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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657
// Copyright (c) Facebook, Inc. and its affiliates.
//
// This source code is licensed under the MIT license found in the
// LICENSE file in the root directory of this source tree.
//! Basic polynomial operations.
//!
//! This module provides a set of function for basic polynomial operations, including:
//! - Polynomial evaluation using Horner method.
//! - Polynomial interpolation using Lagrange method.
//! - Polynomial addition, subtraction, multiplication, and division.
//! - Synthetic polynomial division for efficient division by polynomials of the form
//! `x`^`a` - `b`.
//!
//! In the context of this module any slice of field elements is considered to be a polynomial
//! in reverse coefficient form. A few examples:
//!
//! ```
//! # use winter_math::{fields::{f128::BaseElement}, FieldElement};
//! // p(x) = 2 * x + 1
//! let p = vec![BaseElement::new(1), BaseElement::new(2)];
//!
//! // p(x) = 4 * x^2 + 3
//! let p = [BaseElement::new(3), BaseElement::ZERO, BaseElement::new(4)];
//! ```
use crate::{field::FieldElement, utils::batch_inversion};
use core::mem;
use utils::{collections::Vec, group_vector_elements};
#[cfg(test)]
mod tests;
// POLYNOMIAL EVALUATION
// ================================================================================================
/// Evaluates a polynomial at a single point and returns the result.
///
/// Evaluates polynomial `p` at coordinate `x` using
/// [Horner's method](https://en.wikipedia.org/wiki/Horner%27s_method).
///
/// # Examples
/// ```
/// # use winter_math::polynom::*;
/// # use winter_math::{fields::{f128::BaseElement}, FieldElement};
/// // define polynomial: f(x) = 3 * x^2 + 2 * x + 1
/// let p = (1u32..4).map(BaseElement::from).collect::<Vec<_>>();
///
/// // evaluate the polynomial at point 4
/// let x = BaseElement::new(4);
/// assert_eq!(BaseElement::new(57), eval(&p, x));
/// ```
pub fn eval<B, E>(p: &[B], x: E) -> E
where
B: FieldElement,
E: FieldElement + From<B>,
{
// Horner evaluation
p.iter()
.rev()
.fold(E::ZERO, |acc, &coeff| acc * x + E::from(coeff))
}
/// Evaluates a polynomial at multiple points and returns a vector of results.
///
/// Evaluates polynomial `p` at all coordinates in `xs` slice by repeatedly invoking
/// `polynom::eval()` function.
///
/// # Examples
/// ```
/// # use winter_math::polynom::*;
/// # use winter_math::{fields::{f128::BaseElement}, FieldElement};
/// // define polynomial: f(x) = 3 * x^2 + 2 * x + 1
/// let p = (1_u32..4).map(BaseElement::from).collect::<Vec<_>>();
/// let xs = (3_u32..6).map(BaseElement::from).collect::<Vec<_>>();
///
/// let expected = xs.iter().map(|x| eval(&p, *x)).collect::<Vec<_>>();
/// assert_eq!(expected, eval_many(&p, &xs));
/// ```
pub fn eval_many<B, E>(p: &[B], xs: &[E]) -> Vec<E>
where
B: FieldElement,
E: FieldElement + From<B>,
{
xs.iter().map(|x| eval(p, *x)).collect()
}
// POLYNOMIAL INTERPOLATION
// ================================================================================================
/// Returns a polynomial in coefficient form interpolated from a set of X and Y coordinates.
///
/// Uses [Lagrange interpolation](https://en.wikipedia.org/wiki/Lagrange_polynomial) to build a
/// polynomial from X and Y coordinates. If `remove_leading_zeros = true`, all leading coefficients
/// which are ZEROs will be truncated; otherwise, the length of result will be equal to the number
/// of X coordinates.
///
/// # Panics
/// Panics if number of X and Y coordinates is not the same.
