np.sqrt

np.sqrt(x)

Computes the square root a Galois field array element-wise.

References

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

Notes

This function returns the lexicographically-minimal root \(r\) (the root whose integer representation is smallest). In addition to \(r\), \(-r\) is also a root.

Examples

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

# Only the "quadratic residues" have square roots
In [2]: x = GF.quadratic_residues; x
Out[2]: 
GF([ 0,  1,  2,  4,  5,  7,  8,  9, 10, 14, 16, 18, 19, 20, 25, 28],
   order=31)

In [3]: r = np.sqrt(x)

# Both roots in the finite field
In [4]: r, -r
Out[4]: 
(GF([ 0,  1,  8,  2,  6, 10, 15,  3, 14, 13,  4,  7,  9, 12,  5, 11],
    order=31),
 GF([ 0, 30, 23, 29, 25, 21, 16, 28, 17, 18, 27, 24, 22, 19, 26, 20],
    order=31))

In [5]: r**2
Out[5]: 
GF([ 0,  1,  2,  4,  5,  7,  8,  9, 10, 14, 16, 18, 19, 20, 25, 28],
   order=31)

In [6]: (-r)**2
Out[6]: 
GF([ 0,  1,  2,  4,  5,  7,  8,  9, 10, 14, 16, 18, 19, 20, 25, 28],
   order=31)