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// Copyright 2023 RISC Zero, Inc.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
// TODO: Document better
//! Defines field extension (and base fields) used for finite field-based
//! operations across the RISC Zero zkVM architecture
use alloc::vec::Vec;
use core::{cmp, fmt::Debug, ops};
/// A pair of fields, one of which is an extension field of the other.
pub trait Field {
type Elem: Elem + RootsOfUnity;
type ExtElem: ExtElem<SubElem = Self::Elem>;
}
/// Subfield elements that can be compared, copied, and operated
/// on via multiplication, addition, and subtraction
pub trait Elem:
ops::Mul<Output = Self>
+ ops::MulAssign
+ ops::Add<Output = Self>
+ ops::AddAssign
+ ops::Neg
+ ops::Sub<Output = Self>
+ ops::SubAssign
+ cmp::PartialEq
+ cmp::Eq
+ core::clone::Clone
+ core::marker::Copy
+ Sized
+ bytemuck::Pod
+ core::default::Default
+ Clone
+ Copy
+ Send
+ Sync
+ Debug
+ 'static
{
/// Invalid, a value that is not a member of the field. This
/// should only be used with the "is_valid" or "unwrap_or_zero"
/// methods.
const INVALID: Self;
/// Zero, the additive identity.
const ZERO: Self;
/// One, the multiplicative identity.
const ONE: Self;
/// How many u32 words are required to hold a single element
const WORDS: usize;
/// Compute the multiplicative inverse of `x` (or `1 / x` in finite field
/// terms).
fn inv(self) -> Self;
/// Return an element raised to the given power.
fn pow(self, exp: usize) -> Self {
debug_assert!(self.is_valid());
let mut n = exp;
let mut tot = Self::ONE;
let mut x = self;
while n != 0 {
if n % 2 == 1 {
tot *= x;
}
n = n / 2;
x *= x;
}
tot
}
/// Returns a random valid field element.
fn random(rng: &mut impl rand_core::RngCore) -> Self;
/// Import a number into the field from the natural numbers.
fn from_u64(val: u64) -> Self;
/// Represent a field element as a sequence of u32s
fn to_u32_words(&self) -> Vec<u32>;
/// Interpret a sequence of u32s as a field element
fn from_u32_words(val: &[u32]) -> Self;
/// Returns true if this element is not INVALID. Unlike most
/// methods, this may be called on an INVALID element.
fn is_valid(&self) -> bool;
/// Returns 0 if this element is INVALID, else the value of this
/// element. Unlike most methods, this may be called on an
/// INVALID element.
fn valid_or_zero(&self) -> Self {
if self.is_valid() {
*self
} else {
Self::ZERO
}
}
/// Returns this element, but checks to make sure it's valid.
fn ensure_valid(&self) -> &Self {
debug_assert!(self.is_valid());
self
}
/// Interprets a slice of these elements as u32s. These elements
/// may not be INVALID.
fn as_u32_slice(elems: &[Self]) -> &[u32] {
if cfg!(debug_assertions) {
for elem in elems {
elem.ensure_valid();
}
}
Self::as_u32_slice_unchecked(elems)
}
/// Interprets a slice of these elements as u32s. These elements
/// may potentially be INVALID.
fn as_u32_slice_unchecked(elems: &[Self]) -> &[u32] {
bytemuck::cast_slice(elems)
}
/// Interprets a slice of u32s as a slice of these elements.
/// These elements may not be INVALID.
fn from_u32_slice(u32s: &[u32]) -> &[Self] {
let elems = Self::from_u32_slice_unchecked(u32s);
if cfg!(debug_assertions) {
for elem in elems {
elem.ensure_valid();
}
}
elems
}
/// Interprets a slice of u32s as a slice of these elements.
