vsss-rs 5.4.0

//! Represents Galois Field of 2^8 elements. This uses constant time operations
//! for all operations as related to shamir secret sharing. Too many implementations
//! use lookup tables which help for speed but leak secret information.
//! No lookup tables are used in this implementation because Cryptographic operations should
//!
//! 1. Ensure runtime is independent of secret data
//! 2. Ensure code access patterns are independent of secret data
//! 3. Ensure data access patterns are independent of secret data

use crate::util::{CtIsNotZero, field_bounded_add, uniform_nonzero_u8};
use crate::*;
use core::borrow::Borrow;
use core::{
    fmt::{self, Binary, Display, Formatter, LowerHex, UpperHex},
    iter::{Product, Sum},
    ops::{
        Add, AddAssign, BitAnd, BitAndAssign, BitOr, BitOrAssign, BitXor, BitXorAssign, Deref,
        DerefMut, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign,
    },
};
use elliptic_curve::ff::{Field, PrimeField};
use rand_core::RngCore;
use subtle::{Choice, ConditionallySelectable, ConstantTimeEq, CtOption};

#[cfg(any(feature = "alloc", feature = "std"))]
use crate::ParticipantIdGeneratorType;
use rand_core::CryptoRng;
#[cfg(feature = "zeroize")]
use zeroize::DefaultIsZeroes;

#[cfg(any(feature = "alloc", feature = "std"))]
type GfShare = DefaultShare<IdentifierGf256, IdentifierGf256>;

/// Represents the finite field GF(2^8) with 256 elements.
#[derive(Debug, Default, Clone, Copy, PartialEq, Eq, PartialOrd, Ord, Hash)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[repr(transparent)]
pub struct Gf256(pub u8);

#[cfg(feature = "zeroize")]
impl DefaultIsZeroes for Gf256 {}

impl Display for Gf256 {
    fn fmt(&self, f: &mut Formatter<'_>) -> fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl LowerHex for Gf256 {
    fn fmt(&self, f: &mut Formatter<'_>) -> fmt::Result {
        write!(f, "{:02x}", self.0)
    }
}

impl UpperHex for Gf256 {
    fn fmt(&self, f: &mut Formatter<'_>) -> fmt::Result {
        write!(f, "{:02X}", self.0)
    }
}

impl Binary for Gf256 {
    fn fmt(&self, f: &mut Formatter<'_>) -> fmt::Result {
        write!(f, "{:08b}", self.0)
    }
}

impl ConditionallySelectable for Gf256 {
    fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {
        Gf256(u8::conditional_select(&a.0, &b.0, choice))
    }
}

impl ConstantTimeEq for Gf256 {
    fn ct_eq(&self, other: &Self) -> Choice {
        self.0.ct_eq(&other.0)
    }
}

impl Add for Gf256 {
    type Output = Self;

    fn add(self, rhs: Self) -> Self {
        Gf256(self.0 ^ rhs.0)
    }
}

impl Add<&Gf256> for Gf256 {
    type Output = Gf256;

    fn add(self, rhs: &Gf256) -> Gf256 {
        self + *rhs
    }
}

impl Add<Gf256> for &Gf256 {
    type Output = Gf256;

    fn add(self, rhs: Gf256) -> Gf256 {
        *self + rhs
    }
}

impl Add<&Gf256> for &Gf256 {
    type Output = Gf256;

    fn add(self, rhs: &Gf256) -> Gf256 {
        *self + *rhs
    }
}

impl AddAssign for Gf256 {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}

impl AddAssign<&Gf256> for Gf256 {
    fn add_assign(&mut self, rhs: &Gf256) {
        *self = *self + *rhs;
    }
}

impl Sub for Gf256 {
    type Output = Self;

    fn sub(self, rhs: Self) -> Self {
        Gf256(self.0 ^ rhs.0)
    }
}

impl Sub<&Gf256> for Gf256 {
    type Output = Gf256;

    fn sub(self, rhs: &Gf256) -> Gf256 {
        Gf256(self.0 ^ rhs.0)
    }
}

impl Sub<Gf256> for &Gf256 {
    type Output = Gf256;

    fn sub(self, rhs: Gf256) -> Gf256 {
        Gf256(self.0 ^ rhs.0)
    }
}

impl Sub<&Gf256> for &Gf256 {
    type Output = Gf256;

    fn sub(self, rhs: &Gf256) -> Gf256 {
        Gf256(self.0 ^ rhs.0)
    }
}

impl SubAssign for Gf256 {
    fn sub_assign(&mut self, rhs: Self) {
        self.0 ^= rhs.0;
    }
}

impl SubAssign<&Gf256> for Gf256 {
    fn sub_assign(&mut self, rhs: &Gf256) {
        self.0 ^= rhs.0;
    }
}

impl Mul for Gf256 {
    type Output = Self;

    fn mul(self, rhs: Self) -> Self {
        Self(gf256_mul(self.0, rhs.0))
    }
}

impl Mul<&Gf256> for Gf256 {
    type Output = Gf256;

    fn mul(self, rhs: &Gf256) -> Gf256 {
        self * *rhs
    }
}

impl Mul<Gf256> for &Gf256 {
    type Output = Gf256;

    fn mul(self, rhs: Gf256) -> Gf256 {
        *self * rhs
    }
}

impl Mul<&Gf256> for &Gf256 {
    type Output = Gf256;

    fn mul(self, rhs: &Gf256) -> Gf256 {
        *self * *rhs
    }
}

impl MulAssign for Gf256 {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}

impl MulAssign<&Gf256> for Gf256 {
    fn mul_assign(&mut self, rhs: &Gf256) {
        *self = *self * *rhs;
    }
}

impl Div for Gf256 {
    type Output = Self;

    fn div(self, rhs: Self) -> Self::Output {
        self * rhs.invert().expect("no division by zero")
    }
}

impl Div<&Gf256> for Gf256 {
    type Output = Gf256;

    fn div(self, rhs: &Gf256) -> Gf256 {
        self / *rhs
    }
}

impl Div<Gf256> for &Gf256 {
    type Output = Gf256;

