// Copyright (c) 2020 Apple Inc.
// SPDX-License-Identifier: MPL-2.0
//! Finite field arithmetic.
//!
//! Each field has an associated parameter called the "generator" that generates a multiplicative
//! subgroup of order `2^n` for some `n`.
#[cfg(feature = "crypto-dependencies")]
use crate::prng::{Prng, PrngError};
use crate::{
codec::{CodecError, Decode, Encode},
fp::{FP128, FP32, FP64, FP96},
};
use serde::{
de::{DeserializeOwned, Visitor},
Deserialize, Deserializer, Serialize, Serializer,
};
use std::{
cmp::min,
convert::{TryFrom, TryInto},
fmt::{self, Debug, Display, Formatter},
hash::{Hash, Hasher},
io::{Cursor, Read},
marker::PhantomData,
ops::{Add, AddAssign, BitAnd, Div, DivAssign, Mul, MulAssign, Neg, Shl, Shr, Sub, SubAssign},
};
/// Possible errors from finite field operations.
#[derive(Debug, thiserror::Error)]
pub enum FieldError {
/// Input sizes do not match.
#[error("input sizes do not match")]
InputSizeMismatch,
/// Returned when decoding a `FieldElement` from a short byte string.
#[error("short read from bytes")]
ShortRead,
/// Returned when decoding a `FieldElement` from a byte string encoding an integer larger than
/// or equal to the field modulus.
#[error("read from byte slice exceeds modulus")]
ModulusOverflow,
/// Error while performing I/O.
#[error("I/O error")]
Io(#[from] std::io::Error),
/// Error encoding or decoding a field.
#[error("Codec error")]
Codec(#[from] CodecError),
/// Error converting to `FieldElement::Integer`.
#[error("Integer TryFrom error")]
IntegerTryFrom,
/// Error converting `FieldElement::Integer` into something else.
#[error("Integer TryInto error")]
IntegerTryInto,
}
/// Byte order for encoding FieldElement values into byte sequences.
#[derive(Clone, Copy, Debug)]
enum ByteOrder {
/// Big endian byte order.
BigEndian,
/// Little endian byte order.
LittleEndian,
}
/// Objects with this trait represent an element of `GF(p)` for some prime `p`.
pub trait FieldElement:
Sized
+ Debug
+ Copy
+ PartialEq
+ Eq
+ Add<Output = Self>
+ AddAssign
+ Sub<Output = Self>
+ SubAssign
+ Mul<Output = Self>
+ MulAssign
+ Div<Output = Self>
+ DivAssign
+ Neg<Output = Self>
+ Display
+ From<<Self as FieldElement>::Integer>
+ for<'a> TryFrom<&'a [u8], Error = FieldError>
// NOTE Ideally we would require `Into<[u8; Self::ENCODED_SIZE]>` instead of `Into<Vec<u8>>`,
// since the former avoids a heap allocation and can easily be converted into Vec<u8>, but that
// isn't possible yet[1]. However we can provide the impl on FieldElement implementations.
// [1]: https://github.com/rust-lang/rust/issues/60551
+ Into<Vec<u8>>
+ Serialize
+ DeserializeOwned
+ Encode
+ Decode
+ 'static // NOTE This bound is needed for downcasting a `dyn Gadget<F>>` to a concrete type.
{
/// Size in bytes of the encoding of a value.
const ENCODED_SIZE: usize;
/// The error returned if converting `usize` to an `Integer` fails.
type IntegerTryFromError: std::error::Error;
/// The error returend if converting an `Integer` to a `u64` fails.
type TryIntoU64Error: std::error::Error;
/// The integer representation of the field element.
type Integer: Copy
+ Debug
+ Eq
+ Ord
+ BitAnd<Output = <Self as FieldElement>::Integer>
+ Div<Output = <Self as FieldElement>::Integer>
+ Shl<Output = <Self as FieldElement>::Integer>
+ Shr<Output = <Self as FieldElement>::Integer>
+ Add<Output = <Self as FieldElement>::Integer>
+ Sub<Output = <Self as FieldElement>::Integer>
+ From<Self>
+ TryFrom<usize, Error = Self::IntegerTryFromError>
+ TryInto<u64, Error = Self::TryIntoU64Error>;
/// Modular exponentation, i.e., `self^exp (mod p)`.
