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use crate::{WideningMul, WrappingMul};
/// Modular arithmetic in Montgomery form,
/// with manual modulo handling.
#[derive(Clone, Copy)]
#[repr(transparent)]
pub struct Montgomery<T> {
inner: T,
}
/// Modulo context for Montgomery arithmetic.
#[derive(Clone, Copy)]
pub struct MontgomeryModulo<T> {
// modulo
m: T,
// size in bits
s: usize,
// 2^2s, modulo m
r2: Montgomery<T>,
// inverse of m, modulo 2^s
m_inv: T,
}
impl<
T: Clone
+ num_traits::Zero
+ num_traits::FromPrimitive
+ num_traits::WrappingAdd
+ num_traits::WrappingSub
+ std::ops::Rem<T, Output = T>
+ std::cmp::Ord
+ WideningMul
+ WrappingMul,
> MontgomeryModulo<T>
{
fn inv_s(x: &T, mut s: usize) -> T {
debug_assert!(s != 0);
let lead = s.leading_zeros() + 1;
let mut r = T::from_u64(1).unwrap();
let two = T::from_u64(2).unwrap();
for i in (lead..usize::BITS) {
let bit = s & (1 << (usize::BITS - i - 1));
r = r.wpmul(&two.wrapping_sub(&x.wpmul(&r)));
if bit != 0 {
r = r.wpmul(&r.wpmul(&x));
}
}
r
}
fn montgomery_reduce_s(low: &T, high: &T, m: &T, s: usize, m_inv: &T) -> T {
let q = low.wpmul(m_inv);
let (_, w) = q.wdmul(m);
let r = if high >= &w {
high.wrapping_sub(&w)
} else {
high.wrapping_add(m).wrapping_sub(&w)
};
r
}
fn mul_montgomery_s(a: &T, b: &T, m: &T, s: usize, m_inv: &T) -> T {
let (t_low, t_high) = a.wdmul(b);
let r = Self::montgomery_reduce_s(&t_low, &t_high, m, s, m_inv);
r
}
fn r2_s(x: &T, s: usize, m_inv: &T) -> T {
debug_assert!(s != 0);
let lead = s.leading_zeros() + 1;
let mut r = T::zero().wrapping_sub(x) % x.clone();
let two = T::from_u64(2).unwrap();
r = r.wpmul(&two);
if &r >= x {
r = r.wrapping_sub(x);
}
for i in (lead..usize::BITS) {
let bit = s & (1 << (usize::BITS - i - 1));
r = Self::mul_montgomery_s(&r, &r, x, s, m_inv);
if bit != 0 {
r = r.wpmul(&two);
if &r >= x {
r = r.wrapping_sub(x);
}
}
}
r
}
/// Creates a context for modulo `m`,
/// where `T` is assumed to represent
/// an `s`-bit number.
pub fn new(m: T, s: usize) -> Self {
let m_inv = Self::inv_s(&m, s);
let inner = Self::r2_s(&m, s, &m_inv);
Self {
m: m,
s: s,
r2: Montgomery { inner: inner },
m_inv: m_inv,
}
}
/// Multiplies `a` by `b` modulo `m`.
///
/// # Example
/// ```rust
/// # use crtypes_algebra::*;
/// let modulo = 53_u32;
/// let mg = MontgomeryModulo::new(modulo, u32::BITS as usize);
/// let a = 45_u32;
/// let a_mg = mg.encode(a);
/// let b = 11_u32;
/// let b_mg = mg.encode(b);
/// let p_mg = mg.mul(&a_mg, &b_mg);
/// let p = mg.decode(&p_mg);
/// assert_eq!(p, (a * b) % modulo);
/// ```
pub fn mul(&self, a: &Montgomery<T>, b: &Montgomery<T>) -> Montgomery<T> {
Montgomery {
inner: Self::mul_montgomery_s(&a.inner, &b.inner, &self.m, self.s, &self.m_inv),
}
}
/// Converts `x` to Montgomery form.
///
/// # Example
/// ```rust
/// # use crtypes_algebra::*;
/// let mg = MontgomeryModulo::new(53_u32, u32::BITS as usize);
/// let x = 45_u32;
/// let x_mg = mg.encode(x);
/// assert_eq!(mg.decode(&x_mg), x);
/// ```
pub fn encode(&self, x: T) -> Montgomery<T> {
self.mul(&Montgomery { inner: x }, &self.r2)
}
/// Converts `x` back from Montgomery form.
pub fn decode(&self, x: &Montgomery<T>) -> T {
Self::montgomery_reduce_s(&x.inner, &T::zero(), &self.m, self.s, &self.m_inv)
}
}