ark_ec/models/bw6/
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
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
use crate::{
    models::{short_weierstrass::SWCurveConfig, CurveConfig},
    pairing::{MillerLoopOutput, Pairing, PairingOutput},
};
use ark_ff::{
    fields::{
        fp3::Fp3Config,
        fp6_2over3::{Fp6, Fp6Config},
        Field, PrimeField,
    },
    BitIteratorBE, CyclotomicMultSubgroup,
};
use ark_std::cfg_chunks_mut;
use educe::Educe;
use itertools::Itertools;
use num_traits::One;

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

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

pub enum TwistType {
    M,
    D,
}

pub trait BW6Config: 'static + Eq + Sized {
    const X: <Self::Fp as PrimeField>::BigInt;
    const X_IS_NEGATIVE: bool;
    // [X-1]/3 for X>0, and [(-X)+1]/3 otherwise
    const X_MINUS_1_DIV_3: <Self::Fp as PrimeField>::BigInt;
    const ATE_LOOP_COUNT_1: &'static [u64];
    const ATE_LOOP_COUNT_1_IS_NEGATIVE: bool;
    // X^2 - X - 1
    const ATE_LOOP_COUNT_2: &'static [i8];
    const ATE_LOOP_COUNT_2_IS_NEGATIVE: bool;
    const TWIST_TYPE: TwistType;
    const H_T: i64;
    const H_Y: i64;
    const T_MOD_R_IS_ZERO: bool;

    type Fp: PrimeField + Into<<Self::Fp as PrimeField>::BigInt>;
    type Fp3Config: Fp3Config<Fp = Self::Fp>;
    type Fp6Config: Fp6Config<Fp3Config = Self::Fp3Config>;
    type G1Config: SWCurveConfig<BaseField = Self::Fp>;
    type G2Config: SWCurveConfig<
        BaseField = Self::Fp,
        ScalarField = <Self::G1Config as CurveConfig>::ScalarField,
    >;

    // Computes the exponent of an element of the cyclotomic subgroup,
    // and inverses the result if necessary.
    fn cyclotomic_exp_signed(
        f: &Fp6<Self::Fp6Config>,
        x: impl AsRef<[u64]>,
        invert: bool,
    ) -> Fp6<Self::Fp6Config> {
        let mut f = f.cyclotomic_exp(x);
        if invert {
            f.cyclotomic_inverse_in_place();
        }
        f
    }

    // Computes the exponent of an element of the cyclotomic subgroup by the curve seed
    fn exp_by_x(f: &Fp6<Self::Fp6Config>) -> Fp6<Self::Fp6Config> {
        Self::cyclotomic_exp_signed(f, &Self::X, Self::X_IS_NEGATIVE)
    }

    fn final_exponentiation_hard_part(f: &Fp6<Self::Fp6Config>) -> Fp6<Self::Fp6Config> {
        BW6::<Self>::final_exponentiation_hard_part(f)
    }

    fn final_exponentiation(f: MillerLoopOutput<BW6<Self>>) -> Option<PairingOutput<BW6<Self>>> {
        let easy_part = BW6::<Self>::final_exponentiation_easy_part(f.0);
        Some(Self::final_exponentiation_hard_part(&easy_part)).map(PairingOutput)
    }

    fn multi_miller_loop(
        a: impl IntoIterator<Item = impl Into<G1Prepared<Self>>>,
        b: impl IntoIterator<Item = impl Into<G2Prepared<Self>>>,
    ) -> MillerLoopOutput<BW6<Self>> {
        // Implements unoptimized version of the Miller loop for the optimal ate pairing.
        // See formulas (4.15) and (4.17) from https://yelhousni.github.io/phd.pdf

        let (mut pairs_1, mut pairs_2) = a
            .into_iter()
            .zip_eq(b)
            .filter_map(|(p, q)| {
                let (p, q): (G1Prepared<Self>, G2Prepared<Self>) = (p.into(), q.into());
                match !p.is_zero() && !q.is_zero() {
                    true => Some((
                        (p, q.ell_coeffs_1.into_iter()),
                        (p, q.ell_coeffs_2.into_iter()),
                    )),
                    false => None,
                }
            })
            .unzip::<_, _, Vec<_>, Vec<_>>();

