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1027 lines
34 KiB
1027 lines
34 KiB
use algebra::{
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fields::{
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fp6_3over2::{Fp6, Fp6Parameters},
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Field, Fp2Parameters,
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},
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PrimeField,
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};
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use core::{borrow::Borrow, marker::PhantomData};
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use r1cs_core::{ConstraintSystem, ConstraintVar, SynthesisError};
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use crate::{prelude::*, Assignment, Vec};
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type Fp2Gadget<P, ConstraintF> =
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super::fp2::Fp2Gadget<<P as Fp6Parameters>::Fp2Params, ConstraintF>;
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#[derive(Derivative)]
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#[derivative(Debug(bound = "ConstraintF: PrimeField"))]
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#[must_use]
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pub struct Fp6Gadget<P, ConstraintF: PrimeField>
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where
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P: Fp6Parameters,
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P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
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{
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pub c0: Fp2Gadget<P, ConstraintF>,
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pub c1: Fp2Gadget<P, ConstraintF>,
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pub c2: Fp2Gadget<P, ConstraintF>,
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#[derivative(Debug = "ignore")]
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_params: PhantomData<P>,
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}
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impl<P, ConstraintF: PrimeField> Fp6Gadget<P, ConstraintF>
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where
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P: Fp6Parameters,
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P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
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{
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#[inline]
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pub fn new(
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c0: Fp2Gadget<P, ConstraintF>,
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c1: Fp2Gadget<P, ConstraintF>,
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c2: Fp2Gadget<P, ConstraintF>,
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) -> Self {
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Self {
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c0,
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c1,
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c2,
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_params: PhantomData,
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}
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}
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/// Multiply a Fp2Gadget by cubic nonresidue P::NONRESIDUE.
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#[inline]
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pub fn mul_fp2_gadget_by_nonresidue<CS: ConstraintSystem<ConstraintF>>(
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cs: CS,
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fe: &Fp2Gadget<P, ConstraintF>,
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) -> Result<Fp2Gadget<P, ConstraintF>, SynthesisError> {
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fe.mul_by_constant(cs, &P::NONRESIDUE)
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}
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#[inline]
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pub fn mul_by_0_c1_0<CS: ConstraintSystem<ConstraintF>>(
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&self,
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mut cs: CS,
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c1: &Fp2Gadget<P, ConstraintF>,
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) -> Result<Self, SynthesisError> {
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// Karatsuba multiplication
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// v0 = a0 * b0 = 0
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// v1 = a1 * b1
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let v1 = self.c1.mul(cs.ns(|| "first mul"), c1)?;
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// v2 = a2 * b2 = 0
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let a1_plus_a2 = self.c1.add(cs.ns(|| "a1 + a2"), &self.c2)?;
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let b1_plus_b2 = c1.clone();
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let a0_plus_a1 = self.c0.add(cs.ns(|| "a0 + a1"), &self.c1)?;
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// c0 = (NONRESIDUE * ((a1 + a2)*(b1 + b2) - v1 - v2)) + v0
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// = NONRESIDUE * ((a1 + a2) * b1 - v1)
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let c0 = a1_plus_a2
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.mul(cs.ns(|| "second mul"), &b1_plus_b2)?
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.sub(cs.ns(|| "first sub"), &v1)?
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.mul_by_constant(cs.ns(|| "mul_by_nonresidue"), &P::NONRESIDUE)?;
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// c1 = (a0 + a1) * (b0 + b1) - v0 - v1 + NONRESIDUE * v2
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// = (a0 + a1) * b1 - v1
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let c1 = a0_plus_a1
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.mul(cs.ns(|| "third mul"), &c1)?
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.sub(cs.ns(|| "second sub"), &v1)?;
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// c2 = (a0 + a2) * (b0 + b2) - v0 - v2 + v1
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// = v1
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let c2 = v1;
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Ok(Self::new(c0, c1, c2))
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}
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// #[inline]
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pub fn mul_by_c0_c1_0<CS: ConstraintSystem<ConstraintF>>(
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&self,
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mut cs: CS,
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c0: &Fp2Gadget<P, ConstraintF>,
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c1: &Fp2Gadget<P, ConstraintF>,
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) -> Result<Self, SynthesisError> {
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let v0 = self.c0.mul(cs.ns(|| "v0"), c0)?;
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let v1 = self.c1.mul(cs.ns(|| "v1"), c1)?;
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// v2 = 0.
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let a1_plus_a2 = self.c1.add(cs.ns(|| "a1 + a2"), &self.c2)?;
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let a0_plus_a1 = self.c0.add(cs.ns(|| "a0 + a1"), &self.c1)?;
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let a0_plus_a2 = self.c0.add(cs.ns(|| "a0 + a2"), &self.c2)?;
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let b1_plus_b2 = c1.clone();
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let b0_plus_b1 = c0.add(cs.ns(|| "b0 + b1"), &c1)?;
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let b0_plus_b2 = c0.clone();
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let c0 = {
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let cs = &mut cs.ns(|| "c0");
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a1_plus_a2
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.mul(cs.ns(|| "(a1 + a2) * (b1 + b2)"), &b1_plus_b2)?
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.sub(cs.ns(|| "sub v1"), &v1)?
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.mul_by_constant(cs.ns(|| "First mul_by_nonresidue"), &P::NONRESIDUE)?
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.add(cs.ns(|| "add v0"), &v0)?
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};
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let c1 = {
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let cs = &mut cs.ns(|| "c1");
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a0_plus_a1
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.mul(cs.ns(|| "(a0 + a1) * (b0 + b1)"), &b0_plus_b1)?
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.sub(cs.ns(|| "sub v0"), &v0)?
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.sub(cs.ns(|| "sub v1"), &v1)?
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};
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let c2 = {
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a0_plus_a2
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.mul(cs.ns(|| "(a0 + a2) * (b0 + b2)"), &b0_plus_b2)?
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.sub(cs.ns(|| "sub v0"), &v0)?
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.add(cs.ns(|| "add v1"), &v1)?
