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/*
Copyright 2025 Owain Davies
SPDX-License-Identifier: Apache-2.0 OR MIT
*/
use crate::Arbi;
impl Arbi {
/// Return the inverse of `self` modulo `modulus`, if it exists. Otherwise,
/// return `None`.
///
/// Mathematically, if the inverse exists, the return value, \\( r \\),
/// will be such that \\( 0 \leq r \lt |\text{modulus}| \\) (\\( 0 \\)
/// only if \\( |\text{modulus}| = 1 \\)).
///
/// # Panic
/// Panics if `modulus` is zero.
///
/// # Examples
/// ```
/// use arbi::Arbi;
/// let (a, ma) = (Arbi::from(3), Arbi::from(-3));
/// let (b, mb) = (Arbi::from(11), Arbi::from(-11));
///
/// let i = a.invert_ref(&b).unwrap();
/// assert_eq!(i, 4);
///
/// let i = ma.invert_ref(&b).unwrap();
/// assert_eq!(i, 7);
///
/// let i = a.invert_ref(&mb).unwrap();
/// assert_eq!(i, 4);
///
/// let i = ma.invert_ref(&mb).unwrap();
/// assert_eq!(i, 7);
///
/// assert_eq!(a.invert_ref(&Arbi::one()).unwrap(), 0);
/// assert_eq!(a.invert_ref(&Arbi::neg_one()).unwrap(), 0);
///
/// assert_eq!(Arbi::zero().invert_ref(&a), None);
/// assert_eq!(Arbi::zero().invert_ref(&ma), None);
/// ```
///
/// Panics if `modulus` is zero:
/// ```should_panic
/// use arbi::Arbi;
/// Arbi::from(7).invert_ref(&Arbi::zero());
/// ```
pub fn invert_ref(&self, modulus: &Self) -> Option<Self> {
assert!(!modulus.is_zero(), "modulus cannot be zero");
if modulus.size() == 1 && modulus.vec[0] == 1 {
// Modulus is -1 or 1
return Some(Self::zero());
}
// TODO: analyze how we can minimize allocations
/* Assumes m > 1 */
let (mut q, mut r) = (Arbi::zero(), Arbi::zero());
let mut a = self.clone();
let mut s = Self::one();
let mut t = Self::zero();
let mut m = modulus.clone();
while !m.is_zero() {
// Floor division, treating m as positive
Self::fddivide(&mut q, &mut r, &a.vec, &m.vec, a.signum(), 1);
a = m;
m = core::mem::take(&mut r);
let nt = s - core::mem::take(&mut q) * &t;
s = t;
t = nt;
}
if a != 1 {
// Inverse does not exist
return None;
}
/* At this point, s is an inverse to self modulo |modulus| such that
* |s| < |modulus|. Now adjust for potentially negative modulus
*/
// NOTE: If we are happy with 0 <= |r| < |modulus|, just return this:
// match (modulus.is_negative(), s.is_negative()) {
// (true, false) | (false, true) => Some(s + modulus),
// _ => Some(s),
// }
// However, we want 0 <= r < |modulus|.
