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Implement more number theory functions #10

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115 changes: 113 additions & 2 deletions src/lib.rs
Original file line number Diff line number Diff line change
Expand Up @@ -21,10 +21,11 @@ extern crate std;

extern crate num_traits as traits;

use core::cmp::Ordering;
use core::mem;
use core::ops::Add;
use core::ops::{Add, Neg, Shr};

use traits::{Num, Signed, Zero};
use traits::{Num, NumRef, RefNum, Signed, Zero};

mod roots;
pub use roots::Roots;
Expand Down Expand Up @@ -1013,6 +1014,84 @@ impl_integer_for_usize!(usize, test_integer_usize);
#[cfg(has_i128)]
impl_integer_for_usize!(u128, test_integer_u128);

/// Calculate greatest common divisor and the corresponding coefficients.
pub fn extended_gcd<T: Integer + NumRef>(a: T, b: T) -> ExtendedGcd<T>
where
for<'a> &'a T: RefNum<T>,
{
// Euclid's extended algorithm
let (mut s, mut old_s) = (T::zero(), T::one());
let (mut t, mut old_t) = (T::one(), T::zero());
let (mut r, mut old_r) = (b, a);

while r != T::zero() {
let quotient = &old_r / &r;
old_r = old_r - &quotient * &r;
mem::swap(&mut old_r, &mut r);
old_s = old_s - &quotient * &s;
mem::swap(&mut old_s, &mut s);
old_t = old_t - quotient * &t;
mem::swap(&mut old_t, &mut t);
}

let _quotients = (t, s); // == (a, b) / gcd

ExtendedGcd {
gcd: old_r,
x: old_s,
y: old_t,
_hidden: (),
}
}

/// Find the standard representation of a (mod n).
pub fn normalize<T: Integer + NumRef>(a: T, n: &T) -> T {
let a = a % n;
match a.cmp(&T::zero()) {
Ordering::Less => a + n,
_ => a,
}
}

/// Calculate the inverse of a (mod n).
pub fn inverse<T: Integer + NumRef + Clone>(a: T, n: &T) -> Option<T>
where
for<'a> &'a T: RefNum<T>,
{
let ExtendedGcd { gcd, x: c, .. } = extended_gcd(a, n.clone());
if gcd == T::one() {
Some(normalize(c, n))
} else {
None
}
}

/// Calculate base^exp (mod modulus).
pub fn powm<T>(base: &T, exp: &T, modulus: &T) -> T
where
T: Integer + NumRef + Clone + Neg<Output = T> + Shr<i32, Output = T>,
for<'a> &'a T: RefNum<T>,
{
let zero = T::zero();
let one = T::one();
let two = &one + &one;
let mut exp = exp.clone();
let mut result = one.clone();
let mut base = base % modulus;
if exp < zero {
exp = -exp;
base = inverse(base, modulus).unwrap();
}
while exp > zero {
if &exp % &two == one {
result = (result * &base) % modulus;
}
exp = exp >> 1;
base = (&base * &base) % modulus;
}
result
}

/// An iterator over binomial coefficients.
pub struct IterBinomial<T> {
a: T,
Expand Down Expand Up @@ -1169,6 +1248,38 @@ fn test_lcm_overflow() {
check!(u64, 0x8000_0000_0000_0000, 0x02, 0x8000_0000_0000_0000);
}

#[test]
fn test_extended_gcd() {
assert_eq!(
extended_gcd(240, 46),
ExtendedGcd {
gcd: 2,
x: -9,
y: 47,
_hidden: ()
}
);
}

#[test]
fn test_normalize() {
assert_eq!(normalize(10, &7), 3);
assert_eq!(normalize(7, &7), 0);
assert_eq!(normalize(5, &7), 5);
assert_eq!(normalize(-3, &7), 4);
}

#[test]
fn test_inverse() {
assert_eq!(inverse(5, &7).unwrap(), 3);
}

#[test]
fn test_powm() {
// `i64::pow` would overflow.
assert_eq!(powm(&11, &19, &7), 4);
}

#[test]
fn test_iter_binomial() {
macro_rules! check_simple {
Expand Down