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};
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Ok(Self::new(c0, c1, c2))
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}
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}
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impl<P, ConstraintF: PrimeField> FieldGadget<Fp6<P>, ConstraintF> for Fp6Gadget<P, ConstraintF>
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where
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P: Fp6Parameters,
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P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
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{
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type Variable = (
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(ConstraintVar<ConstraintF>, ConstraintVar<ConstraintF>),
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(ConstraintVar<ConstraintF>, ConstraintVar<ConstraintF>),
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(ConstraintVar<ConstraintF>, ConstraintVar<ConstraintF>),
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);
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#[inline]
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fn get_value(&self) -> Option<Fp6<P>> {
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match (
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self.c0.get_value(),
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self.c1.get_value(),
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self.c2.get_value(),
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) {
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(Some(c0), Some(c1), Some(c2)) => Some(Fp6::new(c0, c1, c2)),
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(..) => None,
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}
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}
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#[inline]
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fn get_variable(&self) -> Self::Variable {
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(
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self.c0.get_variable(),
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self.c1.get_variable(),
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self.c2.get_variable(),
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)
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}
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#[inline]
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fn zero<CS: ConstraintSystem<ConstraintF>>(mut cs: CS) -> Result<Self, SynthesisError> {
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let c0 = Fp2Gadget::<P, ConstraintF>::zero(cs.ns(|| "c0"))?;
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let c1 = Fp2Gadget::<P, ConstraintF>::zero(cs.ns(|| "c1"))?;
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let c2 = Fp2Gadget::<P, ConstraintF>::zero(cs.ns(|| "c2"))?;
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Ok(Self::new(c0, c1, c2))
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}
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#[inline]
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fn one<CS: ConstraintSystem<ConstraintF>>(mut cs: CS) -> Result<Self, SynthesisError> {
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let c0 = Fp2Gadget::<P, ConstraintF>::one(cs.ns(|| "c0"))?;
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let c1 = Fp2Gadget::<P, ConstraintF>::zero(cs.ns(|| "c1"))?;
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let c2 = Fp2Gadget::<P, ConstraintF>::zero(cs.ns(|| "c2"))?;
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Ok(Self::new(c0, c1, c2))
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}
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#[inline]
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fn conditionally_add_constant<CS: ConstraintSystem<ConstraintF>>(
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&self,
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mut cs: CS,
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bit: &Boolean,
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coeff: Fp6<P>,
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) -> Result<Self, SynthesisError> {
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let c0 = self
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.c0
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.conditionally_add_constant(cs.ns(|| "c0"), bit, coeff.c0)?;
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let c1 = self
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.c1
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.conditionally_add_constant(cs.ns(|| "c1"), bit, coeff.c1)?;
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let c2 = self
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.c2
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.conditionally_add_constant(cs.ns(|| "c2"), bit, coeff.c2)?;
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Ok(Self::new(c0, c1, c2))
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}
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#[inline]
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fn add<CS: ConstraintSystem<ConstraintF>>(
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&self,
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mut cs: CS,
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other: &Self,
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) -> Result<Self, SynthesisError> {
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let c0 = self.c0.add(&mut cs.ns(|| "add c0"), &other.c0)?;
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let c1 = self.c1.add(&mut cs.ns(|| "add c1"), &other.c1)?;
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let c2 = self.c2.add(&mut cs.ns(|| "add c2"), &other.c2)?;
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Ok(Self::new(c0, c1, c2))
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}
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#[inline]
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fn sub<CS: ConstraintSystem<ConstraintF>>(
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&self,
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mut cs: CS,
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other: &Self,
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) -> Result<Self, SynthesisError> {
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let c0 = self.c0.sub(&mut cs.ns(|| "sub c0"), &other.c0)?;
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let c1 = self.c1.sub(&mut cs.ns(|| "sub c1"), &other.c1)?;
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let c2 = self.c2.sub(&mut cs.ns(|| "sub c2"), &other.c2)?;
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Ok(Self::new(c0, c1, c2))
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}
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#[inline]
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fn negate<CS: ConstraintSystem<ConstraintF>>(
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&self,
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mut cs: CS,
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) -> Result<Self, SynthesisError> {
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let c0 = self.c0.negate(&mut cs.ns(|| "negate c0"))?;
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let c1 = self.c1.negate(&mut cs.ns(|| "negate c1"))?;
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let c2 = self.c2.negate(&mut cs.ns(|| "negate c2"))?;
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Ok(Self::new(c0, c1, c2))
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}
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#[inline]
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fn negate_in_place<CS: ConstraintSystem<ConstraintF>>(
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&mut self,
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mut cs: CS,
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) -> Result<&mut Self, SynthesisError> {
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self.c0.negate_in_place(&mut cs.ns(|| "negate c0"))?;
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self.c1.negate_in_place(&mut cs.ns(|| "negate c1"))?;
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self.c2.negate_in_place(&mut cs.ns(|| "negate c2"))?;
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Ok(self)
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}
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/// Use the Toom-Cook-3x method to compute multiplication.
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#[inline]
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fn mul<CS: ConstraintSystem<ConstraintF>>(
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&self,
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mut cs: CS,
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other: &Self,
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) -> Result<Self, SynthesisError> {
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// Uses Toom-Cool-3x multiplication from
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//
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// Reference:
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// "Multiplication and Squaring on Pairing-Friendly Fields"
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// Devegili, OhEigeartaigh, Scott, Dahab
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// v0 = a(0)b(0) = a0 * b0
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let v0 = self.c0.mul(&mut cs.ns(|| "Calc v0"), &other.c0)?;
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// v1 = a(1)b(1) = (a0 + a1 + a2)(b0 + b1 + b2)
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let v1 = {
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let mut v1_cs = cs.ns(|| "compute v1");
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let a0_plus_a1_plus_a2 = self
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.c0
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.add(v1_cs.ns(|| "a0 + a1"), &self.c1)?
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.add(v1_cs.ns(|| "a0 + a1 + a2"), &self.c2)?;
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let b0_plus_b1_plus_b2 = other
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.c0
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.add(v1_cs.ns(|| "b0 + b1"), &other.c1)?
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.add(v1_cs.ns(|| "b0 + b1 + b2"), &other.c2)?;
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a0_plus_a1_plus_a2.mul(
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v1_cs.ns(|| "(a0 + a1 + a2)(b0 + b1 + b2)"),
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&b0_plus_b1_plus_b2,
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)?
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};
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// v2 = a(−1)b(−1) = (a0 − a1 + a2)(b0 − b1 + b2)
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let v2 = {
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let mut v2_cs = cs.ns(|| "compute v2");
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let a0_minus_a1_plus_a2 = self
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.c0
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.sub(v2_cs.ns(|| "a0 - a1"), &self.c1)?
