Struct BinaryField16 

pub struct BinaryField16(/* private fields */);

The binary tower field GF(2^16).

A quadratic extension of BinaryField8 generated by a root X of X² + αX + 1, where α is the generator of the level below over the one beneath it. The low half of the backing integer holds the coefficient of 1, the high half the coefficient of X.

The serde encoding is canonical: every field element has exactly one valid byte representation (see the manual Deserialize impl below). #[serde(transparent)] makes the derived Serialize emit the backing integer itself, which is what that impl reads back.

Implementations

impl BinaryField16

pub fn from_le_bytes(bytes: [u8; 2]) -> Self

Construct a field element from its little-endian byte representation.

Every byte string of the right length is a valid input: from_repr masks any bits above the canonical range.

Trait Implementations

impl Add for BinaryField16

type Output = BinaryField16

The resulting type after applying the + operator.

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

Performs the + operation.

impl Add<BinaryField8> for BinaryField16

type Output = BinaryField16

The resulting type after applying the + operator.

fn add(self, rhs: BinaryField8) -> Self

Performs the + operation.

impl Add<BinaryField16> for BinaryField8

type Output = BinaryField16

The resulting type after applying the + operator.

fn add(self, rhs: BinaryField16) -> Self::Output

Performs the + operation.

impl Add<BinaryField16> for BinaryField32

type Output = BinaryField32

The resulting type after applying the + operator.

fn add(self, rhs: BinaryField16) -> Self

Performs the + operation.

impl Add<BinaryField16> for BinaryField64

type Output = BinaryField64

The resulting type after applying the + operator.

fn add(self, rhs: BinaryField16) -> Self

Performs the + operation.

impl Add<BinaryField16> for BinaryField128

type Output = BinaryField128

The resulting type after applying the + operator.

fn add(self, rhs: BinaryField16) -> Self

Performs the + operation.

impl Add<BinaryField16> for Ghash128

type Output = Ghash128

The resulting type after applying the + operator.

fn add(self, rhs: BinaryField16) -> Self

Performs the + operation.

impl Add<BinaryField16> for Poly64

type Output = Poly64

The resulting type after applying the + operator.

fn add(self, rhs: BinaryField16) -> Self

Performs the + operation.

impl Add<BinaryField16> for Gf2

type Output = BinaryField16

The resulting type after applying the + operator.

fn add(self, rhs: BinaryField16) -> Self::Output

Performs the + operation.

impl Add<BinaryField32> for BinaryField16

type Output = BinaryField32

The resulting type after applying the + operator.

fn add(self, rhs: BinaryField32) -> Self::Output

Performs the + operation.

impl Add<BinaryField64> for BinaryField16

type Output = BinaryField64

The resulting type after applying the + operator.

fn add(self, rhs: BinaryField64) -> Self::Output

Performs the + operation.

impl Add<BinaryField128> for BinaryField16

type Output = BinaryField128

The resulting type after applying the + operator.

fn add(self, rhs: BinaryField128) -> Self::Output

Performs the + operation.

impl Add<Gf2> for BinaryField16

type Output = BinaryField16

The resulting type after applying the + operator.

fn add(self, rhs: Gf2) -> Self

Performs the + operation.

impl Add<Ghash128> for BinaryField16

type Output = Ghash128

The resulting type after applying the + operator.

fn add(self, rhs: Ghash128) -> Self::Output

Performs the + operation.

impl Add<Poly64> for BinaryField16

type Output = Poly64

The resulting type after applying the + operator.

fn add(self, rhs: Poly64) -> Self::Output

Performs the + operation.

impl<T: Into<Self>> AddAssign<T> for BinaryField16

fn add_assign(&mut self, rhs: T)

Performs the += operation.

impl Algebra<BinaryField8> for BinaryField16

fn quadratic_extension_square(a: &[Self; 2], w: F) -> [Self; 2]

Square a[0] + a[1] X modulo X^2 - w.

fn mixed_dot_product<const N: usize>(a: &[Self; N], f: &[F; N]) -> Self

where F: Dup,

Dot product between algebra elements and base field scalars.

const BATCHED_LC_CHUNK: usize = 8

Optimal chunk size for batched_linear_combination.

fn batched_linear_combination(values: &[Self], coeffs: &[F]) -> Self

where F: Dup,

Runtime-length linear combination: Σ values[i] * coeffs[i].

impl Algebra<BinaryField16> for BinaryField32

fn quadratic_extension_square(a: &[Self; 2], w: F) -> [Self; 2]

Square a[0] + a[1] X modulo X^2 - w.

fn mixed_dot_product<const N: usize>(a: &[Self; N], f: &[F; N]) -> Self

where F: Dup,

Dot product between algebra elements and base field scalars.

const BATCHED_LC_CHUNK: usize = 8

Optimal chunk size for batched_linear_combination.

fn batched_linear_combination(values: &[Self], coeffs: &[F]) -> Self

where F: Dup,

Runtime-length linear combination: Σ values[i] * coeffs[i].

impl Algebra<BinaryField16> for BinaryField64

fn quadratic_extension_square(a: &[Self; 2], w: F) -> [Self; 2]

Square a[0] + a[1] X modulo X^2 - w.

fn mixed_dot_product<const N: usize>(a: &[Self; N], f: &[F; N]) -> Self

where F: Dup,

Dot product between algebra elements and base field scalars.

const BATCHED_LC_CHUNK: usize = 8

Optimal chunk size for batched_linear_combination.

