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// Copyright © 2026 Mikhail Hogrefe
//
// This file is part of Malachite.
//
// Malachite is free software: you can redistribute it and/or modify it under the terms of the GNU
// Lesser General Public License (LGPL) as published by the Free Software Foundation; either version
// 3 of the License, or (at your option) any later version. See <https://www.gnu.org/licenses/>.
use crate::integer::Integer;
use crate::natural::Natural;
use malachite_base::num::arithmetic::traits::{ExtendedGcd, NegAssign, UnsignedAbs};
impl ExtendedGcd for Integer {
type Gcd = Natural;
type Cofactor = Self;
/// Computes the GCD (greatest common divisor) of two [`Integer`]s $a$ and $b$, and also the
/// coefficients $x$ and $y$ in Bézout's identity $ax+by=\gcd(a,b)$. Both [`Integer`]s are
/// taken by value.
///
/// The are infinitely many $x$, $y$ that satisfy the identity for any $a$, $b$, so the full
/// specification is more detailed:
///
/// - $f(0, 0) = (0, 0, 0)$.
/// - $f(a, ak) = (a, 1, 0)$ if $a > 0$ and $k \neq 1$.
/// - $f(a, ak) = (-a, -1, 0)$ if $a < 0$ and $k \neq 1$.
/// - $f(bk, b) = (b, 0, 1)$ if $b > 0$.
/// - $f(bk, b) = (-b, 0, -1)$ if $b < 0$.
/// - $f(a, b) = (g, x, y)$ if $a \neq 0$ and $b \neq 0$ and $\gcd(a, b) \neq \min(|a|, |b|)$,
/// where $g = \gcd(a, b) \geq 0$, $ax + by = g$, $x \leq \lfloor b/g \rfloor$, and $y \leq
/// \lfloor a/g \rfloor$.
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::ExtendedGcd;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::integer::Integer;
///
/// assert_eq!(
/// Integer::from(3)
/// .extended_gcd(Integer::from(5))
/// .to_debug_string(),
/// "(1, 2, -1)"
/// );
/// assert_eq!(
/// Integer::from(240)
/// .extended_gcd(Integer::from(46))
/// .to_debug_string(),
/// "(2, -9, 47)"
/// );
/// assert_eq!(
/// Integer::from(-111)
/// .extended_gcd(Integer::from(300))
/// .to_debug_string(),
/// "(3, 27, 10)"
/// );
/// ```
fn extended_gcd(self, other: Self) -> (Natural, Self, Self) {
let a_sign = self.sign;
let b_sign = other.sign;
let (gcd, mut x, mut y) = self.unsigned_abs().extended_gcd(other.unsigned_abs());
if !a_sign {
x.neg_assign();
}
if !b_sign {
y.neg_assign();
}
(gcd, x, y)
}
}
impl ExtendedGcd<&Self> for Integer {
type Gcd = Natural;
type Cofactor = Self;
/// Computes the GCD (greatest common divisor) of two [`Integer`]s $a$ and $b$, and also the
/// coefficients $x$ and $y$ in Bézout's identity $ax+by=\gcd(a,b)$. The first [`Integer`] is
/// taken by value and the second by reference.
///
/// The are infinitely many $x$, $y$ that satisfy the identity for any $a$, $b$, so the full
/// specification is more detailed:
///
/// - $f(0, 0) = (0, 0, 0)$.
/// - $f(a, ak) = (a, 1, 0)$ if $a > 0$ and $k \neq 1$.
/// - $f(a, ak) = (-a, -1, 0)$ if $a < 0$ and $k \neq 1$.
/// - $f(bk, b) = (b, 0, 1)$ if $b > 0$.
/// - $f(bk, b) = (-b, 0, -1)$ if $b < 0$.
/// - $f(a, b) = (g, x, y)$ if $a \neq 0$ and $b \neq 0$ and $\gcd(a, b) \neq \min(|a|, |b|)$,
/// where $g = \gcd(a, b) \geq 0$, $ax + by = g$, $x \leq \lfloor b/g \rfloor$, and $y \leq
/// \lfloor a/g \rfloor$.
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::ExtendedGcd;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::integer::Integer;
///
/// assert_eq!(
/// Integer::from(3)
/// .extended_gcd(&Integer::from(5))
/// .to_debug_string(),
/// "(1, 2, -1)"
/// );
/// assert_eq!(
/// Integer::from(240)
/// .extended_gcd(&Integer::from(46))
/// .to_debug_string(),
/// "(2, -9, 47)"
/// );
/// assert_eq!(
/// Integer::from(-111)
/// .extended_gcd(&Integer::from(300))
/// .to_debug_string(),
/// "(3, 27, 10)"
/// );
/// ```
fn extended_gcd(self, other: &Self) -> (Natural, Self, Self) {
let a_sign = self.sign;
let (gcd, mut x, mut y) = self.unsigned_abs().extended_gcd(other.unsigned_abs_ref());
if !a_sign {
x.neg_assign();
}
if !other.sign {
y.neg_assign();
}
(gcd, x, y)
}
}
impl ExtendedGcd<Integer> for &Integer {
type Gcd = Natural;
type Cofactor = Integer;
/// Computes the GCD (greatest common divisor) of two [`Integer`]s $a$ and $b$, and also the
/// coefficients $x$ and $y$ in Bézout's identity $ax+by=\gcd(a,b)$. The first [`Integer`] is
/// taken by reference and the second by value.