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.add(v2_cs.ns(|| "a0 - a1 + a2"), &self.c2)?;
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let b0_minus_b1_plus_b2 = other
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.c0
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.sub(v2_cs.ns(|| "b0 - b1"), &other.c1)?
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.add(v2_cs.ns(|| "b0 - b1 + b2"), &other.c2)?;
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a0_minus_a1_plus_a2.mul(
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v2_cs.ns(|| "(a0 - a1 + a2)(b0 - b1 + b2)"),
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&b0_minus_b1_plus_b2,
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)?
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};
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// v3 = a(2)b(2) = (a0 + 2a1 + 4a2)(b0 + 2b1 + 4b2)
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let v3 = {
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let v3_cs = &mut cs.ns(|| "compute v3");
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let a1_double = self.c1.double(v3_cs.ns(|| "2 * a1"))?;
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let a2_quad = self
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.c2
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.double(v3_cs.ns(|| "2 * a2"))?
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.double(v3_cs.ns(|| "4 * a2"))?;
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let a0_plus_2_a1_plus_4_a2 = self
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.c0
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.add(v3_cs.ns(|| "a0 + 2a1"), &a1_double)?
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.add(v3_cs.ns(|| "a0 + 2a1 + 4a2"), &a2_quad)?;
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let b1_double = other.c1.double(v3_cs.ns(|| "2 * b1"))?;
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let b2_quad = other
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.c2
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.double(v3_cs.ns(|| "2 * b2"))?
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.double(v3_cs.ns(|| "4 * b2"))?;
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let b0_plus_2_b1_plus_4_b2 = other
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.c0
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.add(v3_cs.ns(|| "b0 + 2b1"), &b1_double)?
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.add(v3_cs.ns(|| "b0 + 2b1 + 4b2"), &b2_quad)?;
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a0_plus_2_a1_plus_4_a2.mul(
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v3_cs.ns(|| "(a0 + 2a1 + 4a2)(b0 + 2b1 + 4b2)"),
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&b0_plus_2_b1_plus_4_b2,
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)?
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};
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// v4 = a(∞)b(∞) = a2 * b2
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let v4 = self.c2.mul(cs.ns(|| "v2: a2 * b2"), &other.c2)?;
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let two = <P::Fp2Params as Fp2Parameters>::Fp::one().double();
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let six = two.double() + &two;
|
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let mut two_and_six = [two, six];
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algebra::fields::batch_inversion(&mut two_and_six);
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let (two_inverse, six_inverse) = (two_and_six[0], two_and_six[1]);
|
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let half_v0 = v0.mul_by_fp_constant(cs.ns(|| "half_v0"), &two_inverse)?;
|
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let half_v1 = v1.mul_by_fp_constant(cs.ns(|| "half_v1"), &two_inverse)?;
|
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let one_sixth_v2 = v2.mul_by_fp_constant(cs.ns(|| "v2_by_six"), &six_inverse)?;
|
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let one_sixth_v3 = v3.mul_by_fp_constant(cs.ns(|| "v3_by_six"), &six_inverse)?;
|
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let two_v4 = v4.double(cs.ns(|| "2 * v4"))?;
|
|
|
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// c0 = v0 + β((1/2)v0 − (1/2)v1 − (1/6)v2 + (1/6)v3 − 2v4)
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let c0 = {
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let c0_cs = &mut cs.ns(|| "c0");
|
|
|
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// No constraints, only get a linear combination back.
|
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let temp = half_v0
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.sub(c0_cs.ns(|| "sub1"), &half_v1)?
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.sub(c0_cs.ns(|| "sub2"), &one_sixth_v2)?
|
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.add(c0_cs.ns(|| "add3"), &one_sixth_v3)?
|
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.sub(c0_cs.ns(|| "sub4"), &two_v4)?;
|
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let non_residue_times_inner =
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temp.mul_by_constant(&mut c0_cs.ns(|| "mul5"), &P::NONRESIDUE)?;
|
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v0.add(c0_cs.ns(|| "add6"), &non_residue_times_inner)?
|
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};
|
|
|
|
// −(1/2)v0 + v1 − (1/3)v2 − (1/6)v3 + 2v4 + βv4
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let c1 = {
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let c1_cs = &mut cs.ns(|| "c1");
|
|
let one_third_v2 = one_sixth_v2.double(&mut c1_cs.ns(|| "v2_by_3"))?;
|
|
let non_residue_v4 =
|
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v4.mul_by_constant(&mut c1_cs.ns(|| "mul_by_beta"), &P::NONRESIDUE)?;
|
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|
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let result = half_v0
|
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.negate(c1_cs.ns(|| "neg1"))?
|
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.add(c1_cs.ns(|| "add2"), &v1)?
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.sub(c1_cs.ns(|| "sub3"), &one_third_v2)?
|
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.sub(c1_cs.ns(|| "sub4"), &one_sixth_v3)?
|
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.add(c1_cs.ns(|| "sub5"), &two_v4)?
|
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.add(c1_cs.ns(|| "sub6"), &non_residue_v4)?;
|
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result
|
|
};
|
|
|
|
// -v0 + (1/2)v1 + (1/2)v2 −v4
|
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let c2 = {
|
|
let c2_cs = &mut cs.ns(|| "c2");
|
|
let half_v2 = v2.mul_by_fp_constant(&mut c2_cs.ns(|| "mul1"), &two_inverse)?;
|
|
let result = half_v1
|
|
.add(c2_cs.ns(|| "add1"), &half_v2)?
|
|
.sub(c2_cs.ns(|| "sub1"), &v4)?
|
|
.sub(c2_cs.ns(|| "sub2"), &v0)?;
|
|
result
|
|
};
|
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|
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Ok(Self::new(c0, c1, c2))
|
|
}
|
|
|
|
/// Use the Toom-Cook-3x method to compute multiplication.
|
|
#[inline]
|
|
fn square<CS: ConstraintSystem<ConstraintF>>(
|
|
&self,
|
|
mut cs: CS,
|
|
) -> Result<Self, SynthesisError> {
|
|
// Uses Toom-Cool-3x multiplication from
|
|
//
|
|
// Reference:
|
|
// "Multiplication and Squaring on Pairing-Friendly Fields"
|
|
// Devegili, OhEigeartaigh, Scott, Dahab
|
|
|
|
// v0 = a(0)^2 = a0^2
|
|
let v0 = self.c0.square(&mut cs.ns(|| "Calc v0"))?;
|
|
|
|
// v1 = a(1)^2 = (a0 + a1 + a2)^2
|
|
let v1 = {
|
|
let a0_plus_a1_plus_a2 = self
|
|
.c0
|
|
.add(cs.ns(|| "a0 + a1"), &self.c1)?