fn batched_linear_combination(values: &[Self], coeffs: &[F]) -> Self

where F: Dup,

Runtime-length linear combination: Σ values[i] * coeffs[i].

impl Algebra<BinaryField16> for BinaryField128

fn quadratic_extension_square(a: &[Self; 2], w: F) -> [Self; 2]

Square a[0] + a[1] X modulo X^2 - w.

fn mixed_dot_product<const N: usize>(a: &[Self; N], f: &[F; N]) -> Self

where F: Dup,

Dot product between algebra elements and base field scalars.

const BATCHED_LC_CHUNK: usize = 8

Optimal chunk size for batched_linear_combination.

fn batched_linear_combination(values: &[Self], coeffs: &[F]) -> Self

where F: Dup,

Runtime-length linear combination: Σ values[i] * coeffs[i].

impl Algebra<BinaryField16> for Ghash128

fn quadratic_extension_square(a: &[Self; 2], w: F) -> [Self; 2]

Square a[0] + a[1] X modulo X^2 - w.

fn mixed_dot_product<const N: usize>(a: &[Self; N], f: &[F; N]) -> Self

where F: Dup,

Dot product between algebra elements and base field scalars.

const BATCHED_LC_CHUNK: usize = 8

Optimal chunk size for batched_linear_combination.

fn batched_linear_combination(values: &[Self], coeffs: &[F]) -> Self

where F: Dup,

Runtime-length linear combination: Σ values[i] * coeffs[i].

impl Algebra<BinaryField16> for Poly64

fn quadratic_extension_square(a: &[Self; 2], w: F) -> [Self; 2]

Square a[0] + a[1] X modulo X^2 - w.

fn mixed_dot_product<const N: usize>(a: &[Self; N], f: &[F; N]) -> Self

where F: Dup,

Dot product between algebra elements and base field scalars.

const BATCHED_LC_CHUNK: usize = 8

Optimal chunk size for batched_linear_combination.

fn batched_linear_combination(values: &[Self], coeffs: &[F]) -> Self

where F: Dup,

Runtime-length linear combination: Σ values[i] * coeffs[i].

impl Algebra<Gf2> for BinaryField16

fn quadratic_extension_square(a: &[Self; 2], w: F) -> [Self; 2]

Square a[0] + a[1] X modulo X^2 - w.

fn mixed_dot_product<const N: usize>(a: &[Self; N], f: &[F; N]) -> Self

where F: Dup,

Dot product between algebra elements and base field scalars.

const BATCHED_LC_CHUNK: usize = 8

Optimal chunk size for batched_linear_combination.

fn batched_linear_combination(values: &[Self], coeffs: &[F]) -> Self

where F: Dup,

Runtime-length linear combination: Σ values[i] * coeffs[i].

impl AlgebraIdentity<BinaryField8> for BinaryField16

fn algebra_id() -> Vec<u8>

Return the stable, unambiguous identifier of the defining relations and basis.

impl AlgebraIdentity<BinaryField16> for BinaryField32

fn algebra_id() -> Vec<u8>

Return the stable, unambiguous identifier of the defining relations and basis.

impl AlgebraIdentity<BinaryField16> for BinaryField64

fn algebra_id() -> Vec<u8>

Return the stable, unambiguous identifier of the defining relations and basis.

impl AlgebraIdentity<BinaryField16> for BinaryField128

fn algebra_id() -> Vec<u8>

Return the stable, unambiguous identifier of the defining relations and basis.

impl BasedVectorSpace<BinaryField8> for BinaryField16

fn flatten_to_base(vec: Vec<Self>) -> Vec<BinaryField8>

A whole vector of elements is one contiguous run of coefficients, for the same reason a single element is, so the flattening is a single copy of the buffer.

The inverse direction is not symmetric: a Vec<$lower> is allocated at $lower’s alignment, which is weaker than $upper’s, so freeing it under a Layout whose alignment differs from the one it was made with would be undefined behaviour. reconstitute_from_base below instead copies into a freshly allocated buffer at $upper’s own alignment.

const DIMENSION: usize

The dimension of the vector space, i.e. the number of elements in its basis.

fn as_basis_coefficients_slice(&self) -> &[BinaryField8]

Fixes a basis for the algebra A and uses this to map an element of A to a slice of DIMENSION F elements.

fn from_basis_coefficients_fn<Fn: FnMut(usize) -> BinaryField8>(f: Fn) -> Self

Fixes a basis for the algebra A and uses this to map DIMENSION F elements to an element of A. Similar to core:array::from_fn, the DIMENSION F elements are given by Fn(0), ..., Fn(DIMENSION - 1) called in that order.

fn from_basis_coefficients_iter<I: ExactSizeIterator<Item = BinaryField8>>( iter: I, ) -> Option<Self>

Fixes a basis for the algebra A and uses this to map DIMENSION F elements to an element of A.

fn reconstitute_from_base(vec: Vec<BinaryField8>) -> Vec<Self>

Convert from a vector of F to a vector of Self by combining the basis coefficients.

fn from_basis_coefficients_slice(slice: &[F]) -> Option<Self>

Fixes a basis for the algebra A and uses this to map DIMENSION F elements to an element of A.

fn ith_basis_element(i: usize) -> Option<Self>

Given a basis for the Algebra A, return the i’th basis element.

impl BasedVectorSpace<BinaryField16> for BinaryField32

fn flatten_to_base(vec: Vec<Self>) -> Vec<BinaryField16>

A whole vector of elements is one contiguous run of coefficients, for the same reason a single element is, so the flattening is a single copy of the buffer.