///
/// # Example
/// ```
/// # use winter_math::polynom::*;
/// # use winter_math::{fields::{f128::BaseElement}, FieldElement};
/// # use rand_utils::rand_vector;
/// let xs: Vec<BaseElement> = rand_vector(16);
/// let ys: Vec<BaseElement> = rand_vector(16);
///
/// let p = interpolate(&xs, &ys, false);
/// assert_eq!(ys, eval_many(&p, &xs));
/// ```
pub fn interpolate<E>(xs: &[E], ys: &[E], remove_leading_zeros: bool) -> Vec<E>
where
E: FieldElement,
{
debug_assert!(
xs.len() == ys.len(),
"number of X and Y coordinates must be the same"
);
let roots = get_zero_roots(xs);
let numerators: Vec<Vec<E>> = xs.iter().map(|&x| syn_div(&roots, 1, x)).collect();
let denominators: Vec<E> = numerators
.iter()
.zip(xs)
.map(|(e, &x)| eval(e, x))
.collect();
let denominators = batch_inversion(&denominators);
let mut result = E::zeroed_vector(xs.len());
for i in 0..xs.len() {
let y_slice = ys[i] * denominators[i];
for (j, res) in result.iter_mut().enumerate() {
*res += numerators[i][j] * y_slice;
}
}
if remove_leading_zeros {
crate::polynom::remove_leading_zeros(&result)
} else {
result
}
}
/// Returns a vector of polynomials interpolated from the provided X and Y coordinate batches.
///
/// Uses [Lagrange interpolation](https://en.wikipedia.org/wiki/Lagrange_polynomial) to build a
/// vector of polynomial from X and Y coordinate batches (one polynomial per batch).
///
/// When the number of batches is larger, this function is significantly faster than using
/// `polynom::interpolate()` function individually for each batch of coordinates. The speed-up
/// is primarily due to computing all inversions as a single batch inversion across all
/// coordinate batches.
///
/// # Panics
/// Panics if the number of X coordinate batches and Y coordinate batches is not the same.
///
/// # Examples
/// ```
/// # use core::convert::TryInto;
/// # use winter_math::polynom::*;
/// # use winter_math::{fields::{f128::BaseElement}, FieldElement};
/// # use rand_utils::rand_array;
/// let x_batches: Vec<[BaseElement; 8]> = vec![
/// rand_array(),
/// rand_array(),
/// ];
/// let y_batches: Vec<[BaseElement; 8]> = vec![
/// rand_array(),
/// rand_array(),
/// ];
///
/// let polys = interpolate_batch(&x_batches, &y_batches);
/// for ((p, xs), ys) in polys.iter().zip(x_batches).zip(y_batches) {
/// assert_eq!(ys.to_vec(), eval_many(p, &xs));
/// }
/// ```
pub fn interpolate_batch<E, const N: usize>(xs: &[[E; N]], ys: &[[E; N]]) -> Vec<[E; N]>
where
E: FieldElement,
{
debug_assert!(
xs.len() == ys.len(),
"number of X coordinate batches and Y coordinate batches must be the same"
);
let n = xs.len();
let mut equations = group_vector_elements(E::zeroed_vector(n * N * N));
let mut inverses = E::zeroed_vector(n * N);
// TODO: converting this to an array results in about 5% speed-up, but unfortunately, complex
// generic constraints are not yet supported: https://github.com/rust-lang/rust/issues/76560
let mut roots = vec![E::ZERO; N + 1];
for (i, xs) in xs.iter().enumerate() {
fill_zero_roots(xs, &mut roots);
for (j, &x) in xs.iter().enumerate() {
let equation = &mut equations[i * N + j];
// optimized synthetic division for this context
equation[N - 1] = roots[N];
for k in (0..N - 1).rev() {
equation[k] = roots[k + 1] + equation[k + 1] * x;
}
inverses[i * N + j] = eval(equation, x);
}
}
let equations = group_vector_elements::<[E; N], N>(equations);
let inverses = group_vector_elements::<E, N>(batch_inversion(&inverses));
let mut result = group_vector_elements(E::zeroed_vector(n * N));
for (i, poly) in result.iter_mut().enumerate() {
for j in 0..N {
let inv_y = ys[i][j] * inverses[i][j];
for (res_coeff, &eq_coeff) in poly.iter_mut().zip(equations[i][j].iter()) {
*res_coeff += eq_coeff * inv_y;
}
}
}
result
}
// POLYNOMIAL MATH OPERATIONS
// ================================================================================================
/// Returns a polynomial resulting from adding two polynomials together.
