Struct Felt 

pub struct Felt { /* private fields */ }

The prime field known as Goldilocks, defined as F_p where p = 2^64 - 2^32 + 1.

Note that the safety of deriving Serialize and Deserialize relies on the fact that the internal value can be any u64.

Implementations

impl Goldilocks

pub const TWO_ADIC_GENERATORS: [Goldilocks; 33]

A list of generators for the two-adic subgroups of the goldilocks field.

These satisfy the properties that TWO_ADIC_GENERATORS[0] = 1 and TWO_ADIC_GENERATORS[i+1]^2 = TWO_ADIC_GENERATORS[i].

pub const fn new(value: u64) -> Goldilocks

Create a new field element from any u64.

Any u64 value is accepted. No reduction is performed since Goldilocks uses a non-canonical internal representation.

pub const fn new_array<const N: usize>(input: [u64; N]) -> [Goldilocks; N]

Convert a [u64; N] array to an array of field elements.

Const version of input.map(Goldilocks::new).

Trait Implementations

impl Add for Goldilocks

type Output = Goldilocks

The resulting type after applying the + operator.

fn add(self, rhs: Goldilocks) -> Goldilocks

Performs the + operation.

impl<T> AddAssign<T> for Goldilocks

where T: Into<Goldilocks>,

fn add_assign(&mut self, rhs: T)

Performs the += operation.

impl BinomiallyExtendable<2> for Goldilocks

const W: Goldilocks

The constant coefficient W in the binomial X^D - W.

const DTH_ROOT: Goldilocks

A D-th root of unity derived from W.

const EXT_GENERATOR: [Goldilocks; 2]

A generator for the extension field, expressed as a degree-D polynomial.

impl BinomiallyExtendable<5> for Goldilocks

const W: Goldilocks

The constant coefficient W in the binomial X^D - W.

const DTH_ROOT: Goldilocks

A D-th root of unity derived from W.

const EXT_GENERATOR: [Goldilocks; 5]

A generator for the extension field, expressed as a degree-D polynomial.

impl BinomiallyExtendableAlgebra<Goldilocks, 2> for Goldilocks

fn binomial_mul(a: &[Self; D], b: &[Self; D], res: &mut [Self; D], w: F)

Multiplication in the algebra extension ring A<X> / (X^D - W).

fn binomial_add(a: &[Self; D], b: &[Self; D]) -> [Self; D]

Addition of elements in the algebra extension ring A<X> / (X^D - W).

fn binomial_sub(a: &[Self; D], b: &[Self; D]) -> [Self; D]

Subtraction of elements in the algebra extension ring A<X> / (X^D - W).

fn binomial_base_mul(lhs: [Self; D], rhs: Self) -> [Self; D]

impl BinomiallyExtendableAlgebra<Goldilocks, 5> for Goldilocks

fn binomial_mul(a: &[Self; D], b: &[Self; D], res: &mut [Self; D], w: F)

Multiplication in the algebra extension ring A<X> / (X^D - W).

fn binomial_add(a: &[Self; D], b: &[Self; D]) -> [Self; D]

Addition of elements in the algebra extension ring A<X> / (X^D - W).

fn binomial_sub(a: &[Self; D], b: &[Self; D]) -> [Self; D]

Subtraction of elements in the algebra extension ring A<X> / (X^D - W).

fn binomial_base_mul(lhs: [Self; D], rhs: Self) -> [Self; D]

impl Clone for Goldilocks

fn clone(&self) -> Goldilocks

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for Goldilocks

fn fmt(&self, f: &mut Formatter<'_>) -> Result<(), Error>

Formats the value using the given formatter.

impl Default for Goldilocks

fn default() -> Goldilocks

Returns the “default value” for a type.

impl Deserializable for Goldilocks

fn read_from<R>(source: &mut R) -> Result<Goldilocks, DeserializationError>

where R: ByteReader,

Reads a sequence of bytes from the provided source, attempts to deserialize these bytes into Self, and returns the result.

fn min_serialized_size() -> usize

Returns the minimum serialized size for one instance of this type.

fn read_from_bytes(bytes: &[u8]) -> Result<Self, DeserializationError>

Attempts to deserialize the provided bytes into Self and returns the result.

fn read_from_bytes_with_budget( bytes: &[u8], budget: usize, ) -> Result<Self, DeserializationError>

Deserializes Self from bytes with a byte budget limit.