    fn div(self, rhs: Gf256) -> Gf256 {
        *self / rhs
    }
}

impl Div<&Gf256> for &Gf256 {
    type Output = Gf256;

    fn div(self, rhs: &Gf256) -> Gf256 {
        *self / *rhs
    }
}

impl DivAssign for Gf256 {
    fn div_assign(&mut self, rhs: Self) {
        *self *= rhs.invert().expect("no division by zero");
    }
}

impl DivAssign<&Gf256> for Gf256 {
    fn div_assign(&mut self, rhs: &Gf256) {
        *self *= rhs.invert().expect("no division by zero");
    }
}

impl Neg for Gf256 {
    type Output = Self;

    fn neg(self) -> Self {
        self
    }
}

impl BitAnd for Gf256 {
    type Output = Self;

    fn bitand(self, rhs: Self) -> Self {
        Self(self.0 & rhs.0)
    }
}

impl BitAnd<&Gf256> for Gf256 {
    type Output = Gf256;

    fn bitand(self, rhs: &Gf256) -> Gf256 {
        self & *rhs
    }
}

impl BitAnd<Gf256> for &Gf256 {
    type Output = Gf256;

    fn bitand(self, rhs: Gf256) -> Gf256 {
        *self & rhs
    }
}

impl BitAnd<&Gf256> for &Gf256 {
    type Output = Gf256;

    fn bitand(self, rhs: &Gf256) -> Gf256 {
        *self & *rhs
    }
}

impl BitAndAssign for Gf256 {
    fn bitand_assign(&mut self, rhs: Self) {
        self.0 &= rhs.0;
    }
}

impl BitAndAssign<&Gf256> for Gf256 {
    fn bitand_assign(&mut self, rhs: &Gf256) {
        self.0 &= rhs.0;
    }
}

impl BitOr for Gf256 {
    type Output = Self;

    fn bitor(self, rhs: Self) -> Self {
        Self(self.0 | rhs.0)
    }
}

impl BitOr<&Gf256> for Gf256 {
    type Output = Gf256;

    fn bitor(self, rhs: &Gf256) -> Gf256 {
        self | *rhs
    }
}

impl BitOr<Gf256> for &Gf256 {
    type Output = Gf256;

    fn bitor(self, rhs: Gf256) -> Gf256 {
        *self | rhs
    }
}

impl BitOr<&Gf256> for &Gf256 {
    type Output = Gf256;

    fn bitor(self, rhs: &Gf256) -> Gf256 {
        *self | *rhs
    }
}

impl BitOrAssign for Gf256 {
    fn bitor_assign(&mut self, rhs: Self) {
        self.0 |= rhs.0;
    }
}

impl BitOrAssign<&Gf256> for Gf256 {
    fn bitor_assign(&mut self, rhs: &Gf256) {
        self.0 |= rhs.0;
    }
}

impl BitXor for Gf256 {
    type Output = Self;

    fn bitxor(self, rhs: Self) -> Self {
        Self(self.0 ^ rhs.0)
    }
}

impl BitXor<&Gf256> for Gf256 {
    type Output = Gf256;

    fn bitxor(self, rhs: &Gf256) -> Gf256 {
        self ^ *rhs
    }
}

impl BitXor<Gf256> for &Gf256 {
    type Output = Gf256;

    fn bitxor(self, rhs: Gf256) -> Gf256 {
        *self ^ rhs
    }
}

impl BitXor<&Gf256> for &Gf256 {
    type Output = Gf256;

    fn bitxor(self, rhs: &Gf256) -> Gf256 {
        *self ^ *rhs
    }
}

impl BitXorAssign for Gf256 {
    fn bitxor_assign(&mut self, rhs: Self) {
        self.0 ^= rhs.0;
    }
}

impl BitXorAssign<&Gf256> for Gf256 {
    fn bitxor_assign(&mut self, rhs: &Gf256) {
        self.0 ^= rhs.0;
    }
}

impl<T: Borrow<Gf256>> Sum<T> for Gf256 {
    fn sum<I: Iterator<Item = T>>(iter: I) -> Self {
        iter.fold(Self(0), |acc, x| acc + x.borrow())
    }
}

impl<T: Borrow<Gf256>> Product<T> for Gf256 {
    fn product<I: Iterator<Item = T>>(iter: I) -> Self {
        iter.fold(Self(1), |acc, x| acc * x.borrow())
    }
}

impl Field for Gf256 {
    const ZERO: Self = Self(0);
    const ONE: Self = Self(1);

    fn random(mut rng: impl RngCore) -> Self {
        // Uniform over the full field {0, 1, ..., 255}. The prior
        // `(b & 0xFE) + 1` forced the low bit, producing only odd bytes
        // and leaking entropy out of polynomial coefficients used by
        // Shamir secret sharing (audit finding #1).
        Self(rng.next_u32() as u8)
    }

    fn square(&self) -> Self {
        self * self
    }

    fn double(&self) -> Self {
        self + self
    }

    fn invert(&self) -> CtOption<Self> {
        let mut z = self.0;
        for _ in 0..6 {
            z = gf256_mul(z, z);
            z = gf256_mul(z, self.0);
        }
        CtOption::new(Self(gf256_mul(z, z)), self.0.ct_is_not_zero())
    }

    fn sqrt_ratio(num: &Self, div: &Self) -> (Choice, Self) {
        let p = 0x1bu8; // Prime field characteristic for GF(256)
        let pm1d2 = (p - 1) >> 1;
        let pp2d4 = (p + 2) >> 2;
        let z = (2..=p).find(|z| gf256_pow(*z, pm1d2) != 1).unwrap(); // Find a non-quadratic residue

        let a = gf256_mul(num.0, div.0);
        let mut c = gf256_pow(a, pp2d4);
        let mut t = gf256_pow(a, pm1d2);
        let mut r = gf256_pow(z, pm1d2);

        let mut m = t;