let mut x = s % modulus;
if x.is_negative() {
if modulus.is_negative() {
x -= modulus;
} else {
x += modulus;
}
}
Some(x)
}
}
#[cfg(test)]
mod tests {
use super::*;
use crate::util::test::{get_seedable_rng, get_uniform_die, Distribution};
use crate::{SDDigit, SDigit, SQDigit};
pub fn gcdext(a: i128, b: i128) -> (i128, i128, i128) {
if a == 0 && b == 0 {
return (0, 0, 0);
}
let (mut old_r, mut r) = (a, b);
let (mut old_s, mut s) = (1, 0);
let (mut old_t, mut t) = (0, 1);
while r != 0 {
let quotient = old_r / r;
old_r = old_r - quotient * r;
std::mem::swap(&mut old_r, &mut r);
old_s = old_s - quotient * s;
std::mem::swap(&mut old_s, &mut s);
old_t = old_t - quotient * t;
std::mem::swap(&mut old_t, &mut t);
}
if old_r < 0 {
(-old_r, -old_s, -old_t)
} else {
(old_r, old_s, old_t)
}
}
pub fn modinv(a: i128, m: i128) -> Option<i128> {
if m == 0 {
panic!("modulus cannot be zero");
}
let (gcd, s, _) = gcdext(a, m);
if gcd != 1 {
return None;
}
// Ensure result is in range [0, abs(m))
let abs_m = m.abs();
let mut result = s % abs_m;
if result < 0 {
result += abs_m;
}
Some(result)
}
#[test]
#[should_panic = "modulus cannot be zero"]
fn zero_modulus_panics() {
Arbi::from(1).invert_ref(&Arbi::from(0));
}
#[test]
#[should_panic = "modulus cannot be zero"]
fn zero_modulus_panics_reference() {
modinv(1, 0);
}
#[test]
fn one_modulus() {
assert_eq!(Arbi::from(12345).invert_ref(&Arbi::one()).unwrap(), 0);
assert_eq!(Arbi::from(-12345).invert_ref(&Arbi::one()).unwrap(), 0);
assert_eq!(Arbi::from(12345).invert_ref(&Arbi::neg_one()).unwrap(), 0);
assert_eq!(Arbi::from(-12345).invert_ref(&Arbi::neg_one()).unwrap(), 0);
assert_eq!(Arbi::zero().invert_ref(&Arbi::one()).unwrap(), 0);
assert_eq!(Arbi::zero().invert_ref(&Arbi::neg_one()).unwrap(), 0);
}
#[test]
fn one_modulus_reference() {
assert_eq!(modinv(12345, 1).unwrap(), 0);
assert_eq!(modinv(-12345, 1).unwrap(), 0);
assert_eq!(modinv(12345, -1).unwrap(), 0);
assert_eq!(modinv(-12345, -1).unwrap(), 0);
assert_eq!(modinv(0, 1).unwrap(), 0);
assert_eq!(modinv(0, -1).unwrap(), 0);
}
#[test]
fn smoke() {
let (mut rng, _) = get_seedable_rng();
let small = get_uniform_die(i8::MIN, i8::MAX);
let sd = get_uniform_die(SDigit::MIN, SDigit::MAX);
let sdd = get_uniform_die(SDDigit::MIN, SDDigit::MAX);
let sqd = get_uniform_die(SQDigit::MIN, SQDigit::MAX);
let num_samples = 5000 as usize;
let mut samples: Vec<(i128, i128)> =
Vec::with_capacity(num_samples * 3);
for _ in 0..num_samples {
// (i32, i32)
let a = sd.sample(&mut rng) as i128;
let m = sd.sample(&mut rng) as i128;
if m != 0 {
samples.push((a, m))
};
// (i64, i64)
let a = sdd.sample(&mut rng) as i128;
let m = sdd.sample(&mut rng) as i128;
if m != 0 {
samples.push((a, m))
};
// (i128, i128)
let a = sqd.sample(&mut rng) as i128;
let m = sqd.sample(&mut rng) as i128;
if m != 0 {
samples.push((a, m))
};
// (i32, i128)
let a = sd.sample(&mut rng) as i128;
let m = sqd.sample(&mut rng) as i128;
if m != 0 {
samples.push((a, m))
};
// (i128, i32)
let a = sqd.sample(&mut rng) as i128;
let m = sd.sample(&mut rng) as i128;
if m != 0 {
samples.push((a, m))
};
// (i32, i64)
let a = sd.sample(&mut rng) as i128;
let m = sdd.sample(&mut rng) as i128;
if m != 0 {
samples.push((a, m))
};
// (i8, i8)
let a = small.sample(&mut rng) as i128;
let m = small.sample(&mut rng) as i128;
if m != 0 {
samples.push((a, m))
};
// (i8, i64)
let a = small.sample(&mut rng) as i128;
let m = sdd.sample(&mut rng) as i128;
if m != 0 {
samples.push((a, m))
};
// (i64, i8)
let a = sdd.sample(&mut rng) as i128;
let m = small.sample(&mut rng) as i128;
if m != 0 {
samples.push((a, m))
};
}
for (a, m) in samples {
let arbi_a = Arbi::from(a);
let arbi_m = Arbi::from(m);
assert_eq!(
arbi_a.invert_ref(&arbi_m).unwrap_or(Arbi::neg_one()),
modinv(a, m).unwrap_or(-1)
);
}
}
}