TY - GEN

T1 - A high-speed square root algorithm in extension fields

AU - Katou, Hidehiro

AU - Wang, Feng

AU - Nogami, Yasuyuki

AU - Morikawa, Yoshitaka

PY - 2006

Y1 - 2006

N2 - A square root (SQRT) algorithm in GF(pm) (m = r 0r1⋯ rn-1-12d, ri: odd prime, d > 0: integer) is proposed in this paper. First, the Tonelli-Shanks algorithm is modified to compute the inverse SQRT in GF(p 2d ), where most of the computations are performed in the corresponding subfields GF(p2i ) for 0 ≤ i ≤ d-1. Then the Frobenius mappings with an addition chain are adopted for the proposed SQRT algorithm, in which a lot of computations in a given extension field GF(p m) are also reduce to those in a proper subfield by the norm computations. Those reductions of the field degree increase efficiency in the SQRT implementation. More specifically the Tonelli-Shanks algorithm and the proposed algorithm in GF(p22), GF(p44) and GF(p 88) were implemented on a Pentium4 (2.6 GHz) computer using the C++ programming language. The computer simulations showed that, on average, the proposed algorithm accelerates the SQRT computation by 25 times in GF(p 22), by 45 times in GF(p44), and by 70 times in GF(p 88), compared to the Tonelli-Shanks algorithm, which is supported by the evaluation of the number of computations.

AB - A square root (SQRT) algorithm in GF(pm) (m = r 0r1⋯ rn-1-12d, ri: odd prime, d > 0: integer) is proposed in this paper. First, the Tonelli-Shanks algorithm is modified to compute the inverse SQRT in GF(p 2d ), where most of the computations are performed in the corresponding subfields GF(p2i ) for 0 ≤ i ≤ d-1. Then the Frobenius mappings with an addition chain are adopted for the proposed SQRT algorithm, in which a lot of computations in a given extension field GF(p m) are also reduce to those in a proper subfield by the norm computations. Those reductions of the field degree increase efficiency in the SQRT implementation. More specifically the Tonelli-Shanks algorithm and the proposed algorithm in GF(p22), GF(p44) and GF(p 88) were implemented on a Pentium4 (2.6 GHz) computer using the C++ programming language. The computer simulations showed that, on average, the proposed algorithm accelerates the SQRT computation by 25 times in GF(p 22), by 45 times in GF(p44), and by 70 times in GF(p 88), compared to the Tonelli-Shanks algorithm, which is supported by the evaluation of the number of computations.

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U2 - 10.1007/11927587_10

DO - 10.1007/11927587_10

M3 - Conference contribution

AN - SCOPUS:34547443654

SN - 3540491120

SN - 9783540491125

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 94

EP - 106

BT - Information Security and Cryptology - ICISC 2006

PB - Springer Verlag

T2 - ICISC 2006: 9th International Conference on Information Security and Cryptology

Y2 - 30 November 2006 through 1 December 2006

ER -