        // compute f_u which we can later re-use for the 2nd loop
        let mut f_u = cfg_chunks_mut!(pairs_1, 4)
            .map(|pairs| {
                let mut f = <BW6<Self> as Pairing>::TargetField::one();
                for i in BitIteratorBE::without_leading_zeros(Self::ATE_LOOP_COUNT_1).skip(1) {
                    f.square_in_place();
                    for (p, coeffs) in pairs.iter_mut() {
                        BW6::<Self>::ell(&mut f, &coeffs.next().unwrap(), &p.0);
                    }
                    if i {
                        for (p, coeffs) in pairs.iter_mut() {
                            BW6::<Self>::ell(&mut f, &coeffs.next().unwrap(), &p.0);
                        }
                    }
                }
                f
            })
            .product::<<BW6<Self> as Pairing>::TargetField>();

        let f_u_inv;

        if Self::ATE_LOOP_COUNT_1_IS_NEGATIVE {
            f_u_inv = f_u;
            f_u.cyclotomic_inverse_in_place();
        } else {
            f_u_inv = f_u.cyclotomic_inverse().unwrap();
        }

        // f_1(P) = f_(u+1)(P) = f_u(P) * l([u]q, q)(P)
        let mut f_1 = cfg_chunks_mut!(pairs_1, 4)
            .map(|pairs| {
                pairs.iter_mut().fold(f_u, |mut f, (p, coeffs)| {
                    BW6::<Self>::ell(&mut f, &coeffs.next().unwrap(), &p.0);
                    f
                })
            })
            .product::<<BW6<Self> as Pairing>::TargetField>();

        let mut f_2 = cfg_chunks_mut!(pairs_2, 4)
            .map(|pairs| {
                let mut f = f_u;
                for i in (1..Self::ATE_LOOP_COUNT_2.len()).rev() {
                    f.square_in_place();

                    for (p, ref mut coeffs) in pairs.iter_mut() {
                        BW6::<Self>::ell(&mut f, &coeffs.next().unwrap(), &p.0);
                    }

                    let bit = Self::ATE_LOOP_COUNT_2[i - 1];
                    if bit == 1 {
                        f *= &f_u;
                    } else if bit == -1 {
                        f *= &f_u_inv;
                    } else {
                        continue;
                    }
                    for &mut (p, ref mut coeffs) in pairs.iter_mut() {
                        BW6::<Self>::ell(&mut f, &coeffs.next().unwrap(), &p.0);
                    }
                }
                f
            })
            .product::<<BW6<Self> as Pairing>::TargetField>();

        if Self::ATE_LOOP_COUNT_2_IS_NEGATIVE {
            f_2.cyclotomic_inverse_in_place();
        }

        if Self::T_MOD_R_IS_ZERO {
            f_1.frobenius_map_in_place(1);
        } else {
            f_2.frobenius_map_in_place(1);
        }

        MillerLoopOutput(f_1 * &f_2)
    }
}

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 BW6<P: BW6Config>(PhantomData<fn() -> P>);

impl<P: BW6Config> BW6<P> {
    // Evaluate the line function at point p.
    fn ell(f: &mut Fp6<P::Fp6Config>, coeffs: &(P::Fp, P::Fp, P::Fp), 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 *= &p.y;
                c1 *= &p.x;
                f.mul_by_014(&c0, &c1, &c2);
            },
            TwistType::D => {
                c0 *= &p.y;
                c1 *= &p.x;
                f.mul_by_034(&c0, &c1, &c2);
            },
        }
    }

    fn exp_by_x_plus_1(f: &Fp6<P::Fp6Config>) -> Fp6<P::Fp6Config> {
        P::exp_by_x(f) * f
    }

    fn exp_by_x_minus_1(f: &Fp6<P::Fp6Config>) -> Fp6<P::Fp6Config> {
        P::exp_by_x(f) * &f.cyclotomic_inverse().unwrap()
    }

    fn exp_by_x_minus_1_div_3(f: &Fp6<P::Fp6Config>) -> Fp6<P::Fp6Config> {
        P::cyclotomic_exp_signed(f, &P::X_MINUS_1_DIV_3, P::X_IS_NEGATIVE)
    }

    // f^[(p^3-1)(p+1)]
    fn final_exponentiation_easy_part(f: Fp6<P::Fp6Config>) -> Fp6<P::Fp6Config> {
        // f^(-1)
        let f_inv = f.inverse().unwrap();
        // f^(p^3)
        let f_p3 = {
            let mut f = f;
            f.conjugate_in_place();
            f
        };
        // g := f^(p^3-1) = f^(p^3) * f^(-1)
        let g = f_p3 * f_inv;
        // g^p
        let g_p = {
            let mut g = g;
            g.frobenius_map_in_place(1);
            g
        };
        // g^(p+1) = g^p * g
        g_p * &g
    }

    fn final_exponentiation_hard_part(f: &Fp6<P::Fp6Config>) -> Fp6<P::Fp6Config> {
        // Generic implementation of the hard part of the final exponentiation for the BW6 family.
        // Computes (u+1)*Phi_k(p(u))/r(u)
        if P::T_MOD_R_IS_ZERO {
            // Algorithm 4.3 from https://yelhousni.github.io/phd.pdf
            // Follows the implementation https://gitlab.inria.fr/zk-curves/snark-2-chains/-/blob/master/sage/pairing_bw6_bls12.py#L1036