        let mut i = 1;
        while m != 1 {
            let mut temp = m;
            for _ in 1..i {
                temp = gf256_mul(temp, temp);
                temp %= p;
            }
            let mut j = 0;
            while temp != 1 {
                temp = gf256_mul(temp, temp);
                temp %= p;
                j += 1;
            }
            let b = gf256_pow(r, 1 << (i - j - 1));
            c = gf256_mul(c, b);
            r = gf256_mul(b, b);
            t = gf256_mul(t, r);
            m = t;
            i = j;
        }
        let is_square = gf256_pow(c, 2).ct_eq(&c);
        (is_square, Self(c))
    }
}

impl From<u8> for Gf256 {
    fn from(val: u8) -> Self {
        Gf256(val)
    }
}

impl From<Gf256> for u8 {
    fn from(val: Gf256) -> u8 {
        val.0
    }
}

impl From<u16> for Gf256 {
    fn from(val: u16) -> Self {
        Gf256(val as u8)
    }
}

impl From<Gf256> for u16 {
    fn from(val: Gf256) -> u16 {
        val.0 as u16
    }
}

impl From<u32> for Gf256 {
    fn from(val: u32) -> Self {
        Gf256(val as u8)
    }
}

impl From<Gf256> for u32 {
    fn from(val: Gf256) -> u32 {
        val.0 as u32
    }
}

impl From<u64> for Gf256 {
    fn from(val: u64) -> Self {
        Gf256(val as u8)
    }
}

impl From<Gf256> for u64 {
    fn from(val: Gf256) -> u64 {
        val.0 as u64
    }
}

impl From<u128> for Gf256 {
    fn from(val: u128) -> Self {
        Gf256(val as u8)
    }
}

impl From<Gf256> for u128 {
    fn from(val: Gf256) -> u128 {
        val.0 as u128
    }
}

impl PrimeField for Gf256 {
    type Repr = [u8; 1];

    fn from_repr(repr: Self::Repr) -> CtOption<Self> {
        CtOption::new(Self(repr[0]), Choice::from(1u8))
    }

    fn to_repr(&self) -> Self::Repr {
        [self.0]
    }

    fn is_odd(&self) -> Choice {
        (self.0 & 1).ct_eq(&1)
    }

    const MODULUS: &'static str = "";
    const NUM_BITS: u32 = 8;
    const CAPACITY: u32 = 7;
    const TWO_INV: Self = Self(141);
    const MULTIPLICATIVE_GENERATOR: Self = Self(2);
    const S: u32 = 3;
    const ROOT_OF_UNITY: Self = Self(8);
    const ROOT_OF_UNITY_INV: Self = Self(114);
    const DELTA: Self = Self(67);
}

impl Gf256 {
    /// Raise the element to the power of `exp`.
    pub fn pow(&self, exp: u8) -> Self {
        Self(gf256_pow(self.0, exp))
    }

    #[cfg(any(feature = "alloc", feature = "std"))]
    /// Split a byte array into shares.
    pub fn split_array<B: AsRef<[u8]>>(
        threshold: usize,
        limit: usize,
        secret: B,
        rng: impl RngCore + CryptoRng,
    ) -> VsssResult<Vec<Vec<u8>>> {
        Self::split_array_with_participant_generators(
            threshold,
            limit,
            secret,
            rng,
            &[ParticipantIdGeneratorType::default()],
        )
    }

    #[cfg(any(feature = "alloc", feature = "std"))]
    /// Split a byte array into shares using the participant number generator.
    pub fn split_array_with_participant_generators<B: AsRef<[u8]>>(
        threshold: usize,
        limit: usize,
        secret: B,
        mut rng: impl RngCore + CryptoRng,
        participant_generators: &[ParticipantIdGeneratorType<IdentifierGf256>],
    ) -> VsssResult<Vec<Vec<u8>>> {
        if limit > 255 {
            return Err(Error::InvalidSizeRequest);
        }
        let secret = secret.as_ref();
        if secret.is_empty() {
            return Err(Error::InvalidSecret);
        }
        let mut shares = Vec::with_capacity(limit);

        let collection = ParticipantIdGeneratorCollection::from(participant_generators);
        let mut participant_id_iter = collection.iter();

        for _ in 0..limit {
            let id = participant_id_iter
                .next()
                .ok_or(Error::NotEnoughShareIdentifiers)?;
            let mut inner = Vec::with_capacity(limit + 1);
            inner.push(id.0.0);
            shares.push(inner);
        }
        for b in secret {
            let share = IdentifierGf256(Gf256(*b));
            let inner_shares = shamir::split_secret_with_participant_generator::<GfShare>(
                threshold,
                limit,
                &share,
                &mut rng,
                participant_generators,
            )?;
            for (share, inner_share) in shares.iter_mut().zip(inner_shares.iter()) {
                share.push(inner_share.value.0.0);
            }
        }
        Ok(shares)
    }

    #[cfg(any(feature = "alloc", feature = "std"))]
    /// Combine shares into a byte array.
    pub fn combine_array<B: AsRef<[Vec<u8>]>>(shares: B) -> VsssResult<Vec<u8>> {
        let shares = shares.as_ref();

        Self::are_shares_valid(shares)?;

        let mut secret = Vec::with_capacity(shares[0].len() - 1);
        let mut inner_shares = Vec::<GfShare>::with_capacity(shares[0].len() - 1);

        for share in shares {
            inner_shares.push(DefaultShare {
                identifier: IdentifierGf256(Gf256(share[0])),
                value: IdentifierGf256(Gf256(0u8)),
            });
        }
        for i in 1..shares[0].len() {
            for (inner_share, share) in inner_shares.iter_mut().zip(shares.iter()) {
                inner_share.value = IdentifierGf256(Gf256(share[i]));
            }
            secret.push(inner_shares.combine()?.0.0);
        }
        Ok(secret)
    }

