snarkvm_fields/
fp2.rs

1// Copyright (c) 2019-2025 Provable Inc.
2// This file is part of the snarkVM library.
3
4// Licensed under the Apache License, Version 2.0 (the "License");
5// you may not use this file except in compliance with the License.
6// You may obtain a copy of the License at:
7
8// http://www.apache.org/licenses/LICENSE-2.0
9
10// Unless required by applicable law or agreed to in writing, software
11// distributed under the License is distributed on an "AS IS" BASIS,
12// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
13// See the License for the specific language governing permissions and
14// limitations under the License.
15
16use crate::{Field, LegendreSymbol, One, PrimeField, SquareRootField, Zero};
17use snarkvm_utilities::{
18    FromBytes,
19    ToBits,
20    ToBytes,
21    rand::Uniform,
22    serialize::{SerializationError, *},
23};
24
25use rand::{
26    Rng,
27    distributions::{Distribution, Standard},
28};
29use serde::{Deserialize, Serialize};
30use std::{
31    cmp::{Ord, Ordering, PartialOrd},
32    fmt::Debug,
33    hash::Hash,
34    io::{Read, Result as IoResult, Write},
35    ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign},
36};
37
38pub trait Fp2Parameters:
39    'static + Copy + Clone + Default + Debug + PartialEq + Eq + Hash + Serialize + for<'a> Deserialize<'a> + Send + Sync
40{
41    type Fp: PrimeField;
42
43    /// Coefficients for the Frobenius automorphism.
44    const FROBENIUS_COEFF_FP2_C1: [Self::Fp; 2];
45
46    const NONRESIDUE: Self::Fp;
47
48    const QUADRATIC_NONRESIDUE: (Self::Fp, Self::Fp);
49
50    #[inline(always)]
51    fn mul_fp_by_nonresidue(fe: &Self::Fp) -> Self::Fp {
52        Self::NONRESIDUE * fe
53    }
54}
55
56#[derive(Copy, Clone, Debug, Default, PartialEq, Eq, Hash, Serialize, Deserialize)]
57pub struct Fp2<P: Fp2Parameters> {
58    pub c0: P::Fp,
59    pub c1: P::Fp,
60}
61
62impl<P: Fp2Parameters> Fp2<P> {
63    /// Initializes a `Fp2` element from two `Fp` elements.
64    pub const fn new(c0: P::Fp, c1: P::Fp) -> Self {
65        Fp2 { c0, c1 }
66    }
67}
68
69impl<P: Fp2Parameters> Fp2<P> {
70    /// Norm of Fp2 over Fp: Norm(a) = a.x^2 - beta * a.y^2
71    pub fn norm(&self) -> P::Fp {
72        let t0 = self.c0.square();
73        let mut t1 = self.c1.square();
74        t1 = -P::mul_fp_by_nonresidue(&t1);
75        t1.add_assign(t0);
76        t1
77    }
78
79    pub fn mul_by_fp(&mut self, element: &P::Fp) {
80        self.c0.mul_assign(element);
81        self.c1.mul_assign(element);
82    }
83}
84
85impl<P: Fp2Parameters> Zero for Fp2<P> {
86    fn zero() -> Self {
87        Fp2::new(P::Fp::zero(), P::Fp::zero())
88    }
89
90    fn is_zero(&self) -> bool {
91        self.c0.is_zero() && self.c1.is_zero()
92    }
93}
94impl<P: Fp2Parameters> One for Fp2<P> {
95    fn one() -> Self {
96        Fp2::new(P::Fp::one(), P::Fp::zero())
97    }
98
99    fn is_one(&self) -> bool {
100        self.c0.is_one() && self.c1.is_zero()
101    }
102}
103
104impl<P: Fp2Parameters> Field for Fp2<P> {
105    type BasePrimeField = P::Fp;
106
107    fn from_base_prime_field(other: Self::BasePrimeField) -> Self {
108        Self::new(other, P::Fp::zero())
109    }
110
111    #[inline]
112    fn characteristic<'a>() -> &'a [u64] {
113        P::Fp::characteristic()
114    }
115
116    fn double(&self) -> Self {
117        let mut result = *self;
118        result.double_in_place();
119        result
120    }
121
122    fn double_in_place(&mut self) {
123        self.c0.double_in_place();
124        self.c1.double_in_place();
125    }
126
127    fn square(&self) -> Self {
128        let mut result = *self;
129        result.square_in_place();
