Improved generalized Atkin algorithm for computing square roots in finite fields

Fanyu Kong, Zhun Cai, Jia Yu, Daxing Li

Abstract: Recently, S. Müller developed a generalized Atkin algorithm for computing square roots, which requires two exponentiations in finite fields GF(q) when q≡9(mod16). In this paper, we present a simple improvement to it and the improved algorithm requires only one exponentiation for half of squares in finite fields GF(q) when q≡9(mod16). Furthermore, in finite fields GF(pm), where p≡9(mod16) and m is odd, we reduce the complexity of the algorithm from O(m3log3p) to O(m2log2p(logm+logp)) using the Frobenius map and normal basis representation.