Square root algorithm in Fq for q ≡ 2s+1 (mod 2s+1)

Presented is a square root algorithm in F-q which generalises Atkins's square root algorithm [see reference 6] for q equivalent to 5 (mod 8) and Muller's algorithm [see reference 7] for q equivalent to 9 (mod 16). The presented algorithm precomputes a primitive 2(s)-th root of unity xi where s is the largest positive integer satisfying 2(s)vertical bar q - 1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for square root computation and is favourably compared with the algorithms of Atkin, Muuller and Kong et al.