    #[cfg(any(feature = "alloc", feature = "std"))]
    fn are_shares_valid(shares: &[Vec<u8>]) -> VsssResult<()> {
        if shares.len() < 2 {
            return Err(Error::SharingMinThreshold);
        }
        if shares[0].len() < 2 {
            return Err(Error::InvalidShare);
        }
        if shares[1..].iter().any(|s| s.len() != shares[0].len()) {
            return Err(Error::InvalidShare);
        }
        Ok(())
    }
}

fn gf256_pow(base: u8, exp: u8) -> u8 {
    let mut result = 1;
    for i in 0..8 {
        result *= result;
        let mut tmp = result;
        tmp *= base;
        let allow = ((exp >> i) & 1).ct_eq(&1);
        result.conditional_assign(&tmp, allow);
    }
    result.conditional_assign(&1, exp.ct_eq(&0));
    result
}

fn gf256_mul(a: u8, b: u8) -> u8 {
    let mut a = a as i8;
    let mut b = b as i8;
    let mut r = 0i8;
    for _ in 0..8 {
        r ^= a & -(b & 1);
        b >>= 1;
        let t = a >> 7;
        a <<= 1;
        a ^= 0x1b & t;
    }
    r as u8
}

#[derive(Debug, Copy, Clone, Default, Eq, PartialEq, Ord, PartialOrd, Hash)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[repr(transparent)]
/// Represents an identifier in the Galois Field GF(2^8).
///
/// Used solely for Sequential Participant ID generation,
/// since GF256 addition = xor i.e. identifiers just oscillate between
/// the start number and the incremented number instead of adding.
pub struct IdentifierGf256(pub Gf256);

impl Display for IdentifierGf256 {
    fn fmt(&self, f: &mut Formatter<'_>) -> fmt::Result {
        write!(f, "{}", self.0)
    }
}

#[cfg(feature = "zeroize")]
impl DefaultIsZeroes for IdentifierGf256 {}

impl Deref for IdentifierGf256 {
    type Target = Gf256;

    fn deref(&self) -> &Self::Target {
        &self.0
    }
}

impl DerefMut for IdentifierGf256 {
    fn deref_mut(&mut self) -> &mut Self::Target {
        &mut self.0
    }
}

impl AsRef<Gf256> for IdentifierGf256 {
    fn as_ref(&self) -> &Gf256 {
        &self.0
    }
}

impl AsMut<Gf256> for IdentifierGf256 {
    fn as_mut(&mut self) -> &mut Gf256 {
        &mut self.0
    }
}

impl From<Gf256> for IdentifierGf256 {
    fn from(val: Gf256) -> Self {
        IdentifierGf256(val)
    }
}

impl From<&IdentifierGf256> for IdentifierGf256 {
    fn from(val: &IdentifierGf256) -> Self {
        IdentifierGf256(val.0)
    }
}

impl Mul<&IdentifierGf256> for IdentifierGf256 {
    type Output = IdentifierGf256;

    fn mul(self, rhs: &IdentifierGf256) -> IdentifierGf256 {
        IdentifierGf256(self.0 * rhs.0)
    }
}

impl ShareElement for IdentifierGf256 {
    type Serialization = [u8; 1];

    type Inner = Gf256;

    fn random(mut rng: impl RngCore + CryptoRng) -> Self {
        // x-coordinate of a Shamir evaluation point; zero is reserved
        // for the secret (f(0)), so MUST be non-zero. Uniform over 1..=255.
        Self(Gf256(uniform_nonzero_u8(rng.next_u32(), 255)))
    }

    fn zero() -> Self {
        Self(Gf256::ZERO)
    }

    fn one() -> Self {
        Self(Gf256::ONE)
    }

    fn is_zero(&self) -> Choice {
        self.0.is_zero()
    }

    fn serialize(&self) -> Self::Serialization {
        [self.0.0]
    }

    fn deserialize(serialized: &Self::Serialization) -> VsssResult<Self> {
        Ok(Self(Gf256(serialized[0])))
    }

    fn from_slice(slice: &[u8]) -> VsssResult<Self> {
        if slice.len() != 1 {
            return Err(Error::InvalidShareElement);
        }
        Ok(Self(Gf256(slice[0])))
    }

    #[cfg(any(feature = "alloc", feature = "std"))]
    fn to_vec(&self) -> Vec<u8> {
        vec![self.0.0]
    }
}

impl ShareIdentifier for IdentifierGf256 {
    fn inc(&mut self, increment: &Self) {
        // Zero-on-overflow (audit finding #3). Prior `saturating_add`
        // pinned the cursor at 255, emitting duplicate x-values. The
        // `is_zero()` halt in `ParticipantIdGeneratorCollection::iter`
        // ends the sequential stream cleanly at field exhaustion.
        self.0.0 = field_bounded_add(self.0.0, increment.0.0, 256);
    }

    fn invert(&self) -> VsssResult<Self> {
        Option::from(self.0.invert())
            .map(Self)
            .ok_or(Error::InvalidShareElement)
    }

    fn random_coefficient(rng: impl RngCore + CryptoRng) -> Self {
        // Bypass the "+1" non-zero x-sampler in `ShareElement::random`
        // and draw directly from `Gf256` — uniform over 0..=255.
        Self(Gf256::random(rng))
    }
}

impl IdentifierGf256 {
    /// Returns additive identity.
    pub const ZERO: Self = Self(Gf256(0));
    /// Returns multiplicative identity.
    pub const ONE: Self = Self(Gf256(1));
}

#[cfg(test)]
#[cfg(any(feature = "alloc", feature = "std"))]
mod tests {
    use super::gf256_cmp;
    use super::*;
    use crate::shamir;
    use crate::{ParticipantIdGeneratorCollection, ParticipantIdGeneratorType};
    use rand::{Rng, SeedableRng};
    use rand_chacha::ChaCha8Rng;
    use std::collections::HashSet;
    use std::prelude::v1::Vec;

