galois.GF2

class galois.GF2(array, dtype=None, copy=True, order='K', ndmin=0)

Creates an array over \(\mathrm{GF}(2)\).

This class is a pre-generated galois.FieldArray subclass generated with galois.GF(2) and is included in the API for convenience. See galois.FieldArray and galois.FieldClass for more complete documentation and examples.

Parameters

• array (int, str, tuple, list, numpy.ndarray, galois.FieldArray) –

  The input array-like object to be converted to a Galois field array. See the examples section for demonstations of array creation using each input type. See see galois.FieldClass.display() and galois.FieldClass.display_mode for a description of the “integer” and “polynomial” representation of Galois field elements.

  • int: A single integer, which is the “integer representation” of a Galois field element, creates a 0-D array.

  • str: A single string, which is the “polynomial representation” of a Galois field element, creates a 0-D array.

  • tuple, list: A list or tuple (or nested lists/tuples) of ints or strings (which can be mix-and-matched) creates an array of Galois field elements from their integer or polynomial representations.

  • numpy.ndarray, galois.FieldArray: An array of ints creates a copy of the array over this specific field.

• dtype (numpy.dtype, optional) – The numpy.dtype of the array elements. The default is None which represents the smallest unsigned dtype for this class, i.e. the first element in galois.FieldClass.dtypes.

• copy (bool, optional) – The copy keyword argument from numpy.array(). The default is True which makes a copy of the input array.

• order (str, optional) – The order keyword argument from numpy.array(). Valid values are "K" (default), "A", "C", or "F".

• ndmin (int, optional) – The ndmin keyword argument from numpy.array(). The minimum number of dimensions of the output. The default is 0.

Returns

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

Return type

galois.FieldArray

Examples

This class is equivalent (and, in fact, identical) to the class returned from the Galois field class constructor.

In [1]: print(galois.GF2)
<class 'numpy.ndarray over GF(2)'>

In [2]: GF2 = galois.GF(2); print(GF2)
<class 'numpy.ndarray over GF(2)'>

In [3]: GF2 is galois.GF2
Out[3]: True

The Galois field properties can be viewed by class attributes, see galois.FieldClass.

# View a summary of the field's properties
In [4]: print(galois.GF2.properties)
GF(2):
  characteristic: 2
  degree: 1
  order: 2
  irreducible_poly: x + 1
  is_primitive_poly: True
  primitive_element: 1

# Or access each attribute individually
In [5]: galois.GF2.irreducible_poly
Out[5]: Poly(x + 1, GF(2))

In [6]: galois.GF2.is_prime_field
Out[6]: True

The class’s constructor mimics the call signature of numpy.array().

# Construct a Galois field array from an iterable
In [7]: galois.GF2([1,0,1,1,0,0,0,1])
Out[7]: GF([1, 0, 1, 1, 0, 0, 0, 1], order=2)

# Or an iterable of iterables
In [8]: galois.GF2([[1,0], [1,1]])
Out[8]: 
GF([[1, 0],
    [1, 1]], order=2)

# Or a single integer
In [9]: galois.GF2(1)
Out[9]: GF(1, order=2)

classmethod Elements(dtype=None)

Creates a 1-D 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 unsigned dtype for this class, i.e. the first element in galois.FieldClass.dtypes.

Returns

A 1-D Galois field array of all the field’s elements.

Return type

galois.FieldArray

Examples

In [10]: GF = galois.GF(2**4)

In [11]: GF.Elements()
Out[11]: 
GF([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15],
   order=2^4)

As usual, Galois field elements can be displayed in either the “integer” (default), “polynomial”, or “power” representation. This can be changed by calling galois.FieldClass.display().

# Permanently set the display mode to "poly"
In [12]: GF.display("poly");

In [13]: GF.Elements()
Out[13]: 
GF([0, 1, α, α + 1, α^2, α^2 + 1, α^2 + α, α^2 + α + 1, α^3, α^3 + 1,
    α^3 + α, α^3 + α + 1, α^3 + α^2, α^3 + α^2 + 1, α^3 + α^2 + α,
    α^3 + α^2 + α + 1], order=2^4)

# Temporarily set the display mode to "power"
In [14]: with GF.display("power"):
   ....:     print(GF.Elements())
   ....: 
GF([0, 1, α, α^4, α^2, α^8, α^5, α^10, α^3, α^14, α^9, α^7, α^6, α^13,
    α^11, α^12], order=2^4)

# Reset the display mode to "int"
In [15]: GF.display();

classmethod Identity(size, dtype=None)

Creates an \(n \times n\) Galois field identity matrix.

Parameters

• size (int) – The size \(n\) along one axis of the matrix. The resulting array has shape (size, size).

• dtype (numpy.dtype, optional) – The numpy.dtype of the array elements. The default is None which represents the smallest unsigned dtype for this class, i.e. the first element in galois.FieldClass.dtypes.

Returns

A Galois field identity matrix of shape (size, size).

Return type

galois.FieldArray

Examples

In [16]: GF = galois.GF(31)

In [17]: GF.Identity(4)
Out[17]: 
GF([[1, 0, 0, 0],
    [0, 1, 0, 0],
    [0, 0, 1, 0],
    [0, 0, 0, 1]], order=31)

classmethod Ones(shape, dtype=None)

Creates a Galois field array with all ones.

