miden_crypto/dsa/rpo_falcon512/math/
polynomial.rsuse alloc::vec::Vec;
use core::{
default::Default,
fmt::Debug,
ops::{Add, AddAssign, Div, Mul, MulAssign, Neg, Sub, SubAssign},
};
use num::{One, Zero};
use super::{field::FalconFelt, Inverse};
use crate::{
dsa::rpo_falcon512::{MODULUS, N},
Felt,
};
#[derive(Debug, Clone, Default)]
pub struct Polynomial<F> {
pub coefficients: Vec<F>,
}
impl<F> Polynomial<F>
where
F: Clone,
{
pub fn new(coefficients: Vec<F>) -> Self {
Self { coefficients }
}
}
impl<
F: Mul<Output = F> + Sub<Output = F> + AddAssign + Zero + Div<Output = F> + Clone + Inverse,
> Polynomial<F>
{
pub fn hadamard_mul(&self, other: &Self) -> Self {
Polynomial::new(
self.coefficients
.iter()
.zip(other.coefficients.iter())
.map(|(a, b)| *a * *b)
.collect(),
)
}
pub fn hadamard_div(&self, other: &Self) -> Self {
let other_coefficients_inverse = F::batch_inverse_or_zero(&other.coefficients);
Polynomial::new(
self.coefficients
.iter()
.zip(other_coefficients_inverse.iter())
.map(|(a, b)| *a * *b)
.collect(),
)
}
pub fn hadamard_inv(&self) -> Self {
let coefficients_inverse = F::batch_inverse_or_zero(&self.coefficients);
Polynomial::new(coefficients_inverse)
}
}
impl<F: Zero + PartialEq + Clone> Polynomial<F> {
pub fn degree(&self) -> Option<usize> {
if self.coefficients.is_empty() {
return None;
}
let mut max_index = self.coefficients.len() - 1;
while self.coefficients[max_index] == F::zero() {
if let Some(new_index) = max_index.checked_sub(1) {
max_index = new_index;
} else {
return None;
}
}
Some(max_index)
}
pub fn lc(&self) -> F {
match self.degree() {
Some(non_negative_degree) => self.coefficients[non_negative_degree].clone(),
None => F::zero(),
}
}
}
impl<
F: One
+ Zero
+ Clone
+ Neg<Output = F>
+ MulAssign
+ AddAssign
+ Div<Output = F>
+ Sub<Output = F>
+ PartialEq,
> Polynomial<F>
{
pub fn reduce_by_cyclotomic(&self, n: usize) -> Self {
let mut coefficients = vec![F::zero(); n];
let mut sign = -F::one();
for (i, c) in self.coefficients.iter().cloned().enumerate() {
if i % n == 0 {
sign *= -F::one();
}
coefficients[i % n] += sign.clone() * c;
}
Polynomial::new(coefficients)
}
pub fn field_norm(&self) -> Self {
let n = self.coefficients.len();
let mut f0_coefficients = vec![F::zero(); n / 2];
let mut f1_coefficients = vec![F::zero(); n / 2];
for i in 0..n / 2 {
f0_coefficients[i] = self.coefficients[2 * i].clone();
f1_coefficients[i] = self.coefficients[2 * i + 1].clone();
}
let f0 = Polynomial::new(f0_coefficients);
let f1 = Polynomial::new(f1_coefficients);
let f0_squared = (f0.clone() * f0).reduce_by_cyclotomic(n / 2);
let f1_squared = (f1.clone() * f1).reduce_by_cyclotomic(n / 2);
let x = Polynomial::new(vec![F::zero(), F::one()]);
f0_squared - (x * f1_squared).reduce_by_cyclotomic(n / 2)
}
pub fn lift_next_cyclotomic(&self) -> Self {
let n = self.coefficients.len();
let mut coefficients = vec![F::zero(); n * 2];
for i in 0..n {
coefficients[2 * i] = self.coefficients[i].clone();
}
Self::new(coefficients)
}
pub fn galois_adjoint(&self) -> Self {
Self::new(
self.coefficients
.iter()
.enumerate()
.map(|(i, c)| if i % 2 == 0 { c.clone() } else { c.clone().neg() })
.collect(),
)
}
}
impl<F: Clone + Into<f64>> Polynomial<F> {
pub(crate) fn l2_norm_squared(&self) -> f64 {
self.coefficients
.iter()
.map(|i| Into::<f64>::into(i.clone()))
.map(|i| i * i)
.sum::<f64>()
}
}
impl<F> PartialEq for Polynomial<F>
where
F: Zero + PartialEq + Clone + AddAssign,
{
fn eq(&self, other: &Self) -> bool {
