ark-curves-cherry-picked/bls12_381/src/curves/g2.rs
bls12_381/src/curves/g2.rsRaw
1use ark_std::ops::Neg;
2
3use ark_ec::{
4 bls12,
5 bls12::Bls12Config,
6 hashing::curve_maps::wb::{IsogenyMap, WBConfig},
7 models::CurveConfig,
8 short_weierstrass::{Affine, Projective, SWCurveConfig},
9 AffineRepr, CurveGroup, PrimeGroup,
10};
11use ark_ff::{AdditiveGroup, Field, MontFp, Zero};
12use ark_serialize::{Compress, SerializationError};
13
14use super::{
15 g2_swu_iso,
16 util::{serialize_fq, EncodingFlags, G2_SERIALIZED_SIZE},
17};
18use crate::{
19 util::{read_g2_compressed, read_g2_uncompressed},
20 *,
21};
22
23pub type G2Affine = bls12::G2Affine<crate::Config>;
24pub type G2Projective = bls12::G2Projective<crate::Config>;
25
26#[derive(Clone, Default, PartialEq, Eq)]
27pub struct Config;
28
29impl CurveConfig for Config {
30 type BaseField = Fq2;
31 type ScalarField = Fr;
32
33 /// COFACTOR = (x^8 - 4 x^7 + 5 x^6) - (4 x^4 + 6 x^3 - 4 x^2 - 4 x + 13) //
34 /// 9
35 /// = 305502333931268344200999753193121504214466019254188142667664032982267604182971884026507427359259977847832272839041616661285803823378372096355777062779109
36 #[rustfmt::skip]
37 const COFACTOR: &'static [u64] = &[
38 0xcf1c38e31c7238e5,
39 0x1616ec6e786f0c70,
40 0x21537e293a6691ae,
41 0xa628f1cb4d9e82ef,
42 0xa68a205b2e5a7ddf,
43 0xcd91de4547085aba,
44 0x91d50792876a202,
45 0x5d543a95414e7f1,
46 ];
47
48 /// COFACTOR_INV = COFACTOR^{-1} mod r
49 /// 26652489039290660355457965112010883481355318854675681319708643586776743290055
50 const COFACTOR_INV: Fr =
51 MontFp!("26652489039290660355457965112010883481355318854675681319708643586776743290055");
52}
53
54impl SWCurveConfig for Config {
55 /// COEFF_A = [0, 0]
56 const COEFF_A: Fq2 = Fq2::new(g1::Config::COEFF_A, g1::Config::COEFF_A);
57
58 /// COEFF_B = [4, 4]
59 const COEFF_B: Fq2 = Fq2::new(g1::Config::COEFF_B, g1::Config::COEFF_B);
60
61 /// AFFINE_GENERATOR_COEFFS = (G2_GENERATOR_X, G2_GENERATOR_Y)
62 const GENERATOR: G2Affine = G2Affine::new_unchecked(G2_GENERATOR_X, G2_GENERATOR_Y);
63
64 #[inline(always)]
65 fn mul_by_a(_: Self::BaseField) -> Self::BaseField {
66 Self::BaseField::zero()
67 }
68
69 fn is_in_correct_subgroup_assuming_on_curve(point: &G2Affine) -> bool {
70 // Algorithm from Section 4 of https://eprint.iacr.org/2021/1130.
71 //
72 // Checks that [p]P = [X]P
73
74 let mut x_times_point = point.mul_bigint(crate::Config::X);
75 if crate::Config::X_IS_NEGATIVE {
76 x_times_point = -x_times_point;
77 }
78
79 let p_times_point = p_power_endomorphism(point);
80
81 x_times_point.eq(&p_times_point)
82 }
83
84 #[inline]
85 fn clear_cofactor(p: &G2Affine) -> G2Affine {
86 // Based on Section 4.1 of https://eprint.iacr.org/2017/419.pdf
87 // [h(ψ)]P = [x^2 − x − 1]P + [x − 1]ψ(P) + (ψ^2)(2P)
88
89 // x = -15132376222941642752
90 // When multiplying, use -c1 instead, and then negate the result. That's much
91 // more efficient, since the scalar -c1 has less limbs and a much lower Hamming
92 // weight.