fn pow(&self, exp: Self::Integer) -> Self;
/// Modular inversion, i.e., `self^-1 (mod p)`. If `self` is 0, then the output is undefined.
fn inv(&self) -> Self;
/// Returns the prime modulus `p`.
fn modulus() -> Self::Integer;
/// Interprets the next [`Self::ENCODED_SIZE`] bytes from the input slice as an element of the
/// field. The `m` most significant bits are cleared, where `m` is equal to the length of
/// [`Self::Integer`] in bits minus the length of the modulus in bits.
///
/// # Errors
///
/// An error is returned if the provided slice is too small to encode a field element or if the
/// result encodes an integer larger than or equal to the field modulus.
///
/// # Warnings
///
/// This function should only be used within [`prng::Prng`] to convert a random byte string into
/// a field element. Use [`Self::decode`] to deserialize field elements. Use
/// [`field::rand`] or [`prng::Prng`] to randomly generate field elements.
#[doc(hidden)]
fn try_from_random(bytes: &[u8]) -> Result<Self, FieldError>;
/// Returns the size of the multiplicative subgroup generated by `generator()`.
fn generator_order() -> Self::Integer;
/// Returns the generator of the multiplicative subgroup of size `generator_order()`.
fn generator() -> Self;
/// Returns the `2^l`-th principal root of unity for any `l <= 20`. Note that the `2^0`-th
/// prinicpal root of unity is 1 by definition.
fn root(l: usize) -> Option<Self>;
/// Returns the additive identity.
fn zero() -> Self;
/// Returns the multiplicative identity.
fn one() -> Self;
/// Convert a slice of field elements into a vector of bytes.
///
/// # Notes
///
/// Ideally we would implement `From<&[F: FieldElement]> for Vec<u8>` or the corresponding
/// `Into`, but the orphan rule and the stdlib's blanket implementations of `Into` make this
/// impossible.
fn slice_into_byte_vec(values: &[Self]) -> Vec<u8> {
let mut vec = Vec::with_capacity(values.len() * Self::ENCODED_SIZE);
for elem in values {
vec.append(&mut (*elem).into());
}
vec
}
/// Convert a slice of bytes into a vector of field elements. The slice is interpreted as a
/// sequence of [`Self::ENCODED_SIZE`]-byte sequences.
///
/// # Errors
///
/// Returns an error if the length of the provided byte slice is not a multiple of the size of a
/// field element, or if any of the values in the byte slice are invalid encodings of a field
/// element, because the encoded integer is larger than or equal to the field modulus.
///
/// # Notes
///
/// Ideally we would implement `From<&[u8]> for Vec<F: FieldElement>` or the corresponding
/// `Into`, but the orphan rule and the stdlib's blanket implementations of `Into` make this
/// impossible.
fn byte_slice_into_vec(bytes: &[u8]) -> Result<Vec<Self>, FieldError> {
if bytes.len() % Self::ENCODED_SIZE != 0 {
return Err(FieldError::ShortRead);
}
let mut vec = Vec::with_capacity(bytes.len() / Self::ENCODED_SIZE);
for chunk in bytes.chunks_exact(Self::ENCODED_SIZE) {
vec.push(Self::get_decoded(chunk)?);
}
Ok(vec)
}
}
/// Methods common to all `FieldElement` implementations that are private to the crate.
pub(crate) trait FieldElementExt: FieldElement {
/// Encode `input` as bitvector of elements of `Self`. Output is written into the `output` slice.
/// If `output.len()` is smaller than the number of bits required to respresent `input`,
/// an error is returned.
