1//! 𝕋_<N,q>[X] = ℝ_<N,q>[X] / ℤ_<N,q>[X], polynomials modulo X^N+1 with
  2//! coefficients in 𝕋_Q.
  3//!
  4//! Note: this is not an algebraic ring, since internal-product is not well
  5//! defined. But since we work over the discrete torus 𝕋_q, which we identify as
  6//! 𝕋q = ℤ/qℤ ≈ ℤq, whith q=64. Since we allow product between 𝕋q elements and
  7//! u64, we fit it into the `Ring` trait (from ring.rs) so that we can compose
  8//! the 𝕋_<N,q> implementation with the other objects from the code.
  9
 10use rand::{distributions::Distribution, Rng};
 11use std::array;
 12use std::iter::Sum;
 13use std::ops::{Add, AddAssign, Mul, Sub, SubAssign};
 14
 15use crate::{ring::Ring, torus::T64};
 16
 17/// 𝕋_<N,Q>[X] = 𝕋<Q>[X]/(X^N +1), polynomials modulo X^N+1 with coefficients in
 18/// 𝕋, where Q=2^64.
 19#[derive(Clone, Copy, Debug)]
 20pub struct Tn<const N: usize>(pub [T64; N]);
 21
 22impl<const N: usize> Ring for Tn<N> {
 23    type C = T64;
 24
 25    fn coeffs(&self) -> Vec<T64> {
 26        self.0.to_vec()
 27    }
 28
 29    fn zero() -> Self {
 30        Self(array::from_fn(|_| T64::zero()))
 31    }
 32
 33    fn rand(mut rng: impl Rng, dist: impl Distribution<f64>) -> Self {
 34        Self(array::from_fn(|_| T64::rand_f64(&mut rng, &dist)))
 35    }
 36
 37    fn from_vec(coeffs: Vec<Self::C>) -> Self {
 38        let mut p = coeffs;
 39        modulus::<N>(&mut p);
 40        Self(array::from_fn(|i| p[i]))
 41    }
 42
 43    fn decompose(&self, beta: u32, l: u32) -> Vec<Self> {
 44        let elems: Vec<Vec<T64>> = self.0.iter().map(|r| r.decompose(beta, l)).collect();
 45        // transpose it
 46        let r: Vec<Vec<T64>> = (0..elems[0].len())
 47            .map(|i| (0..elems.len()).map(|j| elems[j][i]).collect())
 48            .collect();
 49        // convert it to Tn<N>
 50        r.iter().map(|a_i| Self::from_vec(a_i.clone())).collect()
 51    }
 52}
 53
 54// apply mod (X^N+1)
 55pub fn modulus<const N: usize>(p: &mut Vec<T64>) {
 56    if p.len() < N {
 57        return;
 58    }
 59    for i in N..p.len() {
 60        p[i - N] = p[i - N].clone() - p[i].clone();
 61        p[i] = T64::zero();
 62    }
 63    p.truncate(N);
 64}
 65
 66impl<const N: usize> Add<Tn<N>> for Tn<N> {
 67    type Output = Self;
 68
 69    fn add(self, rhs: Self) -> Self {
 70        Self(array::from_fn(|i| self.0[i] + rhs.0[i]))
 71    }
 72}
 73impl<const N: usize> Add<&Tn<N>> for &Tn<N> {
 74    type Output = Tn<N>;
 75
 76    fn add(self, rhs: &Tn<N>) -> Self::Output {
 77        Tn(array::from_fn(|i| self.0[i] + rhs.0[i]))
 78    }
 79}
 80impl<const N: usize> AddAssign for Tn<N> {
 81    fn add_assign(&mut self, rhs: Self) {
 82        for i in 0..N {
 83            self.0[i] += rhs.0[i];
 84        }
 85    }
 86}
 87
 88impl<const N: usize> Sum<Tn<N>> for Tn<N> {
 89    fn sum<I>(iter: I) -> Self
 90    where
 91        I: Iterator<Item = Self>,
 92    {
 93        let mut acc = Tn::<N>::zero();
 94        for e in iter {
 95            acc += e;
 96        }
 97        acc
 98    }
 99}
100
101impl<const N: usize> Sub<Tn<N>> for Tn<N> {
102    type Output = Self;
103
104    fn sub(self, rhs: Self) -> Self {
105        Self(array::from_fn(|i| self.0[i] - rhs.0[i]))
106    }
107}
108impl<const N: usize> Sub<&Tn<N>> for &Tn<N> {
109    type Output = Tn<N>;
110
111    fn sub(self, rhs: &Tn<N>) -> Self::Output {
112        Tn(array::from_fn(|i| self.0[i] - rhs.0[i]))
113    }
114}
115
116impl<const N: usize> SubAssign for Tn<N> {
117    fn sub_assign(&mut self, rhs: Self) {
118        for i in 0..N {
119            self.0[i] -= rhs.0[i];
120        }
121    }
122}
123
124impl<const N: usize> PartialEq for Tn<N> {
125    fn eq(&self, other: &Self) -> bool {
126        self.0 == other.0
127    }
128}
129
130impl<const N: usize> Mul<Tn<N>> for Tn<N> {
131    type Output = Self;
132
133    fn mul(self, rhs: Self) -> Self {
134        naive_poly_mul(&self, &rhs)
135    }
136}
137impl<const N: usize> Mul<&Tn<N>> for &Tn<N> {
138    type Output = Tn<N>;
139
140    fn mul(self, rhs: &Tn<N>) -> Self::Output {
141        naive_poly_mul(self, rhs)
142    }
143}
144
145fn naive_poly_mul<const N: usize>(poly1: &Tn<N>, poly2: &Tn<N>) -> Tn<N> {
146    let poly1: Vec<u128> = poly1.0.iter().map(|c| c.0 as u128).collect();
147    let poly2: Vec<u128> = poly2.0.iter().map(|c| c.0 as u128).collect();
148    let mut result: Vec<u128> = vec![0; (N * 2) - 1];
149    for i in 0..N {
150        for j in 0..N {
151            result[i + j] = result[i + j] + poly1[i] * poly2[j];
152        }
153    }
154
155    // apply mod (X^N + 1))
156    modulus_u128::<N>(&mut result);
157
158    // sanity check: check that there are no coeffs > i64_max
159    assert_eq!(
160        result,
161        Tn::<N>(array::from_fn(|i| T64(result[i] as u64)))
162            .coeffs()
163            .iter()
164            .map(|c| c.0 as u128)
165            .collect::<Vec<_>>()
166    );
167    Tn(array::from_fn(|i| T64(result[i] as u64)))
168}
169fn modulus_u128<const N: usize>(p: &mut Vec<u128>) {
170    if p.len() < N {
171        return;
172    }
173    for i in N..p.len() {
174        p[i - N] = p[i - N].clone() - p[i].clone();
175        p[i] = 0;
176    }
177    p.truncate(N);
178}
179
180// mul by u64
181impl<const N: usize> Mul<u64> for Tn<N> {
182    type Output = Self;
183    fn mul(self, s: u64) -> Self {
184        Self(array::from_fn(|i| self.0[i] * s))
185    }
186}
187impl<const N: usize> Mul<&u64> for &Tn<N> {
188    type Output = Tn<N>;
189    fn mul(self, s: &u64) -> Self::Output {
190        Tn::<N>(array::from_fn(|i| self.0[i] * *s))
191    }
192}