Improved generalized Atkin algorithm for computing square roots in finite fields

Authors:
Fanyu Kong;Zhun Cai;Jia Yu;Daxing Li
Affiliations:
Institute of Network Security, Shandong University, Jinan, Shandong, PR China;Institute of Network Security, Shandong University, Jinan, Shandong, PR China;Institute of Network Security, Shandong University, Jinan, Shandong, PR China;Institute of Network Security, Shandong University, Jinan, Shandong, PR China
Venue:
Information Processing Letters
Year:
2006

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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(pm), where p ≡ 9 (mod 16) and m is odd, we reduce the complexity of the algorithm from O(m3 log3 p) to O(m2 log2 p(log m + log p)) using the Frobenius map and normal basis representation.