///
/// # Arguments
///
/// * `input` - The field element to encode
/// * `output` - The slice to write the encoded bits into. Least signicant bit comes first
fn fill_with_bitvector_representation(
input: &Self::Integer,
output: &mut [Self],
) -> Result<(), FieldError> {
// Create a mutable copy of `input`. In each iteration of the following loop we take the
// least significant bit, and shift input to the right by one bit.
let mut i = *input;
let one = Self::Integer::from(Self::one());
for bit in output.iter_mut() {
let w = Self::from(i & one);
*bit = w;
i = i >> one;
}
// If `i` is still not zero, this means that it cannot be encoded by `bits` bits.
if i != Self::Integer::from(Self::zero()) {
return Err(FieldError::InputSizeMismatch);
}
Ok(())
}
/// Encode `input` as `bits`-bit vector of elements of `Self` if it's small enough
/// to be represented with that many bits.
///
/// # Arguments
///
/// * `input` - The field element to encode
/// * `bits` - The number of bits to use for the encoding
fn encode_into_bitvector_representation(
input: &Self::Integer,
bits: usize,
) -> Result<Vec<Self>, FieldError> {
let mut result = vec![Self::zero(); bits];
Self::fill_with_bitvector_representation(input, &mut result)?;
Ok(result)
}
/// Decode the bitvector-represented value `input` into a simple representation as a single
/// field element.
///
/// # Errors
///
/// This function errors if `2^input.len() - 1` does not fit into the field `Self`.
fn decode_from_bitvector_representation(input: &[Self]) -> Result<Self, FieldError> {
let fi_one = Self::Integer::from(Self::one());
if !Self::valid_integer_bitlength(input.len()) {
return Err(FieldError::ModulusOverflow);
}
let mut decoded = Self::zero();
for (l, bit) in input.iter().enumerate() {
let fi_l = Self::Integer::try_from(l).map_err(|_| FieldError::IntegerTryFrom)?;
let w = fi_one << fi_l;
decoded += Self::from(w) * *bit;
}
Ok(decoded)
}
/// Interpret `i` as [`Self::Integer`] if it's representable in that type and smaller than the
/// field modulus.
fn valid_integer_try_from<N>(i: N) -> Result<Self::Integer, FieldError>
where
Self::Integer: TryFrom<N>,
{
let i_int = Self::Integer::try_from(i).map_err(|_| FieldError::IntegerTryFrom)?;
if Self::modulus() <= i_int {
return Err(FieldError::ModulusOverflow);
}
Ok(i_int)
}
/// Check if the largest number representable with `bits` bits (i.e. 2^bits - 1) is
/// representable in this field.
fn valid_integer_bitlength(bits: usize) -> bool {
if bits >= 8 * Self::ENCODED_SIZE {
return false;
}
if let Ok(bits_int) = Self::Integer::try_from(bits) {
if Self::modulus() >> bits_int != Self::Integer::from(Self::zero()) {
return true;
}
}
false
}
}
impl<F: FieldElement> FieldElementExt for F {}
/// serde Visitor implementation used to generically deserialize `FieldElement`
/// values from byte arrays.
struct FieldElementVisitor<F: FieldElement> {
phantom: PhantomData<F>,
}
impl<'de, F: FieldElement> Visitor<'de> for FieldElementVisitor<F> {
type Value = F;
fn expecting(&self, formatter: &mut Formatter) -> fmt::Result {
formatter.write_fmt(format_args!("an array of {} bytes", F::ENCODED_SIZE))
}
fn visit_bytes<E>(self, v: &[u8]) -> Result<Self::Value, E>
where
E: serde::de::Error,
{
Self::Value::try_from(v).map_err(E::custom)
}
fn visit_seq<A>(self, mut seq: A) -> Result<Self::Value, A::Error>
where
A: serde::de::SeqAccess<'de>,
{
let mut bytes = vec![];
while let Some(byte) = seq.next_element()? {
bytes.push(byte);
}
self.visit_bytes(&bytes)
}
}
macro_rules! make_field {
(
$(#[$meta:meta])*
$elem:ident, $int:ident, $fp:ident, $encoding_size:literal, $encoding_order:expr,
) => {
$(#[$meta])*
///
/// This structure represents a field element in a prime order field. The concrete
/// representation of the element is via the Montgomery domain. For an element n in GF(p),
/// we store n * R^-1 mod p (where R is a given power of two). This representation enables
/// using a more efficient (and branchless) multiplication algorithm, at the expense of
/// having to convert elements between their Montgomery domain representation and natural
/// representation. For calculations with many multiplications or exponentiations, this is
/// worthwhile.
