np

Documentation of some native numpy functions when called on Galois field arrays.

General

np.copy(a)	Returns a copy of a given Galois field array.
np.concatenate(arrays[, axis])	Concatenates the input arrays along the given axis.
np.insert(array, object, values[, axis])	Inserts values along the given axis.

Arithmetic

np.add(x, y)	Adds two Galois field arrays element-wise.
np.subtract(x, y)	Subtracts two Galois field arrays element-wise.
np.multiply(x, y)	Multiplies two Galois field arrays element-wise.
np.divide(x, y)	Divides two Galois field arrays element-wise.
np.negative(x)	Returns the element-wise additive inverse of a Galois field array.
np.reciprocal(x)	Returns the element-wise multiplicative inverse of a Galois field array.
np.power(x, y)	Exponentiates a Galois field array element-wise.
np.square(x)	Squares a Galois field array element-wise.
np.log(x)	Computes the logarithm (base GF.primitive_element) of a Galois field array element-wise.
np.matmul(x1, x2)	Computes the matrix multiplication of two Galois field arrays.

Linear Algebra

np.trace(x)	Returns the sum along the diagonal of a Galois field array.
np.matmul(x1, x2)	Computes the matrix multiplication of two Galois field arrays.
np.linalg.matrix_rank(x)	Returns the rank of a Galois field matrix.
np.linalg.matrix_power(x)	Raises a square Galois field matrix to an integer power.
np.linalg.det(A)	Computes the determinant of the matrix.
np.linalg.inv(A)	Computes the inverse of the matrix.
np.linalg.solve(x)	Solves the system of linear equations.

np.add(x, y)

Adds two Galois field arrays element-wise.

References

• https://numpy.org/doc/stable/reference/generated/numpy.add.html

Examples

In [1]: GF = galois.GF(31)

In [2]: x = GF.Random(10); x
Out[2]: GF([25, 23, 29, 14, 16, 10, 17,  4, 27, 13], order=31)

In [3]: y = GF.Random(10); y
Out[3]: GF([28,  4, 15,  6,  6, 26, 26,  6, 24, 30], order=31)

In [4]: np.add(x, y)
Out[4]: GF([22, 27, 13, 20, 22,  5, 12, 10, 20, 12], order=31)

In [5]: x + y
Out[5]: GF([22, 27, 13, 20, 22,  5, 12, 10, 20, 12], order=31)

np.concatenate(arrays, axis=0)

Concatenates the input arrays along the given axis.

See: https://numpy.org/doc/stable/reference/generated/numpy.concatenate.html

Examples

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

In [2]: A = GF.Random((2,2)); A
Out[2]: 
GF([[4, 5],
    [3, 7]], order=2^3)

In [3]: B = GF.Random((2,2)); B
Out[3]: 
GF([[1, 0],
    [5, 5]], order=2^3)

In [4]: np.concatenate((A,B), axis=0)
Out[4]: 
GF([[4, 5],
    [3, 7],
    [1, 0],
    [5, 5]], order=2^3)

In [5]: np.concatenate((A,B), axis=1)
Out[5]: 
GF([[4, 5, 1, 0],
    [3, 7, 5, 5]], order=2^3)

np.divide(x, y)

Divides two Galois field arrays element-wise.

References

• https://numpy.org/doc/stable/reference/generated/numpy.divide.html

Examples

In [1]: GF = galois.GF(31)

In [2]: x = GF.Random(10); x
Out[2]: GF([10, 29, 19, 30, 28, 16,  0, 18,  1,  0], order=31)

In [3]: y = GF.Random(10, low=1); y
Out[3]: GF([30, 30, 27,  1,  6,  9, 30, 25,  1, 19], order=31)

In [4]: z = np.divide(x, y); z
Out[4]: GF([21,  2,  3, 30, 15, 19,  0, 28,  1,  0], order=31)

In [5]: y * z
Out[5]: GF([10, 29, 19, 30, 28, 16,  0, 18,  1,  0], order=31)

In [6]: np.true_divide(x, y)
Out[6]: GF([21,  2,  3, 30, 15, 19,  0, 28,  1,  0], order=31)

In [7]: x / y
Out[7]: GF([21,  2,  3, 30, 15, 19,  0, 28,  1,  0], order=31)

In [8]: np.floor_divide(x, y)
Out[8]: GF([21,  2,  3, 30, 15, 19,  0, 28,  1,  0], order=31)

In [9]: x // y
Out[9]: GF([21,  2,  3, 30, 15, 19,  0, 28,  1,  0], order=31)

np.insert(array, object, values, axis=None)

Inserts values along the given axis.

