galois.GF2

class galois.GF2(array, dtype=None)[source]

Bases: galois.gf.GF, galois.gf.GFArray

Galois field array class for \(\mathrm{GF}(2)\) fields.

Parameters

  • array (array_like) – The input array to be converted to a Galois field array. The input array is copied, so the original array is unmodified by changes to the Galois field array. Valid input array types are numpy.ndarray, list, tuple, or int.

  • dtype (numpy.dtype, optional) – The numpy.dtype of the array elements. The default is numpy.int64.

Returns

The copied input array as a \(\mathrm{GF}(2)\) field array.

Return type

galois.GF2

Examples

GF2 class properties

In [10]: print(galois.GF2)
<Galois Field: GF(2^1), prim_poly = x + 1 (3 decimal)>

In [11]: galois.GF2.characteristic
Out[11]: 2

In [12]: galois.GF2.degree
Out[12]: 1

In [13]: galois.GF2.order
Out[13]: 2

In [14]: galois.GF2.prim_poly
Out[14]: Poly(x + 1, GF2)

Construct arrays in GF2

In [15]: a = galois.GF2([1,0,1,1]); a
Out[15]: GF([1, 0, 1, 1], order=2)

In [16]: b = galois.GF2([1,1,1,1]); b
Out[16]: GF([1, 1, 1, 1], order=2)

Arithmetic with GF2 arrays

# Element-wise addition
In [1]: a + b
Out[1]: GF([0, 1, 0, 0], order=2)

# Element-wise subtraction
In [2]: a - b
Out[2]: GF([0, 1, 0, 0], order=2)

# Element-wise multiplication
In [3]: a * b
Out[3]: GF([1, 0, 1, 1], order=2)

# Element-wise division
In [4]: a / b
Out[4]: GF([1, 0, 1, 1], order=2)
classmethod Elements(dtype=None)

Create a Galois field array of the field’s elements \(\{0, \dots, p^m-1\}\).

Parameters

dtype (numpy.dtype, optional) – The numpy.dtype of the array elements. The default is None which represents the smallest valid dtype for this field class, i.e. cls.dtypes[0].

classmethod Ones(shape, dtype=None)

Create a Galois field array with all ones.

Parameters

  • shape (tuple) – A numpy-compliant shape tuple, see numpy.ndarray.shape. An empty tuple () represents a scalar. A single integer or 1-tuple, e.g. N or (N,), represents the size of a 1-dim array. An n-tuple, e.g. (M,N), represents an n-dim array with each element indicating the size in each dimension.

  • dtype (numpy.dtype, optional) – The numpy.dtype of the array elements. The default is None which represents the smallest valid dtype for this field class, i.e. cls.dtypes[0].

classmethod Random(shape=(), low=0, high=None, dtype=None)

Create a Galois field array with random field elements.

Parameters

  • shape (tuple) – A numpy-compliant shape tuple, see numpy.ndarray.shape. An empty tuple () represents a scalar. A single integer or 1-tuple, e.g. N or (N,), represents the size of a 1-dim array. An n-tuple, e.g. (M,N), represents an n-dim array with each element indicating the size in each dimension.

  • low (int, optional) – The lowest value (inclusive) of a random field element. The default is 0.

  • high (int, optional) – The highest value (exclusive) of a random field element. The default is None which represents the field’s order \(p^m\).

  • dtype (numpy.dtype, optional) – The numpy.dtype of the array elements. The default is None which represents the smallest valid dtype for this field class, i.e. cls.dtypes[0].

classmethod Zeros(shape, dtype=None)

Create a Galois field array with all zeros.

Parameters

  • shape (tuple) – A numpy-compliant shape tuple, see numpy.ndarray.shape. An empty tuple () represents a scalar. A single integer or 1-tuple, e.g. N or (N,), represents the size of a 1-dim array. An n-tuple, e.g. (M,N), represents an n-dim array with each element indicating the size in each dimension.

  • dtype (numpy.dtype, optional) – The numpy.dtype of the array elements. The default is None which represents the smallest valid dtype for this field class, i.e. cls.dtypes[0].

classmethod display(mode='int', poly_var='x')

Sets the printing mode for arrays.

Parameters

  • mode (str, optional) – The field element display mode, either "int" (default) or "poly".

  • poly_var (str, optional) – The polynomial representation’s variable. The default is "x".

Examples

In [1]: GF = galois.GF_factory(2, 3)

In [2]: a = GF.Random(4); a
Out[2]: GF([0, 7, 1, 7], order=2^3)

In [3]: GF.display("poly"); a
Out[3]: GF([0, x^2 + x + 1, 1, x^2 + x + 1], order=2^3)

In [4]: GF.display("poly", "r"); a
Out[4]: GF([0, r^2 + r + 1, 1, r^2 + r + 1], order=2^3)

# Reset the print mode
In [5]: GF.display(); a
Out[5]: GF([0, 7, 1, 7], order=2^3)
classmethod target(target)[source]

Retarget the just-in-time compiled numba ufuncs.

Parameters

target (str) – The target keyword argument from numba.vectorize, either "cpu", "parallel", or "cuda".

alpha = GF(1, order=2)

The primitive element of the Galois field \(\mathrm{GF}(p^m)\). The primitive element is a root of the primitive polynomial \(p(x)\), such that \(p(\alpha) = 0\). The primitive element is also a multiplicative generator of the field, such that \(\mathrm{GF}(p^m) = \{0, 1, \alpha^1, \alpha^2, \dots, \alpha^{p^m - 2}\}\).

Type

int

characteristic = 2

The prime characteristic \(p\) of the Galois field \(\mathrm{GF}(p^m)\). Adding \(p\) copies of any element will always result in \(0\).

Type

int

degree = 1

The prime characteristic’s degree \(m\) of the Galois field \(\mathrm{GF}(p^m)\). The degree is a positive integer.

Type

int

dtypes = [<class 'numpy.uint8'>, <class 'numpy.uint16'>, <class 'numpy.uint32'>, <class 'numpy.int8'>, <class 'numpy.int16'>, <class 'numpy.int32'>, <class 'numpy.int64'>]

List of valid integer numpy.dtype objects that are compatible with this Galois field array class. Valid data types are signed and unsinged integers that can represent decimal values in \([0, p^m)\).

Type

list

order = 2

The order \(p^m\) of the Galois field \(\mathrm{GF}(p^m)\). The order of the field is also equal to the field’s size.

Type

int

prim_poly = Poly(x + 1, GF2)

The primitive polynomial \(p(x)\) of the Galois field \(\mathrm{GF}(p^m)\). The primitive polynomial is of degree \(m\) in \(\mathrm{GF}(p)[x]\).

Type

galois.Poly

ufunc_mode = 'calculate'

The mode for ufunc compilation, either "lookup" or "calculate".

Type

str

ufunc_target = 'cpu'

The numba target for the JIT-compiled ufuncs, either "cpu", "parallel", or "cuda".

Type

str