impl<'de> Deserialize<'de> for Goldilocks

fn deserialize<__D>( __deserializer: __D, ) -> Result<Goldilocks, <__D as Deserializer<'de>>::Error>

where __D: Deserializer<'de>,

Deserialize this value from the given Serde deserializer.

impl Display for Goldilocks

fn fmt(&self, f: &mut Formatter<'_>) -> Result<(), Error>

Formats the value using the given formatter.

impl Div for Goldilocks

type Output = Goldilocks

The resulting type after applying the / operator.

fn div(self, rhs: Goldilocks) -> Goldilocks

Performs the / operation.

impl DivAssign for Goldilocks

fn div_assign(&mut self, rhs: Goldilocks)

Performs the /= operation.

impl Field for Goldilocks

const GENERATOR: Goldilocks

A generator of this field’s multiplicative group.

type Packing = Goldilocks

fn is_zero(&self) -> bool

Check if the given field element is equal to the unique additive identity (ZERO).

fn try_inverse(&self) -> Option<Goldilocks>

The multiplicative inverse of this field element, if it exists.

fn order() -> BigUint

The number of elements in the field.

fn is_one(&self) -> bool

Check if the given field element is equal to the unique multiplicative identity (ONE).

fn inverse(&self) -> Self

The multiplicative inverse of this field element.

fn add_slices(slice_1: &mut [Self], slice_2: &[Self])

Add two slices of field elements together, returning the result in the first slice.

fn bits() -> usize

The number of bits required to define an element of this field.

impl From<Forest> for Felt

fn from(value: Forest) -> Self

Converts to this type from the input type.

impl From<Goldilocks> for Forest

fn from(value: Felt) -> Self

Converts to this type from the input type.

impl HasTwoAdicBinomialExtension<2> for Goldilocks

const EXT_TWO_ADICITY: usize = 33

Two-adicity of the multiplicative group of the extension field.

fn ext_two_adic_generator(bits: usize) -> [Goldilocks; 2]

Returns a two-adic generator for the extension field.

impl HasTwoAdicBinomialExtension<5> for Goldilocks

const EXT_TWO_ADICITY: usize = 32

Two-adicity of the multiplicative group of the extension field.

fn ext_two_adic_generator(bits: usize) -> [Goldilocks; 5]

Returns a two-adic generator for the extension field.

impl Hash for Goldilocks

fn hash<H>(&self, state: &mut H)

where H: Hasher,

Feeds this value into the given Hasher.

fn hash_slice<H>(data: &[Self], state: &mut H)

where H: Hasher, Self: Sized,

Feeds a slice of this type into the given Hasher.

impl InjectiveMonomial<7> for Goldilocks

Degree of the smallest permutation polynomial for Goldilocks.

As p - 1 = 2^32 * 3 * 5 * 17 * … the smallest choice for a degree D satisfying gcd(p - 1, D) = 1 is 7.

fn injective_exp_n(&self) -> Self

Compute x -> x^n for a given n > 1 such that this map is injective.

impl Mul for Goldilocks

type Output = Goldilocks

The resulting type after applying the * operator.

fn mul(self, rhs: Goldilocks) -> Goldilocks

Performs the * operation.

impl<T> MulAssign<T> for Goldilocks

where T: Into<Goldilocks>,

fn mul_assign(&mut self, rhs: T)

Performs the *= operation.

impl Neg for Goldilocks

type Output = Goldilocks

The resulting type after applying the - operator.

fn neg(self) -> <Goldilocks as Neg>::Output

Performs the unary - operation.

impl Ord for Goldilocks

fn cmp(&self, other: &Goldilocks) -> Ordering

This method returns an Ordering between self and other.

fn max(self, other: Self) -> Self

where Self: Sized,

Compares and returns the maximum of two values.

fn min(self, other: Self) -> Self

where Self: Sized,

Compares and returns the minimum of two values.

fn clamp(self, min: Self, max: Self) -> Self

where Self: Sized,

Restrict a value to a certain interval.

impl PartialEq for Goldilocks

fn eq(&self, other: &Goldilocks) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialOrd for Goldilocks

fn partial_cmp(&self, other: &Goldilocks) -> Option<Ordering>

This method returns an ordering between self and other values if one exists.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PermutationMonomial<7> for Goldilocks

fn injective_exp_root_n(&self) -> Goldilocks

In the field Goldilocks, a^{1/7} is equal to a^{10540996611094048183}.

