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use crate::{algorithms, Uint};
// FEATURE: sub_mod, neg_mod, inv_mod, div_mod, root_mod
// See <https://en.wikipedia.org/wiki/Cipolla's_algorithm>
// FEATURE: mul_mod_redc
// and maybe barrett
// See also <https://static1.squarespace.com/static/61f7cacf2d7af938cad5b81c/t/62deb4e0c434f7134c2730ee/1658762465114/modular_multiplication.pdf>
// FEATURE: Modular wrapper class, like Wrapping.
impl<const BITS: usize, const LIMBS: usize> Uint<BITS, LIMBS> {
/// ⚠️ Compute $\mod{\mathtt{self}}_{\mathtt{modulus}}$.
///
/// **Warning.** This function is not part of the stable API.
///
/// Returns zero if the modulus is zero.
// FEATURE: Reduce larger bit-sizes to smaller ones.
#[inline]
#[must_use]
pub fn reduce_mod(mut self, modulus: Self) -> Self {
if modulus.is_zero() {
return Self::ZERO;
}
if self >= modulus {
self %= modulus;
}
self
}
/// Compute $\mod{\mathtt{self} + \mathtt{rhs}}_{\mathtt{modulus}}$.
///
/// Returns zero if the modulus is zero.
#[inline]
#[must_use]
pub fn add_mod(self, rhs: Self, modulus: Self) -> Self {
// Reduce inputs
let lhs = self.reduce_mod(modulus);
let rhs = rhs.reduce_mod(modulus);
// Compute the sum and conditionally subtract modulus once.
let (mut result, overflow) = lhs.overflowing_add(rhs);
if overflow || result >= modulus {
result -= modulus;
}
result
}
/// Compute $\mod{\mathtt{self} ⋅ \mathtt{rhs}}_{\mathtt{modulus}}$.
///
/// Returns zero if the modulus is zero.
///
/// See [`mul_redc`](Self::mul_redc) for a faster variant at the cost of
/// some pre-computation.
#[inline]
#[must_use]
pub fn mul_mod(self, rhs: Self, mut modulus: Self) -> Self {
if modulus.is_zero() {
return Self::ZERO;
}
// Allocate at least `nlimbs(2 * BITS)` limbs to store the product. This array
// casting is a workaround for `generic_const_exprs` not being stable.
let mut product = [[0u64; 2]; LIMBS];
let product_len = crate::nlimbs(2 * BITS);
debug_assert!(2 * LIMBS >= product_len);
// SAFETY: `[[u64; 2]; LIMBS] == [u64; 2 * LIMBS] >= [u64; nlimbs(2 * BITS)]`.
let product = unsafe {
core::slice::from_raw_parts_mut(product.as_mut_ptr().cast::<u64>(), product_len)
};
// Compute full product.
let overflow = algorithms::addmul(product, self.as_limbs(), rhs.as_limbs());
debug_assert!(!overflow);
// Compute modulus using `div_rem`.
// This stores the remainder in the divisor, `modulus`.
algorithms::div(product, &mut modulus.limbs);
modulus
}
/// Compute $\mod{\mathtt{self}^{\mathtt{rhs}}}_{\mathtt{modulus}}$.
///
/// Returns zero if the modulus is zero.
#[inline]
#[must_use]
pub fn pow_mod(mut self, mut exp: Self, modulus: Self) -> Self {
if modulus.is_zero() || modulus <= Self::from(1) {
// Also covers Self::BITS == 0
return Self::ZERO;
}
// Exponentiation by squaring
let mut result = Self::from(1);
while exp > Self::ZERO {
// Multiply by base
if exp.limbs[0] & 1 == 1 {
result = result.mul_mod(self, modulus);
}
// Square base
self = self.mul_mod(self, modulus);
exp >>= 1;
}
result
}
/// Compute $\mod{\mathtt{self}^{-1}}_{\mathtt{modulus}}$.