The inverse direction is not symmetric: a Vec<$lower> is allocated at $lower’s alignment, which is weaker than $upper’s, so freeing it under a Layout whose alignment differs from the one it was made with would be undefined behaviour. reconstitute_from_base below instead copies into a freshly allocated buffer at $upper’s own alignment.

const DIMENSION: usize

The dimension of the vector space, i.e. the number of elements in its basis.

fn as_basis_coefficients_slice(&self) -> &[BinaryField16]

Fixes a basis for the algebra A and uses this to map an element of A to a slice of DIMENSION F elements.

fn from_basis_coefficients_fn<Fn: FnMut(usize) -> BinaryField16>(f: Fn) -> Self

Fixes a basis for the algebra A and uses this to map DIMENSION F elements to an element of A. Similar to core:array::from_fn, the DIMENSION F elements are given by Fn(0), ..., Fn(DIMENSION - 1) called in that order.

fn from_basis_coefficients_iter<I: ExactSizeIterator<Item = BinaryField16>>( iter: I, ) -> Option<Self>

Fixes a basis for the algebra A and uses this to map DIMENSION F elements to an element of A.

fn reconstitute_from_base(vec: Vec<BinaryField16>) -> Vec<Self>

Convert from a vector of F to a vector of Self by combining the basis coefficients.

fn from_basis_coefficients_slice(slice: &[F]) -> Option<Self>

Fixes a basis for the algebra A and uses this to map DIMENSION F elements to an element of A.

fn ith_basis_element(i: usize) -> Option<Self>

Given a basis for the Algebra A, return the i’th basis element.

impl BasedVectorSpace<BinaryField16> for BinaryField64

fn flatten_to_base(vec: Vec<Self>) -> Vec<BinaryField16>

A whole vector of elements is one contiguous run of coefficients, for the same reason a single element is, so the flattening is a single copy of the buffer.

The inverse direction is not symmetric: a Vec<$lower> is allocated at $lower’s alignment, which is weaker than $upper’s, so freeing it under a Layout whose alignment differs from the one it was made with would be undefined behaviour. reconstitute_from_base below instead copies into a freshly allocated buffer at $upper’s own alignment.

const DIMENSION: usize

The dimension of the vector space, i.e. the number of elements in its basis.

fn as_basis_coefficients_slice(&self) -> &[BinaryField16]

Fixes a basis for the algebra A and uses this to map an element of A to a slice of DIMENSION F elements.

fn from_basis_coefficients_fn<Fn: FnMut(usize) -> BinaryField16>(f: Fn) -> Self

Fixes a basis for the algebra A and uses this to map DIMENSION F elements to an element of A. Similar to core:array::from_fn, the DIMENSION F elements are given by Fn(0), ..., Fn(DIMENSION - 1) called in that order.

fn from_basis_coefficients_iter<I: ExactSizeIterator<Item = BinaryField16>>( iter: I, ) -> Option<Self>

Fixes a basis for the algebra A and uses this to map DIMENSION F elements to an element of A.

fn reconstitute_from_base(vec: Vec<BinaryField16>) -> Vec<Self>

Convert from a vector of F to a vector of Self by combining the basis coefficients.

fn from_basis_coefficients_slice(slice: &[F]) -> Option<Self>

Fixes a basis for the algebra A and uses this to map DIMENSION F elements to an element of A.

fn ith_basis_element(i: usize) -> Option<Self>

Given a basis for the Algebra A, return the i’th basis element.

impl BasedVectorSpace<BinaryField16> for BinaryField128

fn flatten_to_base(vec: Vec<Self>) -> Vec<BinaryField16>

A whole vector of elements is one contiguous run of coefficients, for the same reason a single element is, so the flattening is a single copy of the buffer.

The inverse direction is not symmetric: a Vec<$lower> is allocated at $lower’s alignment, which is weaker than $upper’s, so freeing it under a Layout whose alignment differs from the one it was made with would be undefined behaviour. reconstitute_from_base below instead copies into a freshly allocated buffer at $upper’s own alignment.

const DIMENSION: usize

The dimension of the vector space, i.e. the number of elements in its basis.

fn as_basis_coefficients_slice(&self) -> &[BinaryField16]

Fixes a basis for the algebra A and uses this to map an element of A to a slice of DIMENSION F elements.

fn from_basis_coefficients_fn<Fn: FnMut(usize) -> BinaryField16>(f: Fn) -> Self

Fixes a basis for the algebra A and uses this to map DIMENSION F elements to an element of A. Similar to core:array::from_fn, the DIMENSION F elements are given by Fn(0), ..., Fn(DIMENSION - 1) called in that order.

fn from_basis_coefficients_iter<I: ExactSizeIterator<Item = BinaryField16>>( iter: I, ) -> Option<Self>

Fixes a basis for the algebra A and uses this to map DIMENSION F elements to an element of A.

fn reconstitute_from_base(vec: Vec<BinaryField16>) -> Vec<Self>

Convert from a vector of F to a vector of Self by combining the basis coefficients.

fn from_basis_coefficients_slice(slice: &[F]) -> Option<Self>

Fixes a basis for the algebra A and uses this to map DIMENSION F elements to an element of A.

