halo2/halo2_backend/src/plonk/permutation/prover.rs

use group::{
    ff::{BatchInvert, Field},
    Curve,
};
use halo2_middleware::ff::PrimeField;
use rand_core::RngCore;
use std::iter::{self, ExactSizeIterator};

use super::Argument;
use crate::{
    arithmetic::{eval_polynomial, parallelize, CurveAffine},
    plonk::{self, permutation::ProvingKey, ChallengeBeta, ChallengeGamma, ChallengeX, Error},
    poly::{
        commitment::{Blind, Params},
        Coeff, ExtendedLagrangeCoeff, LagrangeCoeff, Polynomial, ProverQuery,
    },
    transcript::{EncodedChallenge, TranscriptWrite},
};
use halo2_middleware::circuit::Any;
use halo2_middleware::poly::Rotation;

// TODO: Document a bit these types
// https://github.com/privacy-scaling-explorations/halo2/issues/264

pub(crate) struct CommittedSet<C: CurveAffine> {
    pub(crate) permutation_product_poly: Polynomial<C::Scalar, Coeff>,
    pub(crate) permutation_product_coset: Polynomial<C::Scalar, ExtendedLagrangeCoeff>,
    permutation_product_blind: Blind<C::Scalar>,
}

pub(crate) struct Committed<C: CurveAffine> {
    pub(crate) sets: Vec<CommittedSet<C>>,
}

pub(crate) struct ConstructedSet<C: CurveAffine> {
    permutation_product_poly: Polynomial<C::Scalar, Coeff>,
    permutation_product_blind: Blind<C::Scalar>,
}

pub(crate) struct Constructed<C: CurveAffine> {
    sets: Vec<ConstructedSet<C>>,
}

pub(crate) struct Evaluated<C: CurveAffine> {
    constructed: Constructed<C>,
}

#[allow(clippy::too_many_arguments)]
pub(in crate::plonk) fn permutation_commit<
    'params,
    C: CurveAffine,
    P: Params<'params, C>,
    E: EncodedChallenge<C>,
    R: RngCore,
    T: TranscriptWrite<C, E>,
>(
    arg: &Argument,
    params: &P,
    pk: &plonk::ProvingKey<C>,
    pkey: &ProvingKey<C>,
    advice: &[Polynomial<C::Scalar, LagrangeCoeff>],
    fixed: &[Polynomial<C::Scalar, LagrangeCoeff>],
    instance: &[Polynomial<C::Scalar, LagrangeCoeff>],
    beta: ChallengeBeta<C>,
    gamma: ChallengeGamma<C>,
    mut rng: R,
    transcript: &mut T,
) -> Result<Committed<C>, Error> {
    let domain = &pk.vk.domain;

    // How many columns can be included in a single permutation polynomial?
    // We need to multiply by z(X) and (1 - (l_last(X) + l_blind(X))). This
    // will never underflow because of the requirement of at least a degree
    // 3 circuit for the permutation argument.
    assert!(pk.vk.cs_degree >= 3);
    let chunk_len = pk.vk.cs_degree - 2;
    let blinding_factors = pk.vk.cs.blinding_factors();

    // Each column gets its own delta power.
    let mut deltaomega = C::Scalar::ONE;

    // Track the "last" value from the previous column set
    let mut last_z = C::Scalar::ONE;

    let mut sets = vec![];

    for (columns, permutations) in arg
        .columns
        .chunks(chunk_len)
        .zip(pkey.permutations.chunks(chunk_len))
    {
        // Goal is to compute the products of fractions
        //
        // (p_j(\omega^i) + \delta^j \omega^i \beta + \gamma) /
        // (p_j(\omega^i) + \beta s_j(\omega^i) + \gamma)
        //
        // where p_j(X) is the jth column in this permutation,
        // and i is the ith row of the column.

        let mut modified_values = vec![C::Scalar::ONE; params.n() as usize];

        // Iterate over each column of the permutation
        for (&column, permuted_column_values) in columns.iter().zip(permutations.iter()) {
            let values = match column.column_type {
                Any::Advice(_) => advice,
                Any::Fixed => fixed,
                Any::Instance => instance,
            };
            parallelize(&mut modified_values, |modified_values, start| {
                for ((modified_values, value), permuted_value) in modified_values
                    .iter_mut()
                    .zip(values[column.index][start..].iter())
                    .zip(permuted_column_values[start..].iter())
                {
                    *modified_values *= *beta * permuted_value + *gamma + value;
                }
            });
        }

        // Invert to obtain the denominator for the permutation product polynomial
        modified_values.batch_invert();

        // Iterate over each column again, this time finishing the computation
        // of the entire fraction by computing the numerators
        for &column in columns.iter() {
            let omega = domain.get_omega();
            let values = match column.column_type {
                Any::Advice(_) => advice,
                Any::Fixed => fixed,
                Any::Instance => instance,
            };
            parallelize(&mut modified_values, |modified_values, start| {
                let mut deltaomega = deltaomega * omega.pow_vartime([start as u64, 0, 0, 0]);
                for (modified_values, value) in modified_values
                    .iter_mut()
                    .zip(values[column.index][start..].iter())
                {
                    // Multiply by p_j(\omega^i) + \delta^j \omega^i \beta
                    *modified_values *= deltaomega * *beta + *gamma + value;
                    deltaomega *= &omega;
                }
            });
            deltaomega *= &<C::Scalar as PrimeField>::DELTA;
        }

        // The modified_values vector is a vector of products of fractions
        // of the form
        //
        // (p_j(\omega^i) + \delta^j \omega^i \beta + \gamma) /
        // (p_j(\omega^i) + \beta s_j(\omega^i) + \gamma)
        //
        // where i is the index into modified_values, for the jth column in
        // the permutation