///
/// Returns `None` if the inverse does not exist.
#[inline]
#[must_use]
pub fn inv_mod(self, modulus: Self) -> Option<Self> {
algorithms::inv_mod(self, modulus)
}
/// Montgomery multiplication.
///
/// Requires `self` and `other` to be less than `modulus`.
///
/// Computes
///
/// $$
/// \mod{\frac{\mathtt{self} ⋅ \mathtt{other}}{ 2^{64 ·
/// \mathtt{LIMBS}}}}_{\mathtt{modulus}} $$
///
/// This is useful because it can be computed notably faster than
/// [`mul_mod`](Self::mul_mod). Many computations can be done by
/// pre-multiplying values with $R = 2^{64 · \mathtt{LIMBS}}$
/// and then using [`mul_redc`](Self::mul_redc) instead of
/// [`mul_mod`](Self::mul_mod).
///
/// For this algorithm to work, it needs an extra parameter `inv` which must
/// be set to
///
/// $$
/// \mathtt{inv} = \mod{\frac{-1}{\mathtt{modulus}} }_{2^{64}}
/// $$
///
/// The `inv` value only exists for odd values of `modulus`. It can be
/// computed using [`inv_ring`](Self::inv_ring) from `U64`.
///
/// ```
/// # use ruint::{uint, Uint, aliases::*};
/// # uint!{
/// # let modulus = 21888242871839275222246405745257275088548364400416034343698204186575808495617_U256;
/// let inv = U64::wrapping_from(modulus).inv_ring().unwrap().wrapping_neg().to();
/// let prod = 5_U256.mul_redc(6_U256, modulus, inv);
/// # assert_eq!(inv.wrapping_mul(modulus.wrapping_to()), u64::MAX);
/// # assert_eq!(inv, 0xc2e1f593efffffff);
/// # }
/// ```
///
/// # Panics
///
/// Panics if `inv` is not correct in debug mode.
#[inline]
#[must_use]
pub fn mul_redc(self, other: Self, modulus: Self, inv: u64) -> Self {
if BITS == 0 {
return Self::ZERO;
}
let result = algorithms::mul_redc(self.limbs, other.limbs, modulus.limbs, inv);
let result = Self::from_limbs(result);
debug_assert!(result < modulus);
result
}
/// Montgomery squaring.
///
/// See [Self::mul_redc].
#[inline]
#[must_use]
pub fn square_redc(self, modulus: Self, inv: u64) -> Self {
if BITS == 0 {
return Self::ZERO;
}
let result = algorithms::square_redc(self.limbs, modulus.limbs, inv);
let result = Self::from_limbs(result);
debug_assert!(result < modulus);
result
}
}
#[cfg(test)]
mod tests {
use super::*;
use crate::{aliases::U64, const_for, nlimbs};
use proptest::{prop_assume, proptest, test_runner::Config};
#[test]
fn test_commutative() {
const_for!(BITS in SIZES {
const LIMBS: usize = nlimbs(BITS);
type U = Uint<BITS, LIMBS>;
proptest!(|(a: U, b: U, m: U)| {
assert_eq!(a.mul_mod(b, m), b.mul_mod(a, m));
});
});
}
#[test]
fn test_associative() {
const_for!(BITS in SIZES {
const LIMBS: usize = nlimbs(BITS);
type U = Uint<BITS, LIMBS>;
proptest!(|(a: U, b: U, c: U, m: U)| {
assert_eq!(a.mul_mod(b.mul_mod(c, m), m), a.mul_mod(b, m).mul_mod(c, m));
});
});
}
#[test]
fn test_distributive() {
const_for!(BITS in SIZES {
