A High - Speed Square Root Algorithm for Extension fields – Especially for Fast Extension Fields –
نویسندگان
چکیده
A square root (SQRT) algorithm in extension field Fpm(m = r0r1 · · · rn−1 · 2, ri : odd prime, d : positive integer) is proposed in this paper. First, a conventional SQRT algorithm, the TonelliShanks algorithm, is modified to compute the inverse SQRT in F p2 , where most of the computations are performed in the corresponding subfields Fp2i for 0 6 i 6 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 Fpm 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 Fp6 and Fp10 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 Fp6 , and by 10 times in Fp10 , compared to the Tonelli-Shanks algorithm.
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A High-Speed Square Root Algorithm in Extension Fields
A square root (SQRT) algorithm inGF (p) (m=r0r1 · · · rn−12, ri: odd prime, d>0: integer) is proposed in this paper. First, the TonelliShanks algorithm is modified to compute the inverse SQRT in GF (p d ), where most of the computations are performed in the corresponding subfields GF (p i ) for 0 i d − 1. Then the Frobenius mappings with an addition chain are adopted for the proposed SQRT algor...
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