///
/// As an invariant, this integer representing the field element in the Montgomery domain
/// must be less than the prime p.
#[derive(Clone, Copy, PartialOrd, Ord, Default)]
pub struct $elem(u128);
impl $elem {
/// Attempts to instantiate an `$elem` from the first `Self::ENCODED_SIZE` bytes in the
/// provided slice. The decoded value will be bitwise-ANDed with `mask` before reducing
/// it using the field modulus.
///
/// # Errors
///
/// An error is returned if the provided slice is not long enough to encode a field
/// element or if the decoded value is greater than the field prime.
///
/// # Notes
///
/// We cannot use `u128::from_le_bytes` or `u128::from_be_bytes` because those functions
/// expect inputs to be exactly 16 bytes long. Our encoding of most field elements is
/// more compact, and does not have to correspond to the size of an integer type. For
/// instance,`Field96`'s encoding is 12 bytes, even though it is a 16 byte `u128` in
/// memory.
fn try_from_bytes(bytes: &[u8], mask: u128) -> Result<Self, FieldError> {
if Self::ENCODED_SIZE > bytes.len() {
return Err(FieldError::ShortRead);
}
let mut int = 0;
for i in 0..Self::ENCODED_SIZE {
let j = match $encoding_order {
ByteOrder::LittleEndian => i,
ByteOrder::BigEndian => Self::ENCODED_SIZE - i - 1,
};
int |= (bytes[j] as u128) << (i << 3);
}
int &= mask;
if int >= $fp.p {
return Err(FieldError::ModulusOverflow);
}
// FieldParameters::montgomery() will return a value that has been fully reduced
// mod p, satisfying the invariant on Self.
Ok(Self($fp.montgomery(int)))
}
}
impl PartialEq for $elem {
fn eq(&self, rhs: &Self) -> bool {
// The fields included in this comparison MUST match the fields
// used in Hash::hash
// https://doc.rust-lang.org/std/hash/trait.Hash.html#hash-and-eq
// Check the invariant that the integer representation is fully reduced.
debug_assert!(self.0 < $fp.p);
debug_assert!(rhs.0 < $fp.p);
self.0 == rhs.0
}
}
impl Hash for $elem {
fn hash<H: Hasher>(&self, state: &mut H) {
// The fields included in this hash MUST match the fields used
// in PartialEq::eq
// https://doc.rust-lang.org/std/hash/trait.Hash.html#hash-and-eq
// Check the invariant that the integer representation is fully reduced.
debug_assert!(self.0 < $fp.p);
self.0.hash(state);
}
}
impl Eq for $elem {}
impl Add for $elem {
type Output = $elem;
fn add(self, rhs: Self) -> Self {
// FieldParameters::add() returns a value that has been fully reduced
// mod p, satisfying the invariant on Self.
Self($fp.add(self.0, rhs.0))
}
}
impl Add for &$elem {
type Output = $elem;
fn add(self, rhs: Self) -> $elem {
*self + *rhs
}
}
impl AddAssign for $elem {
fn add_assign(&mut self, rhs: Self) {
*self = *self + rhs;
}
}
impl Sub for $elem {
type Output = $elem;
fn sub(self, rhs: Self) -> Self {
// We know that self.0 and rhs.0 are both less than p, thus FieldParameters::sub()
// returns a value less than p, satisfying the invariant on Self.
Self($fp.sub(self.0, rhs.0))
}
}
impl Sub for &$elem {
type Output = $elem;
fn sub(self, rhs: Self) -> $elem {
*self - *rhs
}
}
impl SubAssign for $elem {
fn sub_assign(&mut self, rhs: Self) {
*self = *self - rhs;
}
}
impl Mul for $elem {
type Output = $elem;
fn mul(self, rhs: Self) -> Self {
// FieldParameters::mul() always returns a value less than p, so the invariant on
// Self is satisfied.
