Cryptology ePrint Archive: Report 2013/087

Square Root Algorithm in F_q for q=2^s+1 (mod 2^(s+1))

Namhun Koo and Gook Hwa Cho and Soonhak Kwon

Abstract: We present a square root algorithm in F_q which generalizes Atkins's square root algorithm for q=5(mod 8) and Kong et al.'s algorithm for q=9(mod 16) Our algorithm precomputes a primitive 2^s-th root of unity where s is the largest positive integer satisfying 2^s| q-1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for square root computation and is favorably compared with the algorithms of Atkin, Muller and Kong et al.

Category / Keywords: applications / square root algorithm, finite field, Tonelli-Shanks algorithm, Cipolla-Lehmer algorithm