A high-speed square root algorithm in extension fields

Hidehiro Katou, Feng Wang, Yasuyuki Nogami, Yoshitaka Morikawa

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)


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.

Original languageEnglish
Title of host publicationInformation Security and Cryptology - ICISC 2006
Subtitle of host publication9th International Conference, Proceedings
PublisherSpringer Verlag
Number of pages13
ISBN (Print)3540491120, 9783540491125
Publication statusPublished - 2006
EventICISC 2006: 9th International Conference on Information Security and Cryptology - Busan, Korea, Republic of
Duration: Nov 30 2006Dec 1 2006

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4296 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


OtherICISC 2006: 9th International Conference on Information Security and Cryptology
Country/TerritoryKorea, Republic of

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)


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