if self.is_zero() && other.is_zero() {
true
} else if self.is_zero() || other.is_zero() {
false
} else {
let self_degree = self.degree().unwrap();
let other_degree = other.degree().unwrap();
self.coefficients[0..=self_degree] == other.coefficients[0..=other_degree]
}
}
}
impl<F> Eq for Polynomial<F> where F: Zero + PartialEq + Clone + AddAssign {}
impl<F> Add for &Polynomial<F>
where
F: Add<Output = F> + AddAssign + Clone,
{
type Output = Polynomial<F>;
fn add(self, rhs: Self) -> Self::Output {
let coefficients = if self.coefficients.len() >= rhs.coefficients.len() {
let mut coefficients = self.coefficients.clone();
for (i, c) in rhs.coefficients.iter().enumerate() {
coefficients[i] += c.clone();
}
coefficients
} else {
let mut coefficients = rhs.coefficients.clone();
for (i, c) in self.coefficients.iter().enumerate() {
coefficients[i] += c.clone();
}
coefficients
};
Self::Output { coefficients }
}
}
impl<F> Add for Polynomial<F>
where
F: Add<Output = F> + AddAssign + Clone,
{
type Output = Polynomial<F>;
fn add(self, rhs: Self) -> Self::Output {
let coefficients = if self.coefficients.len() >= rhs.coefficients.len() {
let mut coefficients = self.coefficients.clone();
for (i, c) in rhs.coefficients.into_iter().enumerate() {
coefficients[i] += c;
}
coefficients
} else {
let mut coefficients = rhs.coefficients.clone();
for (i, c) in self.coefficients.into_iter().enumerate() {
coefficients[i] += c;
}
coefficients
};
Self::Output { coefficients }
}
}
impl<F> AddAssign for Polynomial<F>
where
F: Add<Output = F> + AddAssign + Clone,
{
fn add_assign(&mut self, rhs: Self) {
if self.coefficients.len() >= rhs.coefficients.len() {
for (i, c) in rhs.coefficients.into_iter().enumerate() {
self.coefficients[i] += c;
}
} else {
let mut coefficients = rhs.coefficients.clone();
for (i, c) in self.coefficients.iter().enumerate() {
coefficients[i] += c.clone();
}
self.coefficients = coefficients;
}
}
}
impl<F> Sub for &Polynomial<F>
where
F: Sub<Output = F> + Clone + Neg<Output = F> + Add<Output = F> + AddAssign,
{
type Output = Polynomial<F>;
fn sub(self, rhs: Self) -> Self::Output {
self + &(-rhs)
}
}
impl<F> Sub for Polynomial<F>
where
F: Sub<Output = F> + Clone + Neg<Output = F> + Add<Output = F> + AddAssign,
{
type Output = Polynomial<F>;
fn sub(self, rhs: Self) -> Self::Output {
self + (-rhs)
}
}
impl<F> SubAssign for Polynomial<F>
where
F: Add<Output = F> + Neg<Output = F> + AddAssign + Clone + Sub<Output = F>,
{
fn sub_assign(&mut self, rhs: Self) {
self.coefficients = self.clone().sub(rhs).coefficients;
}
}
impl<F: Neg<Output = F> + Clone> Neg for &Polynomial<F> {
type Output = Polynomial<F>;
fn neg(self) -> Self::Output {
Self::Output {
coefficients: self.coefficients.iter().cloned().map(|a| -a).collect(),
}
}
}
impl<F: Neg<Output = F> + Clone> Neg for Polynomial<F> {
type Output = Self;
fn neg(self) -> Self::Output {
Self::Output {
coefficients: self.coefficients.iter().cloned().map(|a| -a).collect(),
}
}
}
impl<F> Mul for &Polynomial<F>
where
F: Add + AddAssign + Mul<Output = F> + Sub<Output = F> + Zero + PartialEq + Clone,
{
type Output = Polynomial<F>;
fn mul(self, other: Self) -> Self::Output {
if self.is_zero() || other.is_zero() {
return Polynomial::<F>::zero();
}
let mut coefficients =
vec![F::zero(); self.coefficients.len() + other.coefficients.len() - 1];
for i in 0..self.coefficients.len() {
for j in 0..other.coefficients.len() {
coefficients[i + j] += self.coefficients[i].clone() * other.coefficients[j].clone();
}