    #[test]
    fn compatibility() {
        let mut rng = ChaCha8Rng::from_seed([57u8; 32]);
        for _ in 0..1000 {
            let a = rng.r#gen::<u8>();
            let b = rng.r#gen::<u8>();
            let y = Gf256(a);
            let z = Gf256(b);

            assert_eq!((y * z).0, gf256_cmp::gf256_mul(a, b));
        }
        rng = ChaCha8Rng::from_entropy();
        for _ in 0..1000 {
            let a = rng.r#gen::<u8>();
            let b = rng.r#gen::<u8>();
            let y = Gf256(a);
            let z = Gf256(b);

            assert_eq!((y * z).0, gf256_cmp::gf256_mul(a, b));
        }

        let mut rng = ChaCha8Rng::from_seed([57u8; 32]);
        for _ in 0..1000 {
            let mut a = rng.r#gen::<u8>();
            while a == 0 {
                a = rng.r#gen::<u8>();
            }
            let y = Gf256(a);

            assert_eq!(y.invert().unwrap().0, gf256_cmp::gf256_div(1, a));
        }
    }

    #[test]
    fn shamir() {
        let mut rng = ChaCha8Rng::from_seed([57u8; 32]);
        for i in 1..=255 {
            let secret = IdentifierGf256(Gf256(i));
            let shares = shamir::split_secret::<GfShare>(3, 5, &secret, &mut rng).unwrap();
            assert_eq!(shares[0].identifier.0.0, 1);
            assert_eq!(shares[1].identifier.0.0, 2);
            assert_eq!(shares[2].identifier.0.0, 3);
            assert_eq!(shares[3].identifier.0.0, 4);
            assert_eq!(shares[4].identifier.0.0, 5);
            let res = &shares[0..3].to_vec().combine();
            assert!(
                res.is_ok(),
                "Failed at iteration {}, secret: {}",
                i,
                secret.0.0
            );
            assert_eq!(
                res.unwrap(),
                secret,
                "Failed at iteration {}, secret: {}",
                i,
                secret.0.0
            );
        }
        rng = ChaCha8Rng::from_entropy();
        for i in 1..=255 {
            let secret = IdentifierGf256(Gf256(i));
            let shares = shamir::split_secret::<GfShare>(3, 5, &secret, &mut rng).unwrap();
            assert_eq!(shares[0].identifier.0.0, 1);
            assert_eq!(shares[1].identifier.0.0, 2);
            assert_eq!(shares[2].identifier.0.0, 3);
            assert_eq!(shares[3].identifier.0.0, 4);
            assert_eq!(shares[4].identifier.0.0, 5);
            let res = &shares[2..].to_vec().combine();
            assert_eq!(res.unwrap(), secret);
        }
    }

    #[test]
    fn split_array() {
        let mut rng = ChaCha8Rng::from_seed([57u8; 32]);
        let secret = b"Hello World!";
        let shares = Gf256::split_array(3, 5, secret, &mut rng).unwrap();
        assert_eq!(shares.len(), 5);

        let res = Gf256::combine_array(&shares[..3]);
        assert_eq!(res.unwrap(), secret);

        let p = ParticipantIdGeneratorType::Sequential {
            start: IdentifierGf256(Gf256(10)),
            increment: IdentifierGf256(Gf256(1)),
            count: 5,
        };
        let shares =
            Gf256::split_array_with_participant_generators(3, 5, secret, &mut rng, &[p]).unwrap();
        assert_eq!(shares.len(), 5);

        let res = Gf256::combine_array(&shares[..3]);
        let secret2 = res.unwrap();
        assert_eq!(secret2, secret);

        let res = Gf256::combine_array(&[shares[4].clone(), shares[1].clone(), shares[3].clone()]);
        let secret2 = res.unwrap();
        assert_eq!(secret2, secret);
    }

    #[test]
    fn combine_fuzz() {
        let res = Gf256::combine_array(&[vec![], vec![]]);
        assert!(res.is_err());
        let res = Gf256::combine_array(&[vec![1u8, 8u8], vec![2u8]]);
        assert!(res.is_err());

        let mut rng = ChaCha8Rng::from_entropy();
        for _ in 0..25 {
            let threshold = rng.r#gen::<u8>().saturating_add(1);

            let mut shares = Vec::with_capacity(threshold as usize);
            for i in 0..threshold {
                let share = vec![i; (rng.r#gen::<usize>() % 64) + 1];
                shares.push(share);
            }
            assert!(Gf256::combine_array(shares).is_err());
        }
    }

    /// Audit finding #1 — regression test: `Gf256::random` must be
    /// uniform over the entire field, including even values. The
    /// prior `(b & 0xFE) + 1` only produced odd bytes.
    #[test]
    fn poc1_biased_gf256() {
        // 512 draws comfortably exceeds the ~30 needed to see even/odd/zero
        // each with overwhelming probability under a uniform distribution.
        let mut rng = ChaCha8Rng::from_seed([57u8; 32]);
        let mut seen_even = false;
        let mut seen_odd = false;
        let mut seen_zero = false;
        for _ in 0..512 {
            let x = <Gf256 as Field>::random(&mut rng).0;
            if x % 2 == 0 {
                seen_even = true;
            } else {
                seen_odd = true;
            }
            if x == 0 {
                seen_zero = true;
            }
        }
        assert!(
            seen_even,
            "Gf256::random produced no even values — bias regression"
        );
        assert!(seen_odd, "Gf256::random produced no odd values");
        assert!(
            seen_zero,
            "Gf256::random produced no zero values over 10k draws — field coverage regression"
        );
    }

    /// Audit finding #1 — companion: the x-identifier sampler must
    /// never yield zero (reserved for the secret), across a large
    /// sample.
    #[test]
    fn poc1_identifier_gf256_nonzero() {
        // 2048 draws — large enough that a biased sampler producing zero
        // with any meaningful probability would be caught, small enough
        // to keep the test cheap.
        let mut rng = ChaCha8Rng::from_seed([0xA5u8; 32]);
        for _ in 0..2048 {
            let id = IdentifierGf256::random(&mut rng);
            assert_ne!(id.0.0, 0, "IdentifierGf256::random yielded zero");
        }
    }