Self($fp.mul(self.0, rhs.0))
}
}
impl Mul for &$elem {
type Output = $elem;
fn mul(self, rhs: Self) -> $elem {
*self * *rhs
}
}
impl MulAssign for $elem {
fn mul_assign(&mut self, rhs: Self) {
*self = *self * rhs;
}
}
impl Div for $elem {
type Output = $elem;
#[allow(clippy::suspicious_arithmetic_impl)]
fn div(self, rhs: Self) -> Self {
self * rhs.inv()
}
}
impl Div for &$elem {
type Output = $elem;
fn div(self, rhs: Self) -> $elem {
*self / *rhs
}
}
impl DivAssign for $elem {
fn div_assign(&mut self, rhs: Self) {
*self = *self / rhs;
}
}
impl Neg for $elem {
type Output = $elem;
fn neg(self) -> Self {
// FieldParameters::neg() will return a value less than p because self.0 is less
// than p, and neg() dispatches to sub().
Self($fp.neg(self.0))
}
}
impl Neg for &$elem {
type Output = $elem;
fn neg(self) -> $elem {
-(*self)
}
}
impl From<$int> for $elem {
fn from(x: $int) -> Self {
// FieldParameters::montgomery() will return a value that has been fully reduced
// mod p, satisfying the invariant on Self.
Self($fp.montgomery(u128::try_from(x).unwrap()))
}
}
impl From<$elem> for $int {
fn from(x: $elem) -> Self {
$int::try_from($fp.residue(x.0)).unwrap()
}
}
impl PartialEq<$int> for $elem {
fn eq(&self, rhs: &$int) -> bool {
$fp.residue(self.0) == u128::try_from(*rhs).unwrap()
}
}
impl<'a> TryFrom<&'a [u8]> for $elem {
type Error = FieldError;
fn try_from(bytes: &[u8]) -> Result<Self, FieldError> {
Self::try_from_bytes(bytes, u128::MAX)
}
}
impl From<$elem> for [u8; $elem::ENCODED_SIZE] {
fn from(elem: $elem) -> Self {
let int = $fp.residue(elem.0);
let mut slice = [0; $elem::ENCODED_SIZE];
for i in 0..$elem::ENCODED_SIZE {
let j = match $encoding_order {
ByteOrder::LittleEndian => i,
ByteOrder::BigEndian => $elem::ENCODED_SIZE - i - 1,
};
slice[j] = ((int >> (i << 3)) & 0xff) as u8;
}
slice
}
}
impl From<$elem> for Vec<u8> {
fn from(elem: $elem) -> Self {
<[u8; $elem::ENCODED_SIZE]>::from(elem).to_vec()
}
}
impl Display for $elem {
fn fmt(&self, f: &mut Formatter) -> std::fmt::Result {
write!(f, "{}", $fp.residue(self.0))
}
}
impl Debug for $elem {
fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
write!(f, "{}", $fp.residue(self.0))
}
}
// We provide custom [`serde::Serialize`] and [`serde::Deserialize`] implementations because
// the derived implementations would represent `FieldElement` values as the backing `u128`,
// which is not what we want because (1) we can be more efficient in all cases and (2) in
// some circumstances, [some serializers don't support `u128`](https://github.com/serde-rs/json/issues/625).