///
/// Polynomials `a` and `b` are expected to be in the coefficient form, and the returned
/// polynomial will be in the coefficient form as well. The length of the returned vector
/// will be max(a.len(), b.len()).
///
/// # Examples
/// ```
/// # use winter_math::polynom::*;
/// # use winter_math::{fields::{f128::BaseElement}, FieldElement};
/// // p1(x) = 4 * x^2 + 3 * x + 2
/// let p1 = (2_u32..5).map(BaseElement::from).collect::<Vec<_>>();
/// // p2(x) = 2 * x + 1
/// let p2 = (1_u32..3).map(BaseElement::from).collect::<Vec<_>>();
///
/// // expected result = 4 * x^2 + 5 * x + 3
/// let expected = vec![
/// BaseElement::new(3),
/// BaseElement::new(5),
/// BaseElement::new(4),
/// ];
/// assert_eq!(expected, add(&p1, &p2));
/// ```
pub fn add<E>(a: &[E], b: &[E]) -> Vec<E>
where
E: FieldElement,
{
let result_len = core::cmp::max(a.len(), b.len());
let mut result = Vec::with_capacity(result_len);
for i in 0..result_len {
let c1 = if i < a.len() { a[i] } else { E::ZERO };
let c2 = if i < b.len() { b[i] } else { E::ZERO };
result.push(c1 + c2);
}
result
}
/// Returns a polynomial resulting from subtracting one polynomial from another.
///
/// Specifically, subtracts polynomial `b` from polynomial `a` and returns the result. Both
/// polynomials are expected to be in the coefficient form, and the returned polynomial will
/// be in the coefficient form as well. The length of the returned vector will be
/// max(a.len(), b.len()).
///
/// # Examples
/// ```
/// # use winter_math::polynom::*;
/// # use winter_math::{fields::{f128::BaseElement}, FieldElement};
/// // p1(x) = 4 * x^2 + 3 * x + 2
/// let p1 = (2_u32..5).map(BaseElement::from).collect::<Vec<_>>();
/// // p2(x) = 2 * x + 1
/// let p2 = (1_u32..3).map(BaseElement::from).collect::<Vec<_>>();
///
/// // expected result = 4 * x^2 + x + 1
/// let expected = vec![
/// BaseElement::new(1),
/// BaseElement::new(1),
/// BaseElement::new(4),
/// ];
/// assert_eq!(expected, sub(&p1, &p2));
/// ```
pub fn sub<E>(a: &[E], b: &[E]) -> Vec<E>
where
E: FieldElement,
{
let result_len = core::cmp::max(a.len(), b.len());
let mut result = Vec::with_capacity(result_len);
for i in 0..result_len {
let c1 = if i < a.len() { a[i] } else { E::ZERO };
let c2 = if i < b.len() { b[i] } else { E::ZERO };
result.push(c1 - c2);
}
result
}
/// Returns a polynomial resulting from multiplying two polynomials together.
///
/// Polynomials `a` and `b` are expected to be in the coefficient form, and the returned
/// polynomial will be in the coefficient form as well. The length of the returned vector
/// will be a.len() + b.len() - 1.