This follows from the calculation 7*10540996611094048183 = 4*(2^64 - 2**32) + 1 = 1 mod (p - 1).

impl PrimeCharacteristicRing for Goldilocks

const ZERO: Goldilocks

The additive identity of the ring.

const ONE: Goldilocks

The multiplicative identity of the ring.

const TWO: Goldilocks

The element in the ring given by ONE + ONE.

const NEG_ONE: Goldilocks

The element in the ring given by -ONE.

type PrimeSubfield = Goldilocks

The field ℤ/p where the characteristic of this ring is p.

fn from_prime_subfield( f: <Goldilocks as PrimeCharacteristicRing>::PrimeSubfield, ) -> Goldilocks

Embed an element of the prime field ℤ/p into the ring R.

fn from_bool(b: bool) -> Goldilocks

Return Self::ONE if b is true and Self::ZERO if b is false.

fn halve(&self) -> Goldilocks

The elementary function halve(a) = a/2.

fn mul_2exp_u64(&self, exp: u64) -> Goldilocks

The elementary function mul_2exp_u64(a, exp) = a * 2^{exp}.

fn div_2exp_u64(&self, exp: u64) -> Goldilocks

Divide by a given power of two. div_2exp_u64(a, exp) = a/2^exp

fn sum_array<const N: usize>(input: &[Goldilocks]) -> Goldilocks

Compute the sum of a slice of elements whose length is a compile time constant.

fn dot_product<const N: usize>( lhs: &[Goldilocks; N], rhs: &[Goldilocks; N], ) -> Goldilocks

Compute the dot product of two vectors.

fn zero_vec(len: usize) -> Vec<Goldilocks>

Allocates a vector of zero elements of length len. Many operating systems zero pages before assigning them to a userspace process. In that case, our process should not need to write zeros, which would be redundant. However, the compiler may not always recognize this.

fn from_u8(int: u8) -> Self

Given an integer r, return the sum of r copies of ONE:

fn from_u16(int: u16) -> Self

Given an integer r, return the sum of r copies of ONE:

fn from_u32(int: u32) -> Self

Given an integer r, return the sum of r copies of ONE:

fn from_u64(int: u64) -> Self

Given an integer r, return the sum of r copies of ONE:

fn from_u128(int: u128) -> Self

Given an integer r, return the sum of r copies of ONE:

fn from_usize(int: usize) -> Self

Given an integer r, return the sum of r copies of ONE:

fn from_i8(int: i8) -> Self

Given an integer r, return the sum of r copies of ONE:

fn from_i16(int: i16) -> Self

Given an integer r, return the sum of r copies of ONE:

fn from_i32(int: i32) -> Self

Given an integer r, return the sum of r copies of ONE:

fn from_i64(int: i64) -> Self

Given an integer r, return the sum of r copies of ONE:

fn from_i128(int: i128) -> Self

Given an integer r, return the sum of r copies of ONE:

fn from_isize(int: isize) -> Self

Given an integer r, return the sum of r copies of ONE:

fn double(&self) -> Self

The elementary function double(a) = 2*a.

fn square(&self) -> Self

The elementary function square(a) = a^2.

fn cube(&self) -> Self

The elementary function cube(a) = a^3.

fn xor(&self, y: &Self) -> Self

Computes the arithmetic generalization of boolean xor.

fn xor3(&self, y: &Self, z: &Self) -> Self

Computes the arithmetic generalization of a triple xor.

fn andn(&self, y: &Self) -> Self

Computes the arithmetic generalization of andnot.

fn bool_check(&self) -> Self

The vanishing polynomial for boolean values: x * (x - 1).

fn exp_u64(&self, power: u64) -> Self

Exponentiation by a u64 power.

fn exp_const_u64<const POWER: u64>(&self) -> Self

Exponentiation by a small constant power.

fn exp_power_of_2(&self, power_log: usize) -> Self

The elementary function exp_power_of_2(a, power_log) = a^{2^power_log}.

fn powers(&self) -> Powers<Self>

Construct an iterator which returns powers of self: self^0, self^1, self^2, ....

fn shifted_powers(&self, start: Self) -> Powers<Self>

Construct an iterator which returns powers of self shifted by start: start, start*self^1, start*self^2, ....

impl PrimeField for Goldilocks

fn as_canonical_biguint(&self) -> BigUint

Return the representative of value in canonical form which lies in the range 0 <= x < self.order().

impl PrimeField64 for Goldilocks

const ORDER_U64: u64 = P

fn as_canonical_u64(&self) -> u64

Return the representative of value in canonical form which lies in the range 0 <= x < ORDER_U64.

fn to_unique_u64(&self) -> u64

Convert a field element to a u64 such that any two field elements are converted to the same u64 if and only if they represent the same value.

impl Product for Goldilocks

fn product<I>(iter: I) -> Goldilocks

where I: Iterator<Item = Goldilocks>,

Takes an iterator and generates Self from the elements by multiplying the items.

impl QuotientMap<i128> for Goldilocks

fn from_int(int: i128) -> Goldilocks

Convert a given i128 integer into an element of the Goldilocks field.