130        result
131    }
132
133    #[inline]
134    fn from_random_bytes_with_flags<F: Flags>(bytes: &[u8]) -> Option<(Self, F)> {
135        let split_at = bytes.len() / 2;
136        if let Some(c0) = P::Fp::from_random_bytes(&bytes[..split_at]) {
137            if let Some((c1, flags)) = P::Fp::from_random_bytes_with_flags::<F>(&bytes[split_at..]) {
138                return Some((Fp2::new(c0, c1), flags));
139            }
140        }
141        None
142    }
143
144    #[inline]
145    fn from_random_bytes(bytes: &[u8]) -> Option<Self> {
146        Self::from_random_bytes_with_flags::<EmptyFlags>(bytes).map(|f| f.0)
147    }
148
149    fn square_in_place(&mut self) -> &mut Self {
150        // v0 = c0 - c1
151        let mut v0 = self.c0 - self.c1;
152        // v3 = c0 - beta * c1
153        let v3 = self.c0 - P::mul_fp_by_nonresidue(&self.c1);
154        // v2 = c0 * c1
155        let v2 = self.c0 * self.c1;
156
157        // v0 = (v0 * v3) + v2
158        v0 *= &v3;
159        v0 += &v2;
160
161        self.c1 = v2.double();
162        self.c0 = v0 + P::mul_fp_by_nonresidue(&v2);
163
164        self
165    }
166
167    fn inverse(&self) -> Option<Self> {
168        if self.is_zero() {
169            None
170        } else {
171            // Guide to Pairing-based Cryptography, Algorithm 5.19.
172            // v0 = c0.square()
173            let mut v0 = self.c0.square();
174            // v1 = c1.square()
175            let v1 = self.c1.square();
176            // v0 = v0 - beta * v1
177            v0 -= &P::mul_fp_by_nonresidue(&v1);
178            v0.inverse().map(|v1| {
179                let c0 = self.c0 * v1;
180                let c1 = -(self.c1 * v1);
181                Self::new(c0, c1)
182            })
183        }
184    }
185
186    fn inverse_in_place(&mut self) -> Option<&mut Self> {
187        if let Some(inverse) = self.inverse() {
188            *self = inverse;
189            Some(self)
190        } else {
191            None
192        }
193    }
194
195    fn frobenius_map(&mut self, power: usize) {
196        self.c1.mul_assign(&P::FROBENIUS_COEFF_FP2_C1[power % 2]);
197    }
198}
199
200impl<P: Fp2Parameters> SquareRootField for Fp2<P>
201where
202    P::Fp: SquareRootField,
203{
204    fn legendre(&self) -> LegendreSymbol {
205        self.norm().legendre()
206    }
207
208    fn sqrt(&self) -> Option<Self> {
209        use crate::LegendreSymbol::*;
210        if self.c1.is_zero() {
211            return self.c0.sqrt().map(|c0| Self::new(c0, P::Fp::zero()));
212        }
213        match self.legendre() {
214            // Square root based on the complex method. See
215            // https://eprint.iacr.org/2012/685.pdf (page 15, algorithm 8)
216            Zero => Some(*self),
217            QuadraticNonResidue => None,
218            QuadraticResidue => {
219                let two_inv = P::Fp::half();
220                let alpha = self.norm().sqrt().expect("We are in the QR case, the norm should have a square root");
221                let mut delta = (alpha + self.c0) * two_inv;
222                if delta.legendre().is_qnr() {
223                    delta -= &alpha;
224                }
225                let c0 = delta.sqrt().expect("Delta must have a square root");
226                let c0_inv = c0.inverse().expect("c0 must have an inverse");
227                Some(Self::new(c0, self.c1 * two_inv * c0_inv))
228            }
229        }
230    }
231
232    fn sqrt_in_place(&mut self) -> Option<&mut Self> {
233        (*self).sqrt().map(|sqrt| {
234            *self = sqrt;
235            self
236        })
237    }
238}
239
240/// `Fp2` elements are ordered lexicographically.
