halo2/halo2_backend/src/poly/kzg/multiopen/shplonk/prover.rs

use super::{
    construct_intermediate_sets, ChallengeU, ChallengeV, ChallengeY, Commitment, RotationSet,
};
use crate::arithmetic::{
    eval_polynomial, evaluate_vanishing_polynomial, kate_division, lagrange_interpolate,
    parallelize, powers, CurveAffine,
};
use crate::helpers::SerdeCurveAffine;
use crate::poly::commitment::{Blind, ParamsProver, Prover};
use crate::poly::kzg::commitment::{KZGCommitmentScheme, ParamsKZG};
use crate::poly::query::{PolynomialPointer, ProverQuery};
use crate::poly::{Coeff, Polynomial};
use crate::transcript::{EncodedChallenge, TranscriptWrite};

use crate::multicore::{IntoParallelIterator, ParallelIterator};
use group::Curve;
use halo2_middleware::ff::Field;
use halo2_middleware::zal::traits::MsmAccel;
use halo2curves::pairing::Engine;
use halo2curves::CurveExt;
use rand_core::RngCore;
use std::fmt::Debug;
use std::io;
use std::marker::PhantomData;
use std::ops::MulAssign;

fn div_by_vanishing<F: Field>(poly: Polynomial<F, Coeff>, roots: &[F]) -> Vec<F> {
    let poly = roots
        .iter()
        .fold(poly.values, |poly, point| kate_division(&poly, *point));

    poly
}

struct CommitmentExtension<'a, C: CurveAffine> {
    commitment: Commitment<C::Scalar, PolynomialPointer<'a, C>>,
    low_degree_equivalent: Polynomial<C::Scalar, Coeff>,
}

impl<'a, C: CurveAffine> Commitment<C::Scalar, PolynomialPointer<'a, C>> {
    fn extend(&self, points: &[C::Scalar]) -> CommitmentExtension<'a, C> {
        let poly = lagrange_interpolate(points, &self.evals()[..]);

        let low_degree_equivalent = Polynomial {
            values: poly,
            _marker: PhantomData,
        };

        CommitmentExtension {
            commitment: self.clone(),
            low_degree_equivalent,
        }
    }
}

impl<'a, C: CurveAffine> CommitmentExtension<'a, C> {
    fn linearisation_contribution(&self, u: C::Scalar) -> Polynomial<C::Scalar, Coeff> {
        let p_x = self.commitment.get().poly;
        let r_eval = eval_polynomial(&self.low_degree_equivalent.values[..], u);
        p_x - r_eval
    }

    fn quotient_contribution(&self) -> Polynomial<C::Scalar, Coeff> {
        let len = self.low_degree_equivalent.len();
        let mut p_x = self.commitment.get().poly.clone();
        parallelize(&mut p_x.values[0..len], |lhs, start| {
            for (lhs, rhs) in lhs
                .iter_mut()
                .zip(self.low_degree_equivalent.values[start..].iter())
            {
                *lhs -= *rhs;
            }
        });
        p_x
    }
}

struct RotationSetExtension<'a, C: CurveAffine> {
    commitments: Vec<CommitmentExtension<'a, C>>,
    points: Vec<C::Scalar>,
}

impl<'a, C: CurveAffine> RotationSet<C::Scalar, PolynomialPointer<'a, C>> {
    fn extend(self, commitments: Vec<CommitmentExtension<'a, C>>) -> RotationSetExtension<'a, C> {
        RotationSetExtension {
            commitments,
            points: self.points,
        }
    }
}

/// Concrete KZG prover with SHPLONK variant
#[derive(Debug)]
pub struct ProverSHPLONK<'a, E: Engine> {
    params: &'a ParamsKZG<E>,
}

impl<'a, E: Engine> ProverSHPLONK<'a, E> {
    /// Given parameters creates new prover instance
    pub fn new(params: &'a ParamsKZG<E>) -> Self {
        Self { params }
    }
}

/// Create a multi-opening proof
impl<'params, E: Engine + Debug> Prover<'params, KZGCommitmentScheme<E>>
    for ProverSHPLONK<'params, E>
where
    E::Fr: Ord,
    E::G1Affine: SerdeCurveAffine<ScalarExt = <E as Engine>::Fr, CurveExt = <E as Engine>::G1>,
    E::G1: CurveExt<AffineExt = E::G1Affine>,
    E::G2Affine: SerdeCurveAffine,
{
    const QUERY_INSTANCE: bool = false;

    fn new(params: &'params ParamsKZG<E>) -> Self {
        Self { params }
    }

    /// Create a multi-opening proof
    fn create_proof_with_engine<
        'com,
        Ch: EncodedChallenge<E::G1Affine>,
        T: TranscriptWrite<E::G1Affine, Ch>,
        R,
        I,
    >(
        &self,
        engine: &impl MsmAccel<E::G1Affine>,
        _: R,
        transcript: &mut T,
        queries: I,
    ) -> io::Result<()>
    where
        I: IntoIterator<Item = ProverQuery<'com, E::G1Affine>> + Clone,
        R: RngCore,
    {
        // TODO: explore if it is safe to use same challenge
        // for different sets that are already combined with another challenge
        let y: ChallengeY<_> = transcript.squeeze_challenge_scalar();

        let quotient_contribution = |rotation_set: &RotationSetExtension<E::G1Affine>| {
            // [P_i_0(X) - R_i_0(X), P_i_1(X) - R_i_1(X), ... ]
            #[allow(clippy::needless_collect)]
            let numerators = rotation_set
                .commitments
                .as_slice()
                .into_par_iter()
                .map(|commitment| commitment.quotient_contribution())
                .collect::<Vec<_>>();