This checks the sign and then makes use of the equivalent method for unsigned integers. This should be avoided in performance critical locations.

fn from_canonical_checked(int: i128) -> Option<Goldilocks>

Convert a given u128 integer into an element of the Goldilocks field.

Returns None if the input does not lie in the range:[-(2^63 - 2^31), 2^63 - 2^31].

unsafe fn from_canonical_unchecked(int: i128) -> Goldilocks

Convert a given u128 integer into an element of the Goldilocks field.

Safety

The input must lie in the range:[1 + 2^32 - 2^64, 2^64 - 1].

impl QuotientMap<i16> for Goldilocks

fn from_int(int: i16) -> Goldilocks

Convert a given i16 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

fn from_canonical_checked(int: i16) -> Option<Goldilocks>

Convert a given i16 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

unsafe fn from_canonical_unchecked(int: i16) -> Goldilocks

Convert a given i16 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

impl QuotientMap<i32> for Goldilocks

fn from_int(int: i32) -> Goldilocks

Convert a given i32 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

fn from_canonical_checked(int: i32) -> Option<Goldilocks>

Convert a given i32 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

unsafe fn from_canonical_unchecked(int: i32) -> Goldilocks

Convert a given i32 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

impl QuotientMap<i64> for Goldilocks

fn from_int(int: i64) -> Goldilocks

Convert a given i64 integer into an element of the Goldilocks field.

We simply need to deal with the sign.

fn from_canonical_checked(int: i64) -> Option<Goldilocks>

Convert a given i64 integer into an element of the Goldilocks field.

Returns none if the input does not lie in the range (-(2^63 - 2^31), 2^63 - 2^31).

unsafe fn from_canonical_unchecked(int: i64) -> Goldilocks

Convert a given i64 integer into an element of the Goldilocks field.

Safety

In this case this function is actually always safe as the internal value is allowed to be any u64.

impl QuotientMap<i8> for Goldilocks

fn from_int(int: i8) -> Goldilocks

Convert a given i8 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

fn from_canonical_checked(int: i8) -> Option<Goldilocks>

Convert a given i8 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

unsafe fn from_canonical_unchecked(int: i8) -> Goldilocks

Convert a given i8 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

impl QuotientMap<u128> for Goldilocks

fn from_int(int: u128) -> Goldilocks

Convert a given u128 integer into an element of the Goldilocks field.

Uses a modular reduction to reduce to canonical form. This should be avoided in performance critical locations.

fn from_canonical_checked(int: u128) -> Option<Goldilocks>

Convert a given u128 integer into an element of the Goldilocks field.

Returns None if the input does not lie in the range:[0, 2^64 - 2^32].

unsafe fn from_canonical_unchecked(int: u128) -> Goldilocks

Convert a given u128 integer into an element of the Goldilocks field.

Safety

The input must lie in the range:[0, 2^64 - 1].

impl QuotientMap<u16> for Goldilocks

fn from_int(int: u16) -> Goldilocks

Convert a given u16 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

fn from_canonical_checked(int: u16) -> Option<Goldilocks>

Convert a given u16 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

unsafe fn from_canonical_unchecked(int: u16) -> Goldilocks

Convert a given u16 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

impl QuotientMap<u32> for Goldilocks

fn from_int(int: u32) -> Goldilocks

Convert a given u32 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

fn from_canonical_checked(int: u32) -> Option<Goldilocks>

Convert a given u32 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

unsafe fn from_canonical_unchecked(int: u32) -> Goldilocks

Convert a given u32 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

impl QuotientMap<u64> for Goldilocks

fn from_int(int: u64) -> Goldilocks

Convert a given u64 integer into an element of the Goldilocks field.

No reduction is needed as the internal value is allowed to be any u64.

fn from_canonical_checked(int: u64) -> Option<Goldilocks>

Convert a given u64 integer into an element of the Goldilocks field.

