winter_math/field/f62/mod.rs
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
// 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.
//! An implementation of a 62-bit STARK-friendly prime field with modulus $2^{62} - 111 \cdot 2^{39} + 1$.
//!
//! All operations in this field are implemented using Montgomery arithmetic. It supports very
//! fast modular arithmetic including branchless multiplication and addition. Base elements are
//! stored in the Montgomery form using `u64` as the backing type.
use alloc::{
string::{String, ToString},
vec::Vec,
};
use core::{
fmt::{Debug, Display, Formatter},
mem,
ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign},
slice,
};
#[cfg(feature = "serde")]
use serde::{Deserialize, Serialize};
use utils::{
AsBytes, ByteReader, ByteWriter, Deserializable, DeserializationError, Randomizable,
Serializable,
};
use super::{ExtensibleField, FieldElement, StarkField};
#[cfg(test)]
mod tests;
// CONSTANTS
// ================================================================================================
/// Field modulus = 2^62 - 111 * 2^39 + 1
const M: u64 = 4611624995532046337;
/// 2^128 mod M; this is used for conversion of elements into Montgomery representation.
const R2: u64 = 630444561284293700;
/// 2^192 mod M; this is used during element inversion.
const R3: u64 = 732984146687909319;
/// -M^{-1} mod 2^64; this is used during element multiplication.
const U: u128 = 4611624995532046335;
/// Number of bytes needed to represent field element
const ELEMENT_BYTES: usize = core::mem::size_of::<u64>();
// 2^39 root of unity
const G: u64 = 4421547261963328785;
// FIELD ELEMENT
// ================================================================================================
/// Represents base field element in the field.
///
/// Internal values are stored in Montgomery representation and can be in the range [0; 2M). The
/// backing type is `u64`.
#[derive(Copy, Clone, Default)]
#[cfg_attr(feature = "serde", derive(Deserialize, Serialize))]
#[cfg_attr(feature = "serde", serde(try_from = "u64", into = "u64"))]
pub struct BaseElement(u64);
impl BaseElement {
/// Creates a new field element from the provided `value`; the value is converted into
/// Montgomery representation.
pub const fn new(value: u64) -> BaseElement {
// multiply the value with R2 to convert to Montgomery representation; this is OK because
// given the value of R2, the product of R2 and `value` is guaranteed to be in the range
// [0, 4M^2 - 4M + 1)
let z = mul(value, R2);
BaseElement(z)
}
}
impl FieldElement for BaseElement {
type PositiveInteger = u64;
type BaseField = Self;
const EXTENSION_DEGREE: usize = 1;
const ZERO: Self = BaseElement::new(0);
const ONE: Self = BaseElement::new(1);
const ELEMENT_BYTES: usize = ELEMENT_BYTES;
const IS_CANONICAL: bool = false;
// ALGEBRA
// --------------------------------------------------------------------------------------------
#[inline]
fn double(self) -> Self {
let z = self.0 << 1;
let q = (z >> 62) * M;
Self(z - q)
}
fn exp(self, power: Self::PositiveInteger) -> Self {
let mut b = self;
if power == 0 {
return Self::ONE;
} else if b == Self::ZERO {
return Self::ZERO;
}
let mut r = if power & 1 == 1 { b } else { Self::ONE };
for i in 1..64 - power.leading_zeros() {
b = b.square();
if (power >> i) & 1 == 1 {
r *= b;
}
}
r
}
fn inv(self) -> Self {
BaseElement(inv(self.0))
}
fn conjugate(&self) -> Self {
BaseElement(self.0)
}
// BASE ELEMENT CONVERSIONS
// --------------------------------------------------------------------------------------------
fn base_element(&self, i: usize) -> Self::BaseField {
match i {
0 => *self,
_ => panic!("element index must be 0, but was {i}"),
}
}
fn slice_as_base_elements(elements: &[Self]) -> &[Self::BaseField] {
elements
}
fn slice_from_base_elements(elements: &[Self::BaseField]) -> &[Self] {
elements
}
// SERIALIZATION / DESERIALIZATION
// --------------------------------------------------------------------------------------------
