plonky2/src/util/partial_products.rs

use std::iter;

use itertools::Itertools;

use crate::field::extension_field::target::ExtensionTarget;
use crate::field::extension_field::Extendable;
use crate::field::field_types::{Field, RichField};
use crate::plonk::circuit_builder::CircuitBuilder;
use crate::util::ceil_div_usize;

pub(crate) fn quotient_chunk_products<F: Field>(
    quotient_values: &[F],
    max_degree: usize,
) -> Vec<F> {
    debug_assert!(max_degree > 1);
    assert!(quotient_values.len() > 0);
    let chunk_size = max_degree;
    quotient_values
        .chunks(chunk_size)
        .map(|chunk| chunk.iter().copied().product())
        .collect()
}

/// Compute partial products of the original vector `v` such that all products consist of `max_degree`
/// or less elements. This is done until we've computed the product `P` of all elements in the vector.
pub(crate) fn partial_products_and_z_gx<F: Field>(z_x: F, quotient_chunk_products: &[F]) -> Vec<F> {
    assert!(quotient_chunk_products.len() > 0);
    let mut res = Vec::new();
    let mut acc = z_x;
    for &quotient_chunk_product in quotient_chunk_products {
        acc *= quotient_chunk_product;
        res.push(acc);
    }
    res
}

/// Returns a tuple `(a,b)`, where `a` is the length of the output of `partial_products()` on a
/// vector of length `n`, and `b` is the number of original elements consumed in `partial_products()`.
pub(crate) fn num_partial_products(n: usize, max_degree: usize) -> (usize, usize) {
    debug_assert!(max_degree > 1);
    let chunk_size = max_degree;
    // We'll split the product into `ceil_div_usize(n, chunk_size)` chunks, but the last chunk will
    // be associated with Z(gx) itself. Thus we subtract one to get the chunks associated with
    // partial products.
    let num_chunks = ceil_div_usize(n, chunk_size) - 1;
    (num_chunks, num_chunks * chunk_size)
}

/// Checks the relationship between each pair of partial product accumulators. In particular, this
/// sequence of accumulators starts with `Z(x)`, then contains each partial product polynomials
/// `p_i(x)`, and finally `Z(g x)`. See the partial products section of the Plonky2 paper.
pub(crate) fn check_partial_products<F: Field>(
    numerators: &[F],
    denominators: &[F],
    partials: &[F],
    z_x: F,
    z_gx: F,
    max_degree: usize,
) -> Vec<F> {
    debug_assert!(max_degree > 1);
    let product_accs = iter::once(&z_x)
        .chain(partials.iter())
        .chain(iter::once(&z_gx));
    let chunk_size = max_degree;
    numerators
        .chunks(chunk_size)
        .zip_eq(denominators.chunks(chunk_size))
        .zip_eq(product_accs.tuple_windows())
        .map(|((nume_chunk, deno_chunk), (&prev_acc, &next_acc))| {
            let num_chunk_product = nume_chunk.iter().copied().product();
            let den_chunk_product = deno_chunk.iter().copied().product();
            // Assert that next_acc * deno_product = prev_acc * nume_product.
            prev_acc * num_chunk_product - next_acc * den_chunk_product
        })
        .collect()
}

/// Checks the relationship between each pair of partial product accumulators. In particular, this
/// sequence of accumulators starts with `Z(x)`, then contains each partial product polynomials
/// `p_i(x)`, and finally `Z(g x)`. See the partial products section of the Plonky2 paper.
pub(crate) fn check_partial_products_recursively<F: RichField + Extendable<D>, const D: usize>(
    builder: &mut CircuitBuilder<F, D>,
    numerators: &[ExtensionTarget<D>],
    denominators: &[ExtensionTarget<D>],
    partials: &[ExtensionTarget<D>],
    z_x: ExtensionTarget<D>,
    z_gx: ExtensionTarget<D>,
    max_degree: usize,
) -> Vec<ExtensionTarget<D>> {
    debug_assert!(max_degree > 1);
    let product_accs = iter::once(&z_x)
        .chain(partials.iter())
        .chain(iter::once(&z_gx));
    let chunk_size = max_degree;
    numerators
        .chunks(chunk_size)
        .zip_eq(denominators.chunks(chunk_size))
        .zip_eq(product_accs.tuple_windows())
        .map(|((nume_chunk, deno_chunk), (&prev_acc, &next_acc))| {
            let nume_product = builder.mul_many_extension(nume_chunk);
            let deno_product = builder.mul_many_extension(deno_chunk);
            let next_acc_deno = builder.mul_extension(next_acc, deno_product);
            // Assert that next_acc * deno_product = prev_acc * nume_product.
            builder.mul_sub_extension(prev_acc, nume_product, next_acc_deno)
        })
        .collect()
}

#[cfg(test)]
mod tests {
    use super::*;
    use crate::field::goldilocks_field::GoldilocksField;

    #[test]
    fn test_partial_products() {
        type F = GoldilocksField;
        let denominators = vec![F::ONE; 6];
        let z_x = F::ONE;
        let v = field_vec(&[1, 2, 3, 4, 5, 6]);
        let z_gx = F::from_canonical_u64(720);
        let quotient_chunks_prods = quotient_chunk_products(&v, 2);
        assert_eq!(quotient_chunks_prods, field_vec(&[2, 12, 30]));
        let pps_and_z_gx = partial_products_and_z_gx(z_x, &quotient_chunks_prods);
        let pps = &pps_and_z_gx[..pps_and_z_gx.len() - 1];
        assert_eq!(pps_and_z_gx, field_vec(&[2, 24, 720]));

        let nums = num_partial_products(v.len(), 2);
        assert_eq!(pps.len(), nums.0);
        assert!(check_partial_products(&v, &denominators, pps, z_x, z_gx, 2)
            .iter()
            .all(|x| x.is_zero()));

        let quotient_chunks_prods = quotient_chunk_products(&v, 3);
        assert_eq!(quotient_chunks_prods, field_vec(&[6, 120]));
        let pps_and_z_gx = partial_products_and_z_gx(z_x, &quotient_chunks_prods);
        let pps = &pps_and_z_gx[..pps_and_z_gx.len() - 1];
        assert_eq!(pps_and_z_gx, field_vec(&[6, 720]));
        let nums = num_partial_products(v.len(), 3);
        assert_eq!(pps.len(), nums.0);
        assert!(check_partial_products(&v, &denominators, pps, z_x, z_gx, 3)
            .iter()
            .all(|x| x.is_zero()));
    }

    fn field_vec<F: Field>(xs: &[usize]) -> Vec<F> {
        xs.iter().map(|&x| F::from_canonical_usize(x)).collect()
    }
}