Parameters

• shape (int, 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-D array. A 2-tuple, e.g. (M,N), represents a 2-D 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 unsigned dtype for this class, i.e. the first element in galois.FieldClass.dtypes.

Returns

A Galois field array of ones.

Return type

galois.FieldArray

Examples

In [18]: GF = galois.GF(31)

In [19]: GF.Ones((2,5))
Out[19]: 
GF([[1, 1, 1, 1, 1],
    [1, 1, 1, 1, 1]], order=31)

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

Creates a Galois field array with random field elements.

Parameters

• shape (int, 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-D array. A 2-tuple, e.g. (M,N), represents a 2-D array with each element indicating the size in each dimension.

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

• high (int, optional) – The highest value (exclusive) of a random field element in its integer representation. 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 unsigned dtype for this class, i.e. the first element in galois.FieldClass.dtypes.

Returns

A Galois field array of random field elements.

Return type

galois.FieldArray

Examples

In [20]: GF = galois.GF(31)

In [21]: GF.Random((2,5))
Out[21]: 
GF([[18, 10,  9,  4, 20],
    [12, 25,  6, 20, 12]], order=31)

classmethod Range(start, stop, step=1, dtype=None)

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

Parameters

• start (int) – The starting Galois field value (inclusive) in its integer representation.

• stop (int) – The stopping Galois field value (exclusive) in its integer representation.

• step (int, optional) – The space between values. The default is 1.

• dtype (numpy.dtype, optional) – The numpy.dtype of the array elements. The default is None which represents the smallest unsigned dtype for this class, i.e. the first element in galois.FieldClass.dtypes.

Returns

A 1-D Galois field array of a range of field elements.

Return type

galois.FieldArray

Examples

In [22]: GF = galois.GF(31)

In [23]: GF.Range(10,20)
Out[23]: GF([10, 11, 12, 13, 14, 15, 16, 17, 18, 19], order=31)

classmethod Vandermonde(a, m, n, dtype=None)

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

Parameters

• a (int, galois.FieldArray) – An element of \(\mathrm{GF}(p^m)\).

• m (int) – The number of rows in the Vandermonde matrix.

• n (int) – The number of columns in the Vandermonde matrix.

• dtype (numpy.dtype, optional) – The numpy.dtype of the array elements. The default is None which represents the smallest unsigned dtype for this class, i.e. the first element in galois.FieldClass.dtypes.

Returns

The \(m \times n\) Vandermonde matrix.

Return type

galois.FieldArray

Examples

In [24]: GF = galois.GF(2**3)

In [25]: a = GF.primitive_element

In [26]: V = GF.Vandermonde(a, 7, 7)

In [27]: with GF.display("power"):
   ....:     print(V)
   ....: 
GF([[  1,   1,   1,   1,   1,   1,   1],
    [  1,   α, α^2, α^3, α^4, α^5, α^6],
    [  1, α^2, α^4, α^6,   α, α^3, α^5],
    [  1, α^3, α^6, α^2, α^5,   α, α^4],
    [  1, α^4,   α, α^5, α^2, α^6, α^3],
    [  1, α^5, α^3,   α, α^6, α^4, α^2],
    [  1, α^6, α^5, α^4, α^3, α^2,   α]], order=2^3)

classmethod Vector(array, dtype=None)

Creates a Galois field array over \(\mathrm{GF}(p^m)\) from length-\(m\) vectors over the prime subfield \(\mathrm{GF}(p)\).

This function is the inverse operation of the vector() method.

Parameters

• array (array_like) – The input array with field elements in \(\mathrm{GF}(p)\) to be converted to a Galois field array in \(\mathrm{GF}(p^m)\). The last dimension of the input array must be \(m\). An input array with shape (n1, n2, m) has output shape (n1, n2). By convention, the vectors are ordered from highest degree to 0-th degree.

• dtype (numpy.dtype, optional) – The numpy.dtype of the array elements. The default is None which represents the smallest unsigned dtype for this class, i.e. the first element in galois.FieldClass.dtypes.

Returns

A Galois field array over \(\mathrm{GF}(p^m)\).

Return type

galois.FieldArray

Examples

In [28]: GF = galois.GF(2**6)

In [29]: vec = galois.GF2.Random((3,6)); vec
Out[29]: 
GF([[0, 1, 1, 0, 0, 1],
    [0, 0, 0, 1, 1, 1],
    [0, 0, 1, 0, 1, 1]], order=2)

In [30]: a = GF.Vector(vec); a
Out[30]: GF([25,  7, 11], order=2^6)

In [31]: with GF.display("poly"):
   ....:     print(a)
   ....: 
GF([α^4 + α^3 + 1, α^2 + α + 1, α^3 + α + 1], order=2^6)

In [32]: a.vector()
Out[32]: 
GF([[0, 1, 1, 0, 0, 1],
    [0, 0, 0, 1, 1, 1],
    [0, 0, 1, 0, 1, 1]], order=2)

classmethod Zeros(shape, dtype=None)

Creates a Galois field array with all zeros.

Parameters

• shape (int, 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-D array. A 2-tuple, e.g. (M,N), represents a 2-D 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 unsigned dtype for this class, i.e. the first element in galois.FieldClass.dtypes.

Returns

A Galois field array of zeros.

Return type

galois.FieldArray

Examples

In [33]: GF = galois.GF(31)

In [34]: GF.Zeros((2,5))
Out[34]: 
GF([[0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0]], order=31)