|
|
.add(cs.ns(|| "a0 + a1 + a2"), &self.c2)?;
|
|
a0_plus_a1_plus_a2.square(&mut cs.ns(|| "(a0 + a1 + a2)^2"))?
|
|
};
|
|
|
|
// v2 = a(−1)^2 = (a0 − a1 + a2)^2
|
|
let v2 = {
|
|
let a0_minus_a1_plus_a2 = self
|
|
.c0
|
|
.sub(cs.ns(|| "a0 - a1"), &self.c1)?
|
|
.add(cs.ns(|| "a0 - a2 + a2"), &self.c2)?;
|
|
a0_minus_a1_plus_a2.square(&mut cs.ns(|| "(a0 - a1 + a2)^2"))?
|
|
};
|
|
|
|
// v3 = a(2)^2 = (a0 + 2a1 + 4a2)^2
|
|
let v3 = {
|
|
let a1_double = self.c1.double(cs.ns(|| "2a1"))?;
|
|
let a2_quad = self.c2.double(cs.ns(|| "2a2"))?.double(cs.ns(|| "4a2"))?;
|
|
let a0_plus_2_a1_plus_4_a2 = self
|
|
.c0
|
|
.add(cs.ns(|| "a0 + 2a1"), &a1_double)?
|
|
.add(cs.ns(|| "a0 + 2a1 + 4a2"), &a2_quad)?;
|
|
|
|
a0_plus_2_a1_plus_4_a2.square(&mut cs.ns(|| "(a0 + 2a1 + 4a2)^2"))?
|
|
};
|
|
|
|
// v4 = a(∞)^2 = a2^2
|
|
let v4 = self.c2.square(&mut cs.ns(|| "a2^2"))?;
|
|
|
|
let two = <P::Fp2Params as Fp2Parameters>::Fp::one().double();
|
|
let six = two.double() + &two;
|
|
let mut two_and_six = [two, six];
|
|
algebra::fields::batch_inversion(&mut two_and_six);
|
|
let (two_inverse, six_inverse) = (two_and_six[0], two_and_six[1]);
|
|
|
|
let half_v0 = v0.mul_by_fp_constant(cs.ns(|| "half_v0"), &two_inverse)?;
|
|
let half_v1 = v1.mul_by_fp_constant(cs.ns(|| "half_v1"), &two_inverse)?;
|
|
let one_sixth_v2 = v2.mul_by_fp_constant(cs.ns(|| "one_sixth_v2"), &six_inverse)?;
|
|
let one_sixth_v3 = v3.mul_by_fp_constant(cs.ns(|| "one_sixth_v3"), &six_inverse)?;
|
|
let two_v4 = v4.double(cs.ns(|| "double_v4"))?;
|
|
|
|
// c0 = v0 + β((1/2)v0 − (1/2)v1 − (1/6)v2 + (1/6)v3 − 2v4)
|
|
let c0 = {
|
|
let mut c0_cs = cs.ns(|| "c0");
|
|
// No constraints, only get a linear combination back.
|
|
let inner = half_v0
|
|
.sub(c0_cs.ns(|| "sub1"), &half_v1)?
|
|
.sub(c0_cs.ns(|| "sub2"), &one_sixth_v2)?
|
|
.add(c0_cs.ns(|| "add3"), &one_sixth_v3)?
|
|
.sub(c0_cs.ns(|| "sub4"), &two_v4)?;
|
|
let non_residue_times_inner =
|
|
inner.mul_by_constant(c0_cs.ns(|| "mul_by_res"), &P::NONRESIDUE)?;
|
|
v0.add(c0_cs.ns(|| "add5"), &non_residue_times_inner)?
|
|
};
|
|
|
|
// −(1/2)v0 + v1 − (1/3)v2 − (1/6)v3 + 2v4 + βv4
|
|
let c1 = {
|
|
let mut c1_cs = cs.ns(|| "c1");
|
|
let one_third_v2 = one_sixth_v2.double(c1_cs.ns(|| "v2_by_3"))?;
|
|
let non_residue_v4 = v4.mul_by_constant(c1_cs.ns(|| "mul_by_res"), &P::NONRESIDUE)?;
|
|
|
|
half_v0
|
|
.negate(c1_cs.ns(|| "neg1"))?
|
|
.add(c1_cs.ns(|| "add1"), &v1)?
|
|
.sub(c1_cs.ns(|| "sub2"), &one_third_v2)?
|
|
.sub(c1_cs.ns(|| "sub3"), &one_sixth_v3)?
|
|
.add(c1_cs.ns(|| "add4"), &two_v4)?
|
|
.add(c1_cs.ns(|| "add5"), &non_residue_v4)?
|
|
};
|
|
|
|
// -v0 + (1/2)v1 + (1/2)v2 −v4
|
|
let c2 = {
|
|
let mut c2_cs = cs.ns(|| "c2");
|
|
let half_v2 = v2.mul_by_fp_constant(c2_cs.ns(|| "half_v2"), &two_inverse)?;
|
|
half_v1
|
|
.add(c2_cs.ns(|| "add1"), &half_v2)?
|
|
.sub(c2_cs.ns(|| "sub1"), &v4)?
|
|
.sub(c2_cs.ns(|| "sub2"), &v0)?
|
|
};
|
|
|
|
Ok(Self::new(c0, c1, c2))
|
|
}
|
|
|
|
// 18 constaints, we can probably do better but not sure it's worth it.
