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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::num::arithmetic::traits::{ModPowerOf2IsReduced, Parity};
use crate::num::basic::traits::Zero;
use crate::num::basic::unsigneds::PrimitiveUnsigned;
use crate::polynomial::{ModPowerOf2Integral, ModPowerOf2IntegralAssign};
use crate::unsigned_polynomial::UnsignedPolynomial;
use alloc::vec;
use alloc::vec::Vec;
fn assert_reduced<T: PrimitiveUnsigned>(p: &UnsignedPolynomial<T>, pow: u64) {
assert!(pow <= T::WIDTH);
assert!(
p.mod_power_of_2_is_reduced(pow),
"self must be reduced mod 2^pow, but {p} has a coefficient >= 2^{pow}"
);
}
// The coefficients of the integral modulo $2^k$, where $k$ is `pow`, of the polynomial with
// coefficients `xs`, which is nonempty and reduced modulo $2^k$. The coefficient of $x^{i-1}$,
// divided by $i$, goes to $x^i$; a zero coefficient stays zero. The indices of the nonzero
// coefficients must all be odd, so their product is a unit, and all the divisions share its
// inverse, as in `mod_integral`. Arithmetic modulo $2^\text{W}$, with indices converted by
// wrapping, loses nothing, since $2^k$ divides $2^\text{W}$. The result is not trimmed.
fn mod_power_of_2_integral_coefficients<T: PrimitiveUnsigned>(xs: &[T], pow: u64) -> Vec<T> {
let n = xs.len();
let mut out = vec![T::ZERO; n + 1];
out[1..].copy_from_slice(xs);
// The product of the indices above i whose coefficients are nonzero.
let mut product = T::ONE.mod_power_of_2(pow);
for (i, c) in out.iter_mut().enumerate().skip(2).rev() {
if *c != T::ZERO {
assert!(
i.odd(),
"The integral modulo 2^pow is only defined if every nonzero coefficient belongs to \
an even power of x, but the coefficient of x^{} is nonzero",
i - 1
);
*c = c.wrapping_mul(product).mod_power_of_2(pow);
product = product
.wrapping_mul(T::wrapping_from(i))
.mod_power_of_2(pow);
}
}
if n >= 2 {
// `product` is odd, so it has an inverse.
let mut inverse = product.mod_power_of_2_inverse(pow).unwrap();
for (i, c) in out.iter_mut().enumerate().skip(2) {
if *c != T::ZERO {
*c = c.wrapping_mul(inverse).mod_power_of_2(pow);
inverse = inverse
.wrapping_mul(T::wrapping_from(i))
.mod_power_of_2(pow);
}
}
}
out
}
fn mod_power_of_2_integral_ref<T: PrimitiveUnsigned>(
p: &UnsignedPolynomial<T>,
pow: u64,
) -> UnsignedPolynomial<T> {
assert_reduced(p, pow);
if p.coefficients.is_empty() {
return UnsignedPolynomial::ZERO;
}
let mut q = UnsignedPolynomial {
coefficients: mod_power_of_2_integral_coefficients(&p.coefficients, pow),
};
q.trim();
q
}
impl<T: PrimitiveUnsigned> ModPowerOf2Integral for UnsignedPolynomial<T> {
type Output = Self;
/// Computes the integral modulo $2^k$ of an [`UnsignedPolynomial`] whose constant term is zero,
/// taking the polynomial by value. Its coefficients must already be reduced modulo $2^k$.
///
/// $$
/// f(p, k) = \int_0^x p(t)\,dt \bmod 2^k = \sum_{i=0}^{n-1} \frac{a_i}{i+1}x^{i+1} \bmod 2^k.
/// $$
///
/// The coefficient of $x^{i-1}$ is divided by $i$ and moved to $x^i$. Only odd numbers are
/// units modulo $2^k$, so every nonzero coefficient must belong to an even power of $x$; a zero
/// coefficient stays zero. The divisions share a single inversion modulo $2^k$. The integral of
/// zero is zero, for every $k$.
///
/// # Worst-case complexity
/// $T(n) = O(n)$
///
/// $M(n) = O(n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.len()`.
///
/// # Panics
/// Panics if `pow` is greater than `T::WIDTH`, if any coefficient is greater than or equal to
/// $2^k$, or if the coefficient of an odd power of $x$ is nonzero.
///
/// # Examples
/// ```
/// use core::str::FromStr;
/// use malachite_base::polynomial::ModPowerOf2Integral;
/// use malachite_base::unsigned_polynomial::UnsignedPolynomial;
///
/// // Dividing by 3 is multiplying by 3 modulo 8.
/// assert_eq!(
/// UnsignedPolynomial::<u8>::from_str("x^2")
/// .unwrap()
/// .mod_power_of_2_integral(3)
/// .to_string(),
/// "3*x^3"
/// );
/// // Dividing by 5 is multiplying by 13 modulo 16.
