- 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
Notes¶
An element \(x\) in \(\mathrm{GF}(p^m)\) 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 \(\mathrm{GF}(2^3)\) has characteristic 2, every element has a square root.
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 \(\mathrm{GF}(11)\), the characteristic is greater than 2 so only half of the elements have square roots.
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])