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//! This crate provides traits for working with finite fields.
// Catch documentation errors caused by code changes.
#![no_std]
#![cfg_attr(docsrs, feature(doc_cfg))]
#![deny(rustdoc::broken_intra_doc_links)]
#![forbid(unsafe_code)]
#[cfg(feature = "alloc")]
extern crate alloc;
mod batch;
pub use batch::*;
#[cfg(feature = "derive")]
#[cfg_attr(docsrs, doc(cfg(feature = "derive")))]
pub use ff_derive::PrimeField;
#[cfg(feature = "bits")]
#[cfg_attr(docsrs, doc(cfg(feature = "bits")))]
pub use bitvec::view::BitViewSized;
#[cfg(feature = "bits")]
use bitvec::{array::BitArray, order::Lsb0};
use core::fmt;
use core::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};
use rand_core::RngCore;
use subtle::{Choice, ConditionallySelectable, ConstantTimeEq, CtOption};
/// Bit representation of a field element.
#[cfg(feature = "bits")]
#[cfg_attr(docsrs, doc(cfg(feature = "bits")))]
pub type FieldBits<V> = BitArray<V, Lsb0>;
/// This trait represents an element of a field.
pub trait Field:
Sized
+ Eq
+ Copy
+ Clone
+ Default
+ Send
+ Sync
+ fmt::Debug
+ 'static
+ ConditionallySelectable
+ ConstantTimeEq
+ Add<Output = Self>
+ Sub<Output = Self>
+ Mul<Output = Self>
+ Neg<Output = Self>
+ for<'a> Add<&'a Self, Output = Self>
+ for<'a> Mul<&'a Self, Output = Self>
+ for<'a> Sub<&'a Self, Output = Self>
+ MulAssign
+ AddAssign
+ SubAssign
+ for<'a> MulAssign<&'a Self>
+ for<'a> AddAssign<&'a Self>
+ for<'a> SubAssign<&'a Self>
{
/// Returns an element chosen uniformly at random using a user-provided RNG.
fn random(rng: impl RngCore) -> Self;
/// Returns the zero element of the field, the additive identity.
fn zero() -> Self;
/// Returns the one element of the field, the multiplicative identity.
fn one() -> Self;
/// Returns true iff this element is zero.
fn is_zero(&self) -> Choice {
self.ct_eq(&Self::zero())
}
/// Returns true iff this element is zero.
///
/// # Security
///
/// This method provides **no** constant-time guarantees. Implementors of the
/// `Field` trait **may** optimise this method using non-constant-time logic.
fn is_zero_vartime(&self) -> bool {
self.is_zero().into()
}
/// Squares this element.
#[must_use]
fn square(&self) -> Self;
/// Cubes this element.
#[must_use]
fn cube(&self) -> Self {
self.square() * self
}
/// Doubles this element.
#[must_use]
fn double(&self) -> Self;
/// Computes the multiplicative inverse of this element,
/// failing if the element is zero.
fn invert(&self) -> CtOption<Self>;
/// Returns the square root of the field element, if it is
/// quadratic residue.
fn sqrt(&self) -> CtOption<Self>;
/// Exponentiates `self` by `exp`, where `exp` is a little-endian order
/// integer exponent.
///
/// **This operation is variable time with respect to the exponent.** If the
/// exponent is fixed, this operation is effectively constant time.
fn pow_vartime<S: AsRef<[u64]>>(&self, exp: S) -> Self {
let mut res = Self::one();
for e in exp.as_ref().iter().rev() {
for i in (0..64).rev() {
res = res.square();
if ((*e >> i) & 1) == 1 {
res.mul_assign(self);
}
}
}
res
}
}
/// This represents an element of a prime field.
pub trait PrimeField: Field + From<u64> {
/// The prime field can be converted back and forth into this binary
/// representation.
type Repr: Copy + Default + Send + Sync + 'static + AsRef<[u8]> + AsMut<[u8]>;
/// Interpret a string of numbers as a (congruent) prime field element.
/// Does not accept unnecessary leading zeroes or a blank string.
///
/// # Security
///
/// This method provides **no** constant-time guarantees.
fn from_str_vartime(s: &str) -> Option<Self> {
if s.is_empty() {
return None;
}
if s == "0" {
return Some(Self::zero());
}
let mut res = Self::zero();
let ten = Self::from(10);
let mut first_digit = true;
for c in s.chars() {
match c.to_digit(10) {
Some(c) => {
if first_digit {
if c == 0 {
return None;
}
first_digit = false;
}
res.mul_assign(&ten);
res.add_assign(&Self::from(u64::from(c)));
}
None => {
return None;
}
}
}
Some(res)
}
/// Attempts to convert a byte representation of a field element into an element of
/// this prime field, failing if the input is not canonical (is not smaller than the
/// field's modulus).