93 let x: &'static [u64] = crate::Config::X;
94 let p_projective = p.into_group();
95
96 // [x]P
97 let x_p = Config::mul_affine(p, &x).neg();
98 // ψ(P)
99 let psi_p = p_power_endomorphism(&p);
100 // (ψ^2)(2P)
101 let mut psi2_p2 = double_p_power_endomorphism(&p_projective.double());
102
103 // tmp = [x]P + ψ(P)
104 let mut tmp = x_p.clone();
105 tmp += &psi_p;
106
107 // tmp2 = [x^2]P + [x]ψ(P)
108 let mut tmp2: Projective<Config> = tmp;
109 tmp2 = tmp2.mul_bigint(x).neg();
110
111 // add up all the terms
112 psi2_p2 += tmp2;
113 psi2_p2 -= x_p;
114 psi2_p2 += &-psi_p;
115 (psi2_p2 - p_projective).into_affine()
116 }
117
118 fn deserialize_with_mode<R: ark_serialize::Read>(
119 mut reader: R,
120 compress: ark_serialize::Compress,
121 validate: ark_serialize::Validate,
122 ) -> Result<Affine<Self>, ark_serialize::SerializationError> {
123 let p = if compress == ark_serialize::Compress::Yes {
124 read_g2_compressed(&mut reader)?
125 } else {
126 read_g2_uncompressed(&mut reader)?
127 };
128
129 if validate == ark_serialize::Validate::Yes && !p.is_in_correct_subgroup_assuming_on_curve()
130 {
131 return Err(SerializationError::InvalidData);
132 }
133 Ok(p)
134 }
135
136 fn serialize_with_mode<W: ark_serialize::Write>(
137 item: &Affine<Self>,
138 mut writer: W,
139 compress: ark_serialize::Compress,
140 ) -> Result<(), SerializationError> {
141 let encoding = EncodingFlags {
142 is_compressed: compress == ark_serialize::Compress::Yes,
143 is_infinity: item.is_zero(),
144 is_lexographically_largest: item.y > -item.y,
145 };
146 let mut p = *item;
147 if encoding.is_infinity {
148 p = G2Affine::zero();
149 }
150
151 let mut x_bytes = [0u8; G2_SERIALIZED_SIZE];
152 let c1_bytes = serialize_fq(p.x.c1);
153 let c0_bytes = serialize_fq(p.x.c0);
154 x_bytes[0..48].copy_from_slice(&c1_bytes[..]);
155 x_bytes[48..96].copy_from_slice(&c0_bytes[..]);
156 if encoding.is_compressed {
157 let mut bytes: [u8; G2_SERIALIZED_SIZE] = x_bytes;
158
159 encoding.encode_flags(&mut bytes);
160 writer.write_all(&bytes)?;
161 } else {
162 let mut bytes = [0u8; 2 * G2_SERIALIZED_SIZE];
163
164 let mut y_bytes = [0u8; G2_SERIALIZED_SIZE];
165 let c1_bytes = serialize_fq(p.y.c1);
166 let c0_bytes = serialize_fq(p.y.c0);
167 y_bytes[0..48].copy_from_slice(&c1_bytes[..]);
168 y_bytes[48..96].copy_from_slice(&c0_bytes[..]);
169 bytes[0..G2_SERIALIZED_SIZE].copy_from_slice(&x_bytes);
170 bytes[G2_SERIALIZED_SIZE..].copy_from_slice(&y_bytes);
171
172 encoding.encode_flags(&mut bytes);
173 writer.write_all(&bytes)?;
174 };
175
176 Ok(())
177 }
178
179 fn serialized_size(compress: ark_serialize::Compress) -> usize {
180 if compress == Compress::Yes {
181 G2_SERIALIZED_SIZE
182 } else {
183 2 * G2_SERIALIZED_SIZE
184 }
185 }
186}
187
188pub const G2_GENERATOR_X: Fq2 = Fq2::new(G2_GENERATOR_X_C0, G2_GENERATOR_X_C1);