|
|
#[inline]
|
|
fn inverse<CS: ConstraintSystem<ConstraintF>>(
|
|
&self,
|
|
mut cs: CS,
|
|
) -> Result<Self, SynthesisError> {
|
|
let inverse = Self::alloc(&mut cs.ns(|| "alloc inverse"), || {
|
|
self.get_value().and_then(|val| val.inverse()).get()
|
|
})?;
|
|
let one = Self::one(cs.ns(|| "one"))?;
|
|
inverse.mul_equals(cs.ns(|| "check inverse"), &self, &one)?;
|
|
Ok(inverse)
|
|
}
|
|
|
|
#[inline]
|
|
fn add_constant<CS: ConstraintSystem<ConstraintF>>(
|
|
&self,
|
|
mut cs: CS,
|
|
other: &Fp6<P>,
|
|
) -> Result<Self, SynthesisError> {
|
|
let c0 = self.c0.add_constant(cs.ns(|| "c0"), &other.c0)?;
|
|
let c1 = self.c1.add_constant(cs.ns(|| "c1"), &other.c1)?;
|
|
let c2 = self.c2.add_constant(cs.ns(|| "c2"), &other.c2)?;
|
|
|
|
Ok(Self::new(c0, c1, c2))
|
|
}
|
|
|
|
#[inline]
|
|
fn add_constant_in_place<CS: ConstraintSystem<ConstraintF>>(
|
|
&mut self,
|
|
mut cs: CS,
|
|
other: &Fp6<P>,
|
|
) -> Result<&mut Self, SynthesisError> {
|
|
self.c0.add_constant_in_place(cs.ns(|| "c0"), &other.c0)?;
|
|
self.c1.add_constant_in_place(cs.ns(|| "c1"), &other.c1)?;
|
|
self.c2.add_constant_in_place(cs.ns(|| "c2"), &other.c2)?;
|
|
Ok(self)
|
|
}
|
|
|
|
/// Use the Toom-Cook-3x method to compute multiplication.
|
|
#[inline]
|
|
fn mul_by_constant<CS: ConstraintSystem<ConstraintF>>(
|
|
&self,
|
|
mut cs: CS,
|
|
other: &Fp6<P>,
|
|
) -> Result<Self, SynthesisError> {
|
|
// Uses Toom-Cook-3x multiplication from
|
|
//
|
|
// Reference:
|
|
// "Multiplication and Squaring on Pairing-Friendly Fields"
|
|
// Devegili, OhEigeartaigh, Scott, Dahab
|
|
|
|
// v0 = a(0)b(0) = a0 * b0
|
|
let v0 = self.c0.mul_by_constant(cs.ns(|| "v0"), &other.c0)?;
|
|
|
|
// v1 = a(1)b(1) = (a0 + a1 + a2)(b0 + b1 + b2)
|
|
let v1 = {
|
|
let mut v1_cs = cs.ns(|| "v1");
|
|
let mut a0_plus_a1_plus_a2 = self
|
|
.c0
|
|
.add(v1_cs.ns(|| "a0 + a1"), &self.c1)?
|
|
.add(v1_cs.ns(|| "a0 + a1 + a2"), &self.c2)?;
|
|
let b0_plus_b1_plus_b2 = other.c0 + &other.c1 + &other.c2;
|
|
|
|
a0_plus_a1_plus_a2.mul_by_constant_in_place(
|
|
v1_cs.ns(|| "(a0 + a1 + a2)*(b0 + b1 + b2)"),
|
|
&b0_plus_b1_plus_b2,
|
|
)?;
|
|
a0_plus_a1_plus_a2
|
|
};
|
|
|
|
// v2 = a(−1)b(−1) = (a0 − a1 + a2)(b0 − b1 + b2)
|
|
let mut v2 = {
|
|
let mut v2_cs = cs.ns(|| "v2");
|
|
let mut a0_minus_a1_plus_a2 = self
|
|
.c0
|
|
.sub(v2_cs.ns(|| "sub1"), &self.c1)?
|
|
.add(v2_cs.ns(|| "add2"), &self.c2)?;
|
|
let b0_minus_b1_plus_b2 = other.c0 - &other.c1 + &other.c2;
|
|
a0_minus_a1_plus_a2.mul_by_constant_in_place(
|
|
v2_cs.ns(|| "(a0 - a1 + a2)*(b0 - b1 + b2)"),
|
|
&b0_minus_b1_plus_b2,
|
|
)?;
|
|
a0_minus_a1_plus_a2
|
|
};
|
|
|
|
// v3 = a(2)b(2) = (a0 + 2a1 + 4a2)(b0 + 2b1 + 4b2)
|
|
let mut v3 = {
|
|
let mut v3_cs = cs.ns(|| "v3");
|
|
let a1_double = self.c1.double(v3_cs.ns(|| "2a1"))?;
|
|
let a2_quad = self
|
|
.c2
|
|
.double(v3_cs.ns(|| "2a2"))?
|
|
.double(v3_cs.ns(|| "4a2"))?;
|
|
let mut a0_plus_2_a1_plus_4_a2 = self
|
|
.c0
|
|
.add(v3_cs.ns(|| "a0 + 2a1"), &a1_double)?
|
|
.add(v3_cs.ns(|| "a0 + 2a1 + 4a2"), &a2_quad)?;
|
|
|
|
let b1_double = other.c1.double();
|
|
let b2_quad = other.c2.double().double();
|
|
let b0_plus_2_b1_plus_4_b2 = other.c0 + &b1_double + &b2_quad;
|
|
|
|
a0_plus_2_a1_plus_4_a2.mul_by_constant_in_place(
|
|
v3_cs.ns(|| "(a0 + 2a1 + 4a2)*(b0 + 2b1 + 4b2)"),
|
|
&b0_plus_2_b1_plus_4_b2,
|
|
)?;
|
|
a0_plus_2_a1_plus_4_a2
|
|
};
|
|
|
|
// v4 = a(∞)b(∞) = a2 * b2
|
|
let v4 = self.c2.mul_by_constant(cs.ns(|| "v4"), &other.c2)?;
|
|
|
|
let two = <P::Fp2Params as Fp2Parameters>::Fp::one().double();
|
|
let six = two.double() + &two;
|
|
let mut two_and_six = [two, six];
|
|
algebra::fields::batch_inversion(&mut two_and_six);
|
|
let (two_inverse, six_inverse) = (two_and_six[0], two_and_six[1]);
|
|
|
|
let mut half_v0 = v0.mul_by_fp_constant(cs.ns(|| "half_v0"), &two_inverse)?;
|
|
let half_v1 = v1.mul_by_fp_constant(cs.ns(|| "half_v1"), &two_inverse)?;
|
|
let mut one_sixth_v2 = v2.mul_by_fp_constant(cs.ns(|| "v2_by_6"), &six_inverse)?;
|
|
let one_sixth_v3 = v3.mul_by_fp_constant_in_place(cs.ns(|| "v3_by_6"), &six_inverse)?;
|
|
let two_v4 = v4.double(cs.ns(|| "2v4"))?;
|
|
|
|
// c0 = v0 + β((1/2)v0 − (1/2)v1 − (1/6)v2 + (1/6)v3 − 2v4)
|
|
let c0 = {
|
|
let mut c0_cs = cs.ns(|| "c0");
|
|
|
|
// No constraints, only get a linear combination back.