fn elements_as_bytes(elements: &[Self]) -> &[u8] {
// TODO: take endianness into account
let p = elements.as_ptr();
let len = elements.len() * Self::ELEMENT_BYTES;
unsafe { slice::from_raw_parts(p as *const u8, len) }
}
unsafe fn bytes_as_elements(bytes: &[u8]) -> Result<&[Self], DeserializationError> {
if bytes.len() % Self::ELEMENT_BYTES != 0 {
return Err(DeserializationError::InvalidValue(format!(
"number of bytes ({}) does not divide into whole number of field elements",
bytes.len(),
)));
}
let p = bytes.as_ptr();
let len = bytes.len() / Self::ELEMENT_BYTES;
if (p as usize) % mem::align_of::<u64>() != 0 {
return Err(DeserializationError::InvalidValue(
"slice memory alignment is not valid for this field element type".to_string(),
));
}
Ok(slice::from_raw_parts(p as *const Self, len))
}
}
impl StarkField for BaseElement {
/// sage: MODULUS = 2^62 - 111 * 2^39 + 1 \
/// sage: GF(MODULUS).is_prime_field() \
/// True \
/// sage: GF(MODULUS).order() \
/// 4611624995532046337
const MODULUS: Self::PositiveInteger = M;
const MODULUS_BITS: u32 = 62;
/// sage: GF(MODULUS).primitive_element() \
/// 3
const GENERATOR: Self = BaseElement::new(3);
/// sage: is_odd((MODULUS - 1) / 2^39) \
/// True
const TWO_ADICITY: u32 = 39;
/// sage: k = (MODULUS - 1) / 2^39 \
/// sage: GF(MODULUS).primitive_element()^k \
/// 4421547261963328785
const TWO_ADIC_ROOT_OF_UNITY: Self = BaseElement::new(G);
fn get_modulus_le_bytes() -> Vec<u8> {
Self::MODULUS.to_le_bytes().to_vec()
}
#[inline]
fn as_int(&self) -> Self::PositiveInteger {
// convert from Montgomery representation by multiplying by 1
let result = mul(self.0, 1);
// since the result of multiplication can be in [0, 2M), we need to normalize it
normalize(result)
}
}
impl Randomizable for BaseElement {
const VALUE_SIZE: usize = Self::ELEMENT_BYTES;
fn from_random_bytes(bytes: &[u8]) -> Option<Self> {
Self::try_from(bytes).ok()
}
}
impl Debug for BaseElement {
fn fmt(&self, f: &mut Formatter<'_>) -> core::fmt::Result {
write!(f, "{}", self)
}
}
impl Display for BaseElement {
fn fmt(&self, f: &mut Formatter) -> core::fmt::Result {
write!(f, "{}", self.as_int())
}
}
// EQUALITY CHECKS
// ================================================================================================
impl PartialEq for BaseElement {
#[inline]
fn eq(&self, other: &Self) -> bool {
// since either of the elements can be in [0, 2M) range, we normalize them first to be
// in [0, M) range and then compare them.
normalize(self.0) == normalize(other.0)
}
}
impl Eq for BaseElement {}
// OVERLOADED OPERATORS
// ================================================================================================
impl Add for BaseElement {
type Output = Self;
fn add(self, rhs: Self) -> Self {
Self(add(self.0, rhs.0))
}
}
impl AddAssign for BaseElement {
fn add_assign(&mut self, rhs: Self) {
*self = *self + rhs
}
}
impl Sub for BaseElement {
type Output = Self;
fn sub(self, rhs: Self) -> Self {
Self(sub(self.0, rhs.0))
}
}
impl SubAssign for BaseElement {
fn sub_assign(&mut self, rhs: Self) {
*self = *self - rhs;
}
}
impl Mul for BaseElement {
type Output = Self;
fn mul(self, rhs: Self) -> Self {
Self(mul(self.0, rhs.0))
}
}
impl MulAssign for BaseElement {
fn mul_assign(&mut self, rhs: Self) {
*self = *self * rhs
}
}
impl Div for BaseElement {
type Output = Self;
fn div(self, rhs: Self) -> Self {
Self(mul(self.0, inv(rhs.0)))
}
}
impl DivAssign for BaseElement {
fn div_assign(&mut self, rhs: Self) {
*self = *self / rhs
}
}
impl Neg for BaseElement {
type Output = Self;
fn neg(self) -> Self {
Self(sub(0, self.0))
}
}
// QUADRATIC EXTENSION
// ================================================================================================
/// Defines a quadratic extension of the base field over an irreducible polynomial x<sup>2</sup> -
/// x - 1. Thus, an extension element is defined as α + β * φ, where φ is a root of this polynomial,
/// and α and β are base field elements.
