p3-poseidon2 0.4.3-succinct

An implementation of the Poseidon2 cryptographic permutation for zero-knowledge applications.
Documentation 
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//! As the security analysis of Poseidon2 is identical to that of Poseidon,
//! the relevant constraints regarding the number of full/partial rounds required can be found in
//! the original paper: https://eprint.iacr.org/2019/458.pdf and the associated codebase:
//! https://extgit.iaik.tugraz.at/krypto/hadeshash (See generate_params_poseidon.sage)
//!
//! These constraints are broken down into 6 equations:
//! statistical, interpolation, groebner 1, 2, 3 and
//! an extra constraint coming from the paper https://eprint.iacr.org/2023/537.pdf.
//!
//! For our parameters (M = 128, p > 2^30, WIDTH = t >= 8, D = alpha < 12),
//! the statistical constraint always simplifies to requiring RF >= 6.
//! Additionally p does not appear in Groebner 3 or the constraint coming from https://eprint.iacr.org/2023/537.pdf.
//! The remaining 3 constraints all can be rearranged into the form:
//! F(RF, RP) >= G(p) where G is a function which is non-decreasing with respect to p.
//!
//! Thus, if some tuple (M, p, WIDTH, D, RF, RP) satisfies all constraints, then so will
//! the tuple (M, q, WIDTH, D, RF, RP) for any 2^30 < q < p.
//! Moreover if RF, RP are the "optimal" round numbers (Optimal meaning minimising the number of s-box operations we need to perform)
//! for two tuples (M, p, WIDTH, D) and (M, q, WIDTH, D), then
//! they will also be optimal for (M, r, WIDTH, D) for any q < r < p.
//!
//! We compute the optimal required number of external (full) and internal (partial) rounds using:
//! https://github.com/0xPolygonZero/hash-constants/blob/master/calc_round_numbers.py
//! Using the above analysis we can conclude that the round numbers are equal
//! for all 31 bit primes and 64 bit primes respectively.

use gcd::Gcd;
use p3_field::PrimeField64;

/// Given a field, a width and an D return the number of full and partial rounds needed to achieve 128 bit security.
pub fn poseidon2_round_numbers_128<F: PrimeField64>(width: usize, d: u64) -> (usize, usize) {
    // Start by checking that d is a valid permutation.
    assert_eq!(d.gcd(F::ORDER_U64 - 1), 1);

    // Next compute the number of bits in p.
    let prime_bit_number = F::ORDER_U64.ilog2() + 1;

    match prime_bit_number {
        31 => match (width, d) {
            (16, 3) => (8, 20),
            (16, 5) => (8, 14),
            (16, 7) => (8, 13),
            (16, 9) => (8, 13),
            (16, 11) => (8, 13),
            (24, 3) => (8, 23),
            (24, 5) => (8, 22),
            (24, 7) => (8, 21),
            (24, 9) => (8, 21),
            (24, 11) => (8, 21),
            _ => panic!("The given pair of width and D has not been checked for these fields"),
        },
        64 => match (width, d) {
            (8, 3) => (8, 41),
            (8, 5) => (8, 27),
            (8, 7) => (8, 22),
            (8, 9) => (8, 19),
            (8, 11) => (8, 17),
            (12, 3) => (8, 42),
            (12, 5) => (8, 27),
            (12, 7) => (8, 22),
            (12, 9) => (8, 20),
            (12, 11) => (8, 18),
            (16, 3) => (8, 42),
            (16, 5) => (8, 27),
            (16, 7) => (8, 22),
            (16, 9) => (8, 20),
            (16, 11) => (8, 18),
            _ => panic!("The given pair of width and D has not been checked for these fields"),
        },
        _ => panic!("The optimal parameters for that size of prime have not been computed."),
    }
}