            // A = m**(u-1)
            let a = Self::exp_by_x_minus_1(f);
            // A = A**(u-1)
            let a = Self::exp_by_x_minus_1(&a);
            // A = (m * A).conjugate() * m.frobenius()
            let a = (f * &a).cyclotomic_inverse().unwrap() * f.frobenius_map(1);
            // B = A**(u+1) * m
            let b = Self::exp_by_x_plus_1(&a) * f;
            // A = A**2 * A
            let a = a.square() * &a;
            // A = A.conjugate()
            let a = a.cyclotomic_inverse().unwrap();
            // C = B**((u-1)//3)
            let c = Self::exp_by_x_minus_1_div_3(&b);
            // D = C**(u-1)
            let d = Self::exp_by_x_minus_1(&c);
            // E = (D**(u-1))**(u-1) * D
            let e = Self::exp_by_x_minus_1(&Self::exp_by_x_minus_1(&d)) * &d;
            // F = (E**(u+1) * C).conjugate() * D
            let f = (Self::exp_by_x_plus_1(&e) * &c)
                .cyclotomic_inverse()
                .unwrap()
                * &d;
            // G = ((F * D)**(u+1)).conjugate() * C * B
            let g = Self::exp_by_x_plus_1(&(f * &d))
                .cyclotomic_inverse()
                .unwrap()
                * &c
                * &b;
            // d2 = (ht**2+3*hy**2)//4
            let d2 = ((P::H_T * P::H_T + 3 * P::H_Y * P::H_Y) / 4) as u64;
            // d1 = (ht-hy)//2
            let d1 = (P::H_T - P::H_Y) / 2;
            // H = F**d1 * E
            let h = P::cyclotomic_exp_signed(&f, &[d1 as u64], d1 < 0) * &e;
            // H = H**2 * H * B * G**d2
            let h = h.square() * &h * &b * g.cyclotomic_exp(&[d2]);
            // return A * H
            a * &h
        } else {
            // Algorithm 4.4 from https://yelhousni.github.io/phd.pdf
            // Follows the implementation https://gitlab.inria.fr/zk-curves/snark-2-chains/-/blob/master/sage/pairing_bw6_bls12.py#L969

            // A = m**(u-1)
            let a = Self::exp_by_x_minus_1(f);
            // A = A**(u-1)
            let a = Self::exp_by_x_minus_1(&a);
            // A = A * m.frobenius()
            let a = a * f.frobenius_map(1);
            // B = A**(u+1) * m.conjugate()
            let b = Self::exp_by_x_plus_1(&a) * f.cyclotomic_inverse().unwrap();
            // A = A**2 * A
            let a = a.square() * &a;
            // C = B**((u-1)//3)
            let c = Self::exp_by_x_minus_1_div_3(&b);
            // D = C**(u-1)
            let d = Self::exp_by_x_minus_1(&c);
            // E = (D**(u-1))**(u-1) * D
            let e = Self::exp_by_x_minus_1(&Self::exp_by_x_minus_1(&d)) * &d;
            // D = D.conjugate()
            let d = d.cyclotomic_inverse().unwrap();
            // Fc = D * B
            let fc = d * &b;
            // G = E**(u+1) * Fc
            let g = Self::exp_by_x_plus_1(&e) * &fc;
            // H = G * C
            let h = g * &c;
            // I = (G * D)**(u+1) * Fc.conjugate()
            let i = Self::exp_by_x_plus_1(&(g * &d)) * fc.cyclotomic_inverse().unwrap();
            // d2 = (ht**2+3*hy**2)//4
            let d2 = ((P::H_T * P::H_T + 3 * P::H_Y * P::H_Y) / 4) as u64;
            // d1 = (ht+hy)//2
            let d1 = (P::H_T + P::H_Y) / 2;
            // J = H**d1 * E
            let j = P::cyclotomic_exp_signed(&h, &[d1 as u64], d1 < 0) * &e;
            // K = J**2 * J * B * I**d2
            let k = j.square() * &j * &b * i.cyclotomic_exp(&[d2]);
            // return A * K
            a * &k
        }
    }
}

impl<P: BW6Config> Pairing for BW6<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 = Fp6<P::Fp6Config>;

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

    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)
    }
}