189pub const G2_GENERATOR_Y: Fq2 = Fq2::new(G2_GENERATOR_Y_C0, G2_GENERATOR_Y_C1);
190
191/// G2_GENERATOR_X_C0 =
192/// 352701069587466618187139116011060144890029952792775240219908644239793785735715026873347600343865175952761926303160
193pub const G2_GENERATOR_X_C0: Fq = MontFp!("352701069587466618187139116011060144890029952792775240219908644239793785735715026873347600343865175952761926303160");
194
195/// G2_GENERATOR_X_C1 =
196/// 3059144344244213709971259814753781636986470325476647558659373206291635324768958432433509563104347017837885763365758
197pub const G2_GENERATOR_X_C1: Fq = MontFp!("3059144344244213709971259814753781636986470325476647558659373206291635324768958432433509563104347017837885763365758");
198
199/// G2_GENERATOR_Y_C0 =
200/// 1985150602287291935568054521177171638300868978215655730859378665066344726373823718423869104263333984641494340347905
201pub const G2_GENERATOR_Y_C0: Fq = MontFp!("1985150602287291935568054521177171638300868978215655730859378665066344726373823718423869104263333984641494340347905");
202
203/// G2_GENERATOR_Y_C1 =
204/// 927553665492332455747201965776037880757740193453592970025027978793976877002675564980949289727957565575433344219582
205pub const G2_GENERATOR_Y_C1: Fq = MontFp!("927553665492332455747201965776037880757740193453592970025027978793976877002675564980949289727957565575433344219582");
206
207// PSI_X = 1/(u+1)^((p-1)/3)
208const P_POWER_ENDOMORPHISM_COEFF_0 : Fq2 = Fq2::new(
209 Fq::ZERO,
210 MontFp!(
211 "4002409555221667392624310435006688643935503118305586438271171395842971157480381377015405980053539358417135540939437"
212 )
213);
214
215// PSI_Y = 1/(u+1)^((p-1)/2)
216const P_POWER_ENDOMORPHISM_COEFF_1: Fq2 = Fq2::new(
217 MontFp!(
218 "2973677408986561043442465346520108879172042883009249989176415018091420807192182638567116318576472649347015917690530"),
219 MontFp!(
220 "1028732146235106349975324479215795277384839936929757896155643118032610843298655225875571310552543014690878354869257")
221);
222
223// PSI_2_X = (u+1)^((1-p^2)/3)
224const DOUBLE_P_POWER_ENDOMORPHISM_COEFF_0: Fq2 = Fq2::new(
225 MontFp!("4002409555221667392624310435006688643935503118305586438271171395842971157480381377015405980053539358417135540939436"),
226 Fq::ZERO
227);
228
229/// psi(P) is the untwist-Frobenius-twist endomorhism on E'(Fq2)
230fn p_power_endomorphism(p: &Affine<Config>) -> Affine<Config> {
231 // The p-power endomorphism for G2 is defined as follows:
232 // 1. Note that G2 is defined on curve E': y^2 = x^3 + 4(u+1).
233 // To map a point (x, y) in E' to (s, t) in E,
234 // set s = x / ((u+1) ^ (1/3)), t = y / ((u+1) ^ (1/2)),
235 // because E: y^2 = x^3 + 4.
236 // 2. Apply the Frobenius endomorphism (s, t) => (s', t'),
237 // another point on curve E, where s' = s^p, t' = t^p.
