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Improved Computation of Square Roots in Specific Finite Fields
February 2009 (vol. 58 no. 2)
pp. 188-196
Dong-Guk Han, Electronics and Telecommunications Research Institute, Daejeon
Dooho Choi, Electronics and Telecommunications Research Institute, Daejeon
Howon Kim, Pusan National University, Korea
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.

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Index Terms:
Square roots, finite fields, efficient computation, cryptography.
Dong-Guk Han, Dooho Choi, Howon Kim, "Improved Computation of Square Roots in Specific Finite Fields," IEEE Transactions on Computers, vol. 58, no. 2, pp. 188-196, Feb. 2009, doi:10.1109/TC.2008.201
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