241impl<P: Fp2Parameters> Ord for Fp2<P> {
242    #[inline(always)]
243    fn cmp(&self, other: &Self) -> Ordering {
244        match self.c1.cmp(&other.c1) {
245            Ordering::Greater => Ordering::Greater,
246            Ordering::Less => Ordering::Less,
247            Ordering::Equal => self.c0.cmp(&other.c0),
248        }
249    }
250}
251
252impl<P: Fp2Parameters> PartialOrd for Fp2<P> {
253    #[inline(always)]
254    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
255        Some(self.cmp(other))
256    }
257}
258
259impl<P: Fp2Parameters> From<u128> for Fp2<P> {
260    fn from(other: u128) -> Self {
261        Self::new(other.into(), P::Fp::zero())
262    }
263}
264
265impl<P: Fp2Parameters> From<u64> for Fp2<P> {
266    fn from(other: u64) -> Self {
267        Self::new(other.into(), P::Fp::zero())
268    }
269}
270
271impl<P: Fp2Parameters> From<u32> for Fp2<P> {
272    fn from(other: u32) -> Self {
273        Self::new(other.into(), P::Fp::zero())
274    }
275}
276
277impl<P: Fp2Parameters> From<u16> for Fp2<P> {
278    fn from(other: u16) -> Self {
279        Self::new(other.into(), P::Fp::zero())
280    }
281}
282
283impl<P: Fp2Parameters> From<u8> for Fp2<P> {
284    fn from(other: u8) -> Self {
285        Self::new(other.into(), P::Fp::zero())
286    }
287}
288
289impl<P: Fp2Parameters> ToBits for Fp2<P> {
290    fn write_bits_le(&self, vec: &mut Vec<bool>) {
291        self.c0.write_bits_le(vec);
292        self.c1.write_bits_le(vec);
293    }
294
295    fn write_bits_be(&self, vec: &mut Vec<bool>) {
296        self.c0.write_bits_be(vec);
297        self.c1.write_bits_be(vec);
298    }
299}
300
301impl<P: Fp2Parameters> ToBytes for Fp2<P> {
302    #[inline]
303    fn write_le<W: Write>(&self, mut writer: W) -> IoResult<()> {
304        self.c0.write_le(&mut writer)?;
305        self.c1.write_le(writer)
306    }
307}
308
309impl<P: Fp2Parameters> FromBytes for Fp2<P> {
310    #[inline]
311    fn read_le<R: Read>(mut reader: R) -> IoResult<Self> {
312        let c0 = P::Fp::read_le(&mut reader)?;
313        let c1 = P::Fp::read_le(reader)?;
314        Ok(Fp2::new(c0, c1))
315    }
316}
317
318impl<P: Fp2Parameters> Neg for Fp2<P> {
319    type Output = Self;
320
321    #[inline]
322    fn neg(self) -> Self {
323        let mut res = self;
324        res.c0 = res.c0.neg();
325        res.c1 = res.c1.neg();
326        res
327    }
328}
329
330impl<P: Fp2Parameters> Distribution<Fp2<P>> for Standard {
331    #[inline]
332    fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> Fp2<P> {
333        Fp2::new(Uniform::rand(rng), Uniform::rand(rng))
334    }
335}
336
337impl_add_sub_from_field_ref!(Fp2, Fp2Parameters);
338impl_mul_div_from_field_ref!(Fp2, Fp2Parameters);
339
340impl<P: Fp2Parameters> Add<&'_ Fp2<P>> for Fp2<P> {
341    type Output = Self;
342
343    #[inline]
344    fn add(self, other: &Self) -> Self {
345        let mut result = self;
346        result.add_assign(other);
347        result
348    }
349}
350
351impl<P: Fp2Parameters> Sub<&'_ Fp2<P>> for Fp2<P> {
352    type Output = Self;
353
354    #[inline]
355    fn sub(self, other: &Self) -> Self {
356        let mut result = self;
357        result.sub_assign(&other);
358        result
359    }
360}
361
362impl<P: Fp2Parameters> Mul<&'_ Fp2<P>> for Fp2<P> {
363    type Output = Self;
364
365    #[inline]
366    fn mul(self, other: &Self) -> Self {
367        let mut result = self;
368        result.mul_assign(&other);
369        result
370    }
371}
372
373impl<P: Fp2Parameters> Div<&'_ Fp2<P>> for Fp2<P> {
374    type Output = Self;
375
376    #[inline]
377    fn div(self, other: &Self) -> Self {
378        let mut result = self;