/// assert_eq!(
/// UnsignedPolynomial::<u8>::from_str("x^4+1")
/// .unwrap()
/// .mod_power_of_2_integral(4)
/// .to_string(),
/// "13*x^5+x"
/// );
/// ```
///
/// This is `nmod_poly_integral` from `nmod_poly/integral.c`, FLINT 3.6.0, with the modulus
/// $2^k$, except that FLINT needs every index from 1 to the degree plus 1 to be a unit, even
/// when its coefficient is zero, and so cannot integrate any polynomial of degree at least 1
/// modulo $2^k$.
#[inline]
fn mod_power_of_2_integral(self, pow: u64) -> Self {
mod_power_of_2_integral_ref(&self, pow)
}
}
impl<T: PrimitiveUnsigned> ModPowerOf2Integral for &UnsignedPolynomial<T> {
type Output = UnsignedPolynomial<T>;
/// Computes the integral modulo $2^k$ of an [`UnsignedPolynomial`] whose constant term is zero,
/// taking the polynomial by reference. Its coefficients must already be reduced modulo $2^k$.
///
/// $$
/// f(p, k) = \int_0^x p(t)\,dt \bmod 2^k = \sum_{i=0}^{n-1} \frac{a_i}{i+1}x^{i+1} \bmod 2^k.
/// $$
///
/// The coefficient of $x^{i-1}$ is divided by $i$ and moved to $x^i$. Only odd numbers are
/// units modulo $2^k$, so every nonzero coefficient must belong to an even power of $x$; a zero
/// coefficient stays zero. The divisions share a single inversion modulo $2^k$. The integral of
/// zero is zero, for every $k$.
///
/// # Worst-case complexity
/// $T(n) = O(n)$
///
/// $M(n) = O(n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.len()`.
///
/// # Panics
/// Panics if `pow` is greater than `T::WIDTH`, if any coefficient is greater than or equal to
/// $2^k$, or if the coefficient of an odd power of $x$ is nonzero.
///
/// # Examples
/// ```
/// use core::str::FromStr;
/// use malachite_base::polynomial::ModPowerOf2Integral;
/// use malachite_base::unsigned_polynomial::UnsignedPolynomial;
///
/// // Dividing by 3 is multiplying by 3 modulo 8.
/// assert_eq!(
/// (&UnsignedPolynomial::<u8>::from_str("x^2").unwrap())
/// .mod_power_of_2_integral(3)
/// .to_string(),
/// "3*x^3"
/// );
/// // Dividing by 5 is multiplying by 13 modulo 16.
/// assert_eq!(
/// (&UnsignedPolynomial::<u8>::from_str("x^4+1").unwrap())
/// .mod_power_of_2_integral(4)
/// .to_string(),
/// "13*x^5+x"
/// );
/// ```
///
/// This is `nmod_poly_integral` from `nmod_poly/integral.c`, FLINT 3.6.0, with the modulus
/// $2^k$, except that FLINT needs every index from 1 to the degree plus 1 to be a unit, even
/// when its coefficient is zero, and so cannot integrate any polynomial of degree at least 1
/// modulo $2^k$.
#[inline]
fn mod_power_of_2_integral(self, pow: u64) -> UnsignedPolynomial<T> {
mod_power_of_2_integral_ref(self, pow)
}
}
impl<T: PrimitiveUnsigned> ModPowerOf2IntegralAssign for UnsignedPolynomial<T> {
/// Replaces an [`UnsignedPolynomial`] with its integral modulo $2^k$ whose constant term is
/// zero. Its coefficients must already be reduced modulo $2^k$.
///
/// $$
/// p \gets \int_0^x p(t)\,dt \bmod 2^k = \sum_{i=0}^{n-1} \frac{a_i}{i+1}x^{i+1} \bmod 2^k.
/// $$
///
/// The coefficient of $x^{i-1}$ is divided by $i$ and moved to $x^i$. Only odd numbers are
/// units modulo $2^k$, so every nonzero coefficient must belong to an even power of $x$; a zero
/// coefficient stays zero. The divisions share a single inversion modulo $2^k$. The integral of
/// zero is zero, for every $k$.
///
/// # Worst-case complexity
/// $T(n) = O(n)$
///
/// $M(n) = O(n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.len()`.
///
/// # Panics
/// Panics if `pow` is greater than `T::WIDTH`, if any coefficient is greater than or equal to
/// $2^k$, or if the coefficient of an odd power of $x$ is nonzero.
///
/// # Examples
/// ```
/// use core::str::FromStr;
/// use malachite_base::polynomial::ModPowerOf2IntegralAssign;
/// use malachite_base::unsigned_polynomial::UnsignedPolynomial;
///
/// let mut p = UnsignedPolynomial::<u8>::from_str("x^2").unwrap();
/// p.mod_power_of_2_integral_assign(3);
/// assert_eq!(p.to_string(), "3*x^3");
/// ```
///
/// This is `nmod_poly_integral` from `nmod_poly/integral.c`, FLINT 3.6.0, with the modulus
/// $2^k$, except that FLINT needs every index from 1 to the degree plus 1 to be a unit, even
/// when its coefficient is zero, and so cannot integrate any polynomial of degree at least 1
/// modulo $2^k$.
#[inline]
fn mod_power_of_2_integral_assign(&mut self, pow: u64) {
*self = mod_power_of_2_integral_ref(self, pow);
}
}