See: https://numpy.org/doc/stable/reference/generated/numpy.insert.html

Examples

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

In [2]: x = GF.Random(5); x
Out[2]: GF([7, 2, 7, 2, 5], order=2^3)

In [3]: np.insert(x, 1, [0,1,2,3])
Out[3]: GF([7, 0, 1, 2, 3, 2, 7, 2, 5], order=2^3)

np.log(x)

Computes the logarithm (base GF.primitive_element) of a Galois field array element-wise.

Calling np.log() implicitly uses base galois.GFMeta.primitive_element. See galois.GFArray.log() for logarithm with arbitrary base.

References

• https://numpy.org/doc/stable/reference/generated/numpy.log.html

Examples

In [1]: GF = galois.GF(31)

In [2]: alpha = GF.primitive_element; alpha
Out[2]: GF(3, order=31)

In [3]: x = GF.Random(10, low=1); x
Out[3]: GF([14,  6, 27, 24,  3, 26,  9,  2,  5,  2], order=31)

In [4]: y = np.log(x); y
Out[4]: array([22, 25,  3, 13,  1,  5,  2, 24, 20, 24])

In [5]: alpha ** y
Out[5]: GF([14,  6, 27, 24,  3, 26,  9,  2,  5,  2], order=31)

np.matmul(x1, x2)

Computes the matrix multiplication of two Galois field arrays.

References

• https://numpy.org/doc/stable/reference/generated/numpy.log.html

Examples

In [1]: GF = galois.GF(31)

In [2]: x1 = GF.Random((3,4)); x1
Out[2]: 
GF([[19, 16, 14, 25],
    [26,  3, 11, 14],
    [22,  8, 12, 27]], order=31)

In [3]: x2 = GF.Random((4,5)); x2
Out[3]: 
GF([[14,  4, 19, 18, 18],
    [13, 13, 18,  1,  2],
    [ 7,  1, 21, 26,  8],
    [21, 21,  4, 11, 13]], order=31)

In [4]: np.matmul(x1, x2)
Out[4]: 
GF([[12, 17, 20,  5,  5],
    [30, 14, 29, 12,  0],
    [ 9, 27, 23, 21, 22]], order=31)

In [5]: x1 @ x2
Out[5]: 
GF([[12, 17, 20,  5,  5],
    [30, 14, 29, 12,  0],
    [ 9, 27, 23, 21, 22]], order=31)

np.multiply(x, y)

Multiplies two Galois field arrays element-wise.

References

• https://numpy.org/doc/stable/reference/generated/numpy.multiply.html

Examples

Multiplying two Galois field arrays results in field multiplication.

In [1]: GF = galois.GF(31)

In [2]: x = GF.Random(10); x
Out[2]: GF([17, 26,  6, 17, 12,  4, 24,  9, 15, 16], order=31)

In [3]: y = GF.Random(10); y
Out[3]: GF([ 7, 27, 15, 25, 16, 22, 15,  2,  6,  3], order=31)

In [4]: np.multiply(x, y)
Out[4]: GF([26, 20, 28, 22,  6, 26, 19, 18, 28, 17], order=31)

In [5]: x * y
Out[5]: GF([26, 20, 28, 22,  6, 26, 19, 18, 28, 17], order=31)

Multiplying a Galois field array with an integer results in scalar multiplication.

In [6]: GF = galois.GF(31)

In [7]: x = GF.Random(10); x
Out[7]: GF([ 8,  7, 27,  0, 20,  6,  5, 11, 17,  8], order=31)

In [8]: np.multiply(x, 3)
Out[8]: GF([24, 21, 19,  0, 29, 18, 15,  2, 20, 24], order=31)

In [9]: x * 3
Out[9]: GF([24, 21, 19,  0, 29, 18, 15,  2, 20, 24], order=31)

In [10]: print(GF.properties)
GF(31):
  characteristic: 31
  degree: 1
  order: 31
  irreducible_poly: Poly(x + 28, GF(31))
  is_primitive_poly: True
  primitive_element: GF(3, order=31)

# Adding `characteristic` copies of any element always results in zero
In [11]: x * GF.characteristic
Out[11]: GF([0, 0, 0, 0, 0, 0, 0, 0, 0, 0], order=31)

np.negative(x)

Returns the element-wise additive inverse of a Galois field array.