238 // 3. Map the point from E back to E'; that is,
239 // set x' = s' * ((u+1) ^ (1/3)), y' = t' * ((u+1) ^ (1/2)).
240 //
241 // To sum up, it maps
242 // (x,y) -> (x^p / ((u+1)^((p-1)/3)), y^p / ((u+1)^((p-1)/2)))
243 // as implemented in the code as follows.
244
245 let mut res = *p;
246 res.x.frobenius_map_in_place(1);
247 res.y.frobenius_map_in_place(1);
248
249 let tmp_x = res.x.clone();
250 res.x.c0 = -P_POWER_ENDOMORPHISM_COEFF_0.c1 * &tmp_x.c1;
251 res.x.c1 = P_POWER_ENDOMORPHISM_COEFF_0.c1 * &tmp_x.c0;
252 res.y *= P_POWER_ENDOMORPHISM_COEFF_1;
253
254 res
255}
256
257/// For a p-power endomorphism psi(P), compute psi(psi(P))
258fn double_p_power_endomorphism(p: &Projective<Config>) -> Projective<Config> {
259 let mut res = *p;
260
261 res.x *= DOUBLE_P_POWER_ENDOMORPHISM_COEFF_0;
262 res.y = res.y.neg();
263
264 res
265}
266
267// Parametres from the [IETF draft v16, section E.3](https://www.ietf.org/archive/id/draft-irtf-cfrg-hash-to-curve-16.html#name-3-isogeny-map-for-bls12-381).
268impl WBConfig for Config {
269 type IsogenousCurve = g2_swu_iso::SwuIsoConfig;
270
271 const ISOGENY_MAP: IsogenyMap<'static, Self::IsogenousCurve, Self> =
272 g2_swu_iso::ISOGENY_MAP_TO_G2;
273}
274
275#[cfg(test)]
276mod test {
277
278 use super::*;
279 use ark_std::{rand::Rng, UniformRand};
280
281 fn sample_unchecked() -> Affine<g2::Config> {
282 let mut rng = ark_std::test_rng();
283 loop {
284 let x1 = Fq::rand(&mut rng);
285 let x2 = Fq::rand(&mut rng);
286 let greatest = rng.gen();
287 let x = Fq2::new(x1, x2);
288
289 if let Some(p) = Affine::get_point_from_x_unchecked(x, greatest) {
290 return p;
291 }
292 }
293 }
294
295 #[test]
296 fn test_psi_2() {
297 let p = sample_unchecked();
298 let psi_p = p_power_endomorphism(&p);
299 let psi2_p_composed = p_power_endomorphism(&psi_p);
300 let psi2_p_optimised = double_p_power_endomorphism(&p.into());
301
302 assert_eq!(psi2_p_composed, psi2_p_optimised);
303 }
304
305 #[test]
306 fn test_cofactor_clearing() {
307 // multiplying by h_eff and clearing the cofactor by the efficient
308 // endomorphism-based method should yield the same result.
309 let h_eff: &'static [u64] = &[
310 0xe8020005aaa95551,
311 0x59894c0adebbf6b4,
312 0xe954cbc06689f6a3,
313 0x2ec0ec69d7477c1a,
314 0x6d82bf015d1212b0,
315 0x329c2f178731db95,
316 0x9986ff031508ffe1,
317 0x88e2a8e9145ad768,
318 0x584c6a0ea91b3528,
319 0xbc69f08f2ee75b3,
320 ];
321
322 const SAMPLES: usize = 10;
323 for _ in 0..SAMPLES {
324 let p: Affine<g2::Config> = sample_unchecked();
325 let optimised = p.clear_cofactor();
326 let naive = g2::Config::mul_affine(&p, h_eff);
327 assert_eq!(optimised.into_group(), naive);
328 assert!(optimised.is_on_curve());
329 assert!(optimised.is_in_correct_subgroup_assuming_on_curve());
330 }
331 }
332}