            // define numerator polynomial as
            // N_i_j(X) = (P_i_j(X) - R_i_j(X))
            // and combine polynomials with same evaluation point set
            // N_i(X) = linear_combination(y, N_i_j(X))
            // where y is random scalar to combine numerator polynomials
            let n_x = numerators
                .into_iter()
                .zip(powers(*y))
                .map(|(numerator, power_of_y)| numerator * power_of_y)
                .reduce(|acc, numerator| acc + &numerator)
                .unwrap();

            let points = &rotation_set.points[..];

            // quotient contribution of this evaluation set is
            // Q_i(X) = N_i(X) / Z_i(X) where
            // Z_i(X) = (x - r_i_0) * (x - r_i_1) * ...
            let mut poly = div_by_vanishing(n_x, points);
            poly.resize(self.params.n as usize, E::Fr::ZERO);

            Polynomial {
                values: poly,
                _marker: PhantomData,
            }
        };

        let intermediate_sets = construct_intermediate_sets(queries);
        let (rotation_sets, super_point_set) = (
            intermediate_sets.rotation_sets,
            intermediate_sets.super_point_set,
        );

        let rotation_sets: Vec<RotationSetExtension<E::G1Affine>> = rotation_sets
            .into_par_iter()
            .map(|rotation_set| {
                let commitments: Vec<CommitmentExtension<E::G1Affine>> = rotation_set
                    .commitments
                    .as_slice()
                    .into_par_iter()
                    .map(|commitment_data| commitment_data.extend(&rotation_set.points))
                    .collect();
                rotation_set.extend(commitments)
            })
            .collect();

        let v: ChallengeV<_> = transcript.squeeze_challenge_scalar();

        #[allow(clippy::needless_collect)]
        let quotient_polynomials = rotation_sets
            .as_slice()
            .into_par_iter()
            .map(quotient_contribution)
            .collect::<Vec<_>>();

        let h_x: Polynomial<E::Fr, Coeff> = quotient_polynomials
            .into_iter()
            .zip(powers(*v))
            .map(|(poly, power_of_v)| poly * power_of_v)
            .reduce(|acc, poly| acc + &poly)
            .unwrap();

        let h = self
            .params
            .commit(engine, &h_x, Blind::default())
            .to_affine();
        transcript.write_point(h)?;
        let u: ChallengeU<_> = transcript.squeeze_challenge_scalar();

        let linearisation_contribution = |rotation_set: RotationSetExtension<E::G1Affine>| {
            let mut diffs = super_point_set.clone();
            for point in rotation_set.points.iter() {
                diffs.remove(point);
            }
            let diffs = diffs.into_iter().collect::<Vec<_>>();

            // calculate difference vanishing polynomial evaluation
            let z_i = evaluate_vanishing_polynomial(&diffs[..], *u);

            // inner linearisation contributions are
            // [P_i_0(X) - r_i_0, P_i_1(X) - r_i_1, ... ] where
            // r_i_j = R_i_j(u) is the evaluation of low degree equivalent polynomial
            // where u is random evaluation point
            #[allow(clippy::needless_collect)]
            let inner_contributions = rotation_set
                .commitments
                .as_slice()
                .into_par_iter()
                .map(|commitment| commitment.linearisation_contribution(*u))
                .collect::<Vec<_>>();

            // define inner contributor polynomial as
            // L_i_j(X) = (P_i_j(X) - r_i_j)
            // and combine polynomials with same evaluation point set
            // L_i(X) = linear_combination(y, L_i_j(X))
            // where y is random scalar to combine inner contributors
            let l_x: Polynomial<E::Fr, Coeff> = inner_contributions
                .into_iter()
                .zip(powers(*y))
                .map(|(poly, power_of_y)| poly * power_of_y)
                .reduce(|acc, poly| acc + &poly)
                .unwrap();

            // finally scale l_x by difference vanishing polynomial evaluation z_i
            (l_x * z_i, z_i)
        };

        #[allow(clippy::type_complexity)]
        let (linearisation_contributions, z_diffs): (
            Vec<Polynomial<E::Fr, Coeff>>,
            Vec<E::Fr>,
        ) = rotation_sets
            .into_par_iter()
            .map(linearisation_contribution)
            .unzip();

        let l_x: Polynomial<E::Fr, Coeff> = linearisation_contributions
            .into_iter()
            .zip(powers(*v))
            .map(|(poly, power_of_v)| poly * power_of_v)
            .reduce(|acc, poly| acc + &poly)
            .unwrap();

        let super_point_set = super_point_set.into_iter().collect::<Vec<_>>();
        let zt_eval = evaluate_vanishing_polynomial(&super_point_set[..], *u);
        let l_x = l_x - &(h_x * zt_eval);

        // sanity check
        #[cfg(debug_assertions)]
        {
            let must_be_zero = eval_polynomial(&l_x.values[..], *u);
            assert_eq!(must_be_zero, E::Fr::ZERO);
        }

        let mut h_x = div_by_vanishing(l_x, &[*u]);

        // normalize coefficients by the coefficient of the first polynomial
        let z_0_diff_inv = z_diffs[0].invert().unwrap();
        for h_i in h_x.iter_mut() {
            h_i.mul_assign(z_0_diff_inv)
        }

        let h_x = Polynomial {
            values: h_x,
            _marker: PhantomData,
        };

        let h = self
            .params
            .commit(engine, &h_x, Blind::default())
            .to_affine();
        transcript.write_point(h)?;

        Ok(())
    }
}