///
/// The byte representation is interpreted with the same endianness as elements
/// returned by [`PrimeField::to_repr`].
fn from_repr(repr: Self::Repr) -> CtOption<Self>;
/// Attempts to convert a byte representation of a field element into an element of
/// this prime field, failing if the input is not canonical (is not smaller than the
/// field's modulus).
///
/// The byte representation is interpreted with the same endianness as elements
/// returned by [`PrimeField::to_repr`].
///
/// # Security
///
/// This method provides **no** constant-time guarantees. Implementors of the
/// `PrimeField` trait **may** optimise this method using non-constant-time logic.
fn from_repr_vartime(repr: Self::Repr) -> Option<Self> {
Self::from_repr(repr).into()
}
/// Converts an element of the prime field into the standard byte representation for
/// this field.
///
/// The endianness of the byte representation is implementation-specific. Generic
/// encodings of field elements should be treated as opaque.
fn to_repr(&self) -> Self::Repr;
/// Returns true iff this element is odd.
fn is_odd(&self) -> Choice;
/// Returns true iff this element is even.
#[inline(always)]
fn is_even(&self) -> Choice {
!self.is_odd()
}
/// How many bits are needed to represent an element of this field.
const NUM_BITS: u32;
/// How many bits of information can be reliably stored in the field element.
///
/// This is usually `Self::NUM_BITS - 1`.
const CAPACITY: u32;
/// Returns a fixed multiplicative generator of `modulus - 1` order. This element must
/// also be a quadratic nonresidue.
///
/// It can be calculated using [SageMath] as `GF(modulus).primitive_element()`.
///
/// Implementations of this method MUST ensure that this is the generator used to
/// derive `Self::root_of_unity`.
///
/// [SageMath]: https://www.sagemath.org/
fn multiplicative_generator() -> Self;
/// An integer `s` satisfying the equation `2^s * t = modulus - 1` with `t` odd.
///
/// This is the number of leading zero bits in the little-endian bit representation of
/// `modulus - 1`.
const S: u32;
/// Returns the `2^s` root of unity.
///
/// It can be calculated by exponentiating `Self::multiplicative_generator` by `t`,
/// where `t = (modulus - 1) >> Self::S`.
fn root_of_unity() -> Self;
}
/// This represents the bits of an element of a prime field.
#[cfg(feature = "bits")]
#[cfg_attr(docsrs, doc(cfg(feature = "bits")))]
pub trait PrimeFieldBits: PrimeField {
/// The backing store for a bit representation of a prime field element.
type ReprBits: BitViewSized + Send + Sync;
/// Converts an element of the prime field into a little-endian sequence of bits.
fn to_le_bits(&self) -> FieldBits<Self::ReprBits>;
/// Returns the bits of the field characteristic (the modulus) in little-endian order.
fn char_le_bits() -> FieldBits<Self::ReprBits>;
}
/// Functions and re-exported crates used by the [`PrimeField`] derive macro.
#[cfg(feature = "derive")]
#[cfg_attr(docsrs, doc(cfg(feature = "derive")))]
pub mod derive {
pub use crate::arith_impl::*;
pub use {byteorder, rand_core, subtle};
#[cfg(feature = "bits")]
pub use bitvec;
}
#[cfg(feature = "derive")]
mod arith_impl {
/// Computes `a - (b + borrow)`, returning the result and the new borrow.
#[inline(always)]
pub const fn sbb(a: u64, b: u64, borrow: u64) -> (u64, u64) {
let ret = (a as u128).wrapping_sub((b as u128) + ((borrow >> 63) as u128));
(ret as u64, (ret >> 64) as u64)
}
/// Computes `a + b + carry`, returning the result and the new carry over.
#[inline(always)]
pub const fn adc(a: u64, b: u64, carry: u64) -> (u64, u64) {
let ret = (a as u128) + (b as u128) + (carry as u128);
(ret as u64, (ret >> 64) as u64)
}
/// Computes `a + (b * c) + carry`, returning the result and the new carry over.
#[inline(always)]
pub const fn mac(a: u64, b: u64, c: u64, carry: u64) -> (u64, u64) {
let ret = (a as u128) + ((b as u128) * (c as u128)) + (carry as u128);
(ret as u64, (ret >> 64) as u64)
}
}