Memoirs of the Faculty of Engineering, Okayama University
Published by Faculty of Enginerring, Okayama University

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Memoirs of the School of Engineering, Okayama University

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A High-Speed Square Root Algorithm for Extension fields -Especially for Fast Extension Fields-

Kato, Hidehiro Graduate School of Natural Science and Technology, Okayama University
Nogami, Yasuyuki Graduate School of Natural Science and Technology, Okayama University
Morikawa, Yoshitaka Graduate School of Natural Science and Technology, Okayama University
抄録
A square root (SQRT) algorithm in extension field F(p(m))(m = r(0)r(1)・・・r(n−1)・2(d), r(i) : odd prime, d : positive integer) is proposed in this paper. First, a conventional SQRT algorithm, the Tonelli-Shanks algorithm, is modified to compute the inverse SQRT in F(p(2d)), where most of the computations are performed in the corresponding subfields F(p(2i)) for 0 ≤ i ≤ d-1. Then the Frobenius mappings with addition chain are adopted for the proposed SQRT algorithm, in which a lot of computations in a given extension field F(p(m)) are also reduced to those in a proper subfield by the norm computations. Those reductions of the field degree increase efficiency in the SQRT implementation. The Tonelli-Shanks algorithm and the proposed algorithm in F(p(6)) and F(p(10)) were implemented on a Core2 (2.66 GHz) using the C++ programming language. The computer simulations showed that, on average, the proposed algorithm accelerated the SQRT computation by 6 times in F(p(6)), and by 10 times in F(p(10)), compared to the Tonelli-Shanks algorithm.
ISSN
1349-6115
NCID
AA12014085
NAID