Improved generalized Atkin algorithm for computing square roots in finite fields

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 ( mod 16 ) . 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 ( mod 16 ) . Furthermore, in finite fields GF ( p m ) , where p ≡ 9 ( mod 16 ) and m is odd, we reduce the complexity of the algorithm from O ( m 3 log 3 p ) to O ( m 2 log 2 p ( log m + log p ) ) using the Frobenius map and normal basis representation.