impl Serialize for $elem {
fn serialize<S: Serializer>(&self, serializer: S) -> Result<S::Ok, S::Error> {
let bytes: [u8; $elem::ENCODED_SIZE] = (*self).into();
serializer.serialize_bytes(&bytes)
}
}
impl<'de> Deserialize<'de> for $elem {
fn deserialize<D: Deserializer<'de>>(deserializer: D) -> Result<$elem, D::Error> {
deserializer.deserialize_bytes(FieldElementVisitor { phantom: PhantomData })
}
}
impl Encode for $elem {
fn encode(&self, bytes: &mut Vec<u8>) {
let slice = <[u8; $elem::ENCODED_SIZE]>::from(*self);
bytes.extend_from_slice(&slice);
}
}
impl Decode for $elem {
fn decode(bytes: &mut Cursor<&[u8]>) -> Result<Self, CodecError> {
let mut value = [0u8; $elem::ENCODED_SIZE];
bytes.read_exact(&mut value)?;
$elem::try_from_bytes(&value, u128::MAX).map_err(|e| {
CodecError::Other(Box::new(e) as Box<dyn std::error::Error + 'static + Send + Sync>)
})
}
}
impl FieldElement for $elem {
const ENCODED_SIZE: usize = $encoding_size;
type Integer = $int;
type IntegerTryFromError = <Self::Integer as TryFrom<usize>>::Error;
type TryIntoU64Error = <Self::Integer as TryInto<u64>>::Error;
fn pow(&self, exp: Self::Integer) -> Self {
// FieldParameters::pow() relies on mul(), and will always return a value less
// than p.
Self($fp.pow(self.0, u128::try_from(exp).unwrap()))
}
fn inv(&self) -> Self {
// FieldParameters::inv() ultimately relies on mul(), and will always return a
// value less than p.
Self($fp.inv(self.0))
}
fn modulus() -> Self::Integer {
$fp.p as $int
}
fn try_from_random(bytes: &[u8]) -> Result<Self, FieldError> {
$elem::try_from_bytes(bytes, $fp.bit_mask)
}
fn generator() -> Self {
Self($fp.g)
}
fn generator_order() -> Self::Integer {
1 << (Self::Integer::try_from($fp.num_roots).unwrap())
}
fn root(l: usize) -> Option<Self> {
if l < min($fp.roots.len(), $fp.num_roots+1) {
Some(Self($fp.roots[l]))
} else {
None
}
}
fn zero() -> Self {
Self(0)
}
fn one() -> Self {
Self($fp.roots[0])
}
}
};
}
make_field!(
/// `GF(4293918721)`, a 32-bit field.
Field32,
u32,
FP32,
4,
ByteOrder::BigEndian,
);
make_field!(
/// Same as Field32, but encoded in little endian for compatibility with Prio v2.
FieldPrio2,
u32,
FP32,
4,
ByteOrder::LittleEndian,
);
make_field!(
/// `GF(18446744069414584321)`, a 64-bit field.
Field64,
u64,
FP64,
8,
ByteOrder::BigEndian,
);
make_field!(
/// `GF(79228148845226978974766202881)`, a 96-bit field.
Field96,
u128,
FP96,
12,
ByteOrder::BigEndian,
);
make_field!(
/// `GF(340282366920938462946865773367900766209)`, a 128-bit field.
Field128,
u128,
FP128,
16,
ByteOrder::BigEndian,
);
/// Merge two vectors of fields by summing other_vector into accumulator.
///
/// # Errors
///
/// Fails if the two vectors do not have the same length.
#[cfg(any(test, feature = "prio2"))]
pub(crate) fn merge_vector<F: FieldElement>(
accumulator: &mut [F],
other_vector: &[F],
) -> Result<(), FieldError> {
if accumulator.len() != other_vector.len() {
return Err(FieldError::InputSizeMismatch);
}
for (a, o) in accumulator.iter_mut().zip(other_vector.iter()) {
*a += *o;
}
Ok(())
}
/// Outputs an additive secret sharing of the input.
#[cfg(feature = "crypto-dependencies")]
pub(crate) fn split_vector<F: FieldElement>(
inp: &[F],
num_shares: usize,
) -> Result<Vec<Vec<F>>, PrngError> {
if num_shares == 0 {
return Ok(vec![]);
}
let mut outp = Vec::with_capacity(num_shares);
outp.push(inp.to_vec());
for _ in 1..num_shares {
let share: Vec<F> = random_vector(inp.len())?;
for (x, y) in outp[0].iter_mut().zip(&share) {
*x -= *y;
}
outp.push(share);
}
Ok(outp)
}
/// Generate a vector of uniform random field elements.