/// These elements may be INVALID.
fn from_u32_slice_unchecked(u32s: &[u32]) -> &[Self] {
bytemuck::cast_slice(u32s)
}
}
/// A field extension which can be constructed from a subfield element [Elem]
pub trait ExtElem:
Elem
+ ops::Add<Output = Self>
+ ops::AddAssign
+ ops::Neg<Output = Self>
+ ops::Mul<Self, Output = Self>
+ ops::Mul<Self::SubElem, Output = Self>
+ ops::MulAssign<Self>
+ ops::MulAssign<Self::SubElem>
+ ops::Sub<Output = Self>
+ ops::SubAssign
+ cmp::PartialEq
+ cmp::Eq
{
type SubElem: Elem;
const EXT_SIZE: usize;
/// Construct a field element
fn from_subfield(elem: &Self::SubElem) -> Self;
fn from_subelems(elems: impl IntoIterator<Item = Self::SubElem>) -> Self;
fn subelems(&self) -> &[Self::SubElem];
}
/// Roots of unity for the field whose elements are represented by [ExtElem] and
/// whose subfield elements are represented by [Elem]
pub trait RootsOfUnity: Sized + 'static {
/// Maximum root of unity which is a power of 2 (i.e., there is a
/// 2^MAX_ROU_PO2th root of unity, but no 2^(MAX_ROU_PO2+1)th root.
const MAX_ROU_PO2: usize;
/// For each power of 2, the 'forward' root of unity for
/// the po2. That is, this list satisfies ROU_FWD\[i+1\] ^ 2 =
/// ROU_FWD\[i\] in the prime field, which implies ROU_FWD\[i\] ^
/// (2 ^ i) = 1.
const ROU_FWD: &'static [Self];
/// For each power of 2, the 'reverse' root of unity for
/// the po2. This list satisfies ROU_FWD\[i\] * ROU_REV\[i\] = 1
/// in the prime field F_2013265921.
const ROU_REV: &'static [Self];
}
#[cfg(test)]
mod tests {
use core::fmt::Debug;
use rand::Rng;
use super::{Elem, RootsOfUnity};
pub fn test_roots_of_unity<F: Elem + RootsOfUnity + Debug>() {
let mut cur: Option<F> = None;
for &rou in F::ROU_FWD.iter().rev() {
if let Some(ref mut curval) = &mut cur {
*curval *= *curval;
assert_eq!(*curval, rou);
} else {
cur = Some(rou);
}
}
assert_eq!(cur, Some(F::ONE));
for (&fwd, &rev) in F::ROU_FWD.iter().zip(F::ROU_REV.iter()) {
assert_eq!(fwd * rev, F::ONE);
}
}
fn non_zero_rand<F: Elem>(r: &mut impl Rng) -> F {
loop {
let val = F::random(r);
if val != F::ZERO {
return val;
}
}
}
pub fn test_field_ops<F: Elem>(p_u64: u64)
where
F: Into<u64> + From<u64> + Debug,
{
// For testng, we do 128-bit arithmetic so we don't have to worry about
// overflows.
let p: u128 = p_u64 as _;
assert_eq!(F::from(0), F::ZERO);
assert_eq!(F::from(p_u64), F::ZERO);
assert_eq!(F::from(1), F::ONE);
assert_eq!(F::from(p_u64 - 1) + F::from(1), F::ZERO);
assert_eq!(F::ZERO.inv(), F::ZERO);
assert_eq!(F::ONE.inv(), F::ONE);
// Compare against many randomly generated numbers to make sure results match
// the expected results for regular modular arithmetic.
let mut rng = rand::thread_rng();
for _ in 0..1000 {
let x: F = non_zero_rand(&mut rng);
let y: F = non_zero_rand(&mut rng);
let xi: u128 = x.into() as _;
let yi: u128 = y.into() as _;
assert_eq!((x + y).into() as u128, (&xi + &yi) % p);
assert_eq!((x * y).into() as u128, (&xi * &yi) % p);
assert_eq!((x - y).into() as u128, (&xi + p - &yi) % p);
let xinv = x.inv();
if x != F::ONE {
assert!(xinv != x);
}
assert_eq!(xinv * x, F::ONE);
}
}
}
/// The field extension whose subfield is order 15*2^27 + 1;
/// this field choice allows 32-bit addition without overflow
pub mod baby_bear;
/// The field extension whose subfield is order 2^64 - 2^32 + 1;
/// this field choice allows for fast reduction
pub mod goldilocks;