}
Polynomial { coefficients }
}
}
impl<F> Mul for Polynomial<F>
where
F: Add + AddAssign + Mul<Output = F> + Zero + PartialEq + Clone,
{
type Output = Self;
fn mul(self, other: Self) -> Self::Output {
if self.is_zero() || other.is_zero() {
return Self::zero();
}
let mut coefficients =
vec![F::zero(); self.coefficients.len() + other.coefficients.len() - 1];
for i in 0..self.coefficients.len() {
for j in 0..other.coefficients.len() {
coefficients[i + j] += self.coefficients[i].clone() * other.coefficients[j].clone();
}
}
Self { coefficients }
}
}
impl<F: Add + Mul<Output = F> + Zero + Clone> Mul<F> for &Polynomial<F> {
type Output = Polynomial<F>;
fn mul(self, other: F) -> Self::Output {
Polynomial {
coefficients: self.coefficients.iter().cloned().map(|i| i * other.clone()).collect(),
}
}
}
impl<F: Add + Mul<Output = F> + Zero + Clone> Mul<F> for Polynomial<F> {
type Output = Polynomial<F>;
fn mul(self, other: F) -> Self::Output {
Polynomial {
coefficients: self.coefficients.iter().cloned().map(|i| i * other.clone()).collect(),
}
}
}
impl<F: Mul<Output = F> + Sub<Output = F> + AddAssign + Zero + Div<Output = F> + Clone>
Polynomial<F>
{
pub fn karatsuba(&self, other: &Self) -> Self {
Polynomial::new(vector_karatsuba(&self.coefficients, &other.coefficients))
}
}
impl<F> One for Polynomial<F>
where
F: Clone + One + PartialEq + Zero + AddAssign,
{
fn one() -> Self {
Self { coefficients: vec![F::one()] }
}
}
impl<F> Zero for Polynomial<F>
where
F: Zero + PartialEq + Clone + AddAssign,
{
fn zero() -> Self {
Self { coefficients: vec![] }
}
fn is_zero(&self) -> bool {
self.degree().is_none()
}
}
impl<F: Zero + Clone> Polynomial<F> {
pub fn shift(&self, shamt: usize) -> Self {
Self {
coefficients: [vec![F::zero(); shamt], self.coefficients.clone()].concat(),
}
}
pub fn constant(f: F) -> Self {
Self { coefficients: vec![f] }
}
pub fn map<G: Clone, C: FnMut(&F) -> G>(&self, closure: C) -> Polynomial<G> {
Polynomial::<G>::new(self.coefficients.iter().map(closure).collect())
}
pub fn fold<G, C: FnMut(G, &F) -> G + Clone>(&self, mut initial_value: G, closure: C) -> G {
for c in self.coefficients.iter() {
initial_value = (closure.clone())(initial_value, c);
}
initial_value
}
}
impl<F> Div<Polynomial<F>> for Polynomial<F>
where
F: Zero
+ One
+ PartialEq
+ AddAssign
+ Clone
+ Mul<Output = F>
+ MulAssign
+ Div<Output = F>
+ Neg<Output = F>
+ Sub<Output = F>,
{
type Output = Polynomial<F>;
fn div(self, denominator: Self) -> Self::Output {
if denominator.is_zero() {
panic!();
}
if self.is_zero() {
Self::zero();
}
let mut remainder = self.clone();
let mut quotient = Polynomial::<F>::zero();
while remainder.degree().unwrap() >= denominator.degree().unwrap() {
let shift = remainder.degree().unwrap() - denominator.degree().unwrap();
let quotient_coefficient = remainder.lc() / denominator.lc();
let monomial = Self::constant(quotient_coefficient).shift(shift);
quotient += monomial.clone();
remainder -= monomial * denominator.clone();
if remainder.is_zero() {
break;
}
}
quotient
}
}
fn vector_karatsuba<
F: Zero + AddAssign + Mul<Output = F> + Sub<Output = F> + Div<Output = F> + Clone,
>(
left: &[F],
right: &[F],
) -> Vec<F> {
let n = left.len();
if n <= 8 {
let mut product = vec![F::zero(); left.len() + right.len() - 1];
for (i, l) in left.iter().enumerate() {
for (j, r) in right.iter().enumerate() {
product[i + j] += l.clone() * r.clone();
}
}
return product;
}
let n_over_2 = n / 2;
let mut product = vec![F::zero(); 2 * n - 1];
