class galois.GF2(galois.FieldArray)

A FieldArray subclass over $$\mathrm{GF}(2)$$.

Info

This class is a pre-generated FieldArray subclass generated with galois.GF(2) and is included in the API for convenience.

Examples

This class is equivalent, and in fact identical, to the FieldArray subclass returned from the class factory GF().

In : galois.GF2 is galois.GF(2)
Out: True

In : issubclass(galois.GF2, galois.FieldArray)
Out: True

In : print(galois.GF2.properties)
Galois Field:
name: GF(2)
characteristic: 2
degree: 1
order: 2
irreducible_poly: x + 1
is_primitive_poly: True
primitive_element: 1


Create a FieldArray instance using GF2’s constructor.

In : x = galois.GF2([1, 0, 1, 1]); x
Out: GF([1, 0, 1, 1], order=2)

In : isinstance(x, galois.GF2)
Out: True


## Constructors¶

GF2(x: , dtype: = None, ...)

Creates an array over $$\mathrm{GF}(p^m)$$.

classmethod Identity(size: int, ...) Self

Creates an $$n \times n$$ identity matrix.

classmethod Ones(...) Self

Creates an array of all ones.

classmethod Random(shape: ShapeLike = (), ...) Self

Creates an array with random elements.

classmethod Range(stop, ...) Self

Creates a 1-D array with a range of elements.

classmethod Vandermonde(rows, ...) Self

Creates an $$m \times n$$ Vandermonde matrix of $$a \in \mathrm{GF}(q)$$.

classmethod Zeros(...) Self

Creates an array of all zeros.

## Conversions¶

classmethod Vector(...)

Converts length-$$m$$ vectors over the prime subfield $$\mathrm{GF}(p)$$ to an array over $$\mathrm{GF}(p^m)$$.

vector(dtype: = None)

Converts an array over $$\mathrm{GF}(p^m)$$ to length-$$m$$ vectors over the prime subfield $$\mathrm{GF}(p)$$.

## Elements¶

class property elements : FieldArray

All of the finite field’s elements $$\{0, \dots, p^m-1\}$$.

class property non_squares : FieldArray

All non-squares in the Galois field.

class property primitive_element : FieldArray

A primitive element $$\alpha$$ of the Galois field $$\mathrm{GF}(p^m)$$.

class property primitive_elements : FieldArray

All primitive elements $$\alpha$$ of the Galois field $$\mathrm{GF}(p^m)$$.

classmethod primitive_root_of_unity(n: int) Self

Finds a primitive $$n$$-th root of unity in the finite field.

classmethod primitive_roots_of_unity(n: int) Self

Finds all primitive $$n$$-th roots of unity in the finite field.

class property squares : FieldArray

All squares in the finite field.

class property units : FieldArray

All of the finite field’s units $$\{1, \dots, p^m-1\}$$.

## String representation¶

__repr__() str

Displays the array specifying the class and finite field order.

__str__() str

Displays the array without specifying the class or finite field order.

classmethod arithmetic_table(...) str

Generates the specified arithmetic table for the finite field.

class property properties : str

A formatted string of relevant properties of the Galois field.

classmethod repr_table(...) str

Generates a finite field element representation table comparing the power, polynomial, vector, and integer representations.

## Element representation¶

class property element_repr : Literal[int] | Literal[poly] | Literal[power]

The current finite field element representation.

classmethod repr(...)

Sets the element representation for all arrays from this FieldArray subclass.

## Arithmetic compilation¶

class property default_ufunc_mode : Literal[jit - lookup] | typing.Literal[jit - calculate] | typing.Literal[python - calculate]

The default ufunc compilation mode for this FieldArray subclass.

class property dtypes : list[np.dtype]

List of valid integer numpy.dtype values that are compatible with this finite field.

class property ufunc_mode : Literal[jit - lookup] | typing.Literal[jit - calculate] | typing.Literal[python - calculate]

The current ufunc compilation mode for this FieldArray subclass.

class property ufunc_modes : list[str]

All supported ufunc compilation modes for this FieldArray subclass.

classmethod compile(mode)

Recompile the just-in-time compiled ufuncs for a new calculation mode.

## Methods¶

Computes the additive order of each element in $$x$$.

characteristic_poly() Poly

Computes the characteristic polynomial of a finite field element $$a$$ or a square matrix $$\mathbf{A}$$.

field_norm()

Computes the field norm $$\mathrm{N}_{L / K}(x)$$ of the elements of $$x$$.

field_trace()

Computes the field trace $$\mathrm{Tr}_{L / K}(x)$$ of the elements of $$x$$.

log(base: = None)

Computes the discrete logarithm of the array $$x$$ base $$\beta$$.

minimal_poly() Poly

Computes the minimal polynomial of a finite field element $$a$$.

multiplicative_order()

Computes the multiplicative order $$\textrm{ord}(x)$$ of each element in $$x$$.

## Linear algebra¶

column_space() Self

Computes the column space of the matrix $$\mathbf{A}$$.

left_null_space() Self

Computes the left null space of the matrix $$\mathbf{A}$$.

lu_decompose() tuple[Self, Self]

Decomposes the input array into the product of lower and upper triangular matrices.

null_space() Self

Computes the null space of the matrix $$\mathbf{A}$$.

plu_decompose() tuple[Self, Self, Self]

Decomposes the input array into the product of lower and upper triangular matrices using partial pivoting.

row_reduce(ncols: = None, ...) Self

Performs Gaussian elimination on the matrix to achieve reduced row echelon form (RREF).

row_space() Self

Computes the row space of the matrix $$\mathbf{A}$$.

## Properties¶

class property characteristic : int

The prime characteristic $$p$$ of the Galois field $$\mathrm{GF}(p^m)$$.

class property degree : int

The extension degree $$m$$ of the Galois field $$\mathrm{GF}(p^m)$$.

class property irreducible_poly : Poly

The irreducible polynomial $$f(x)$$ of the Galois field $$\mathrm{GF}(p^m)$$.

class property name : str

The finite field’s name as a string GF(p) or GF(p^m).

class property order : int

The order $$p^m$$ of the Galois field $$\mathrm{GF}(p^m)$$.

class property prime_subfield :

The prime subfield $$\mathrm{GF}(p)$$ of the extension field $$\mathrm{GF}(p^m)$$.

## Attributes¶

class property is_extension_field : bool

Indicates if the finite field is an extension field, having prime power order.

class property is_prime_field : bool

Indicates if the finite field is a prime field, having prime order.

class property is_primitive_poly : bool

Indicates whether the irreducible_poly is a primitive polynomial.

is_square()

Determines if the elements of $$x$$ are squares in the finite field.