///
/// The are infinitely many $x$, $y$ that satisfy the identity for any $a$, $b$, so the full
/// specification is more detailed:
///
/// - $f(0, 0) = (0, 0, 0)$.
/// - $f(a, ak) = (a, 1, 0)$ if $a > 0$ and $k \neq 1$.
/// - $f(a, ak) = (-a, -1, 0)$ if $a < 0$ and $k \neq 1$.
/// - $f(bk, b) = (b, 0, 1)$ if $b > 0$.
/// - $f(bk, b) = (-b, 0, -1)$ if $b < 0$.
/// - $f(a, b) = (g, x, y)$ if $a \neq 0$ and $b \neq 0$ and $\gcd(a, b) \neq \min(|a|, |b|)$,
/// where $g = \gcd(a, b) \geq 0$, $ax + by = g$, $x \leq \lfloor b/g \rfloor$, and $y \leq
/// \lfloor a/g \rfloor$.
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::ExtendedGcd;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::integer::Integer;
///
/// assert_eq!(
/// (&Integer::from(3))
/// .extended_gcd(Integer::from(5))
/// .to_debug_string(),
/// "(1, 2, -1)"
/// );
/// assert_eq!(
/// (&Integer::from(240))
/// .extended_gcd(Integer::from(46))
/// .to_debug_string(),
/// "(2, -9, 47)"
/// );
/// assert_eq!(
/// (&Integer::from(-111))
/// .extended_gcd(Integer::from(300))
/// .to_debug_string(),
/// "(3, 27, 10)"
/// );
/// ```
fn extended_gcd(self, other: Integer) -> (Natural, Integer, Integer) {
let b_sign = other.sign;
let (gcd, mut x, mut y) = self.unsigned_abs_ref().extended_gcd(other.unsigned_abs());
if !self.sign {
x.neg_assign();
}
if !b_sign {
y.neg_assign();
}
(gcd, x, y)
}
}
impl ExtendedGcd<&Integer> for &Integer {
type Gcd = Natural;
type Cofactor = Integer;
/// Computes the GCD (greatest common divisor) of two [`Integer`]s $a$ and $b$, and also the
/// coefficients $x$ and $y$ in Bézout's identity $ax+by=\gcd(a,b)$. Both [`Integer`]s are
/// taken by reference.
///
/// The are infinitely many $x$, $y$ that satisfy the identity for any $a$, $b$, so the full
/// specification is more detailed:
///
/// - $f(0, 0) = (0, 0, 0)$.
/// - $f(a, ak) = (a, 1, 0)$ if $a > 0$ and $k \neq 1$.
/// - $f(a, ak) = (-a, -1, 0)$ if $a < 0$ and $k \neq 1$.
/// - $f(bk, b) = (b, 0, 1)$ if $b > 0$.
/// - $f(bk, b) = (-b, 0, -1)$ if $b < 0$.
/// - $f(a, b) = (g, x, y)$ if $a \neq 0$ and $b \neq 0$ and $\gcd(a, b) \neq \min(|a|, |b|)$,
/// where $g = \gcd(a, b) \geq 0$, $ax + by = g$, $x \leq \lfloor b/g \rfloor$, and $y \leq
/// \lfloor a/g \rfloor$.
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::ExtendedGcd;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::integer::Integer;
///
/// assert_eq!(
/// (&Integer::from(3))
/// .extended_gcd(&Integer::from(5))
/// .to_debug_string(),
/// "(1, 2, -1)"
/// );
/// assert_eq!(
/// (&Integer::from(240))
/// .extended_gcd(&Integer::from(46))
/// .to_debug_string(),
/// "(2, -9, 47)"
/// );
/// assert_eq!(
/// (&Integer::from(-111))
/// .extended_gcd(&Integer::from(300))
/// .to_debug_string(),
/// "(3, 27, 10)"
/// );
/// ```
fn extended_gcd(self, other: &Integer) -> (Natural, Integer, Integer) {
let (gcd, mut x, mut y) = self
.unsigned_abs_ref()
.extended_gcd(other.unsigned_abs_ref());
if !self.sign {
x.neg_assign();
}
if !other.sign {
y.neg_assign();
}
(gcd, x, y)
}
}