fn ith_basis_element(i: usize) -> Option<Self>

Given a basis for the Algebra A, return the i’th basis element.

impl Clone for BinaryField16

fn clone(&self) -> Self

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Copy for BinaryField16

impl Debug for BinaryField16

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

Formats the value using the given formatter.

impl Default for BinaryField16

fn default() -> Self

Returns the “default value” for a type.

impl<'de> Deserialize<'de> for BinaryField16

fn deserialize<D: Deserializer<'de>>(d: D) -> Result<Self, D::Error>

Deserialize this value from the given Serde deserializer.

impl Display for BinaryField16

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

Formats the value using the given formatter.

impl Distribution<BinaryField16> for StandardUniform

fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> BinaryField16

Generate a random value of T, using rng as the source of randomness.

fn sample_iter<R>(self, rng: R) -> Iter<Self, R, T>

where R: Rng, Self: Sized,

Create an iterator that generates random values of T, using rng as the source of randomness.

fn map<F, S>(self, func: F) -> Map<Self, F, T, S>

where F: Fn(T) -> S, Self: Sized,

Map sampled values to type S

impl Div for BinaryField16

type Output = BinaryField16

The resulting type after applying the / operator.

fn div(self, rhs: BinaryField16) -> Self

Performs the / operation.

impl DivAssign for BinaryField16

fn div_assign(&mut self, rhs: BinaryField16)

Performs the /= operation.

impl Eq for BinaryField16

impl ExtensionField<BinaryField8> for BinaryField16

type ExtensionPacking = BinaryField16

fn is_in_basefield(&self) -> bool

Determine if the given element lies in the base field.

fn as_base(&self) -> Option<BinaryField8>

If the element lies in the base field project it down. Otherwise return None.

fn from_ext_basis_coefficients(coeffs: &[Self]) -> Option<Self>

Reassemble an element of Self from D = DIMENSION coefficients in Self via Σⱼ basisⱼ · coeffsⱼ. Returns None if coeffs.len() != Self::DIMENSION.

impl ExtensionField<BinaryField16> for BinaryField32

type ExtensionPacking = BinaryField32

fn is_in_basefield(&self) -> bool

Determine if the given element lies in the base field.

fn as_base(&self) -> Option<BinaryField16>

If the element lies in the base field project it down. Otherwise return None.

fn from_ext_basis_coefficients(coeffs: &[Self]) -> Option<Self>

Reassemble an element of Self from D = DIMENSION coefficients in Self via Σⱼ basisⱼ · coeffsⱼ. Returns None if coeffs.len() != Self::DIMENSION.

impl ExtensionField<BinaryField16> for BinaryField64

type ExtensionPacking = BinaryField64

fn is_in_basefield(&self) -> bool

Determine if the given element lies in the base field.

fn as_base(&self) -> Option<BinaryField16>

If the element lies in the base field project it down. Otherwise return None.

fn from_ext_basis_coefficients(coeffs: &[Self]) -> Option<Self>

Reassemble an element of Self from D = DIMENSION coefficients in Self via Σⱼ basisⱼ · coeffsⱼ. Returns None if coeffs.len() != Self::DIMENSION.

impl ExtensionField<BinaryField16> for BinaryField128

type ExtensionPacking = BinaryField128

fn is_in_basefield(&self) -> bool

Determine if the given element lies in the base field.

fn as_base(&self) -> Option<BinaryField16>

If the element lies in the base field project it down. Otherwise return None.

fn from_ext_basis_coefficients(coeffs: &[Self]) -> Option<Self>

Reassemble an element of Self from D = DIMENSION coefficients in Self via Σⱼ basisⱼ · coeffsⱼ. Returns None if coeffs.len() != Self::DIMENSION.

impl Field for BinaryField16

fn interpolation_node(i: usize) -> Self

The enumeration by bit pattern: interpolation_node(i) is the element whose coefficients over GF(2) are the bits of i. It is injective for i < 2^BITS.

Panics

Panics if i >= 2^BITS. from_repr masks, so an out-of-range index would otherwise alias an earlier node in release builds and break an interpolation argument silently; the check is paid once per node at setup, never in a loop.

fn bits() -> usize

An element of GF(2^n) is exactly n bits wide.

const GENERATOR: Self

A generator of this field’s multiplicative group.

type Packing = BinaryField16

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

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

fn try_sqrt(&self) -> Option<Self>

A square root of this field element, if one exists.

fn order() -> BigUint

The number of elements in the field.

const BENEFITS_FROM_LOCKSTEP_EVALUATION: bool = false

Whether evaluating multiple packed vectors of this field in lockstep (to overlap independent dependency chains and hide packed-multiplication latency) is expected to help throughput for this field.

fn is_zero(&self) -> bool

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

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 batched_columnwise_dot_product<EF, R, I, const N: usize>( acc: &mut [<EF as ExtensionField<Self>>::ExtensionPacking], items: I, )

where EF: ExtensionField<Self>, R: Iterator<Item = Self::Packing>, I: Iterator<Item = (R, [EF; N])>,

Accumulate acc[c * N + j] += scales[j] * row[c] over a stream of packed rows.

impl From<BinaryField8> for BinaryField16

fn from(x: BinaryField8) -> Self

The lower level occupies the low BITS bits; the other coefficients vanish.

impl From<BinaryField16> for BinaryField32

fn from(x: BinaryField16) -> Self

The lower level occupies the low BITS bits; the other coefficients vanish.

