1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
// 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;
use crate::num::basic::unsigneds::PrimitiveUnsigned;
use crate::polynomial::{ModPowerOf2Derivative, ModPowerOf2DerivativeAssign};
use crate::unsigned_polynomial::UnsignedPolynomial;
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}"
);
}
// Multiplies the coefficient of x^i, for i at least 1, by i, reduces it modulo 2^pow, and moves it
// to x^(i-1). Arithmetic modulo 2^WIDTH, with a wrapping running counter for i, loses nothing,
// since 2^pow divides 2^WIDTH. The products can be zero, so the result is trimmed.
fn mod_power_of_2_derivative_ref<T: PrimitiveUnsigned>(
p: &UnsignedPolynomial<T>,
pow: u64,
) -> UnsignedPolynomial<T> {
assert_reduced(p, pow);
let mut i = T::ZERO;
let mut q = UnsignedPolynomial {
coefficients: p
.coefficients
.iter()
.skip(1)
.map(|&c| {
i.wrapping_add_assign(T::ONE);
c.wrapping_mul(i).mod_power_of_2(pow)
})
.collect(),
};
q.trim();
q
}
fn mod_power_of_2_derivative_in_place<T: PrimitiveUnsigned>(
p: &mut UnsignedPolynomial<T>,
pow: u64,
) {
assert_reduced(p, pow);
if p.coefficients.is_empty() {
return;
}
p.coefficients.remove(0);
let mut i = T::ZERO;
for c in &mut p.coefficients {
i.wrapping_add_assign(T::ONE);
*c = c.wrapping_mul(i).mod_power_of_2(pow);
}
p.trim();
}
impl<T: PrimitiveUnsigned> ModPowerOf2Derivative for UnsignedPolynomial<T> {
type Output = Self;
/// Computes the derivative of an [`UnsignedPolynomial`] modulo $2^k$, taking the polynomial by
/// value. The coefficients must already be reduced modulo $2^k$.
///
/// $$
/// f(p, k) = p' \bmod 2^k.
/// $$
///
/// The coefficient of $x^i$ is multiplied by $i$, reduced, and moved to $x^{i-1}$. Since $ia_i$
/// can be divisible by the modulus even when $a_i$ is not zero, the derivative can lose any
/// number of degrees. A constant polynomial, including zero, has derivative zero.
///
/// # 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`, or if any coefficient of `self` is greater than
/// or equal to $2^k$.
///
/// # Examples
/// ```
/// use core::str::FromStr;
/// use malachite_base::polynomial::ModPowerOf2Derivative;
/// use malachite_base::unsigned_polynomial::UnsignedPolynomial;
///
/// let p = UnsignedPolynomial::<u8>::from_str("x^3+3*x^2+2*x+1").unwrap();
/// assert_eq!(p.mod_power_of_2_derivative(2).to_string(), "3*x^2+2*x+2");
///
/// let p = UnsignedPolynomial::<u8>::from_str("x^5+x").unwrap();
/// assert_eq!(p.mod_power_of_2_derivative(1).to_string(), "x^4+1");
///
/// // The derivative can lose more than one degree.
/// let p = UnsignedPolynomial::<u8>::from_str("x^4+x^3+1").unwrap();
/// assert_eq!(p.mod_power_of_2_derivative(1).to_string(), "x^2");
/// ```
///
/// This is equivalent to `nmod_poly_derivative` from `nmod_poly/derivative.c`, FLINT 3.6.0,
/// with the modulus $2^k$.
#[inline]
fn mod_power_of_2_derivative(mut self, pow: u64) -> Self {
mod_power_of_2_derivative_in_place(&mut self, pow);
self
}
}
impl<T: PrimitiveUnsigned> ModPowerOf2Derivative for &UnsignedPolynomial<T> {
type Output = UnsignedPolynomial<T>;
/// Computes the derivative of an [`UnsignedPolynomial`] modulo $2^k$, taking the polynomial by
/// reference. The coefficients must already be reduced modulo $2^k$.
