Struct Felt 

pub struct Felt(/* private fields */);

A Felt backed by Plonky3’s Goldilocks field element.

Implementations

impl Felt

pub const ORDER: u64 = <Goldilocks as PrimeField64>::ORDER_U64

Order of the field.

pub const ZERO: Felt

pub const ONE: Felt

pub const NUM_BYTES: usize = Goldilocks::NUM_BYTES

The number of bytes which this field element occupies in memory.

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

Creates 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 fn from_u8(value: u8) -> Felt

pub fn from_u16(value: u16) -> Felt

pub fn from_u32(value: u32) -> Felt

pub fn double(&self) -> Felt

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

pub fn square(&self) -> Felt

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

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

Exponentiation by a u64 power.

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

Exponentiation by a small constant power.

pub fn as_canonical_u64(&self) -> u64

Return the representative of element in canonical form which lies in the range 0 <= x < ORDER.

pub fn as_canonical_u64_ct(&self) -> u64

Constant-time equivalent of as_canonical_u64() using the same reduction logic as Plonky3’s Goldilocks implementation.

Methods from Deref<Target = Goldilocks>

pub const TWO_ADIC_GENERATORS: [Goldilocks; 33]

Trait Implementations

impl Add for Felt

type Output = Felt

The resulting type after applying the + operator.

fn add(self, other: Felt) -> Felt

Performs the + operation.

impl AddAssign for Felt

fn add_assign(&mut self, other: Felt)

Performs the += operation.

impl BinomiallyExtendable<2> for Felt

const W: Felt

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

const DTH_ROOT: Felt

A D-th root of unity derived from W.

const EXT_GENERATOR: [Felt; 2]

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

impl BinomiallyExtendable<5> for Felt

const W: Felt

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

const DTH_ROOT: Felt

A D-th root of unity derived from W.

const EXT_GENERATOR: [Felt; 5]

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

impl BinomiallyExtendableAlgebra<Felt, 2> for Felt

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<Felt, 5> for Felt

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 Felt

fn clone(&self) -> Felt

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for Felt

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

Formats the value using the given formatter.

impl Default for Felt

fn default() -> Felt

Returns the “default value” for a type.

impl Deref for Felt

type Target = Goldilocks

The resulting type after dereferencing.

fn deref(&self) -> &<Felt as Deref>::Target

Dereferences the value.

impl DerefMut for Felt

fn deref_mut(&mut self) -> &mut <Felt as Deref>::Target

Mutably dereferences the value.

impl Deserializable for Felt

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

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

where __D: Deserializer<'de>,

Deserialize this value from the given Serde deserializer.

impl Display for Felt

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

Formats the value using the given formatter.

impl Div for Felt

type Output = Felt

The resulting type after applying the / operator.

fn div(self, other: Felt) -> Felt

Performs the / operation.

impl DivAssign for Felt

fn div_assign(&mut self, other: Felt)

Performs the /= operation.

impl Field for Felt

const GENERATOR: Felt

A generator of this field’s multiplicative group.

type Packing = Felt

fn is_zero(&self) -> bool

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

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

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<ContextId> for Felt

fn from(context_id: ContextId) -> Felt

Converts to this type from the input type.

impl From<Felt> for Goldilocks

fn from(value: Felt) -> Goldilocks

Converts to this type from the input type.

impl From<Forest> for Felt

fn from(value: Forest) -> Felt

Converts to this type from the input type.

impl From<Goldilocks> for Felt

fn from(value: Goldilocks) -> Felt

Converts to this type from the input type.

impl From<RowIndex> for Felt

fn from(step: RowIndex) -> Felt

Converts to this type from the input type.

impl From<u16> for Felt

fn from(int: u16) -> Felt

Converts to this type from the input type.

impl From<u32> for Felt

fn from(int: u32) -> Felt

Converts to this type from the input type.

impl From<u8> for Felt

fn from(int: u8) -> Felt

Converts to this type from the input type.

impl HasTwoAdicBinomialExtension<2> for Felt

const EXT_TWO_ADICITY: usize

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

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

Returns a two-adic generator for the extension field.

impl HasTwoAdicBinomialExtension<5> for Felt

const EXT_TWO_ADICITY: usize

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

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

Returns a two-adic generator for the extension field.

impl Hash for Felt

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 Felt

fn injective_exp_n(&self) -> Self

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

impl Mul for Felt

type Output = Felt

The resulting type after applying the * operator.

fn mul(self, other: Felt) -> Felt

Performs the * operation.

impl MulAssign for Felt

fn mul_assign(&mut self, other: Felt)

Performs the *= operation.

impl Neg for Felt

type Output = Felt

The resulting type after applying the - operator.

fn neg(self) -> Felt

Performs the unary - operation.

impl Ord for Felt

fn cmp(&self, other: &Felt) -> 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<Goldilocks> for Felt

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 PartialEq for Felt

fn eq(&self, other: &Felt) -> 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 Felt

fn partial_cmp(&self, other: &Felt) -> 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 Felt

fn injective_exp_root_n(&self) -> Felt

Compute x -> x^K for a given K > 1 such that x^{NK} = x for all elements x.

impl PrimeCharacteristicRing for Felt

const ZERO: Felt

The additive identity of the ring.

const ONE: Felt

The multiplicative identity of the ring.

const TWO: Felt

The element in the ring given by ONE + ONE.

const NEG_ONE: Felt

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: <Felt as PrimeCharacteristicRing>::PrimeSubfield, ) -> Felt

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

fn from_bool(value: bool) -> Felt

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

fn halve(&self) -> Felt

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

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

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

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

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

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

Exponentiation by a u64 power.

