galois.FieldArray.is_square

galois.FieldArray.is_square() → bool | ndarray

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

Returns:¶: A boolean array indicating if each element in $x$ is a square. The return value is a single boolean if the input array $x$ is a scalar.

See also

squares, non_squares

Notes

An element $x$ in $GF (p^{m})$ is a square if there exists a $y$ such that $y^{2} = x$ in the field.

In fields with characteristic 2, every element is a square (with two identical square roots). In fields with characteristic greater than 2, exactly half of the nonzero elements are squares (with two unique square roots).

References

Section 3.5.1 from https://cacr.uwaterloo.ca/hac/about/chap3.pdf.

Examples

Since $GF (2^{3})$ has characteristic 2, every element has a square root.

Integer Polynomial Power

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

In [2]: x = GF.elements; x
Out[2]: GF([0, 1, 2, 3, 4, 5, 6, 7], order=2^3)

In [3]: x.is_square()
Out[3]: array([ True,  True,  True,  True,  True,  True,  True,  True])

In [4]: GF = galois.GF(2**3, repr="poly")

In [5]: x = GF.elements; x
Out[5]: 
GF([          0,           1,           α,       α + 1,         α^2,
        α^2 + 1,     α^2 + α, α^2 + α + 1], order=2^3)

In [6]: x.is_square()
Out[6]: array([ True,  True,  True,  True,  True,  True,  True,  True])

In [7]: GF = galois.GF(2**3, repr="power")

In [8]: x = GF.elements; x
Out[8]: GF([  0,   1,   α, α^3, α^2, α^6, α^4, α^5], order=2^3)

In [9]: x.is_square()
Out[9]: array([ True,  True,  True,  True,  True,  True,  True,  True])

In $GF (11)$ , the characteristic is greater than 2 so only half of the elements have square roots.

Integer Power

In [10]: GF = galois.GF(11)

In [11]: x = GF.elements; x
Out[11]: GF([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10], order=11)

In [12]: x.is_square()
Out[12]: 
array([ True,  True, False,  True,  True,  True, False, False, False,
        True, False])

In [13]: GF = galois.GF(11, repr="power")

In [14]: x = GF.elements; x
Out[14]: GF([  0,   1,   α, α^8, α^2, α^4, α^9, α^7, α^3, α^6, α^5], order=11)

In [15]: x.is_square()
Out[15]: 
array([ True,  True, False,  True,  True,  True, False, False, False,
        True, False])