        // Compute the evaluations of the permutation product polynomial
        // over our domain, starting with z[0] = 1
        let mut z = vec![last_z];
        for row in 1..(params.n() as usize) {
            let mut tmp = z[row - 1];

            tmp *= &modified_values[row - 1];
            z.push(tmp);
        }
        let mut z = domain.lagrange_from_vec(z);
        // Set blinding factors
        for z in &mut z[params.n() as usize - blinding_factors..] {
            *z = C::Scalar::random(&mut rng);
        }
        // Set new last_z
        last_z = z[params.n() as usize - (blinding_factors + 1)];

        let blind = Blind(C::Scalar::random(&mut rng));

        let permutation_product_commitment_projective = params.commit_lagrange(&z, blind);
        let permutation_product_blind = blind;
        let z = domain.lagrange_to_coeff(z);
        let permutation_product_poly = z.clone();

        let permutation_product_coset = domain.coeff_to_extended(z.clone());

        let permutation_product_commitment = permutation_product_commitment_projective.to_affine();

        // Hash the permutation product commitment
        transcript.write_point(permutation_product_commitment)?;

        sets.push(CommittedSet {
            permutation_product_poly,
            permutation_product_coset,
            permutation_product_blind,
        });
    }

    Ok(Committed { sets })
}

impl<C: CurveAffine> Committed<C> {
    pub(in crate::plonk) fn construct(self) -> Constructed<C> {
        Constructed {
            sets: self
                .sets
                .iter()
                .map(|set| ConstructedSet {
                    permutation_product_poly: set.permutation_product_poly.clone(),
                    permutation_product_blind: set.permutation_product_blind,
                })
                .collect(),
        }
    }
}

impl<C: CurveAffine> super::ProvingKey<C> {
    pub(in crate::plonk) fn open(
        &self,
        x: ChallengeX<C>,
    ) -> impl Iterator<Item = ProverQuery<'_, C>> + Clone {
        self.polys.iter().map(move |poly| ProverQuery {
            point: *x,
            poly,
            blind: Blind::default(),
        })
    }

    pub(in crate::plonk) fn evaluate<E: EncodedChallenge<C>, T: TranscriptWrite<C, E>>(
        &self,
        x: ChallengeX<C>,
        transcript: &mut T,
    ) -> Result<(), Error> {
        // Hash permutation evals
        for eval in self.polys.iter().map(|poly| eval_polynomial(poly, *x)) {
            transcript.write_scalar(eval)?;
        }

        Ok(())
    }
}

impl<C: CurveAffine> Constructed<C> {
    pub(in crate::plonk) fn evaluate<E: EncodedChallenge<C>, T: TranscriptWrite<C, E>>(
        self,
        pk: &plonk::ProvingKey<C>,
        x: ChallengeX<C>,
        transcript: &mut T,
    ) -> Result<Evaluated<C>, Error> {
        let domain = &pk.vk.domain;
        let blinding_factors = pk.vk.cs.blinding_factors();

        {
            let mut sets = self.sets.iter();

            while let Some(set) = sets.next() {
                let permutation_product_eval = eval_polynomial(&set.permutation_product_poly, *x);

                let permutation_product_next_eval = eval_polynomial(
                    &set.permutation_product_poly,
                    domain.rotate_omega(*x, Rotation::next()),
                );

                // Hash permutation product evals
                for eval in iter::empty()
                    .chain(Some(&permutation_product_eval))
                    .chain(Some(&permutation_product_next_eval))
                {
                    transcript.write_scalar(*eval)?;
                }

                // If we have any remaining sets to process, evaluate this set at omega^u
                // so we can constrain the last value of its running product to equal the
                // first value of the next set's running product, chaining them together.
                if sets.len() > 0 {
                    let permutation_product_last_eval = eval_polynomial(
                        &set.permutation_product_poly,
                        domain.rotate_omega(*x, Rotation(-((blinding_factors + 1) as i32))),
                    );

                    transcript.write_scalar(permutation_product_last_eval)?;
                }
            }
        }

        Ok(Evaluated { constructed: self })
    }
}

impl<C: CurveAffine> Evaluated<C> {
    pub(in crate::plonk) fn open<'a>(
        &'a self,
        pk: &'a plonk::ProvingKey<C>,
        x: ChallengeX<C>,
    ) -> impl Iterator<Item = ProverQuery<'a, C>> + Clone {
        let blinding_factors = pk.vk.cs.blinding_factors();
        let x_next = pk.vk.domain.rotate_omega(*x, Rotation::next());
        let x_last = pk
            .vk
            .domain
            .rotate_omega(*x, Rotation(-((blinding_factors + 1) as i32)));

        iter::empty()
            .chain(self.constructed.sets.iter().flat_map(move |set| {
                iter::empty()
                    // Open permutation product commitments at x and \omega x
                    .chain(Some(ProverQuery {
                        point: *x,
                        poly: &set.permutation_product_poly,
                        blind: set.permutation_product_blind,
                    }))
                    .chain(Some(ProverQuery {
                        point: x_next,
                        poly: &set.permutation_product_poly,
                        blind: set.permutation_product_blind,
                    }))
            }))
            // Open it at \omega^{last} x for all but the last set. This rotation is only
            // sensical for the first row, but we only use this rotation in a constraint
            // that is gated on l_0.
            .chain(
                self.constructed
                    .sets
                    .iter()
                    .rev()
                    .skip(1)
                    .flat_map(move |set| {
                        Some(ProverQuery {
                            point: x_last,
                            poly: &set.permutation_product_poly,
                            blind: set.permutation_product_blind,
                        })
                    }),
            )
    }
}