impl ExtensibleField<2> for BaseElement {
#[inline(always)]
fn mul(a: [Self; 2], b: [Self; 2]) -> [Self; 2] {
let z = a[0] * b[0];
[z + a[1] * b[1], (a[0] + a[1]) * (b[0] + b[1]) - z]
}
#[inline(always)]
fn mul_base(a: [Self; 2], b: Self) -> [Self; 2] {
[a[0] * b, a[1] * b]
}
#[inline(always)]
fn frobenius(x: [Self; 2]) -> [Self; 2] {
[x[0] + x[1], -x[1]]
}
}
// CUBIC EXTENSION
// ================================================================================================
/// Defines a cubic extension of the base field over an irreducible polynomial x<sup>3</sup> +
/// 2x + 2. Thus, an extension element is defined as α + β * φ + γ * φ^2, where φ is a root of this
/// polynomial, and α, β and γ are base field elements.
impl ExtensibleField<3> for BaseElement {
#[inline(always)]
fn mul(a: [Self; 3], b: [Self; 3]) -> [Self; 3] {
// performs multiplication in the extension field using 6 multiplications, 8 additions,
// and 9 subtractions in the base field. overall, a single multiplication in the extension
// field is roughly equal to 12 multiplications in the base field.
let a0b0 = a[0] * b[0];
let a1b1 = a[1] * b[1];
let a2b2 = a[2] * b[2];
let a0b0_a0b1_a1b0_a1b1 = (a[0] + a[1]) * (b[0] + b[1]);
let minus_a0b0_a0b2_a2b0_minus_a2b2 = (a[0] - a[2]) * (b[2] - b[0]);
let a1b1_minus_a1b2_minus_a2b1_a2b2 = (a[1] - a[2]) * (b[1] - b[2]);
let a0b0_a1b1 = a0b0 + a1b1;
let minus_2a1b2_minus_2a2b1 = (a1b1_minus_a1b2_minus_a2b1_a2b2 - a1b1 - a2b2).double();
let a0b0_minus_2a1b2_minus_2a2b1 = a0b0 + minus_2a1b2_minus_2a2b1;
let a0b1_a1b0_minus_2a1b2_minus_2a2b1_minus_2a2b2 =
a0b0_a0b1_a1b0_a1b1 + minus_2a1b2_minus_2a2b1 - a2b2.double() - a0b0_a1b1;
let a0b2_a1b1_a2b0_minus_2a2b2 = minus_a0b0_a0b2_a2b0_minus_a2b2 + a0b0_a1b1 - a2b2;
[
a0b0_minus_2a1b2_minus_2a2b1,
a0b1_a1b0_minus_2a1b2_minus_2a2b1_minus_2a2b2,
a0b2_a1b1_a2b0_minus_2a2b2,
]
}
#[inline(always)]
fn mul_base(a: [Self; 3], b: Self) -> [Self; 3] {
[a[0] * b, a[1] * b, a[2] * b]
}
#[inline(always)]
fn frobenius(x: [Self; 3]) -> [Self; 3] {
// coefficients were computed using SageMath
[
x[0] + BaseElement::new(2061766055618274781) * x[1]
+ BaseElement::new(786836585661389001) * x[2],
BaseElement::new(2868591307402993000) * x[1]
+ BaseElement::new(3336695525575160559) * x[2],
BaseElement::new(2699230790596717670) * x[1]
+ BaseElement::new(1743033688129053336) * x[2],
]
}
}
// TYPE CONVERSIONS
// ================================================================================================
impl From<u32> for BaseElement {
/// Converts a 32-bit value into a field element.
fn from(value: u32) -> Self {
BaseElement::new(value as u64)
}
}
impl From<u16> for BaseElement {
/// Converts a 16-bit value into a field element.
fn from(value: u16) -> Self {
BaseElement::new(value as u64)
}
}
impl From<u8> for BaseElement {
/// Converts an 8-bit value into a field element.
fn from(value: u8) -> Self {
BaseElement::new(value as u64)
}
}
impl From<BaseElement> for u128 {
fn from(value: BaseElement) -> Self {
value.as_int() as u128
}
}
impl From<BaseElement> for u64 {
fn from(value: BaseElement) -> Self {
value.as_int()
}
}
impl TryFrom<u64> for BaseElement {
type Error = String;
fn try_from(value: u64) -> Result<Self, Self::Error> {
if value >= M {
Err(format!(
"invalid field element: value {value} is greater than or equal to the field modulus"
))
} else {
Ok(Self::new(value))
}
}
}
impl TryFrom<u128> for BaseElement {
type Error = String;
fn try_from(value: u128) -> Result<Self, Self::Error> {
if value >= M as u128 {
Err(format!(
"invalid field element: value {value} is greater than or equal to the field modulus"
))
} else {
Ok(Self::new(value as u64))
}
}
}
impl TryFrom<[u8; 8]> for BaseElement {
type Error = String;
fn try_from(bytes: [u8; 8]) -> Result<Self, Self::Error> {
let value = u64::from_le_bytes(bytes);
Self::try_from(value)
}
}
impl<'a> TryFrom<&'a [u8]> for BaseElement {
type Error = DeserializationError;
/// Converts a slice of bytes into a field element; returns error if the value encoded in bytes
/// is not a valid field element. The bytes are assumed to encode the element in the canonical
/// representation in little-endian byte order.