Return None if the given integer is greater than p = 2^64 - 2^32 + 1.

unsafe fn from_canonical_unchecked(int: u64) -> Goldilocks

Convert a given u64 integer into an element of the Goldilocks field.

Safety

In this case this function is actually always safe as the internal value is allowed to be any u64.

impl QuotientMap<u8> for Goldilocks

fn from_int(int: u8) -> Goldilocks

Convert a given u8 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

fn from_canonical_checked(int: u8) -> Option<Goldilocks>

Convert a given u8 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

unsafe fn from_canonical_unchecked(int: u8) -> Goldilocks

Convert a given u8 integer into an element of the Goldilocks field.

Due to the integer type, the input value is always canonical.

impl Randomizable for Felt

const VALUE_SIZE: usize = 8

Size of Self in bytes.

fn from_random_bytes(source: &[u8]) -> Option<Self>

Returns Self if the set of bytes forms a valid value, otherwise returns None.

impl RawDataSerializable for Goldilocks

const NUM_BYTES: usize = 8

The number of bytes which this field element occupies in memory. Must be equal to the length of self.into_bytes().

fn into_bytes(self) -> [u8; 8]

Convert a field element into a collection of bytes.

fn into_u32_stream( input: impl IntoIterator<Item = Goldilocks>, ) -> impl IntoIterator<Item = u32>

Convert an iterator of field elements into an iterator of u32s.

fn into_u64_stream( input: impl IntoIterator<Item = Goldilocks>, ) -> impl IntoIterator<Item = u64>

Convert an iterator of field elements into an iterator of u64s.

fn into_parallel_byte_streams<const N: usize>( input: impl IntoIterator<Item = [Goldilocks; N]>, ) -> impl IntoIterator<Item = [u8; N]>

Convert an iterator of field element arrays into an iterator of byte arrays.

fn into_parallel_u32_streams<const N: usize>( input: impl IntoIterator<Item = [Goldilocks; N]>, ) -> impl IntoIterator<Item = [u32; N]>

Convert an iterator of field element arrays into an iterator of u32 arrays.

fn into_parallel_u64_streams<const N: usize>( input: impl IntoIterator<Item = [Goldilocks; N]>, ) -> impl IntoIterator<Item = [u64; N]>

Convert an iterator of field element arrays into an iterator of u64 arrays.

fn into_byte_stream( input: impl IntoIterator<Item = Self>, ) -> impl IntoIterator<Item = u8>

Convert an iterator of field elements into an iterator of bytes.

impl Serializable for Goldilocks

fn write_into<W>(&self, target: &mut W)

where W: ByteWriter,

Serializes self into bytes and writes these bytes into the target.

fn get_size_hint(&self) -> usize

Returns an estimate of how many bytes are needed to represent self.

fn to_bytes(&self) -> Vec<u8>

Serializes self into a vector of bytes.

impl Serialize for Goldilocks

fn serialize<__S>( &self, __serializer: __S, ) -> Result<<__S as Serializer>::Ok, <__S as Serializer>::Error>

where __S: Serializer,

Serialize this value into the given Serde serializer.

impl Sub for Goldilocks

type Output = Goldilocks

The resulting type after applying the - operator.

fn sub(self, rhs: Goldilocks) -> Goldilocks

Performs the - operation.

impl<T> SubAssign<T> for Goldilocks

where T: Into<Goldilocks>,

fn sub_assign(&mut self, rhs: T)

Performs the -= operation.

impl Sum for Goldilocks

fn sum<I>(iter: I) -> Goldilocks

where I: Iterator<Item = Goldilocks>,

Takes an iterator and generates Self from the elements by “summing up” the items.

impl TwoAdicField for Goldilocks

const TWO_ADICITY: usize = 32

The number of factors of two in this field’s multiplicative group.

fn two_adic_generator(bits: usize) -> Goldilocks

Returns a generator of the multiplicative group of order 2^bits. Assumes bits <= TWO_ADICITY, otherwise the result is undefined.

impl UniformSamplingField for Goldilocks

const MAX_SINGLE_SAMPLE_BITS: usize = 24

Maximum number of bits we can sample at negligible (~1/field prime) probability of triggering an error / requiring a resample.

const SAMPLING_BITS_M: [u64; 64]

An array storing the largest value m_k for each k in [0, 31], such that m_k is a multiple of 2^k and less than P. m_k is defined as:

impl Copy for Goldilocks

impl Eq for Goldilocks

impl Packable for Goldilocks