A high-speed square root algorithm in extension fields

Hidehiro Katou; Feng Wang; Yasuyuki Nogami; Yoshitaka Morikawa

doi:10.1007/11927587_10

A high-speed square root algorithm in extension fields

Hidehiro Katou, Feng Wang, Yasuyuki Nogami, Yoshitaka Morikawa

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

1 Citation (Scopus)

Abstract

A square root (SQRT) algorithm in GF(p^m) (m = r ₀r₁⋯ r_n-1-12^d, r_i: 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.

Original language	English
Title of host publication	Information Security and Cryptology - ICISC 2006
Subtitle of host publication	9th International Conference, Proceedings
Publisher	Springer Verlag
Pages	94-106
Number of pages	13
ISBN (Print)	3540491120, 9783540491125
DOIs	https://doi.org/10.1007/11927587_10
Publication status	Published - 2006
Event	ICISC 2006: 9th International Conference on Information Security and Cryptology - Busan, Korea, Republic of Duration: Nov 30 2006 → Dec 1 2006

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	4296 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Other

Other	ICISC 2006: 9th International Conference on Information Security and Cryptology
Country/Territory	Korea, Republic of
City	Busan
Period	11/30/06 → 12/1/06

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science(all)

Access to Document

10.1007/11927587_10

Cite this

Katou, H., Wang, F., Nogami, Y., & Morikawa, Y. (2006). A high-speed square root algorithm in extension fields. In Information Security and Cryptology - ICISC 2006: 9th International Conference, Proceedings (pp. 94-106). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4296 LNCS). Springer Verlag. https://doi.org/10.1007/11927587_10

A high-speed square root algorithm in extension fields. / Katou, Hidehiro; Wang, Feng; Nogami, Yasuyuki; Morikawa, Yoshitaka.

Information Security and Cryptology - ICISC 2006: 9th International Conference, Proceedings. Springer Verlag, 2006. p. 94-106 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4296 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Katou, H, Wang, F, Nogami, Y & Morikawa, Y 2006, A high-speed square root algorithm in extension fields. in Information Security and Cryptology - ICISC 2006: 9th International Conference, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4296 LNCS, Springer Verlag, pp. 94-106, ICISC 2006: 9th International Conference on Information Security and Cryptology, Busan, Korea, Republic of, 11/30/06. https://doi.org/10.1007/11927587_10

Katou H, Wang F, Nogami Y, Morikawa Y. A high-speed square root algorithm in extension fields. In Information Security and Cryptology - ICISC 2006: 9th International Conference, Proceedings. Springer Verlag. 2006. p. 94-106. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/11927587_10

Katou, Hidehiro ; Wang, Feng ; Nogami, Yasuyuki ; Morikawa, Yoshitaka. / A high-speed square root algorithm in extension fields. Information Security and Cryptology - ICISC 2006: 9th International Conference, Proceedings. Springer Verlag, 2006. pp. 94-106 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "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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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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A high-speed square root algorithm in extension fields

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