fn try_from(bytes: &[u8]) -> Result<Self, Self::Error> {
if bytes.len() < ELEMENT_BYTES {
return Err(DeserializationError::InvalidValue(format!(
"not enough bytes for a full field element; expected {} bytes, but was {} bytes",
ELEMENT_BYTES,
bytes.len(),
)));
}
if bytes.len() > ELEMENT_BYTES {
return Err(DeserializationError::InvalidValue(format!(
"too many bytes for a field element; expected {} bytes, but was {} bytes",
ELEMENT_BYTES,
bytes.len(),
)));
}
let value = bytes
.try_into()
.map(u64::from_le_bytes)
.map_err(|error| DeserializationError::UnknownError(format!("{error}")))?;
if value >= M {
return Err(DeserializationError::InvalidValue(format!(
"invalid field element: value {value} is greater than or equal to the field modulus"
)));
}
Ok(BaseElement::new(value))
}
}
impl AsBytes for BaseElement {
fn as_bytes(&self) -> &[u8] {
// TODO: take endianness into account
let self_ptr: *const BaseElement = self;
unsafe { slice::from_raw_parts(self_ptr as *const u8, ELEMENT_BYTES) }
}
}
// SERIALIZATION / DESERIALIZATION
// ------------------------------------------------------------------------------------------------
impl Serializable for BaseElement {
fn write_into<W: ByteWriter>(&self, target: &mut W) {
// convert from Montgomery representation into canonical representation
target.write_bytes(&self.as_int().to_le_bytes());
}
fn get_size_hint(&self) -> usize {
self.as_int().get_size_hint()
}
}
impl Deserializable for BaseElement {
fn read_from<R: ByteReader>(source: &mut R) -> Result<Self, DeserializationError> {
let value = source.read_u64()?;
if value >= M {
return Err(DeserializationError::InvalidValue(format!(
"invalid field element: value {value} is greater than or equal to the field modulus"
)));
}
Ok(BaseElement::new(value))
}
}
// FINITE FIELD ARITHMETIC
// ================================================================================================
/// Computes (a + b) reduced by M such that the output is in [0, 2M) range; a and b are assumed to
/// be in [0, 2M).
#[inline(always)]
fn add(a: u64, b: u64) -> u64 {
let z = a + b;
let q = (z >> 62) * M;
z - q
}
/// Computes (a - b) reduced by M such that the output is in [0, 2M) range; a and b are assumed to
/// be in [0, 2M).
#[inline(always)]
fn sub(a: u64, b: u64) -> u64 {
if a < b {
2 * M - b + a
} else {
a - b
}
}
/// Computes (a * b) reduced by M such that the output is in [0, 2M) range; a and b are assumed to
/// be in [0, 2M).
#[inline(always)]
const fn mul(a: u64, b: u64) -> u64 {
let z = (a as u128) * (b as u128);
let q = (((z as u64) as u128) * U) as u64;
let z = z + (q as u128) * (M as u128);
(z >> 64) as u64
}
/// Computes y such that (x * y) % M = 1 except for when when x = 0; in such a case, 0 is returned;
/// x is assumed to in [0, 2M) range, and the output will also be in [0, 2M) range.
#[inline(always)]
#[allow(clippy::many_single_char_names)]
fn inv(x: u64) -> u64 {
if x == 0 {
return 0;
};
let mut a: u128 = 0;
let mut u: u128 = if x & 1 == 1 {
x as u128
} else {
(x as u128) + (M as u128)
};
let mut v: u128 = M as u128;
let mut d = (M as u128) - 1;
while v != 1 {
while v < u {
u -= v;
d += a;
while u & 1 == 0 {
if d & 1 == 1 {
d += M as u128;
}
u >>= 1;
d >>= 1;
}
}
v -= u;
a += d;
while v & 1 == 0 {
if a & 1 == 1 {
a += M as u128;
}
v >>= 1;
a >>= 1;
}
}
while a > (M as u128) {
a -= M as u128;
}
mul(a as u64, R3)
}
// HELPER FUNCTIONS
// ================================================================================================
/// Reduces any value in [0, 2M) range to [0, M) range
#[inline(always)]
fn normalize(value: u64) -> u64 {
if value >= M {
value - M
} else {
value
}
}