_exponentiation_last_chunk(
        elt: &Fp4<P::Fp4Config>,
        elt_inv: &Fp4<P::Fp4Config>,
    ) -> Fp4<P::Fp4Config> {
        let elt_clone = *elt;
        let elt_inv_clone = *elt_inv;

        let mut elt_q = *elt;
        elt_q.frobenius_map_in_place(1);

        let w1_part = elt_q.cyclotomic_exp(P::FINAL_EXPONENT_LAST_CHUNK_1);
        let w0_part = if P::FINAL_EXPONENT_LAST_CHUNK_W0_IS_NEG {
            elt_inv_clone.cyclotomic_exp(P::FINAL_EXPONENT_LAST_CHUNK_ABS_OF_W0)
        } else {
            elt_clone.cyclotomic_exp(P::FINAL_EXPONENT_LAST_CHUNK_ABS_OF_W0)
        };

        w1_part * &w0_part
    }
}

impl<P: MNT4Config> Pairing for MNT4<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 = Fp4<P::Fp4Config>;

    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)
    }
}