References

• https://numpy.org/doc/stable/reference/generated/numpy.negative.html

Examples

In [1]: GF = galois.GF(31)

In [2]: x = GF.Random(10); x
Out[2]: GF([18, 15, 29, 25,  1,  3, 23,  9,  5, 28], order=31)

In [3]: y = np.negative(x); y
Out[3]: GF([13, 16,  2,  6, 30, 28,  8, 22, 26,  3], order=31)

In [4]: x + y
Out[4]: GF([0, 0, 0, 0, 0, 0, 0, 0, 0, 0], order=31)

In [5]: -x
Out[5]: GF([13, 16,  2,  6, 30, 28,  8, 22, 26,  3], order=31)

In [6]: -1*x
Out[6]: GF([13, 16,  2,  6, 30, 28,  8, 22, 26,  3], order=31)

np.power(x, y)

Exponentiates a Galois field array element-wise.

References

• https://numpy.org/doc/stable/reference/generated/numpy.power.html

Examples

In [1]: GF = galois.GF(31)

In [2]: x = GF.Random(10); x
Out[2]: GF([25, 22, 29,  1,  1, 29,  6, 14,  4,  8], order=31)

In [3]: np.power(x, 3)
Out[3]: GF([ 1, 15, 23,  1,  1, 23, 30, 16,  2, 16], order=31)

In [4]: x ** 3
Out[4]: GF([ 1, 15, 23,  1,  1, 23, 30, 16,  2, 16], order=31)

In [5]: x * x * x
Out[5]: GF([ 1, 15, 23,  1,  1, 23, 30, 16,  2, 16], order=31)

In [6]: x = GF.Random(10, low=1); x
Out[6]: GF([27, 21, 14,  6,  7,  9, 12, 14,  3,  6], order=31)

In [7]: y = np.random.randint(-10, 10, 10); y
Out[7]: array([-8, -4,  1, -6,  8, -3, -8,  1,  8, -4])

In [8]: np.power(x, y)
Out[8]: GF([16, 19, 14,  1, 10,  2,  7, 14, 20,  5], order=31)

In [9]: x ** y
Out[9]: GF([16, 19, 14,  1, 10,  2,  7, 14, 20,  5], order=31)

np.reciprocal(x)

Returns the element-wise multiplicative inverse of a Galois field array.

References

• https://numpy.org/doc/stable/reference/generated/numpy.reciprocal.html

Examples

In [1]: GF = galois.GF(31)

In [2]: x = GF.Random(5, low=1); x
Out[2]: GF([21,  8,  3, 14, 18], order=31)

In [3]: y = np.reciprocal(x); y
Out[3]: GF([ 3,  4, 21, 20, 19], order=31)

In [4]: x * y
Out[4]: GF([1, 1, 1, 1, 1], order=31)

In [5]: x ** -1
Out[5]: GF([ 3,  4, 21, 20, 19], order=31)

In [6]: GF(1) / x
Out[6]: GF([ 3,  4, 21, 20, 19], order=31)

In [7]: GF(1) // x
Out[7]: GF([ 3,  4, 21, 20, 19], order=31)

np.square(x)

Squares a Galois field array element-wise.

References

• https://numpy.org/doc/stable/reference/generated/numpy.square.html

Examples

In [1]: GF = galois.GF(31)

In [2]: x = GF.Random(10); x
Out[2]: GF([24, 10, 21, 13, 10, 16,  1,  3,  6, 18], order=31)

In [3]: np.square(x)
Out[3]: GF([18,  7,  7, 14,  7,  8,  1,  9,  5, 14], order=31)

In [4]: x ** 2
Out[4]: GF([18,  7,  7, 14,  7,  8,  1,  9,  5, 14], order=31)

In [5]: x * x
Out[5]: GF([18,  7,  7, 14,  7,  8,  1,  9,  5, 14], order=31)

np.subtract(x, y)

Subtracts two Galois field arrays element-wise.

References

• https://numpy.org/doc/stable/reference/generated/numpy.subtract.html

Examples

In [1]: GF = galois.GF(31)

In [2]: x = GF.Random(10); x
Out[2]: GF([20, 10, 25, 27, 13, 29, 14,  2,  7,  2], order=31)

In [3]: y = GF.Random(10); y
Out[3]: GF([ 5,  4, 29,  0, 13, 20, 10, 24, 11,  0], order=31)

In [4]: np.subtract(x, y)
Out[4]: GF([15,  6, 27, 27,  0,  9,  4,  9, 27,  2], order=31)

In [5]: x - y
Out[5]: GF([15,  6, 27, 27,  0,  9,  4,  9, 27,  2], order=31)