    /// Audit finding #2 — regression test: polynomial coefficients
    /// must be sampled uniformly over the entire field, including
    /// zero. The prior fill loop rejected zeros, biasing the
    /// distribution and enabling a re-sharing attack on small fields.
    #[test]
    fn poc2_reject_zero_bias_reshare_attack() {
        let mut rng = ChaCha8Rng::from_seed([0x42u8; 32]);
        let secret = IdentifierGf256(Gf256(7));
        // 200 splits of a degree-10 polynomial over GF(256) with 15
        // evaluation points yields ~1 - (255/256)^(200*15) ≈ 0.99998
        // probability of observing at least one zero share value
        // under uniform coefficients. With the ChaCha seed below the
        // outcome is of course deterministic; the count is sized so
        // that any seed has high safety margin.
        let threshold = 11;
        let limit = 15;
        let runs = 200;
        let mut seen_zero_share_value = false;
        for _ in 0..runs {
            let shares =
                shamir::split_secret::<GfShare>(threshold, limit, &secret, &mut rng).unwrap();
            let res = shares[0..threshold].to_vec().combine();
            assert!(res.is_ok(), "combine failed — zero-coefficient path broken");
            if shares.iter().any(|s| s.value.0.0 == 0) {
                seen_zero_share_value = true;
            }
        }
        assert!(
            seen_zero_share_value,
            "No zero share value over {runs} splits × {limit} shares — coefficient distribution still biased away from zero",
        );
    }

    /// Audit finding #3 — regression test: sequential x-identifier
    /// generation must not emit duplicates past field exhaustion.
    /// The prior `saturating_add` pinned the cursor at u8::MAX so
    /// every subsequent id was 255.
    #[test]
    fn poc3_gf256_inc_saturating_add() {
        // Start very close to the field boundary so the fix can be
        // observed: old code would emit [253, 254, 255, 255, 255];
        // fixed code emits [253, 254, 255] then halts.
        let start = IdentifierGf256(Gf256(253));
        let inc = IdentifierGf256(Gf256(1));
        let seq = ParticipantIdGeneratorType::Sequential {
            start,
            increment: inc,
            count: 10,
        };
        let generators = [seq];
        let collection = ParticipantIdGeneratorCollection::from(&generators[..]);
        let ids: Vec<_> = collection.iter().collect();

        // All emitted ids must be unique.
        let mut seen = HashSet::new();
        for id in &ids {
            assert!(
                seen.insert(id.0.0),
                "duplicate identifier emitted: {}",
                id.0.0
            );
        }
        // And the stream must have halted rather than silently returning
        // saturated duplicates.
        assert!(
            ids.len() <= 3,
            "generator emitted {} ids past field boundary — saturating_add regression",
            ids.len()
        );
    }

    /// Direct observation of audit finding #2: `Polynomial::fill` must
    /// produce zero coefficients across many runs. Probes the polynomial
    /// state directly rather than inferring via share values.
    #[test]
    fn zero_coefficients_actually_occur() {
        use crate::Polynomial;
        let mut rng = ChaCha8Rng::from_seed([0x7Fu8; 32]);
        let intercept = IdentifierGf256(Gf256(0xA5));
        let threshold = 10usize;
        let runs = 200;
        let mut zero_coef_count = 0usize;
        for _ in 0..runs {
            let mut poly: Vec<GfShare> = <Vec<GfShare> as Polynomial<GfShare>>::create(threshold);
            poly.fill(&intercept, &mut rng, threshold).unwrap();
            // Slots 1..threshold are random coefficients; slot 0 is the intercept.
            for coef in &poly[1..threshold] {
                if coef.identifier.0.0 == 0 {
                    zero_coef_count += 1;
                }
            }
        }
        // 200 runs × 9 coefficients = 1800 draws; under uniform sampling
        // expected zeros ≈ 1800/256 ≈ 7. Prior biased fill would produce 0.
        assert!(
            zero_coef_count > 0,
            "No zero coefficient across {runs} fills × {} slots — coefficient sampling still biased against zero",
            threshold - 1,
        );
    }

    /// A secret of zero must split and reconstruct correctly. The
    /// existing `shamir` test only iterates secrets 1..=255, so this
    /// covers the f(0) = 0 edge case.
    #[test]
    fn zero_secret_round_trip() {
        let mut rng = ChaCha8Rng::from_seed([0xC3u8; 32]);
        let zero_secret = IdentifierGf256(Gf256(0));
        let shares = shamir::split_secret::<GfShare>(3, 5, &zero_secret, &mut rng).unwrap();
        let recovered = shares[..3].to_vec().combine().unwrap();
        assert_eq!(recovered, zero_secret, "zero-secret round-trip failed");
        // Also combine from a different threshold subset.
        let recovered2 = shares[2..].to_vec().combine().unwrap();
        assert_eq!(recovered2, zero_secret);
    }

    /// Shares whose value byte is zero (the polynomial happened to
    /// evaluate to zero at that x) must still round-trip. With a
    /// multi-byte secret and many reps, zero-valued share bytes occur
    /// frequently; combine must accept them.
    #[test]
    fn zero_valued_shares_round_trip() {
        let mut rng = ChaCha8Rng::from_seed([0x5Au8; 32]);
        let secret = b"The quick brown fox jumps over the lazy dog";
        let runs = 50;
        let mut saw_zero_share_byte = false;
        for _ in 0..runs {
            let shares = Gf256::split_array(5, 8, secret, &mut rng).unwrap();
            // Inspect share bytes (index 0 is the identifier; 1.. are values).
            for s in &shares {
                if s[1..].iter().any(|&b| b == 0) {
                    saw_zero_share_byte = true;
                }
            }
            let recovered = Gf256::combine_array(&shares[..5]).unwrap();
            assert_eq!(
                &recovered[..],
                secret,
                "combine failed with zero-valued shares in set"
            );
        }
        assert!(
            saw_zero_share_byte,
            "No zero byte appeared in any share value across {runs} runs — statistical regression",
        );
    }

    /// Every share identifier emitted by `split_secret` must be
    /// non-zero (the zero element is reserved for the secret f(0)).
    #[test]
    fn no_share_identifier_is_zero() {
        let mut rng = ChaCha8Rng::from_seed([0xDEu8; 32]);
        for _ in 0..100 {
            let secret = IdentifierGf256(Gf256(rng.r#gen::<u8>()));
            let shares = shamir::split_secret::<GfShare>(3, 5, &secret, &mut rng).unwrap();
            for s in &shares {
                assert_ne!(
                    s.identifier.0.0, 0,
                    "zero identifier produced by split_secret",
                );
            }
        }
    }
}