impl From<BinaryField16> for BinaryField64

fn from(x: BinaryField16) -> Self

The lower level occupies the low BITS bits; the other coefficients vanish.

impl From<BinaryField16> for BinaryField128

fn from(x: BinaryField16) -> Self

The lower level occupies the low BITS bits; the other coefficients vanish.

impl From<BinaryField16> for Ghash128

fn from(x: BinaryField16) -> Self

Converts to this type from the input type.

impl From<BinaryField16> for Poly64

fn from(x: BinaryField16) -> Self

Converts to this type from the input type.

impl From<Gf2> for BinaryField16

fn from(x: Gf2) -> Self

GF(2) is the prime subfield of every level, embedded as {ZERO, ONE}.

impl HasFrobenius<BinaryField8> for BinaryField16

fn frobenius(&self) -> Self

x ↦ x^n for n = 2^BITS the order of the base field.

fn repeated_frobenius(&self, count: usize) -> Self

Apply the Frobenius automorphism count times.

fn pseudo_inv(&self) -> Self

Computes the pseudo inverse of the given field element.

fn galois_orbit(self) -> Vec<Self>

Returns the full Galois orbit of the element under Frobenius.

impl HasFrobenius<BinaryField16> for BinaryField32

fn frobenius(&self) -> Self

x ↦ x^n for n = 2^BITS the order of the base field.

fn repeated_frobenius(&self, count: usize) -> Self

Apply the Frobenius automorphism count times.

fn pseudo_inv(&self) -> Self

Computes the pseudo inverse of the given field element.

fn galois_orbit(self) -> Vec<Self>

Returns the full Galois orbit of the element under Frobenius.

impl HasFrobenius<BinaryField16> for BinaryField64

fn frobenius(&self) -> Self

x ↦ x^n for n = 2^BITS the order of the base field.

fn repeated_frobenius(&self, count: usize) -> Self

Apply the Frobenius automorphism count times.

fn pseudo_inv(&self) -> Self

Computes the pseudo inverse of the given field element.

fn galois_orbit(self) -> Vec<Self>

Returns the full Galois orbit of the element under Frobenius.

impl HasFrobenius<BinaryField16> for BinaryField128

fn frobenius(&self) -> Self

x ↦ x^n for n = 2^BITS the order of the base field.

fn repeated_frobenius(&self, count: usize) -> Self

Apply the Frobenius automorphism count times.

fn pseudo_inv(&self) -> Self

Computes the pseudo inverse of the given field element.

fn galois_orbit(self) -> Vec<Self>

Returns the full Galois orbit of the element under Frobenius.

impl Hash for BinaryField16

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

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 Mul for BinaryField16

type Output = BinaryField16

The resulting type after applying the * operator.

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

Performs the * operation.

impl Mul<BinaryField8> for BinaryField16

type Output = BinaryField16

The resulting type after applying the * operator.

fn mul(self, rhs: BinaryField8) -> Self

Performs the * operation.

impl Mul<BinaryField16> for BinaryField8

type Output = BinaryField16

The resulting type after applying the * operator.

fn mul(self, rhs: BinaryField16) -> BinaryField16

Performs the * operation.

impl Mul<BinaryField16> for BinaryField32

type Output = BinaryField32

The resulting type after applying the * operator.

fn mul(self, rhs: BinaryField16) -> Self

Performs the * operation.

impl Mul<BinaryField16> for BinaryField64

type Output = BinaryField64

The resulting type after applying the * operator.

fn mul(self, rhs: BinaryField16) -> Self

Performs the * operation.

impl Mul<BinaryField16> for BinaryField128

type Output = BinaryField128

The resulting type after applying the * operator.

fn mul(self, rhs: BinaryField16) -> Self

Performs the * operation.

impl Mul<BinaryField16> for Ghash128

type Output = Ghash128

The resulting type after applying the * operator.

fn mul(self, rhs: BinaryField16) -> Self

Performs the * operation.

impl Mul<BinaryField16> for Poly64

type Output = Poly64

The resulting type after applying the * operator.

fn mul(self, rhs: BinaryField16) -> Self

Performs the * operation.

impl Mul<BinaryField16> for Gf2

type Output = BinaryField16

The resulting type after applying the * operator.

fn mul(self, rhs: BinaryField16) -> BinaryField16

Performs the * operation.

impl Mul<BinaryField32> for BinaryField16

type Output = BinaryField32

The resulting type after applying the * operator.

fn mul(self, rhs: BinaryField32) -> BinaryField32

Performs the * operation.

impl Mul<BinaryField64> for BinaryField16

type Output = BinaryField64

The resulting type after applying the * operator.

fn mul(self, rhs: BinaryField64) -> BinaryField64

Performs the * operation.

impl Mul<BinaryField128> for BinaryField16

type Output = BinaryField128

The resulting type after applying the * operator.

fn mul(self, rhs: BinaryField128) -> BinaryField128

Performs the * operation.

impl Mul<Gf2> for BinaryField16

type Output = BinaryField16

The resulting type after applying the * operator.

fn mul(self, rhs: Gf2) -> Self

Performs the * operation.

impl Mul<Ghash128> for BinaryField16

type Output = Ghash128

The resulting type after applying the * operator.

fn mul(self, rhs: Ghash128) -> Self::Output

Performs the * operation.

impl Mul<Poly64> for BinaryField16

type Output = Poly64

The resulting type after applying the * operator.

