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https://github.com/arnaucube/ark-r1cs-std.git
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MNT4/6 curves and recursive SNARKs (#150)
* Add mnt6_753 curve Generalize mnt6 curve model * Add mnt4 curves * Use resampled generators * Calculate correct G2 cofactors * Add fields to r1cs-std * Add pairings * Improve reusing of Fq/Fr among MNT curves * Add instantiations of curves Fix Fp6_2over3 Rebase code to current master * Add test for recursive NIZK proof verification * Address comments in PR * Improve test case and port to GM17 Also fix a minor bug in to_field_vec
This commit is contained in:
Pascal Berrang
GitHub
338
r1cs-std/src/pairing/mnt4/mod.rs
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338
r1cs-std/src/pairing/mnt4/mod.rs
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@@ -0,0 +1,338 @@
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use r1cs_core::{ConstraintSystem, SynthesisError};
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use super::PairingGadget as PG;
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use crate::{
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fields::{fp::FpGadget, fp2::Fp2Gadget, fp4::Fp4Gadget, FieldGadget},
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groups::mnt4::{
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AteAdditionCoefficientsGadget, AteDoubleCoefficientsGadget, G1Gadget, G1PreparedGadget,
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G2Gadget, G2PreparedGadget, G2ProjectiveExtendedGadget,
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},
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};
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use algebra::{
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curves::mnt4::{MNT4Parameters, MNT4},
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fields::BitIterator,
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};
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use core::marker::PhantomData;
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pub struct PairingGadget<P: MNT4Parameters>(PhantomData<P>);
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type Fp2G<P> = Fp2Gadget<<P as MNT4Parameters>::Fp2Params, <P as MNT4Parameters>::Fp>;
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type Fp4G<P> = Fp4Gadget<<P as MNT4Parameters>::Fp4Params, <P as MNT4Parameters>::Fp>;
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pub type GTGadget<P> = Fp4G<P>;
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impl<P: MNT4Parameters> PairingGadget<P> {
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pub(crate) fn doubling_step_for_flipped_miller_loop<CS: ConstraintSystem<P::Fp>>(
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mut cs: CS,
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r: &G2ProjectiveExtendedGadget<P>,
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) -> Result<
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(
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G2ProjectiveExtendedGadget<P>,
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AteDoubleCoefficientsGadget<P>,
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),
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SynthesisError,
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> {
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let a = r.t.square(cs.ns(|| "r.t^2"))?;
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let b = r.x.square(cs.ns(|| "r.x^2"))?;
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let c = r.y.square(cs.ns(|| "r.y^2"))?;
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let d = c.square(cs.ns(|| "c^2"))?;
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let mut e = r.x.add(cs.ns(|| "r.x + c"), &c)?;
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e.square_in_place(cs.ns(|| "(r.x + c)^2"))?;
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e.sub_in_place(cs.ns(|| "(r.x + c)^2 - b"), &b)?;
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e.sub_in_place(cs.ns(|| "(r.x + c)^2 - b - d"), &d)?;
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let mut f = b.double(cs.ns(|| "b + b"))?;
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f.add_in_place(cs.ns(|| "b + b + b"), &b)?;
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let twist_a = a.mul_by_constant(cs.ns(|| "TWIST_COEFF_A * a"), &P::TWIST_COEFF_A)?;
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f.add_in_place(cs.ns(|| "(b + b + b) + (TWIST_COEFF_A * a)"), &twist_a)?;
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let g = f.square(cs.ns(|| "f^2"))?;
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let d_eight = d
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.double(cs.ns(|| "2 * d"))?
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.double(cs.ns(|| "4 * d"))?
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.double(cs.ns(|| "8 * d"))?;
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let e2 = e.double(cs.ns(|| "2 * e"))?;
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let e4 = e2.double(cs.ns(|| "4 * e"))?;
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let x = g.sub(cs.ns(|| "- (e + e + e + e) + g"), &e4)?;
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let mut y = e2.sub(cs.ns(|| "e + e - x"), &x)?;
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y.mul_in_place(cs.ns(|| "f * (e + e - x)"), &f)?;
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y.sub_in_place(cs.ns(|| "- d_eight + f * (e + e - x)"), &d_eight)?;
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let mut z = r.y.add(cs.ns(|| "r.y + r.z"), &r.z)?;
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z.square_in_place(cs.ns(|| "(r.y + r.z)^2"))?;
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z.sub_in_place(cs.ns(|| "(r.y + r.z)^2 - c"), &c)?;
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let z2 = r.z.square(cs.ns(|| "r.z^2"))?;
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z.sub_in_place(cs.ns(|| "(r.y + r.z)^2 - c - r.z^2"), &z2)?;
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let t = z.square(cs.ns(|| "z^2"))?;
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let r2 = G2ProjectiveExtendedGadget { x, y, z, t };
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let c_h =
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r2.z.add(cs.ns(|| "r2.z + r.t"), &r.t)?