#[cfg(test)]
#[cfg(any(feature = "alloc", feature = "std"))]
mod gf256_cmp {
    // Ref https://github.com/veracruz-project/veracruz/blob/main/sdk/data-generators/shamir-secret-sharing/src/main.rs

    #[rustfmt::skip]
    const GF256_LOG: [u8; 256] = [
        0xff, 0x00, 0x19, 0x01, 0x32, 0x02, 0x1a, 0xc6,
        0x4b, 0xc7, 0x1b, 0x68, 0x33, 0xee, 0xdf, 0x03,
        0x64, 0x04, 0xe0, 0x0e, 0x34, 0x8d, 0x81, 0xef,
        0x4c, 0x71, 0x08, 0xc8, 0xf8, 0x69, 0x1c, 0xc1,
        0x7d, 0xc2, 0x1d, 0xb5, 0xf9, 0xb9, 0x27, 0x6a,
        0x4d, 0xe4, 0xa6, 0x72, 0x9a, 0xc9, 0x09, 0x78,
        0x65, 0x2f, 0x8a, 0x05, 0x21, 0x0f, 0xe1, 0x24,
        0x12, 0xf0, 0x82, 0x45, 0x35, 0x93, 0xda, 0x8e,
        0x96, 0x8f, 0xdb, 0xbd, 0x36, 0xd0, 0xce, 0x94,
        0x13, 0x5c, 0xd2, 0xf1, 0x40, 0x46, 0x83, 0x38,
        0x66, 0xdd, 0xfd, 0x30, 0xbf, 0x06, 0x8b, 0x62,
        0xb3, 0x25, 0xe2, 0x98, 0x22, 0x88, 0x91, 0x10,
        0x7e, 0x6e, 0x48, 0xc3, 0xa3, 0xb6, 0x1e, 0x42,
        0x3a, 0x6b, 0x28, 0x54, 0xfa, 0x85, 0x3d, 0xba,
        0x2b, 0x79, 0x0a, 0x15, 0x9b, 0x9f, 0x5e, 0xca,
        0x4e, 0xd4, 0xac, 0xe5, 0xf3, 0x73, 0xa7, 0x57,
        0xaf, 0x58, 0xa8, 0x50, 0xf4, 0xea, 0xd6, 0x74,
        0x4f, 0xae, 0xe9, 0xd5, 0xe7, 0xe6, 0xad, 0xe8,
        0x2c, 0xd7, 0x75, 0x7a, 0xeb, 0x16, 0x0b, 0xf5,
        0x59, 0xcb, 0x5f, 0xb0, 0x9c, 0xa9, 0x51, 0xa0,
        0x7f, 0x0c, 0xf6, 0x6f, 0x17, 0xc4, 0x49, 0xec,
        0xd8, 0x43, 0x1f, 0x2d, 0xa4, 0x76, 0x7b, 0xb7,
        0xcc, 0xbb, 0x3e, 0x5a, 0xfb, 0x60, 0xb1, 0x86,
        0x3b, 0x52, 0xa1, 0x6c, 0xaa, 0x55, 0x29, 0x9d,
        0x97, 0xb2, 0x87, 0x90, 0x61, 0xbe, 0xdc, 0xfc,
        0xbc, 0x95, 0xcf, 0xcd, 0x37, 0x3f, 0x5b, 0xd1,
        0x53, 0x39, 0x84, 0x3c, 0x41, 0xa2, 0x6d, 0x47,
        0x14, 0x2a, 0x9e, 0x5d, 0x56, 0xf2, 0xd3, 0xab,
        0x44, 0x11, 0x92, 0xd9, 0x23, 0x20, 0x2e, 0x89,
        0xb4, 0x7c, 0xb8, 0x26, 0x77, 0x99, 0xe3, 0xa5,
        0x67, 0x4a, 0xed, 0xde, 0xc5, 0x31, 0xfe, 0x18,
        0x0d, 0x63, 0x8c, 0x80, 0xc0, 0xf7, 0x70, 0x07,
    ];

    #[rustfmt::skip]
    const GF256_EXP: [u8; 2*255] = [
        0x01, 0x03, 0x05, 0x0f, 0x11, 0x33, 0x55, 0xff,
        0x1a, 0x2e, 0x72, 0x96, 0xa1, 0xf8, 0x13, 0x35,
        0x5f, 0xe1, 0x38, 0x48, 0xd8, 0x73, 0x95, 0xa4,
        0xf7, 0x02, 0x06, 0x0a, 0x1e, 0x22, 0x66, 0xaa,
        0xe5, 0x34, 0x5c, 0xe4, 0x37, 0x59, 0xeb, 0x26,
        0x6a, 0xbe, 0xd9, 0x70, 0x90, 0xab, 0xe6, 0x31,
        0x53, 0xf5, 0x04, 0x0c, 0x14, 0x3c, 0x44, 0xcc,
        0x4f, 0xd1, 0x68, 0xb8, 0xd3, 0x6e, 0xb2, 0xcd,
        0x4c, 0xd4, 0x67, 0xa9, 0xe0, 0x3b, 0x4d, 0xd7,
        0x62, 0xa6, 0xf1, 0x08, 0x18, 0x28, 0x78, 0x88,
        0x83, 0x9e, 0xb9, 0xd0, 0x6b, 0xbd, 0xdc, 0x7f,
        0x81, 0x98, 0xb3, 0xce, 0x49, 0xdb, 0x76, 0x9a,
        0xb5, 0xc4, 0x57, 0xf9, 0x10, 0x30, 0x50, 0xf0,
        0x0b, 0x1d, 0x27, 0x69, 0xbb, 0xd6, 0x61, 0xa3,
        0xfe, 0x19, 0x2b, 0x7d, 0x87, 0x92, 0xad, 0xec,
        0x2f, 0x71, 0x93, 0xae, 0xe9, 0x20, 0x60, 0xa0,
        0xfb, 0x16, 0x3a, 0x4e, 0xd2, 0x6d, 0xb7, 0xc2,
        0x5d, 0xe7, 0x32, 0x56, 0xfa, 0x15, 0x3f, 0x41,
        0xc3, 0x5e, 0xe2, 0x3d, 0x47, 0xc9, 0x40, 0xc0,
        0x5b, 0xed, 0x2c, 0x74, 0x9c, 0xbf, 0xda, 0x75,
        0x9f, 0xba, 0xd5, 0x64, 0xac, 0xef, 0x2a, 0x7e,
        0x82, 0x9d, 0xbc, 0xdf, 0x7a, 0x8e, 0x89, 0x80,
        0x9b, 0xb6, 0xc1, 0x58, 0xe8, 0x23, 0x65, 0xaf,
        0xea, 0x25, 0x6f, 0xb1, 0xc8, 0x43, 0xc5, 0x54,
        0xfc, 0x1f, 0x21, 0x63, 0xa5, 0xf4, 0x07, 0x09,
        0x1b, 0x2d, 0x77, 0x99, 0xb0, 0xcb, 0x46, 0xca,
        0x45, 0xcf, 0x4a, 0xde, 0x79, 0x8b, 0x86, 0x91,
        0xa8, 0xe3, 0x3e, 0x42, 0xc6, 0x51, 0xf3, 0x0e,
        0x12, 0x36, 0x5a, 0xee, 0x29, 0x7b, 0x8d, 0x8c,
        0x8f, 0x8a, 0x85, 0x94, 0xa7, 0xf2, 0x0d, 0x17,
        0x39, 0x4b, 0xdd, 0x7c, 0x84, 0x97, 0xa2, 0xfd,
        0x1c, 0x24, 0x6c, 0xb4, 0xc7, 0x52, 0xf6,