let left_lo = &left[0..n_over_2];
let right_lo = &right[0..n_over_2];
let left_hi = &left[n_over_2..];
let right_hi = &right[n_over_2..];
let left_sum: Vec<F> =
left_lo.iter().zip(left_hi).map(|(a, b)| a.clone() + b.clone()).collect();
let right_sum: Vec<F> =
right_lo.iter().zip(right_hi).map(|(a, b)| a.clone() + b.clone()).collect();
let prod_lo = vector_karatsuba(left_lo, right_lo);
let prod_hi = vector_karatsuba(left_hi, right_hi);
let prod_mid: Vec<F> = vector_karatsuba(&left_sum, &right_sum)
.iter()
.zip(prod_lo.iter().zip(prod_hi.iter()))
.map(|(s, (l, h))| s.clone() - (l.clone() + h.clone()))
.collect();
for (i, l) in prod_lo.into_iter().enumerate() {
product[i] = l;
}
for (i, m) in prod_mid.into_iter().enumerate() {
product[i + n_over_2] += m;
}
for (i, h) in prod_hi.into_iter().enumerate() {
product[i + n] += h
}
product
}
impl From<Polynomial<FalconFelt>> for Polynomial<Felt> {
fn from(item: Polynomial<FalconFelt>) -> Self {
let res: Vec<Felt> =
item.coefficients.iter().map(|a| Felt::from(a.value() as u16)).collect();
Polynomial::new(res)
}
}
impl From<&Polynomial<FalconFelt>> for Polynomial<Felt> {
fn from(item: &Polynomial<FalconFelt>) -> Self {
let res: Vec<Felt> =
item.coefficients.iter().map(|a| Felt::from(a.value() as u16)).collect();
Polynomial::new(res)
}
}
impl From<Polynomial<i16>> for Polynomial<FalconFelt> {
fn from(item: Polynomial<i16>) -> Self {
let res: Vec<FalconFelt> = item.coefficients.iter().map(|&a| FalconFelt::new(a)).collect();
Polynomial::new(res)
}
}
impl From<&Polynomial<i16>> for Polynomial<FalconFelt> {
fn from(item: &Polynomial<i16>) -> Self {
let res: Vec<FalconFelt> = item.coefficients.iter().map(|&a| FalconFelt::new(a)).collect();
Polynomial::new(res)
}
}
impl From<Vec<i16>> for Polynomial<FalconFelt> {
fn from(item: Vec<i16>) -> Self {
let res: Vec<FalconFelt> = item.iter().map(|&a| FalconFelt::new(a)).collect();
Polynomial::new(res)
}
}
impl From<&Vec<i16>> for Polynomial<FalconFelt> {
fn from(item: &Vec<i16>) -> Self {
let res: Vec<FalconFelt> = item.iter().map(|&a| FalconFelt::new(a)).collect();
Polynomial::new(res)
}
}
impl Polynomial<FalconFelt> {
pub fn norm_squared(&self) -> u64 {
self.coefficients
.iter()
.map(|&i| i.balanced_value() as i64)
.map(|i| (i * i) as u64)
.sum::<u64>()
}
pub fn to_elements(&self) -> Vec<Felt> {
self.coefficients.iter().map(|&a| Felt::from(a.value() as u16)).collect()
}
pub fn mul_modulo_p(a: &Self, b: &Self) -> [u64; 1024] {
let mut c = [0; 2 * N];
for i in 0..N {
for j in 0..N {
c[i + j] += a.coefficients[i].value() as u64 * b.coefficients[j].value() as u64;
}
}
c
}
pub fn reduce_negacyclic(a: &[u64; 1024]) -> Self {
let mut c = [FalconFelt::zero(); N];
let modulus = MODULUS as u16;
for i in 0..N {
let ai = a[N + i] % modulus as u64;
let neg_ai = (modulus - ai as u16) % modulus;
let bi = (a[i] % modulus as u64) as u16;
c[i] = FalconFelt::new(((neg_ai + bi) % modulus) as i16);
}
Self::new(c.to_vec())
}
}
#[cfg(test)]
mod tests {
use super::{FalconFelt, Polynomial, N};
#[test]
fn test_negacyclic_reduction() {
let coef1: [u8; N] = rand_utils::rand_array();
let coef2: [u8; N] = rand_utils::rand_array();
let poly1 = Polynomial::new(coef1.iter().map(|&a| FalconFelt::new(a as i16)).collect());
let poly2 = Polynomial::new(coef2.iter().map(|&a| FalconFelt::new(a as i16)).collect());
let prod = poly1.clone() * poly2.clone();
assert_eq!(
prod.reduce_by_cyclotomic(N),
Polynomial::reduce_negacyclic(&Polynomial::mul_modulo_p(&poly1, &poly2))
);
}
}