#[cfg(feature = "crypto-dependencies")]
pub fn random_vector<F: FieldElement>(len: usize) -> Result<Vec<F>, PrngError> {
Ok(Prng::new()?.take(len).collect())
}
#[cfg(test)]
mod tests {
use super::*;
use crate::fp::MAX_ROOTS;
use crate::prng::Prng;
use assert_matches::assert_matches;
use std::collections::hash_map::DefaultHasher;
#[test]
fn test_endianness() {
let little_endian_encoded: [u8; FieldPrio2::ENCODED_SIZE] =
FieldPrio2(0x12_34_56_78).into();
let mut big_endian_encoded: [u8; Field32::ENCODED_SIZE] = Field32(0x12_34_56_78).into();
big_endian_encoded.reverse();
assert_eq!(little_endian_encoded, big_endian_encoded);
}
#[test]
fn test_accumulate() {
let mut lhs = vec![Field32(1); 10];
let rhs = vec![Field32(2); 10];
merge_vector(&mut lhs, &rhs).unwrap();
lhs.iter().for_each(|f| assert_eq!(*f, Field32(3)));
rhs.iter().for_each(|f| assert_eq!(*f, Field32(2)));
let wrong_len = vec![Field32::zero(); 9];
let result = merge_vector(&mut lhs, &wrong_len);
assert_matches!(result, Err(FieldError::InputSizeMismatch));
}
fn hash_helper<H: Hash>(input: H) -> u64 {
let mut hasher = DefaultHasher::new();
input.hash(&mut hasher);
hasher.finish()
}
// Some of the checks in this function, like `assert_eq!(one - one, zero)`
// or `assert_eq!(two / two, one)` trip this clippy lint for tautological
// comparisons, but we have a legitimate need to verify these basics. We put
// the #[allow] on the whole function since "attributes on expressions are
// experimental" https://github.com/rust-lang/rust/issues/15701
#[allow(clippy::eq_op)]
fn field_element_test<F: FieldElement + Hash>() {
let mut prng: Prng<F, _> = Prng::new().unwrap();
let int_modulus = F::modulus();
let int_one = F::Integer::try_from(1).unwrap();
let zero = F::zero();
let one = F::one();
let two = F::from(F::Integer::try_from(2).unwrap());
let four = F::from(F::Integer::try_from(4).unwrap());
// add
assert_eq!(F::from(int_modulus - int_one) + one, zero);
assert_eq!(one + one, two);
assert_eq!(two + F::from(int_modulus), two);
// sub
assert_eq!(zero - one, F::from(int_modulus - int_one));
assert_eq!(one - one, zero);
assert_eq!(two - F::from(int_modulus), two);
assert_eq!(one - F::from(int_modulus - int_one), two);
// add + sub
for _ in 0..100 {
let f = prng.get();
let g = prng.get();
assert_eq!(f + g - f - g, zero);
assert_eq!(f + g - g, f);
assert_eq!(f + g - f, g);
}
// mul
assert_eq!(two * two, four);
assert_eq!(two * one, two);
assert_eq!(two * zero, zero);
assert_eq!(one * F::from(int_modulus), zero);
// div
assert_eq!(four / two, two);
assert_eq!(two / two, one);
assert_eq!(zero / two, zero);
assert_eq!(two / zero, zero); // Undefined behavior
assert_eq!(zero.inv(), zero); // Undefined behavior
// mul + div
for _ in 0..100 {
let f = prng.get();
if f == zero {
continue;
}
assert_eq!(f * f.inv(), one);
assert_eq!(f.inv() * f, one);
}
// pow
assert_eq!(two.pow(F::Integer::try_from(0).unwrap()), one);
assert_eq!(two.pow(int_one), two);
assert_eq!(two.pow(F::Integer::try_from(2).unwrap()), four);
assert_eq!(two.pow(int_modulus - int_one), one);
assert_eq!(two.pow(int_modulus), two);
// roots
let mut int_order = F::generator_order();
for l in 0..MAX_ROOTS + 1 {
assert_eq!(
F::generator().pow(int_order),
F::root(l).unwrap(),
"failure for F::root({})",
l
);
int_order = int_order >> int_one;
}
// serialization
let test_inputs = vec![zero, one, prng.get(), F::from(int_modulus - int_one)];
for want in test_inputs.iter() {
let mut bytes = vec![];
want.encode(&mut bytes);
assert_eq!(bytes.len(), F::ENCODED_SIZE);
let got = F::get_decoded(&bytes).unwrap();
assert_eq!(got, *want);
}
let serialized_vec = F::slice_into_byte_vec(&test_inputs);
let deserialized = F::byte_slice_into_vec(&serialized_vec).unwrap();
assert_eq!(deserialized, test_inputs);
// equality and hash: Generate many elements, confirm they are not equal, and confirm
// various products that should be equal have the same hash. Three is chosen as a generator
// here because it happens to generate fairly large subgroups of (Z/pZ)* for all four
// primes.