|
|
let mut inner = half_v0
|
|
.sub(c0_cs.ns(|| "sub1"), &half_v1)?
|
|
.sub(c0_cs.ns(|| "sub2"), &one_sixth_v2)?
|
|
.add(c0_cs.ns(|| "add3"), &one_sixth_v3)?
|
|
.sub(c0_cs.ns(|| "sub4"), &two_v4)?;
|
|
let non_residue_times_inner =
|
|
inner.mul_by_constant_in_place(&mut c0_cs, &P::NONRESIDUE)?;
|
|
v0.add(c0_cs.ns(|| "add5"), non_residue_times_inner)?
|
|
};
|
|
|
|
// −(1/2)v0 + v1 − (1/3)v2 − (1/6)v3 + 2v4 + βv4
|
|
let c1 = {
|
|
let mut c1_cs = cs.ns(|| "c1");
|
|
let one_third_v2 = one_sixth_v2.double_in_place(c1_cs.ns(|| "double1"))?;
|
|
let non_residue_v4 =
|
|
v4.mul_by_constant(c1_cs.ns(|| "mul_by_const1"), &P::NONRESIDUE)?;
|
|
|
|
half_v0
|
|
.negate_in_place(c1_cs.ns(|| "neg1"))?
|
|
.add(c1_cs.ns(|| "add1"), &v1)?
|
|
.sub(c1_cs.ns(|| "sub2"), one_third_v2)?
|
|
.sub(c1_cs.ns(|| "sub3"), &one_sixth_v3)?
|
|
.add(c1_cs.ns(|| "add4"), &two_v4)?
|
|
.add(c1_cs.ns(|| "add5"), &non_residue_v4)?
|
|
};
|
|
|
|
// -v0 + (1/2)v1 + (1/2)v2 −v4
|
|
let c2 = {
|
|
let mut c2_cs = cs.ns(|| "c2");
|
|
let half_v2 = v2.mul_by_fp_constant_in_place(c2_cs.ns(|| "half_v2"), &two_inverse)?;
|
|
half_v1
|
|
.add(c2_cs.ns(|| "add1"), half_v2)?
|
|
.sub(c2_cs.ns(|| "sub2"), &v4)?
|
|
.sub(c2_cs.ns(|| "sub3"), &v0)?
|
|
};
|
|
|
|
Ok(Self::new(c0, c1, c2))
|
|
}
|
|
|
|
fn frobenius_map<CS: ConstraintSystem<ConstraintF>>(
|
|
&self,
|
|
cs: CS,
|
|
power: usize,
|
|
) -> Result<Self, SynthesisError> {
|
|
let mut result = self.clone();
|
|
result.frobenius_map_in_place(cs, power)?;
|
|
Ok(result)
|
|
}
|
|
|
|
fn frobenius_map_in_place<CS: ConstraintSystem<ConstraintF>>(
|
|
&mut self,
|
|
mut cs: CS,
|
|
power: usize,
|
|
) -> Result<&mut Self, SynthesisError> {
|
|
self.c0.frobenius_map_in_place(&mut cs.ns(|| "c0"), power)?;
|
|
self.c1.frobenius_map_in_place(&mut cs.ns(|| "c1"), power)?;
|
|
self.c2.frobenius_map_in_place(&mut cs.ns(|| "c2"), power)?;
|
|
|
|
self.c1.mul_by_constant_in_place(
|
|
cs.ns(|| "c1_power"),
|
|
&P::FROBENIUS_COEFF_FP6_C1[power % 6],
|
|
)?;
|
|
self.c2.mul_by_constant_in_place(
|
|
cs.ns(|| "c2_power"),
|
|
&P::FROBENIUS_COEFF_FP6_C2[power % 6],
|
|
)?;
|
|
|
|
Ok(self)
|
|
}
|
|
|
|
fn cost_of_mul() -> usize {
|
|
5 * Fp2Gadget::<P, ConstraintF>::cost_of_mul()
|
|
}
|
|
|
|
fn cost_of_inv() -> usize {
|
|
Self::cost_of_mul() + <Self as EqGadget<ConstraintF>>::cost()
|
|
}
|
|
}
|
|
|
|
impl<P, ConstraintF: PrimeField> PartialEq for Fp6Gadget<P, ConstraintF>
|
|
where
|
|
P: Fp6Parameters,
|
|
P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
|
|
{
|
|
fn eq(&self, other: &Self) -> bool {
|
|
self.c0 == other.c0 && self.c1 == other.c1 && self.c2 == other.c2
|
|
}
|
|
}
|
|
|
|
impl<P, ConstraintF: PrimeField> Eq for Fp6Gadget<P, ConstraintF>
|
|
where
|
|
P: Fp6Parameters,
|
|
P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
|
|
{
|
|
}
|
|
|
|
impl<P, ConstraintF: PrimeField> EqGadget<ConstraintF> for Fp6Gadget<P, ConstraintF>
|
|
where
|
|
P: Fp6Parameters,
|
|
P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
|
|
{
|
|
}
|
|
|
|
impl<P, ConstraintF: PrimeField> ConditionalEqGadget<ConstraintF> for Fp6Gadget<P, ConstraintF>
|
|
where
|
|
P: Fp6Parameters,
|
|
P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
|
|
{
|
|
#[inline]
|
|
fn conditional_enforce_equal<CS: ConstraintSystem<ConstraintF>>(
|
|
&self,
|
|
mut cs: CS,
|
|
other: &Self,
|
|
condition: &Boolean,
|
|
) -> Result<(), SynthesisError> {
|
|
self.c0
|
|
.conditional_enforce_equal(&mut cs.ns(|| "c0"), &other.c0, condition)?;
|
|
self.c1
|
|
.conditional_enforce_equal(&mut cs.ns(|| "c1"), &other.c1, condition)?;
|
|
self.c2
|
|
.conditional_enforce_equal(&mut cs.ns(|| "c2"), &other.c2, condition)?;
|
|
Ok(())
|
|
}
|
|
|
|
fn cost() -> usize {
|
|
3 * <Fp2Gadget<P, ConstraintF> as ConditionalEqGadget<ConstraintF>>::cost()
|
|
}
|
|
}
|
|
|
|
impl<P, ConstraintF: PrimeField> NEqGadget<ConstraintF> for Fp6Gadget<P, ConstraintF>