379        result.mul_assign(&other.inverse().unwrap());
380        result
381    }
382}
383
384impl<P: Fp2Parameters> AddAssign<&'_ Self> for Fp2<P> {
385    #[inline]
386    fn add_assign(&mut self, other: &Self) {
387        self.c0.add_assign(other.c0);
388        self.c1.add_assign(other.c1);
389    }
390}
391
392impl<P: Fp2Parameters> SubAssign<&'_ Self> for Fp2<P> {
393    #[inline]
394    fn sub_assign(&mut self, other: &Self) {
395        self.c0.sub_assign(&other.c0);
396        self.c1.sub_assign(&other.c1);
397    }
398}
399
400impl<P: Fp2Parameters> MulAssign<&'_ Self> for Fp2<P> {
401    #[inline]
402    #[allow(clippy::suspicious_op_assign_impl)]
403    fn mul_assign(&mut self, other: &Self) {
404        *self = Self::new(
405            P::Fp::sum_of_products([self.c0, P::mul_fp_by_nonresidue(&self.c1)].iter(), [other.c0, other.c1].iter()),
406            P::Fp::sum_of_products([self.c0, self.c1].iter(), [other.c1, other.c0].iter()),
407        )
408    }
409}
410
411impl<P: Fp2Parameters> DivAssign<&'_ Self> for Fp2<P> {
412    #[inline]
413    fn div_assign(&mut self, other: &Self) {
414        self.mul_assign(&other.inverse().unwrap());
415    }
416}
417
418impl<P: Fp2Parameters> std::fmt::Display for Fp2<P> {
419    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
420        write!(f, "Fp2({} + {} * u)", self.c0, self.c1)
421    }
422}
423
424impl<P: Fp2Parameters> CanonicalSerializeWithFlags for Fp2<P> {
425    #[inline]
426    fn serialize_with_flags<W: Write, F: Flags>(&self, mut writer: W, flags: F) -> Result<(), SerializationError> {
427        CanonicalSerialize::serialize_uncompressed(&self.c0, &mut writer)?;
428        self.c1.serialize_with_flags(&mut writer, flags)?;
429        Ok(())
430    }
431
432    fn serialized_size_with_flags<F: Flags>(&self) -> usize {
433        self.c0.uncompressed_size() + self.c1.serialized_size_with_flags::<F>()
434    }
435}
436
437impl<P: Fp2Parameters> CanonicalSerialize for Fp2<P> {
438    #[inline]
439    fn serialize_with_mode<W: Write>(&self, writer: W, _compress: Compress) -> Result<(), SerializationError> {
440        self.serialize_with_flags(writer, EmptyFlags)
441    }
442
443    #[inline]
444    fn serialized_size(&self, compress: Compress) -> usize {
445        self.c0.serialized_size(compress) + self.c1.serialized_size(compress)
446    }
447}
448
449impl<P: Fp2Parameters> CanonicalDeserializeWithFlags for Fp2<P> {
450    #[inline]
451    fn deserialize_with_flags<R: Read, F: Flags>(mut reader: R) -> Result<(Self, F), SerializationError> {
452        let c0: P::Fp = CanonicalDeserialize::deserialize_uncompressed(&mut reader)?;
453        let (c1, flags): (P::Fp, _) = CanonicalDeserializeWithFlags::deserialize_with_flags(&mut reader)?;
454        Ok((Fp2::new(c0, c1), flags))
455    }
456}
457impl<P: Fp2Parameters> Valid for Fp2<P> {
458    fn check(&self) -> Result<(), snarkvm_utilities::SerializationError> {
459        Ok(())
460    }
461
462    fn batch_check<'a>(
463        _batch: impl Iterator<Item = &'a Self> + Send,
464    ) -> Result<(), snarkvm_utilities::SerializationError>
465    where
466        Self: 'a,
467    {
468        Ok(())
469    }
470}
471
472impl<P: Fp2Parameters> CanonicalDeserialize for Fp2<P> {
473    #[inline]
474    fn deserialize_with_mode<R: Read>(
475        mut reader: R,
476        compress: Compress,
477        validate: Validate,
478    ) -> Result<Self, SerializationError> {
479        let c0 = P::Fp::deserialize_with_mode(&mut reader, compress, validate)?;
480        let c1 = P::Fp::deserialize_with_mode(&mut reader, compress, validate)?;
481        Ok(Fp2::new(c0, c1))
482    }
483}
snarkvm_fields/fp2.rs

snarkvm_fields/
fp2.rs