///
/// # Examples
/// ```
/// # use winter_math::polynom::*;
/// # use winter_math::{fields::{f128::BaseElement}, FieldElement};
/// // p1(x) = x + 1
/// let p1 = [BaseElement::ONE, BaseElement::ONE];
/// // p2(x) = x^2 + 2
/// let p2 = [BaseElement::new(2), BaseElement::ZERO, BaseElement::ONE];
///
/// // expected result = x^3 + x^2 + 2 * x + 2
/// let expected = vec![
/// BaseElement::new(2),
/// BaseElement::new(2),
/// BaseElement::new(1),
/// BaseElement::new(1),
/// ];
/// assert_eq!(expected, mul(&p1, &p2));
/// ```
pub fn mul<E>(a: &[E], b: &[E]) -> Vec<E>
where
E: FieldElement,
{
let result_len = a.len() + b.len() - 1;
let mut result = E::zeroed_vector(result_len);
for i in 0..a.len() {
for j in 0..b.len() {
let s = a[i] * b[j];
result[i + j] += s;
}
}
result
}
/// Returns a polynomial resulting from multiplying a given polynomial by a scalar value.
///
/// Specifically, multiplies every coefficient of polynomial `p` by constant `k` and returns
/// the resulting vector.
///
/// # Examples
/// ```
/// # use winter_math::polynom::*;
/// # use winter_math::{fields::{f128::BaseElement}, FieldElement};
/// let p = [
/// BaseElement::new(1),
/// BaseElement::new(2),
/// BaseElement::new(3),
/// ];
/// let k = BaseElement::new(2);
///
/// let expected = vec![
/// BaseElement::new(2),
/// BaseElement::new(4),
/// BaseElement::new(6),
/// ];
/// assert_eq!(expected, mul_by_scalar(&p, k));
/// ```
pub fn mul_by_scalar<E>(p: &[E], k: E) -> Vec<E>
where
E: FieldElement,
{
let mut result = Vec::with_capacity(p.len());
for coeff in p {
result.push(*coeff * k);
}
result
}
/// Returns a polynomial resulting from dividing one polynomial by another.
///
/// Specifically, divides polynomial `a` by polynomial `b` and returns the result. If the
/// polynomials don't divide evenly, the remainder is ignored. Both polynomials are expected to
/// be in the coefficient form, and the returned polynomial will be in the coefficient form as
/// well. The length of the returned vector will be a.len() - b.len() + 1.
///
/// # Panics
/// Panics if:
/// * Polynomial `b` is empty.
/// * Degree of polynomial `b` is zero and the constant coefficient is ZERO.
/// * The degree of polynomial `b` is greater than the degree of polynomial `a`.
///
/// # Examples
/// ```
/// # use winter_math::polynom::*;
/// # use winter_math::{fields::{f128::BaseElement}, FieldElement};
/// // p1(x) = x^3 + x^2 + 2 * x + 2
/// let p1 = [
/// BaseElement::new(2),
/// BaseElement::new(2),
/// BaseElement::new(1),
/// BaseElement::new(1),
/// ];
/// // p2(x) = x^2 + 2
/// let p2 = [BaseElement::new(2), BaseElement::ZERO, BaseElement::ONE];
///
/// // expected result = x + 1
/// let expected = vec![BaseElement::ONE, BaseElement::ONE];
/// assert_eq!(expected, div(&p1, &p2));
/// ```
pub fn div<E>(a: &[E], b: &[E]) -> Vec<E>
where
E: FieldElement,
{
let mut apos = degree_of(a);
let mut a = a.to_vec();
let bpos = degree_of(b);
assert!(apos >= bpos, "cannot divide by polynomial of higher degree");
if bpos == 0 {
assert!(!b.is_empty(), "cannot divide by empty polynomial");
assert!(b[0] != E::ZERO, "cannot divide polynomial by zero");
}
let mut result = E::zeroed_vector(apos - bpos + 1);
for i in (0..result.len()).rev() {
let quot = a[apos] / b[bpos];
result[i] = quot;
for j in (0..bpos).rev() {
a[i + j] -= b[j] * quot;
}
apos = apos.wrapping_sub(1);
}
result
}
/// Returns a polynomial resulting from dividing a polynomial by a polynomial of special form.