let three = F::from(F::Integer::try_from(3).unwrap());
let mut powers_of_three = Vec::with_capacity(500);
let mut power = one;
for _ in 0..500 {
powers_of_three.push(power);
power *= three;
}
// Check all these elements are mutually not equal.
for i in 0..powers_of_three.len() {
let first = &powers_of_three[i];
for second in &powers_of_three[0..i] {
assert_ne!(first, second);
}
}
// Check that 3^i is the same whether it's calculated with pow() or repeated
// multiplication, with both equality and hash equality.
for (i, power) in powers_of_three.iter().enumerate() {
let result = three.pow(F::Integer::try_from(i).unwrap());
assert_eq!(result, *power);
let hash1 = hash_helper(power);
let hash2 = hash_helper(result);
assert_eq!(hash1, hash2);
}
// Check that 3^n = (3^i)*(3^(n-i)), via both equality and hash equality.
let expected_product = powers_of_three[powers_of_three.len() - 1];
let expected_hash = hash_helper(expected_product);
for i in 0..powers_of_three.len() {
let a = powers_of_three[i];
let b = powers_of_three[powers_of_three.len() - 1 - i];
let product = a * b;
assert_eq!(product, expected_product);
assert_eq!(hash_helper(product), expected_hash);
}
// Construct an element from a number that needs to be reduced, and test comparisons on it,
// confirming that FieldParameters::montgomery() reduced it correctly.
let p = F::from(int_modulus);
assert_eq!(p, zero);
assert_eq!(hash_helper(p), hash_helper(zero));
let p_plus_one = F::from(int_modulus + F::Integer::try_from(1).unwrap());
assert_eq!(p_plus_one, one);
assert_eq!(hash_helper(p_plus_one), hash_helper(one));
}
#[test]
fn test_field32() {
field_element_test::<Field32>();
}
#[test]
fn test_field_priov2() {
field_element_test::<FieldPrio2>();
}
#[test]
fn test_field64() {
field_element_test::<Field64>();
}
#[test]
fn test_field96() {
field_element_test::<Field96>();
}
#[test]
fn test_field128() {
field_element_test::<Field128>();
}
#[test]
fn test_encode_into_bitvector() {
let zero = Field128::zero();
let one = Field128::one();
let zero_enc = Field128::encode_into_bitvector_representation(&0, 4).unwrap();
let one_enc = Field128::encode_into_bitvector_representation(&1, 4).unwrap();
let fifteen_enc = Field128::encode_into_bitvector_representation(&15, 4).unwrap();
assert_eq!(zero_enc, [zero; 4]);
assert_eq!(one_enc, [one, zero, zero, zero]);
assert_eq!(fifteen_enc, [one; 4]);
Field128::encode_into_bitvector_representation(&16, 4).unwrap_err();
}
#[test]
fn test_fill_bitvector() {
let zero = Field128::zero();
let one = Field128::one();
let mut output: Vec<Field128> = vec![zero; 6];
Field128::fill_with_bitvector_representation(&9, &mut output[1..5]).unwrap();
assert_eq!(output, [zero, one, zero, zero, one, zero]);
Field128::fill_with_bitvector_representation(&16, &mut output[1..5]).unwrap_err();
}
}