|
|
where
|
|
P: Fp6Parameters,
|
|
P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
|
|
{
|
|
#[inline]
|
|
fn enforce_not_equal<CS: ConstraintSystem<ConstraintF>>(
|
|
&self,
|
|
mut cs: CS,
|
|
other: &Self,
|
|
) -> Result<(), SynthesisError> {
|
|
self.c0.enforce_not_equal(&mut cs.ns(|| "c0"), &other.c0)?;
|
|
self.c1.enforce_not_equal(&mut cs.ns(|| "c1"), &other.c1)?;
|
|
self.c2.enforce_not_equal(&mut cs.ns(|| "c2"), &other.c2)?;
|
|
Ok(())
|
|
}
|
|
|
|
fn cost() -> usize {
|
|
3 * <Fp2Gadget<P, ConstraintF> as NEqGadget<ConstraintF>>::cost()
|
|
}
|
|
}
|
|
|
|
impl<P, ConstraintF: PrimeField> ToBitsGadget<ConstraintF> for Fp6Gadget<P, ConstraintF>
|
|
where
|
|
P: Fp6Parameters,
|
|
P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
|
|
{
|
|
fn to_bits<CS: ConstraintSystem<ConstraintF>>(
|
|
&self,
|
|
mut cs: CS,
|
|
) -> Result<Vec<Boolean>, SynthesisError> {
|
|
let mut c0 = self.c0.to_bits(cs.ns(|| "c0"))?;
|
|
let mut c1 = self.c1.to_bits(cs.ns(|| "c1"))?;
|
|
let mut c2 = self.c2.to_bits(cs.ns(|| "c2"))?;
|
|
|
|
c0.append(&mut c1);
|
|
c0.append(&mut c2);
|
|
|
|
Ok(c0)
|
|
}
|
|
|
|
fn to_non_unique_bits<CS: ConstraintSystem<ConstraintF>>(
|
|
&self,
|
|
mut cs: CS,
|
|
) -> Result<Vec<Boolean>, SynthesisError> {
|
|
let mut c0 = self.c0.to_non_unique_bits(cs.ns(|| "c0"))?;
|
|
let mut c1 = self.c1.to_non_unique_bits(cs.ns(|| "c1"))?;
|
|
let mut c2 = self.c2.to_non_unique_bits(cs.ns(|| "c2"))?;
|
|
|
|
c0.append(&mut c1);
|
|
c0.append(&mut c2);
|
|
|
|
Ok(c0)
|
|
}
|
|
}
|
|
|
|
impl<P, ConstraintF: PrimeField> ToBytesGadget<ConstraintF> for Fp6Gadget<P, ConstraintF>
|
|
where
|
|
P: Fp6Parameters,
|
|
P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
|
|
{
|
|
fn to_bytes<CS: ConstraintSystem<ConstraintF>>(
|
|
&self,
|
|
mut cs: CS,
|
|
) -> Result<Vec<UInt8>, SynthesisError> {
|
|
let mut c0 = self.c0.to_bytes(cs.ns(|| "c0"))?;
|
|
let mut c1 = self.c1.to_bytes(cs.ns(|| "c1"))?;
|
|
let mut c2 = self.c2.to_bytes(cs.ns(|| "c2"))?;
|
|
|
|
c0.append(&mut c1);
|
|
c0.append(&mut c2);
|
|
|
|
Ok(c0)
|
|
}
|
|
|
|
fn to_non_unique_bytes<CS: ConstraintSystem<ConstraintF>>(
|
|
&self,
|
|
mut cs: CS,
|
|
) -> Result<Vec<UInt8>, SynthesisError> {
|
|
let mut c0 = self.c0.to_non_unique_bytes(cs.ns(|| "c0"))?;
|
|
let mut c1 = self.c1.to_non_unique_bytes(cs.ns(|| "c1"))?;
|
|
let mut c2 = self.c2.to_non_unique_bytes(cs.ns(|| "c2"))?;
|
|
|
|
c0.append(&mut c1);
|
|
c0.append(&mut c2);
|
|
|
|
Ok(c0)
|
|
}
|
|
}
|
|
|
|
impl<P, ConstraintF: PrimeField> Clone for Fp6Gadget<P, ConstraintF>
|
|
where
|
|
P: Fp6Parameters,
|
|
P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
|
|
{
|
|
fn clone(&self) -> Self {
|
|
Self::new(self.c0.clone(), self.c1.clone(), self.c2.clone())
|
|
}
|
|
}
|
|
|
|
impl<P, ConstraintF: PrimeField> CondSelectGadget<ConstraintF> for Fp6Gadget<P, ConstraintF>
|
|
where
|
|
P: Fp6Parameters,
|
|
P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
|
|
{
|
|
#[inline]
|
|
fn conditionally_select<CS: ConstraintSystem<ConstraintF>>(
|
|
mut cs: CS,
|
|
cond: &Boolean,
|
|
true_value: &Self,
|
|
false_value: &Self,
|
|
) -> Result<Self, SynthesisError> {
|
|
let c0 = Fp2Gadget::<P, ConstraintF>::conditionally_select(
|
|
&mut cs.ns(|| "c0"),
|
|
cond,
|
|
&true_value.c0,
|
|
&false_value.c0,
|
|
)?;
|
|
let c1 = Fp2Gadget::<P, ConstraintF>::conditionally_select(
|
|
&mut cs.ns(|| "c1"),
|
|
cond,
|
|
&true_value.c1,
|
|
&false_value.c1,
|
|
)?;
|
|
let c2 = Fp2Gadget::<P, ConstraintF>::conditionally_select(
|
|
&mut cs.ns(|| "c2"),
|
|
cond,
|
|
&true_value.c2,
|
|
&false_value.c2,
|
|
)?;
|
|
|
|
Ok(Self::new(c0, c1, c2))
|
|
}
|
|
|
|
fn cost() -> usize {
|
|
3 * <Fp2Gadget<P, ConstraintF> as CondSelectGadget<ConstraintF>>::cost()
|
|
}
|
|
}
|
|
|
|
impl<P, ConstraintF: PrimeField> TwoBitLookupGadget<ConstraintF> for Fp6Gadget<P, ConstraintF>