///
/// Specifically, divides polynomial `p` by polynomial (x^`a` - `b`) using
/// [synthetic division](https://en.wikipedia.org/wiki/Synthetic_division) method; if the
/// polynomials don't divide evenly, the remainder is ignored. Polynomial `p` is expected
/// to be in the coefficient form, and the result will be in the coefficient form as well.
/// The length of the resulting polynomial will be equal to `p.len()`.
///
/// This function is significantly faster than the generic `polynom::div()` function.
///
/// # Panics
/// Panics if:
/// * `a` is zero;
/// * `b` is zero;
/// * `p.len()` is smaller than or equal to `a`.
///
/// # Examples
/// ```
/// # use winter_math::polynom::*;
/// # use winter_math::{fields::{f128::BaseElement}, FieldElement};
/// // p(x) = x^3 + x^2 + 2 * x + 2
/// let p = [
/// BaseElement::new(2),
/// BaseElement::new(2),
/// BaseElement::new(1),
/// BaseElement::new(1),
/// ];
///
/// // expected result = x^2 + 2
/// let expected = vec![
/// BaseElement::new(2),
/// BaseElement::ZERO,
/// BaseElement::new(1),
/// BaseElement::ZERO,
/// ];
///
/// // divide by x + 1
/// assert_eq!(expected, syn_div(&p, 1, -BaseElement::ONE));
/// ```
pub fn syn_div<E>(p: &[E], a: usize, b: E) -> Vec<E>
where
E: FieldElement,
{
let mut result = p.to_vec();
syn_div_in_place(&mut result, a, b);
result
}
/// Divides a polynomial by a polynomial of special form and saves the result into the original
/// polynomial.
///
/// Specifically, divides polynomial `p` by polynomial (x^`a` - `b`) using
/// [synthetic division](https://en.wikipedia.org/wiki/Synthetic_division) method and saves the
/// result into `p`. If the polynomials don't divide evenly, the remainder is ignored. Polynomial
/// `p` is expected to be in the coefficient form, and the result will be in coefficient form as
/// well.
///
/// This function is significantly faster than the generic `polynom::div()` function, and as
/// compared to `polynom::syn_div()` function, this function does not allocate any additional
/// memory.
///
/// # Panics
/// Panics if:
/// * `a` is zero;
/// * `b` is zero;
/// * `p.len()` is smaller than or equal to `a`.
///
/// # Examples
/// ```
/// # use winter_math::polynom::*;
/// # use winter_math::{fields::{f128::BaseElement}, FieldElement};
/// // p(x) = x^3 + x^2 + 2 * x + 2
/// let mut p = [
/// BaseElement::new(2),
/// BaseElement::new(2),
/// BaseElement::new(1),
/// BaseElement::new(1),
/// ];
///
/// // divide by x + 1
/// syn_div_in_place(&mut p, 1, -BaseElement::ONE);
///
/// // expected result = x^2 + 2
/// let expected = [
/// BaseElement::new(2),
/// BaseElement::ZERO,
/// BaseElement::new(1),
/// BaseElement::ZERO,
/// ];
///
/// assert_eq!(expected, p);
pub fn syn_div_in_place<E>(p: &mut [E], a: usize, b: E)
where
E: FieldElement,
{
assert!(a != 0, "divisor degree cannot be zero");
assert!(b != E::ZERO, "constant cannot be zero");
assert!(
p.len() > a,
"divisor degree cannot be greater than dividend size"
);
if a == 1 {
// if we are dividing by (x - `b`), we can use a single variable to keep track
// of the remainder; this way, we can avoid shifting the values in the slice later
let mut c = E::ZERO;
for coeff in p.iter_mut().rev() {
*coeff += b * c;
mem::swap(coeff, &mut c);
}
} else {
// if we are dividing by a polynomial of higher power, we need to keep track of the
// full remainder. we do that in place, but then need to shift the values at the end
// to discard the remainder
let degree_offset = p.len() - a;
if b == E::ONE {
// if `b` is 1, no need to multiply by `b` in every iteration of the loop
for i in (0..degree_offset).rev() {
p[i] += p[i + a];
}
} else {
for i in (0..degree_offset).rev() {
p[i] += p[i + a] * b;
}
}
// discard the remainder
p.copy_within(a.., 0);
p[degree_offset..].fill(E::ZERO);
}
}
// DEGREE INFERENCE
// ================================================================================================
/// Returns the degree of the provided polynomial.
