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// 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.
use core::{
convert::TryFrom,
fmt::{Debug, Display},
ops::{
Add, AddAssign, BitAnd, Div, DivAssign, Mul, MulAssign, Neg, Shl, Shr, ShrAssign, Sub,
SubAssign,
},
};
use utils::{
collections::Vec, AsBytes, Deserializable, DeserializationError, Randomizable, Serializable,
};
// FIELD ELEMENT
// ================================================================================================
/// Defines an element in a finite field.
///
/// This trait defines basic arithmetic operations for elements in
/// [finite fields](https://en.wikipedia.org/wiki/Finite_field) (e.g. addition subtraction,
/// multiplication, division) as well as several convenience functions (e.g. double, square cube).
/// Moreover, it defines interfaces for serializing and deserializing field elements.
///
/// The elements could be in a prime field or an extension of a prime field. Currently, only
/// quadratic field extensions are supported.
pub trait FieldElement:
Copy
+ Clone
+ Debug
+ Display
+ Default
+ Send
+ Sync
+ Eq
+ PartialEq
+ Sized
+ Add<Self, Output = Self>
+ Sub<Self, Output = Self>
+ Mul<Self, Output = Self>
+ Div<Self, Output = Self>
+ AddAssign<Self>
+ SubAssign<Self>
+ MulAssign<Self>
+ DivAssign<Self>
+ Neg<Output = Self>
+ From<<Self as FieldElement>::BaseField>
+ From<u128>
+ From<u64>
+ From<u32>
+ From<u16>
+ From<u8>
+ for<'a> TryFrom<&'a [u8]>
+ AsBytes
+ Randomizable
+ Serializable
+ Deserializable
{
/// A type defining positive integers big enough to describe a field modulus for
/// `Self::BaseField` with no loss of precision.
type PositiveInteger: Debug
+ Copy
+ PartialEq
+ PartialOrd
+ ShrAssign
+ Shl<u32, Output = Self::PositiveInteger>
+ Shr<u32, Output = Self::PositiveInteger>
+ BitAnd<Output = Self::PositiveInteger>
+ From<u32>
+ From<u64>;
/// Base field type for this finite field. For prime fields, `BaseField` should be set
/// to `Self`.
type BaseField: StarkField;
/// Number of bytes needed to encode an element
const ELEMENT_BYTES: usize;
/// True if internal representation of the element is the same as its canonical representation.
const IS_CANONICAL: bool;
/// The additive identity.
const ZERO: Self;
/// The multiplicative identity.
const ONE: Self;
// ALGEBRA
// --------------------------------------------------------------------------------------------
/// Returns this field element added to itself.
fn double(self) -> Self {
self + self
}
/// Returns this field element raised to power 2.
fn square(self) -> Self {
self * self
}
/// Returns this field element raised to power 3.
fn cube(self) -> Self {
self * self * self
}
/// Exponentiates this field element by `power` parameter.
fn exp(self, power: Self::PositiveInteger) -> Self {
let mut r = Self::ONE;
let mut b = self;
let mut p = power;
let int_zero = Self::PositiveInteger::from(0u32);
let int_one = Self::PositiveInteger::from(1u32);
if p == int_zero {
return Self::ONE;
} else if b == Self::ZERO {
return Self::ZERO;
}
while p > int_zero {
if p & int_one == int_one {
r *= b;
}
p >>= int_one;
b = b.square();
}
r
}
/// Returns a multiplicative inverse of this field element. If this element is ZERO, ZERO is
/// returned.
fn inv(self) -> Self;
/// Returns a conjugate of this field element.
fn conjugate(&self) -> Self;
// SERIALIZATION / DESERIALIZATION
// --------------------------------------------------------------------------------------------
/// Converts a list of elements into a list of bytes.
///
/// The elements may be in the internal representation rather than in the canonical
/// representation. This conversion is intended to be zero-copy (i.e. by re-interpreting the
/// underlying memory).
fn elements_as_bytes(elements: &[Self]) -> &[u8];
/// Converts a list of bytes into a list of field elements.
///
/// The elements are assumed to encoded in the internal representation rather than in the
/// canonical representation. The conversion is intended to be zero-copy (i.e. by
/// re-interpreting the underlying memory).
///
/// # Errors
/// An error is returned if:
/// * Memory alignment of `bytes` does not match memory alignment of field element data.
/// * Length of `bytes` does not divide into whole number of elements.
///
/// # Safety
/// This function is unsafe because it does not check whether underlying bytes represent valid
/// field elements according to their internal representation.
unsafe fn bytes_as_elements(bytes: &[u8]) -> Result<&[Self], DeserializationError>;
// UTILITIES
// --------------------------------------------------------------------------------------------
/// Returns a vector of length `n` initialized with all ZERO elements.
///
/// Specialized implementations of this function may be faster than the generic implementation.
fn zeroed_vector(n: usize) -> Vec<Self> {
vec![Self::ZERO; n]
}
/// Converts a list of field elements into a list of elements in the underlying base field.
///
/// For base STARK fields, the input and output lists are the same. For extension field, the
/// output list will contain decompositions of each extension element into underlying base
/// elements.
fn as_base_elements(elements: &[Self]) -> &[Self::BaseField];
}
// STARK FIELD
// ================================================================================================
/// Defines an element in a STARK-friendly finite field.
///
/// A STARK-friendly field is defined as a prime field with high two-addicity. That is, the
/// the modulus of the field should be a prime number of the form `k` * 2^`n` + 1 (a Proth prime),
/// where `n` is relatively larger (e.g., greater than 32).
pub trait StarkField: FieldElement<BaseField = Self> {
/// Type describing quadratic extension of this StarkField.
type QuadExtension: FieldElement<BaseField = Self>;
/// Prime modulus of the field. Must be of the form `k` * 2^`n` + 1 (a Proth prime).
/// This ensures that the field has high 2-adicity.
const MODULUS: Self::PositiveInteger;
/// The number of bits needed to represents `Self::MODULUS`.
const MODULUS_BITS: u32;
/// A multiplicative generator of the field.
const GENERATOR: Self;
/// Let Self::MODULUS = `k` * 2^`n` + 1; then, TWO_ADICITY is `n`.
const TWO_ADICITY: u32;
/// Let Self::MODULUS = `k` * 2^`n` + 1; then, TWO_ADIC_ROOT_OF_UNITY is 2^`n` root of unity
/// computed as Self::GENERATOR^`k`.
const TWO_ADIC_ROOT_OF_UNITY: Self;
/// Returns the root of unity of order 2^`n`.
///
/// # Panics
/// Panics if the root of unity for the specified order does not exist in this field.
fn get_root_of_unity(n: u32) -> Self {
assert!(n != 0, "cannot get root of unity for n = 0");
assert!(
n <= Self::TWO_ADICITY,
"order cannot exceed 2^{}",
Self::TWO_ADICITY
);
let power = Self::PositiveInteger::from(1u32) << (Self::TWO_ADICITY - n);
Self::TWO_ADIC_ROOT_OF_UNITY.exp(power)
}
/// Returns byte representation of the field modulus in little-endian byte order.
fn get_modulus_le_bytes() -> Vec<u8>;
/// Returns a canonical integer representation of the field element.
fn as_int(&self) -> Self::PositiveInteger;
}