|
|
where
|
|
P: Fp6Parameters,
|
|
P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
|
|
{
|
|
type TableConstant = Fp6<P>;
|
|
fn two_bit_lookup<CS: ConstraintSystem<ConstraintF>>(
|
|
mut cs: CS,
|
|
b: &[Boolean],
|
|
c: &[Self::TableConstant],
|
|
) -> Result<Self, SynthesisError> {
|
|
let c0s = c.iter().map(|f| f.c0).collect::<Vec<_>>();
|
|
let c1s = c.iter().map(|f| f.c1).collect::<Vec<_>>();
|
|
let c2s = c.iter().map(|f| f.c2).collect::<Vec<_>>();
|
|
let c0 = Fp2Gadget::<P, ConstraintF>::two_bit_lookup(cs.ns(|| "Lookup c0"), b, &c0s)?;
|
|
let c1 = Fp2Gadget::<P, ConstraintF>::two_bit_lookup(cs.ns(|| "Lookup c1"), b, &c1s)?;
|
|
let c2 = Fp2Gadget::<P, ConstraintF>::two_bit_lookup(cs.ns(|| "Lookup c2"), b, &c2s)?;
|
|
Ok(Self::new(c0, c1, c2))
|
|
}
|
|
|
|
fn cost() -> usize {
|
|
3 * <Fp2Gadget<P, ConstraintF> as TwoBitLookupGadget<ConstraintF>>::cost()
|
|
}
|
|
}
|
|
|
|
impl<P, ConstraintF: PrimeField> ThreeBitCondNegLookupGadget<ConstraintF>
|
|
for Fp6Gadget<P, ConstraintF>
|
|
where
|
|
P: Fp6Parameters,
|
|
P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
|
|
{
|
|
type TableConstant = Fp6<P>;
|
|
|
|
fn three_bit_cond_neg_lookup<CS: ConstraintSystem<ConstraintF>>(
|
|
mut cs: CS,
|
|
b: &[Boolean],
|
|
b0b1: &Boolean,
|
|
c: &[Self::TableConstant],
|
|
) -> Result<Self, SynthesisError> {
|
|
let c0s = c.iter().map(|f| f.c0).collect::<Vec<_>>();
|
|
let c1s = c.iter().map(|f| f.c1).collect::<Vec<_>>();
|
|
let c2s = c.iter().map(|f| f.c2).collect::<Vec<_>>();
|
|
let c0 = Fp2Gadget::<P, ConstraintF>::three_bit_cond_neg_lookup(
|
|
cs.ns(|| "Lookup c0"),
|
|
b,
|
|
b0b1,
|
|
&c0s,
|
|
)?;
|
|
let c1 = Fp2Gadget::<P, ConstraintF>::three_bit_cond_neg_lookup(
|
|
cs.ns(|| "Lookup c1"),
|
|
b,
|
|
b0b1,
|
|
&c1s,
|
|
)?;
|
|
let c2 = Fp2Gadget::<P, ConstraintF>::three_bit_cond_neg_lookup(
|
|
cs.ns(|| "Lookup c2"),
|
|
b,
|
|
b0b1,
|
|
&c2s,
|
|
)?;
|
|
Ok(Self::new(c0, c1, c2))
|
|
}
|
|
|
|
fn cost() -> usize {
|
|
3 * <Fp2Gadget<P, ConstraintF> as ThreeBitCondNegLookupGadget<ConstraintF>>::cost()
|
|
}
|
|
}
|
|
|
|
impl<P, ConstraintF: PrimeField> AllocGadget<Fp6<P>, ConstraintF> for Fp6Gadget<P, ConstraintF>
|
|
where
|
|
P: Fp6Parameters,
|
|
P::Fp2Params: Fp2Parameters<Fp = ConstraintF>,
|
|
{
|
|
#[inline]
|
|
fn alloc<F, T, CS: ConstraintSystem<ConstraintF>>(
|
|
mut cs: CS,
|
|
value_gen: F,
|
|
) -> Result<Self, SynthesisError>
|
|
where
|
|
F: FnOnce() -> Result<T, SynthesisError>,
|
|
T: Borrow<Fp6<P>>,
|
|
{
|
|
let (c0, c1, c2) = match value_gen() {
|
|
Ok(fe) => {
|
|
let fe = *fe.borrow();
|
|
(Ok(fe.c0), Ok(fe.c1), Ok(fe.c2))
|
|
},
|
|
_ => (
|
|
Err(SynthesisError::AssignmentMissing),
|
|
Err(SynthesisError::AssignmentMissing),
|
|
Err(SynthesisError::AssignmentMissing),
|
|
),
|
|
};
|
|
|
|
let c0 = Fp2Gadget::<P, ConstraintF>::alloc(&mut cs.ns(|| "c0"), || c0)?;
|
|
let c1 = Fp2Gadget::<P, ConstraintF>::alloc(&mut cs.ns(|| "c1"), || c1)?;
|
|
let c2 = Fp2Gadget::<P, ConstraintF>::alloc(&mut cs.ns(|| "c2"), || c2)?;
|
|
Ok(Self::new(c0, c1, c2))
|
|
}
|
|
|
|
#[inline]
|
|
fn alloc_input<F, T, CS: ConstraintSystem<ConstraintF>>(
|
|
mut cs: CS,
|
|
value_gen: F,
|
|
) -> Result<Self, SynthesisError>
|
|
where
|
|
F: FnOnce() -> Result<T, SynthesisError>,
|
|
T: Borrow<Fp6<P>>,
|
|
{
|
|
let (c0, c1, c2) = match value_gen() {
|
|
Ok(fe) => {
|
|
let fe = *fe.borrow();
|
|
(Ok(fe.c0), Ok(fe.c1), Ok(fe.c2))
|
|
},
|
|
_ => (
|
|
Err(SynthesisError::AssignmentMissing),
|
|
Err(SynthesisError::AssignmentMissing),
|
|
Err(SynthesisError::AssignmentMissing),
|
|
),
|
|
};
|
|
|
|
let c0 = Fp2Gadget::<P, ConstraintF>::alloc_input(&mut cs.ns(|| "c0"), || c0)?;
|
|
let c1 = Fp2Gadget::<P, ConstraintF>::alloc_input(&mut cs.ns(|| "c1"), || c1)?;
|
|
let c2 = Fp2Gadget::<P, ConstraintF>::alloc_input(&mut cs.ns(|| "c2"), || c2)?;
|
|
Ok(Self::new(c0, c1, c2))
|
|
}
|
|
}
|