///
/// $$
/// f(p, k) = p' \bmod 2^k.
/// $$
///
/// The coefficient of $x^i$ is multiplied by $i$, reduced, and moved to $x^{i-1}$. Since $ia_i$
/// can be divisible by the modulus even when $a_i$ is not zero, the derivative can lose any
/// number of degrees. A constant polynomial, including zero, has derivative zero.
///
/// # 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`, or if any coefficient of `self` is greater than
/// or equal to $2^k$.
///
/// # Examples
/// ```
/// use core::str::FromStr;
/// use malachite_base::polynomial::ModPowerOf2Derivative;
/// use malachite_base::unsigned_polynomial::UnsignedPolynomial;
///
/// let p = UnsignedPolynomial::<u8>::from_str("x^3+3*x^2+2*x+1").unwrap();
/// assert_eq!((&p).mod_power_of_2_derivative(2).to_string(), "3*x^2+2*x+2");
///
/// let p = UnsignedPolynomial::<u8>::from_str("x^5+x").unwrap();
/// assert_eq!((&p).mod_power_of_2_derivative(1).to_string(), "x^4+1");
///
/// // The derivative can lose more than one degree.
/// let p = UnsignedPolynomial::<u8>::from_str("x^4+x^3+1").unwrap();
/// assert_eq!((&p).mod_power_of_2_derivative(1).to_string(), "x^2");
/// ```
///
/// This is equivalent to `nmod_poly_derivative` from `nmod_poly/derivative.c`, FLINT 3.6.0,
/// with the modulus $2^k$.
#[inline]
fn mod_power_of_2_derivative(self, pow: u64) -> UnsignedPolynomial<T> {
mod_power_of_2_derivative_ref(self, pow)
}
}
impl<T: PrimitiveUnsigned> ModPowerOf2DerivativeAssign for UnsignedPolynomial<T> {
/// Replaces an [`UnsignedPolynomial`] with its derivative modulo $2^k$, in place. The
/// coefficients must already be reduced modulo $2^k$.
///
/// $$
/// p \gets p' \bmod 2^k.
/// $$
///
/// The coefficient of $x^i$ is multiplied by $i$, reduced, and moved to $x^{i-1}$. Since $ia_i$
/// can be divisible by the modulus even when $a_i$ is not zero, the derivative can lose any
/// number of degrees. A constant polynomial, including zero, has derivative zero.
///
/// # 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`, or if any coefficient of `self` is greater than
/// or equal to $2^k$.
///
/// # Examples
/// ```
/// use core::str::FromStr;
/// use malachite_base::polynomial::ModPowerOf2DerivativeAssign;
/// use malachite_base::unsigned_polynomial::UnsignedPolynomial;
///
/// let mut p = UnsignedPolynomial::<u8>::from_str("x^3+3*x^2+2*x+1").unwrap();
/// p.mod_power_of_2_derivative_assign(2);
/// assert_eq!(p.to_string(), "3*x^2+2*x+2");
///
/// let mut p = UnsignedPolynomial::<u8>::from_str("x^5+x").unwrap();
/// p.mod_power_of_2_derivative_assign(1);
/// assert_eq!(p.to_string(), "x^4+1");
///
/// // The derivative can lose more than one degree.
/// let mut p = UnsignedPolynomial::<u8>::from_str("x^4+x^3+1").unwrap();
/// p.mod_power_of_2_derivative_assign(1);
/// assert_eq!(p.to_string(), "x^2");
/// ```
///
/// This is equivalent to `nmod_poly_derivative` from `nmod_poly/derivative.c`, FLINT 3.6.0,
/// with the modulus $2^k$.
#[inline]
fn mod_power_of_2_derivative_assign(&mut self, pow: u64) {
mod_power_of_2_derivative_in_place(self, pow);
}
}