Finite Fields

Module Contents

Classes

`Field`	A type protocol for defining fields.
`PrimeField`	Superclass for integers modulo a prime. Not intended to be used
`GaloisField`	Superclass for defining finite fields. Not intended to be used

Attributes

`F`
`T`
`U`

Module Details

F

F

F = TypeVar("F", bound="Field")

Field

A type protocol for defining fields.

class Field

Bases: typing_extensions.Protocol

__slots__ = []

classmethod zero() → F 

classmethod from_int(n: int) → F 

__radd__(left: F) → F 

__add__(right: F) → F 

__iadd__(right: F) → F 

__sub__(right: F) → F 

__rsub__(left: F) → F 

__mul__(right: F) → F 

__rmul__(left: F) → F 

__imul__(right: F) → F 

__pow__(exponent: int) → F 

__ipow__(right: int) → F 

__neg__() → F 

__truediv__(right: F) → F 

T

T

T = TypeVar("T", bound="PrimeField")

PrimeField

Superclass for integers modulo a prime. Not intended to be used directly, but rather to be subclassed.

class PrimeField

Bases: int, Field

__slots__ = []

PRIME :int

__floordiv__

__rfloordiv__

__ifloordiv__

__divmod__

__rdivmod__

__rpow__

__and__

__or__

__xor__

__rxor__

__ixor__

__rshift__

__lshift__

__irshift__

__ilshift__

classmethod from_be_bytes(buffer: ethereum.base_types.Bytes) → T 

Converts a sequence of bytes into a element of the field. :param buffer: Bytes to decode.

Returns: self – Unsigned integer decoded from buffer.
Return type: T

classmethod zero() → T 

classmethod from_int(n: int) → T 

__radd__(left: T) → T : Return value+self.

__add__(right: T) → T : Return self+value.

__iadd__(right: T) → T 

__sub__(right: T) → T : Return self-value.

__rsub__(left: T) → T : Return value-self.

__mul__(right: T) → T : Return self*value.

__rmul__(left: T) → T : Return value*self.

__imul__(right: T) → T 

__pow__(exponent: int) → T : Return pow(self, value, mod).

__ipow__(right: int) → T 

__neg__() → T : -self

__truediv__(right: T) → T : Return self/value.

multiplicative_inverse() → T 

to_be_bytes32() → ethereum.base_types.Bytes32 : Converts this arbitrarily sized unsigned integer into its big endian representation with exactly 32 bytes. :returns: big_endian – Big endian (most significant bits first) representation. :rtype: Bytes32

U

U

U = TypeVar("U", bound="GaloisField")

GaloisField

Superclass for defining finite fields. Not intended to be used directly, but rather to be subclassed.

Fields are represented as F_p[x]/(x^n + …) where the MODULUS is a tuple of the non-leading coefficients of the defining polynomial. For example x^3 + 2x^2 + 3x + 4 is (2, 3, 4).

In practice the polynomial is likely to be be sparse and you should overload the __mul__() function to take advantage of this fact.

class GaloisField

Bases: tuple, Field

__slots__ = []

PRIME :int

MODULUS :Tuple[int, Ellipsis]

FROBENIUS_COEFFICIENTS :Tuple[GaloisField, Ellipsis]

classmethod zero() → U 

classmethod from_int(n: int) → U 

__add__(right: U) → U : Return self+value.

__radd__(left: U) → U 

__iadd__(right: U) → U 

__sub__(right: U) → U 

__rsub__(left: U) → U 

__mul__(right: U) → U : Return self*value.

__rmul__(left: U) → U : Return value*self.

__imul__(right: U) → U 

__truediv__(right: U) → U 

__neg__() → U 

scalar_mul(x: int) → U : Multiply a field element by a integer. This is faster than using from_int() and field multiplication.

deg() → int: This is a support function for multiplicative_inverse().

multiplicative_inverse() → U : Calculate the multiplicative inverse. Uses the Euclidian algorithm.

__pow__(exponent: int) → U 

classmethod calculate_frobenius_coefficients() → Tuple[U, Ellipsis]: Calculate the coefficients needed by frobenius().

frobenius() → U : Returns self ** p. This function is known as the Frobenius endomorphism and has many special mathematical properties. In particular it is extremely cheap to compute compared to other exponentiations.