A Complete Generalization of Atkin's Square Root Algorithm
Fundamenta Informaticae
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In this paper, we study exponentiation in the specific finite fields {\bf F}_{q} with very special exponents such as those that occur in algorithms for computing square roots. Here, q is a prime power, q = p^{k}, where k 1, and k is odd. Our algorithmic approach improves the corresponding exponentiation resulted from the better rewritten exponent. To the best of our knowledge, it is the first major improvement to the Tonelli-Shanks algorithm, for example, the number of multiplications can be reduced to at least 60 percent on the average when p \equiv 1 (mod 16). Several numerical examples are given that show the speedup of the proposed methods.