fn mul(self, rhs: Poly64) -> Self::Output

Performs the * operation.

impl<T: Into<Self>> MulAssign<T> for BinaryField16

fn mul_assign(&mut self, rhs: T)

Performs the *= operation.

impl Neg for BinaryField16

type Output = BinaryField16

The resulting type after applying the - operator.

fn neg(self) -> Self::Output

Performs the unary - operation.

impl Ord for BinaryField16

fn cmp(&self, other: &Self) -> 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.

fn clamp_to<R>(self, range: R) -> Self

where Self: Sized, R: ClampBounds<Self>,

🔬This is a nightly-only experimental API. (clamp_to)

Restrict a value to a certain range.

impl Packable for BinaryField16

impl PackedFieldExtension<BinaryField8, BinaryField16> for BinaryField16

A tower level packs one element per vector, so every lane index is 0.

fn from_ext_fn(f: impl Fn(usize) -> Self) -> Self

Construct a packed extension by applying f to each lane.

fn from_ext_slice(slice: &[Self]) -> Self

Pack a length-WIDTH slice of extension field elements into one packed extension.

fn extract(&self, lane: usize) -> Self

Extract the extension field element at the given SIMD lane.

fn add_assign_lane(&mut self, lane: usize, value: Self)

Accumulate value into a single SIMD lane, leaving the other W - 1 lanes unchanged.

fn packed_ext_powers(base: Self) -> Powers<Self>

Similar to packed_powers, construct an iterator which returns powers of base packed into PackedFieldExtension elements.

fn pack_ext_columns<const N: usize>(rows: &[[ExtField; N]]) -> [Self; N]

Pack N columns from W rows of extension field elements into N packed extensions.

fn pack_ext_columns_fn<const N: usize>( row_fn: impl Fn(usize) -> [ExtField; N], ) -> [Self; N]

Pack N columns using a closure that produces each row.

fn to_ext_slice(&self, out: &mut [ExtField])

Write all W lanes into the given slice.

fn unpack_ext_into<const N: usize>( packed: &[Self; N], rows: &mut [[ExtField; N]], )

Unpack N packed extensions into W rows of N extension elements.

fn unpack_ext_iter<const N: usize>( packed: [Self; N], ) -> impl Iterator<Item = [ExtField; N]>

Iterator equivalent of PackedFieldExtension::unpack_ext_into.

fn to_ext_iter( iter: impl IntoIterator<Item = Self>, ) -> impl Iterator<Item = ExtField>

Convert an iterator of packed extension field elements to an iterator of extension field elements (flat — one ExtField per lane per packed value).

fn unpack_transpose_into(src: &[Self], dst: &mut [ExtField], src_width: usize)

Unpacks a packed row-major matrix into its scalar transpose, in one pass.

fn packed_ext_powers_capped( base: ExtField, unpacked_len: usize, ) -> impl Iterator<Item = Self>

Similar to packed_ext_powers but only returns unpacked_len powers of base.

impl PackedFieldExtension<BinaryField16, BinaryField32> for BinaryField32

A tower level packs one element per vector, so every lane index is 0.

fn from_ext_fn(f: impl Fn(usize) -> Self) -> Self

Construct a packed extension by applying f to each lane.

fn from_ext_slice(slice: &[Self]) -> Self

Pack a length-WIDTH slice of extension field elements into one packed extension.

fn extract(&self, lane: usize) -> Self

Extract the extension field element at the given SIMD lane.

fn add_assign_lane(&mut self, lane: usize, value: Self)

Accumulate value into a single SIMD lane, leaving the other W - 1 lanes unchanged.

fn packed_ext_powers(base: Self) -> Powers<Self>

Similar to packed_powers, construct an iterator which returns powers of base packed into PackedFieldExtension elements.

fn pack_ext_columns<const N: usize>(rows: &[[ExtField; N]]) -> [Self; N]

Pack N columns from W rows of extension field elements into N packed extensions.

fn pack_ext_columns_fn<const N: usize>( row_fn: impl Fn(usize) -> [ExtField; N], ) -> [Self; N]

Pack N columns using a closure that produces each row.

fn to_ext_slice(&self, out: &mut [ExtField])

Write all W lanes into the given slice.

fn unpack_ext_into<const N: usize>( packed: &[Self; N], rows: &mut [[ExtField; N]], )

Unpack N packed extensions into W rows of N extension elements.

fn unpack_ext_iter<const N: usize>( packed: [Self; N], ) -> impl Iterator<Item = [ExtField; N]>

Iterator equivalent of PackedFieldExtension::unpack_ext_into.

fn to_ext_iter( iter: impl IntoIterator<Item = Self>, ) -> impl Iterator<Item = ExtField>

Convert an iterator of packed extension field elements to an iterator of extension field elements (flat — one ExtField per lane per packed value).

fn unpack_transpose_into(src: &[Self], dst: &mut [ExtField], src_width: usize)

Unpacks a packed row-major matrix into its scalar transpose, in one pass.

fn packed_ext_powers_capped( base: ExtField, unpacked_len: usize, ) -> impl Iterator<Item = Self>

Similar to packed_ext_powers but only returns unpacked_len powers of base.

impl PackedFieldExtension<BinaryField16, BinaryField64> for BinaryField64

A tower level packs one element per vector, so every lane index is 0.

fn from_ext_fn(f: impl Fn(usize) -> Self) -> Self

Construct a packed extension by applying f to each lane.