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.square(cs.ns(|| "(r2.z + r.t)^2"))?
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.sub(cs.ns(|| "(r2.z + r.t)^2 - r2.t"), &r2.t)?
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.sub(cs.ns(|| "(r2.z + r.t)^2 - r2.t - a"), &a)?;
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let c_4c = c.double(cs.ns(|| "2 * c"))?.double(cs.ns(|| "4 * c"))?;
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let mut c_j = f.add(cs.ns(|| "f + r.t"), &r.t)?;
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c_j.square_in_place(cs.ns(|| "(f + r.t)^2"))?;
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c_j.sub_in_place(cs.ns(|| "(f + r.t)^2 - g"), &g)?;
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c_j.sub_in_place(cs.ns(|| "(f + r.t)^2 - g - a"), &a)?;
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let mut c_l = f.add(cs.ns(|| "f + r.x"), &r.x)?;
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c_l.square_in_place(cs.ns(|| "(f + r.x)^2"))?;
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c_l.sub_in_place(cs.ns(|| "(f + r.x)^2 - g"), &g)?;
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c_l.sub_in_place(cs.ns(|| "(f + r.x)^2 - g - b"), &b)?;
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let coeff = AteDoubleCoefficientsGadget {
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c_h,
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c_4c,
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c_j,
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c_l,
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};
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Ok((r2, coeff))
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}
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pub(crate) fn mixed_addition_step_for_flipped_miller_loop<CS: ConstraintSystem<P::Fp>>(
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mut cs: CS,
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x: &Fp2G<P>,
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y: &Fp2G<P>,
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r: &G2ProjectiveExtendedGadget<P>,
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) -> Result<
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(
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G2ProjectiveExtendedGadget<P>,
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AteAdditionCoefficientsGadget<P>,
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),
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SynthesisError,
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> {
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let a = y.square(cs.ns(|| "y^2"))?;
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let b = r.t.mul(cs.ns(|| "r.t * x"), &x)?;
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let mut d = r.z.add(cs.ns(|| "r.z + y"), &y)?;
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d.square_in_place(cs.ns(|| "(r.z + y)^2"))?;
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d.sub_in_place(cs.ns(|| "(r.z + y)^2 - a"), &a)?;
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d.sub_in_place(cs.ns(|| "(r.z + y)^2 - a - r.t"), &r.t)?;
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d.mul_in_place(cs.ns(|| "((r.z + y)^2 - a - r.t) * r.t"), &r.t)?;
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let h = b.sub(cs.ns(|| "b - r.x"), &r.x)?;
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let i = h.square(cs.ns(|| "h^2"))?;
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let e = i.double(cs.ns(|| "2 * i"))?.double(cs.ns(|| "4 * i"))?;
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let j = h.mul(cs.ns(|| "h * e"), &e)?;
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let v = r.x.mul(cs.ns(|| "r.x * e"), &e)?;
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let ry2 = r.y.double(cs.ns(|| "r.y + r.y"))?;
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let l1 = d.sub(cs.ns(|| "d - (r.y + r.y)"), &ry2)?;
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let v2 = v.double(cs.ns(|| "v + v"))?;
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let x = l1
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.square(cs.ns(|| "l1^2"))?
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.sub(cs.ns(|| "l1^2 - j"), &j)?
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.sub(cs.ns(|| "l1^2 - j - (v + v)"), &v2)?;
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let v_minus_x = v.sub(cs.ns(|| "v - x"), &x)?;
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let j_ry2 = j.mul(cs.ns(|| "j * (r.y + r.y)"), &ry2)?;
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let y = l1
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.mul(cs.ns(|| "l1 * (v - x)"), &v_minus_x)?