const LIMBS: usize = nlimbs(BITS);
type U = Uint<BITS, LIMBS>;
proptest!(|(a: U, b: U, c: U, m: U)| {
assert_eq!(a.mul_mod(b.add_mod(c, m), m), a.mul_mod(b, m).add_mod(a.mul_mod(c, m), m));
});
});
}
#[test]
fn test_add_identity() {
const_for!(BITS in NON_ZERO {
const LIMBS: usize = nlimbs(BITS);
type U = Uint<BITS, LIMBS>;
proptest!(|(value: U, m: U)| {
assert_eq!(value.add_mod(U::from(0), m), value.reduce_mod(m));
});
});
}
#[test]
fn test_mul_identity() {
const_for!(BITS in NON_ZERO {
const LIMBS: usize = nlimbs(BITS);
type U = Uint<BITS, LIMBS>;
proptest!(|(value: U, m: U)| {
assert_eq!(value.mul_mod(U::from(0), m), U::ZERO);
assert_eq!(value.mul_mod(U::from(1), m), value.reduce_mod(m));
});
});
}
#[test]
fn test_pow_identity() {
const_for!(BITS in NON_ZERO {
const LIMBS: usize = nlimbs(BITS);
type U = Uint<BITS, LIMBS>;
proptest!(|(a: U, m: U)| {
assert_eq!(a.pow_mod(U::from(0), m), U::from(1).reduce_mod(m));
assert_eq!(a.pow_mod(U::from(1), m), a.reduce_mod(m));
});
});
}
#[test]
fn test_pow_rules() {
const_for!(BITS in NON_ZERO {
const LIMBS: usize = nlimbs(BITS);
type U = Uint<BITS, LIMBS>;
// Too slow.
if LIMBS > 8 {
return;
}
let config = Config { cases: 5, ..Default::default() };
proptest!(config, |(a: U, b: U, c: U, m: U)| {
// TODO: a^(b+c) = a^b * a^c. Which requires carmichael fn.
// TODO: (a^b)^c = a^(b * c). Which requires carmichael fn.
assert_eq!(a.mul_mod(b, m).pow_mod(c, m), a.pow_mod(c, m).mul_mod(b.pow_mod(c, m), m));
});
});
}
#[test]
fn test_inv() {
const_for!(BITS in NON_ZERO {
const LIMBS: usize = nlimbs(BITS);
type U = Uint<BITS, LIMBS>;
proptest!(|(a: U, m: U)| {
if let Some(inv) = a.inv_mod(m) {
assert_eq!(a.mul_mod(inv, m), U::from(1));
}
});
});
}
#[test]
fn test_mul_redc() {
const_for!(BITS in NON_ZERO if (BITS >= 16) {
const LIMBS: usize = nlimbs(BITS);
type U = Uint<BITS, LIMBS>;
proptest!(|(a: U, b: U, m: U)| {
prop_assume!(m >= U::from(2));
if let Some(inv) = U64::from(m.as_limbs()[0]).inv_ring() {
let inv = (-inv).as_limbs()[0];
let r = U::from(2).pow_mod(U::from(64 * LIMBS), m);
let ar = a.mul_mod(r, m);
let br = b.mul_mod(r, m);
// TODO: Test for larger (>= m) values of a, b.
let expected = a.mul_mod(b, m).mul_mod(r, m);
assert_eq!(ar.mul_redc(br, m, inv), expected);
}
});
});
}
#[test]
fn test_square_redc() {
const_for!(BITS in NON_ZERO if (BITS >= 16) {
const LIMBS: usize = nlimbs(BITS);
type U = Uint<BITS, LIMBS>;
proptest!(|(a: U, m: U)| {
prop_assume!(m >= U::from(2));
if let Some(inv) = U64::from(m.as_limbs()[0]).inv_ring() {
let inv = (-inv).as_limbs()[0];
let r = U::from(2).pow_mod(U::from(64 * LIMBS), m);
let ar = a.mul_mod(r, m);
// TODO: Test for larger (>= m) values of a, b.
let expected = a.mul_mod(a, m).mul_mod(r, m);
assert_eq!(ar.square_redc(m, inv), expected);
}
});
});
}
}