fn from_ext_slice(slice: &[Self]) -> Self

Pack a length-WIDTH slice of extension field elements into one packed extension.

fn extract(&self, lane: usize) -> Self

Extract the extension field element at the given SIMD lane.

fn add_assign_lane(&mut self, lane: usize, value: Self)

Accumulate value into a single SIMD lane, leaving the other W - 1 lanes unchanged.

fn packed_ext_powers(base: Self) -> Powers<Self>

Similar to packed_powers, construct an iterator which returns powers of base packed into PackedFieldExtension elements.

fn pack_ext_columns<const N: usize>(rows: &[[ExtField; N]]) -> [Self; N]

Pack N columns from W rows of extension field elements into N packed extensions.

fn pack_ext_columns_fn<const N: usize>( row_fn: impl Fn(usize) -> [ExtField; N], ) -> [Self; N]

Pack N columns using a closure that produces each row.

fn to_ext_slice(&self, out: &mut [ExtField])

Write all W lanes into the given slice.

fn unpack_ext_into<const N: usize>( packed: &[Self; N], rows: &mut [[ExtField; N]], )

Unpack N packed extensions into W rows of N extension elements.

fn unpack_ext_iter<const N: usize>( packed: [Self; N], ) -> impl Iterator<Item = [ExtField; N]>

Iterator equivalent of PackedFieldExtension::unpack_ext_into.

fn to_ext_iter( iter: impl IntoIterator<Item = Self>, ) -> impl Iterator<Item = ExtField>

Convert an iterator of packed extension field elements to an iterator of extension field elements (flat — one ExtField per lane per packed value).

fn unpack_transpose_into(src: &[Self], dst: &mut [ExtField], src_width: usize)

Unpacks a packed row-major matrix into its scalar transpose, in one pass.

fn packed_ext_powers_capped( base: ExtField, unpacked_len: usize, ) -> impl Iterator<Item = Self>

Similar to packed_ext_powers but only returns unpacked_len powers of base.

impl PackedFieldExtension<BinaryField16, BinaryField128> for BinaryField128

A tower level packs one element per vector, so every lane index is 0.

fn from_ext_fn(f: impl Fn(usize) -> Self) -> Self

Construct a packed extension by applying f to each lane.

fn from_ext_slice(slice: &[Self]) -> Self

Pack a length-WIDTH slice of extension field elements into one packed extension.

fn extract(&self, lane: usize) -> Self

Extract the extension field element at the given SIMD lane.

fn add_assign_lane(&mut self, lane: usize, value: Self)

Accumulate value into a single SIMD lane, leaving the other W - 1 lanes unchanged.

fn packed_ext_powers(base: Self) -> Powers<Self>

Similar to packed_powers, construct an iterator which returns powers of base packed into PackedFieldExtension elements.

fn pack_ext_columns<const N: usize>(rows: &[[ExtField; N]]) -> [Self; N]

Pack N columns from W rows of extension field elements into N packed extensions.

fn pack_ext_columns_fn<const N: usize>( row_fn: impl Fn(usize) -> [ExtField; N], ) -> [Self; N]

Pack N columns using a closure that produces each row.

fn to_ext_slice(&self, out: &mut [ExtField])

Write all W lanes into the given slice.

fn unpack_ext_into<const N: usize>( packed: &[Self; N], rows: &mut [[ExtField; N]], )

Unpack N packed extensions into W rows of N extension elements.

fn unpack_ext_iter<const N: usize>( packed: [Self; N], ) -> impl Iterator<Item = [ExtField; N]>

Iterator equivalent of PackedFieldExtension::unpack_ext_into.

fn to_ext_iter( iter: impl IntoIterator<Item = Self>, ) -> impl Iterator<Item = ExtField>

Convert an iterator of packed extension field elements to an iterator of extension field elements (flat — one ExtField per lane per packed value).

fn unpack_transpose_into(src: &[Self], dst: &mut [ExtField], src_width: usize)

Unpacks a packed row-major matrix into its scalar transpose, in one pass.

fn packed_ext_powers_capped( base: ExtField, unpacked_len: usize, ) -> impl Iterator<Item = Self>

Similar to packed_ext_powers but only returns unpacked_len powers of base.

impl PartialEq for BinaryField16

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

Equality operator ==.

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

Inequality operator !=.

impl PartialOrd for BinaryField16

fn partial_cmp(&self, other: &Self) -> 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 PrimeCharacteristicRing for BinaryField16

fn halve(&self) -> Self

Panics

Always panics: 2 is not invertible in characteristic 2.

fn div_2exp_u64(&self, _exp: u64) -> Self

Panics

Always panics: 2 is not invertible in characteristic 2.

fn bool_check(&self) -> Self

x·(x - 1) = x² - x = x² + x in characteristic 2, and squaring is a linear map here, so this avoids the tower/polynomial basis changes a general product pays for.

const ZERO: Self

The additive identity of the ring.

const ONE: Self

The multiplicative identity of the ring.

const TWO: Self

The element in the ring given by ONE + ONE.

const NEG_ONE: Self

The element in the ring given by -ONE.

type PrimeSubfield = Gf2

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

fn from_prime_subfield(f: Self::PrimeSubfield) -> Self

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

fn from_bool(b: bool) -> Self

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

fn double(&self) -> Self

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

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

Computes the arithmetic generalization of boolean xor.

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

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

fn square(&self) -> Self

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

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

Compute the dot product of two vectors.