///
/// If the size of the provided slice is much larger than the degree of the polynomial (i.e.,
/// a large number of leading coefficients is ZERO), this operation can be quite inefficient.
///
/// # Examples
/// ```
/// # use winter_math::polynom::*;
/// # use winter_math::{fields::{f128::BaseElement}, FieldElement};
/// assert_eq!(0, degree_of::<BaseElement>(&[]));
/// assert_eq!(0, degree_of(&[BaseElement::ONE]));
/// assert_eq!(1, degree_of(&[BaseElement::ONE, BaseElement::new(2)]));
/// assert_eq!(1, degree_of(&[BaseElement::ONE, BaseElement::new(2), BaseElement::ZERO]));
/// assert_eq!(2, degree_of(&[BaseElement::ONE, BaseElement::new(2), BaseElement::new(3)]));
/// assert_eq!(
/// 2,
/// degree_of(&[
/// BaseElement::ONE,
/// BaseElement::new(2),
/// BaseElement::new(3),
/// BaseElement::ZERO
/// ])
/// );
/// ```
pub fn degree_of<E>(poly: &[E]) -> usize
where
E: FieldElement,
{
for i in (0..poly.len()).rev() {
if poly[i] != E::ZERO {
return i;
}
}
0
}
/// Returns a polynomial with all leading ZERO coefficients removed.
///
/// # Examples
/// ```
/// # use winter_math::polynom::*;
/// # use winter_math::{fields::{f128::BaseElement}, FieldElement};
/// let a = vec![1u128, 2, 3, 4, 5, 6, 0, 0]
/// .into_iter()
/// .map(BaseElement::new)
/// .collect::<Vec<_>>();
/// let b = remove_leading_zeros(&a);
/// assert_eq!(6, b.len());
/// assert_eq!(a[..6], b);
///
/// let a = vec![0u128, 0, 0, 0]
/// .into_iter()
/// .map(BaseElement::new)
/// .collect::<Vec<_>>();
/// let b = remove_leading_zeros(&a);
/// assert_eq!(0, b.len());
/// ```
pub fn remove_leading_zeros<E>(values: &[E]) -> Vec<E>
where
E: FieldElement,
{
for i in (0..values.len()).rev() {
if values[i] != E::ZERO {
return values[..(i + 1)].to_vec();
}
}
vec![]
}
// HELPER FUNCTIONS
// ================================================================================================
fn get_zero_roots<E: FieldElement>(xs: &[E]) -> Vec<E> {
let mut result = unsafe { utils::uninit_vector(xs.len() + 1) };
fill_zero_roots(xs, &mut result);
result
}
fn fill_zero_roots<E: FieldElement>(xs: &[E], result: &mut [E]) {
let mut n = result.len();
n -= 1;
result[n] = E::ONE;
for i in 0..xs.len() {
n -= 1;
result[n] = E::ZERO;
#[allow(clippy::assign_op_pattern)]
for j in n..xs.len() {
result[j] = result[j] - result[j + 1] * xs[i];
}
}
}