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_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, ....

fn dot_product<const N: usize>(u: &[Self; N], v: &[Self; N]) -> Self

Compute the dot product of two vectors.

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

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

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

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.

impl PrimeField for Felt

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 Felt

const ORDER_U64: u64 = <Goldilocks as PrimeField64>::ORDER_U64

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<'a> Product<&'a Felt> for Felt

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

where I: Iterator<Item = &'a Felt>,

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

impl Product for Felt

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

where I: Iterator<Item = Felt>,

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

impl QuotientMap<i128> for Felt

fn from_int(int: i128) -> Felt

Convert a given i128 integer into an element of the Felt 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<Felt>

Convert a given u128 integer into an element of the Felt 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) -> Felt

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

Safety

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

impl QuotientMap<i16> for Felt

fn from_int(int: i16) -> Felt

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

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

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

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

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

unsafe fn from_canonical_unchecked(int: i16) -> Felt

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

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

impl QuotientMap<i32> for Felt

fn from_int(int: i32) -> Felt

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

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

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

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

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

unsafe fn from_canonical_unchecked(int: i32) -> Felt

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

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

impl QuotientMap<i64> for Felt

fn from_int(int: i64) -> Felt

Convert a given integer into an element of the field ℤ/p.

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

Convert a given integer into an element of the field ℤ/p. The input is checked to ensure it lies within a given range.

unsafe fn from_canonical_unchecked(int: i64) -> Felt

Convert a given integer into an element of the field ℤ/p. The input is guaranteed to lie within a specific range depending on p. If the input lies outside of this range, the output is undefined.

impl QuotientMap<i8> for Felt

fn from_int(int: i8) -> Felt

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

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

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

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

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

unsafe fn from_canonical_unchecked(int: i8) -> Felt

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

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

impl QuotientMap<u128> for Felt

fn from_int(int: u128) -> Felt

Convert a given u128 integer into an element of the Felt 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<Felt>

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

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

unsafe fn from_canonical_unchecked(int: u128) -> Felt

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

Safety

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

impl QuotientMap<u16> for Felt

fn from_int(int: u16) -> Felt

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

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

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

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

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

unsafe fn from_canonical_unchecked(int: u16) -> Felt

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

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

impl QuotientMap<u32> for Felt

fn from_int(int: u32) -> Felt

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

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

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

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

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

unsafe fn from_canonical_unchecked(int: u32) -> Felt

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

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

impl QuotientMap<u64> for Felt

fn from_int(int: u64) -> Felt

Convert a given integer into an element of the field ℤ/p.

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

Convert a given integer into an element of the field ℤ/p. The input is checked to ensure it lies within a given range.

unsafe fn from_canonical_unchecked(int: u64) -> Felt

Convert a given integer into an element of the field ℤ/p. The input is guaranteed to lie within a specific range depending on p. If the input lies outside of this range, the output is undefined.

impl QuotientMap<u8> for Felt

fn from_int(int: u8) -> Felt

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

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

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

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

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

unsafe fn from_canonical_unchecked(int: u8) -> Felt

Convert a given u8 integer into an element of the Felt 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<Felt>

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

impl RawDataSerializable for Felt

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 = Felt>, ) -> impl IntoIterator<Item = u32>

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

fn into_u64_stream( input: impl IntoIterator<Item = Felt>, ) -> 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 = [Felt; 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 = [Felt; 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 = [Felt; 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 Felt

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 Felt

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 Felt

type Output = Felt

The resulting type after applying the - operator.

fn sub(self, other: Felt) -> Felt

Performs the - operation.

impl SubAssign for Felt

fn sub_assign(&mut self, other: Felt)

Performs the -= operation.

impl<'a> Sum<&'a Felt> for Felt

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

where I: Iterator<Item = &'a Felt>,

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

impl Sum for Felt

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

where I: Iterator<Item = Felt>,

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

impl TryFrom<u64> for Felt

type Error = FeltFromIntError

The type returned in the event of a conversion error.

fn try_from(int: u64) -> Result<Felt, <Felt as TryFrom<u64>>::Error>

Performs the conversion.

impl TwoAdicField for Felt

const TWO_ADICITY: usize = <Goldilocks as TwoAdicField>::TWO_ADICITY

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

fn two_adic_generator(bits: usize) -> Felt

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

impl UniformSamplingField for Felt

const MAX_SINGLE_SAMPLE_BITS: usize = <Goldilocks as UniformSamplingField>::MAX_SINGLE_SAMPLE_BITS

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] = <Goldilocks as UniformSamplingField>::SAMPLING_BITS_M

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 Felt

impl Eq for Felt

impl Packable for Felt