        0x01, 0x03, 0x05, 0x0f, 0x11, 0x33, 0x55, 0xff,
        0x1a, 0x2e, 0x72, 0x96, 0xa1, 0xf8, 0x13, 0x35,
        0x5f, 0xe1, 0x38, 0x48, 0xd8, 0x73, 0x95, 0xa4,
        0xf7, 0x02, 0x06, 0x0a, 0x1e, 0x22, 0x66, 0xaa,
        0xe5, 0x34, 0x5c, 0xe4, 0x37, 0x59, 0xeb, 0x26,
        0x6a, 0xbe, 0xd9, 0x70, 0x90, 0xab, 0xe6, 0x31,
        0x53, 0xf5, 0x04, 0x0c, 0x14, 0x3c, 0x44, 0xcc,
        0x4f, 0xd1, 0x68, 0xb8, 0xd3, 0x6e, 0xb2, 0xcd,
        0x4c, 0xd4, 0x67, 0xa9, 0xe0, 0x3b, 0x4d, 0xd7,
        0x62, 0xa6, 0xf1, 0x08, 0x18, 0x28, 0x78, 0x88,
        0x83, 0x9e, 0xb9, 0xd0, 0x6b, 0xbd, 0xdc, 0x7f,
        0x81, 0x98, 0xb3, 0xce, 0x49, 0xdb, 0x76, 0x9a,
        0xb5, 0xc4, 0x57, 0xf9, 0x10, 0x30, 0x50, 0xf0,
        0x0b, 0x1d, 0x27, 0x69, 0xbb, 0xd6, 0x61, 0xa3,
        0xfe, 0x19, 0x2b, 0x7d, 0x87, 0x92, 0xad, 0xec,
        0x2f, 0x71, 0x93, 0xae, 0xe9, 0x20, 0x60, 0xa0,
        0xfb, 0x16, 0x3a, 0x4e, 0xd2, 0x6d, 0xb7, 0xc2,
        0x5d, 0xe7, 0x32, 0x56, 0xfa, 0x15, 0x3f, 0x41,
        0xc3, 0x5e, 0xe2, 0x3d, 0x47, 0xc9, 0x40, 0xc0,
        0x5b, 0xed, 0x2c, 0x74, 0x9c, 0xbf, 0xda, 0x75,
        0x9f, 0xba, 0xd5, 0x64, 0xac, 0xef, 0x2a, 0x7e,
        0x82, 0x9d, 0xbc, 0xdf, 0x7a, 0x8e, 0x89, 0x80,
        0x9b, 0xb6, 0xc1, 0x58, 0xe8, 0x23, 0x65, 0xaf,
        0xea, 0x25, 0x6f, 0xb1, 0xc8, 0x43, 0xc5, 0x54,
        0xfc, 0x1f, 0x21, 0x63, 0xa5, 0xf4, 0x07, 0x09,
        0x1b, 0x2d, 0x77, 0x99, 0xb0, 0xcb, 0x46, 0xca,
        0x45, 0xcf, 0x4a, 0xde, 0x79, 0x8b, 0x86, 0x91,
        0xa8, 0xe3, 0x3e, 0x42, 0xc6, 0x51, 0xf3, 0x0e,
        0x12, 0x36, 0x5a, 0xee, 0x29, 0x7b, 0x8d, 0x8c,
        0x8f, 0x8a, 0x85, 0x94, 0xa7, 0xf2, 0x0d, 0x17,
        0x39, 0x4b, 0xdd, 0x7c, 0x84, 0x97, 0xa2, 0xfd,
        0x1c, 0x24, 0x6c, 0xb4, 0xc7, 0x52, 0xf6,
    ];

    /// Multiply in GF(256).
    pub fn gf256_mul(a: u8, b: u8) -> u8 {
        if a == 0 || b == 0 {
            0
        } else {
            GF256_EXP
                [usize::from(GF256_LOG[usize::from(a)]) + usize::from(GF256_LOG[usize::from(b)])]
        }
    }

    /// Divide in GF(256)/
    pub fn gf256_div(a: u8, b: u8) -> u8 {
        // multiply `a` against inverse `b`
        gf256_mul(a, GF256_EXP[usize::from(255 - GF256_LOG[usize::from(b)])])
    }
}