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.sub(cs.ns(|| "l1 * (v - x) - (j * (r.y + r.y)"), &j_ry2)?;
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let mut z = r.z.add(cs.ns(|| "r.z + h"), &h)?;
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z.square_in_place(cs.ns(|| "(r.z + h)^2"))?;
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z.sub_in_place(cs.ns(|| "(r.z + h)^2 - r.t"), &r.t)?;
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z.sub_in_place(cs.ns(|| "(r.z + h)^2 - r.t - i"), &i)?;
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let t = z.square(cs.ns(|| "z^2"))?;
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let r2 = G2ProjectiveExtendedGadget {
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x,
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y,
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z: z.clone(),
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t,
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};
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let coeff = AteAdditionCoefficientsGadget { c_l1: l1, c_rz: z };
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Ok((r2, coeff))
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}
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pub fn ate_miller_loop<CS: ConstraintSystem<P::Fp>>(
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mut cs: CS,
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p: &G1PreparedGadget<P>,
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q: &G2PreparedGadget<P>,
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) -> Result<Fp4G<P>, SynthesisError> {
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let mut l1_coeff = Fp2G::<P>::new(p.x.clone(), FpGadget::<P::Fp>::zero(cs.ns(|| "zero"))?);
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l1_coeff.sub_in_place(cs.ns(|| "l1_coeff"), &q.x_over_twist)?;
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let mut f = Fp4G::<P>::one(cs.ns(|| "one"))?;
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let mut dbl_idx: usize = 0;
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let mut add_idx: usize = 0;
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let mut found_one = false;
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for (j, bit) in BitIterator::new(P::ATE_LOOP_COUNT).enumerate() {
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// code below gets executed for all bits (EXCEPT the MSB itself) of
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// mnt6_param_p (skipping leading zeros) in MSB to LSB order
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if !found_one && bit {
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found_one = true;
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continue;
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} else if !found_one {
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continue;
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}
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let mut cs = cs.ns(|| format!("bit {}", j));
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let dc = &q.double_coefficients[dbl_idx];
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dbl_idx += 1;
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let c_j_x_twist = dc.c_j.mul(cs.ns(|| "dc.c_j * p.x_twist"), &p.x_twist)?;
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let c0 = dc.c_l.sub(cs.ns(|| "-dc.c_4c + dc.c_l"), &dc.c_4c)?.sub(
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cs.ns(|| "-dc.c_4c - (dc.c_j * p.x_twist) + dc.c_l"),
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&c_j_x_twist,
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)?;
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let c1 = dc.c_h.mul(cs.ns(|| "dc.c_h * p.y_twist"), &p.y_twist)?;
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let g_rr_at_p = Fp4G::<P>::new(c0, c1);
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f = f
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.square(cs.ns(|| "f^2"))?
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.mul(cs.ns(|| "f^2 * g_rr_at_p"), &g_rr_at_p)?;
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if bit {
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let ac = &q.addition_coefficients[add_idx];
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add_idx += 1;
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let l1_coeff_c_l1 = l1_coeff.mul(cs.ns(|| "l1_coeff * ac.c_l1"), &ac.c_l1)?;
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let g_rq_at_p = Fp4G::<P>::new(
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ac.c_rz.mul(cs.ns(|| "ac.c_rz * p.y_twist"), &p.y_twist)?,
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q.y_over_twist
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.mul(cs.ns(|| "q.y_over_twist * ac.c_rz"), &ac.c_rz)?
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.add(
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cs.ns(|| "q.y_over_twist * ac.c_rz + (l1_coeff * ac.c_l1)"),
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&l1_coeff_c_l1,
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)?
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.negate(cs.ns(|| "-(q.y_over_twist * ac.c_rz + (l1_coeff * ac.c_l1))"))?,
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);
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f.mul_in_place(cs.ns(|| "f *= g_rq_at_p"), &g_rq_at_p)?;
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}
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}
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if P::ATE_IS_LOOP_COUNT_NEG {
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let ac = &q.addition_coefficients[add_idx];
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let l1_coeff_c_l1 = l1_coeff.mul(cs.ns(|| "l1_coeff * ac.c_l1"), &ac.c_l1)?;
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let g_rnegr_at_p = Fp4G::<P>::new(
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ac.c_rz.mul(cs.ns(|| "ac.c_rz * p.y_twist"), &p.y_twist)?,
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q.y_over_twist
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.mul(cs.ns(|| "q.y_over_twist * ac.c_rz"), &ac.c_rz)?
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.add(
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cs.ns(|| "q.y_over_twist * ac.c_rz + (l1_coeff * ac.c_l1)"),
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&l1_coeff_c_l1,
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)?
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.negate(cs.ns(|| "-(q.y_over_twist * ac.c_rz + (l1_coeff * ac.c_l1))"))?,
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);
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f = f
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.mul(cs.ns(|| "f * g_rnegr_at_p"), &g_rnegr_at_p)?