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.

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 cube(&self) -> Self

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

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

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

impl Product for BinaryField16

fn product<I: Iterator<Item = Self>>(iter: I) -> Self

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

impl RawDataSerializable for BinaryField16

const NUM_BYTES: usize

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; 2]

Convert a field element into a collection of bytes.

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

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

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

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

fn into_parallel_u64_streams<const N: usize>( input: impl IntoIterator<Item = [Self; 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.

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

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

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

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

impl Serialize for BinaryField16

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

where __S: Serializer,

Serialize this value into the given Serde serializer.

impl StructuralPartialEq for BinaryField16

impl Sub for BinaryField16

type Output = BinaryField16

The resulting type after applying the - operator.

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

Performs the - operation.

impl Sub<BinaryField8> for BinaryField16

type Output = BinaryField16

The resulting type after applying the - operator.

fn sub(self, rhs: BinaryField8) -> Self

Performs the - operation.

impl Sub<BinaryField16> for BinaryField8

type Output = BinaryField16

The resulting type after applying the - operator.

fn sub(self, rhs: BinaryField16) -> Self::Output

Performs the - operation.

impl Sub<BinaryField16> for BinaryField32

type Output = BinaryField32

The resulting type after applying the - operator.

fn sub(self, rhs: BinaryField16) -> Self

Performs the - operation.

impl Sub<BinaryField16> for BinaryField64

type Output = BinaryField64

The resulting type after applying the - operator.

fn sub(self, rhs: BinaryField16) -> Self

Performs the - operation.

impl Sub<BinaryField16> for BinaryField128

type Output = BinaryField128

The resulting type after applying the - operator.

fn sub(self, rhs: BinaryField16) -> Self

Performs the - operation.

impl Sub<BinaryField16> for Ghash128

type Output = Ghash128

The resulting type after applying the - operator.

fn sub(self, rhs: BinaryField16) -> Self

Performs the - operation.

impl Sub<BinaryField16> for Poly64

type Output = Poly64

The resulting type after applying the - operator.

fn sub(self, rhs: BinaryField16) -> Self

Performs the - operation.

impl Sub<BinaryField16> for Gf2

type Output = BinaryField16

The resulting type after applying the - operator.

fn sub(self, rhs: BinaryField16) -> Self::Output

Performs the - operation.

impl Sub<BinaryField32> for BinaryField16

type Output = BinaryField32

The resulting type after applying the - operator.

fn sub(self, rhs: BinaryField32) -> Self::Output

Performs the - operation.

impl Sub<BinaryField64> for BinaryField16

type Output = BinaryField64

The resulting type after applying the - operator.

fn sub(self, rhs: BinaryField64) -> Self::Output

Performs the - operation.

impl Sub<BinaryField128> for BinaryField16

type Output = BinaryField128

The resulting type after applying the - operator.

fn sub(self, rhs: BinaryField128) -> Self::Output

Performs the - operation.

impl Sub<Gf2> for BinaryField16

type Output = BinaryField16

The resulting type after applying the - operator.

fn sub(self, rhs: Gf2) -> Self

Performs the - operation.

impl Sub<Ghash128> for BinaryField16

type Output = Ghash128

The resulting type after applying the - operator.

fn sub(self, rhs: Ghash128) -> Self::Output

Performs the - operation.

impl Sub<Poly64> for BinaryField16

type Output = Poly64

The resulting type after applying the - operator.

fn sub(self, rhs: Poly64) -> Self::Output

Performs the - operation.

impl<T: Into<Self>> SubAssign<T> for BinaryField16

fn sub_assign(&mut self, rhs: T)

Performs the -= operation.

impl Sum for BinaryField16

fn sum<I: Iterator<Item = Self>>(iter: I) -> Self

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

impl TowerLevel for BinaryField16

fn mul_alpha(self) -> Self

X·(c0 + c1·X) = c1 + (c0 + α·c1)·X, using X² = αX + 1.

const LOG_BITS: usize

The k in GF(2^(2^k)).

type Repr = u16

The backing integer of an element.

fn from_repr(r: Self::Repr) -> Self

Build an element from a bit pattern, discarding the bits above 2^LOG_BITS.

fn to_repr(self) -> Self::Repr

The bit pattern of this element, zero above 2^LOG_BITS.

fn from_le_byte_iter(bytes: impl Iterator<Item = u8>) -> Self

Build an element from the next RawDataSerializable::NUM_BYTES bytes of a little-endian byte stream, discarding the bits above 2^LOG_BITS.

fn cantor_basis(i: usize) -> Self

The i-th vector of the Cantor basis of this level.

impl TranscriptField for BinaryField16

fn algebra_tag(degree: usize, basis: [u8; 32]) -> TypeTag

Stable identity of an algebra with degree coefficients over this field. basis is Keccak-256 of its p3_field::AlgebraIdentity bytes and must be included unchanged in the returned tag.

fn observe_seed<C: CanObserve<Self>>(challenger: &mut C, bytes: &[u8])

Absorb a byte string injectively into this field’s sponge alphabet.

fn wire_len() -> usize

Byte length of the canonical encoding of one element.

fn encode(value: &Self, out: &mut Vec<u8>)

Append exactly Self::wire_len() bytes identifying this element uniquely.

fn decode(bytes: &[u8]) -> Result<Self, TranscriptError>

Decode the first Self::wire_len() bytes, rejecting short or noncanonical inputs.