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.inverse(cs.ns(|| "inverse f"))?;
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}
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Ok(f)
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}
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pub fn final_exponentiation<CS: ConstraintSystem<P::Fp>>(
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mut cs: CS,
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value: &Fp4G<P>,
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) -> Result<GTGadget<P>, SynthesisError> {
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let value_inv = value.inverse(cs.ns(|| "value inverse"))?;
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let value_to_first_chunk = Self::final_exponentiation_first_chunk(
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cs.ns(|| "value_to_first_chunk"),
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value,
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&value_inv,
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)?;
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let value_inv_to_first_chunk = Self::final_exponentiation_first_chunk(
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cs.ns(|| "value_inv_to_first_chunk"),
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&value_inv,
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value,
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)?;
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Self::final_exponentiation_last_chunk(
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cs.ns(|| "final_exp_last_chunk"),
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&value_to_first_chunk,
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&value_inv_to_first_chunk,
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)
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}
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fn final_exponentiation_first_chunk<CS: ConstraintSystem<P::Fp>>(
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mut cs: CS,
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elt: &Fp4G<P>,
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elt_inv: &Fp4G<P>,
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) -> Result<Fp4G<P>, SynthesisError> {
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// (q^2-1)
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// elt_q2 = elt^(q^2)
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let mut elt_q2 = elt.clone();
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elt_q2.frobenius_map_in_place(cs.ns(|| "frobenius 2"), 2)?;
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// elt_q2_over_elt = elt^(q^2-1)
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elt_q2.mul(cs.ns(|| "elt_q2 * elt_inv"), elt_inv)
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}
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fn final_exponentiation_last_chunk<CS: ConstraintSystem<P::Fp>>(
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mut cs: CS,
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elt: &Fp4G<P>,
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elt_inv: &Fp4G<P>,
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) -> Result<Fp4G<P>, SynthesisError> {
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let elt_clone = elt.clone();
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let elt_inv_clone = elt_inv.clone();
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let mut elt_q = elt.clone();
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elt_q.frobenius_map_in_place(cs.ns(|| "frobenius 1"), 1)?;
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let w1_part = elt_q.cyclotomic_exp(cs.ns(|| "w1_part"), &P::FINAL_EXPONENT_LAST_CHUNK_1)?;
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let w0_part;
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if P::FINAL_EXPONENT_LAST_CHUNK_W0_IS_NEG {
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w0_part = elt_inv_clone
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.cyclotomic_exp(cs.ns(|| "w0_part"), &P::FINAL_EXPONENT_LAST_CHUNK_ABS_OF_W0)?;
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} else {
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w0_part = elt_clone
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.cyclotomic_exp(cs.ns(|| "w0_part"), &P::FINAL_EXPONENT_LAST_CHUNK_ABS_OF_W0)?;
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}
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w1_part.mul(cs.ns(|| "w1_part * w0_part"), &w0_part)
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}
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}
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impl<P: MNT4Parameters> PG<MNT4<P>, P::Fp> for PairingGadget<P> {
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type G1Gadget = G1Gadget<P>;
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type G2Gadget = G2Gadget<P>;
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type G1PreparedGadget = G1PreparedGadget<P>;
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type G2PreparedGadget = G2PreparedGadget<P>;
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type GTGadget = GTGadget<P>;
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fn miller_loop<CS: ConstraintSystem<P::Fp>>(
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mut cs: CS,
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ps: &[Self::G1PreparedGadget],
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qs: &[Self::G2PreparedGadget],
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) -> Result<Self::GTGadget, SynthesisError> {
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let mut result = Fp4G::<P>::one(cs.ns(|| "one"))?;
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for (i, (p, q)) in ps.iter().zip(qs.iter()).enumerate() {
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let miller =
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Self::ate_miller_loop(cs.ns(|| format!("ate miller loop iteration {}", i)), p, q)?;
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result.mul_in_place(
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cs.ns(|| format!("mul ate miller loop iteration {}", i)),
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&miller,
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)?;
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}
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Ok(result)
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}
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fn final_exponentiation<CS: ConstraintSystem<P::Fp>>(
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cs: CS,
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r: &Self::GTGadget,
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) -> Result<Self::GTGadget, SynthesisError> {
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Self::final_exponentiation(cs, r)
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}
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fn prepare_g1<CS: ConstraintSystem<P::Fp>>(
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cs: CS,
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p: &Self::G1Gadget,
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) -> Result<Self::G1PreparedGadget, SynthesisError> {
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Self::G1PreparedGadget::from_affine(cs, p)
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}
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fn prepare_g2<CS: ConstraintSystem<P::Fp>>(
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cs: CS,
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q: &Self::G2Gadget,
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) -> Result<Self::G2PreparedGadget, SynthesisError> {
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Self::G2PreparedGadget::from_affine(cs, q)
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}
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}
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