Cyclic vector multiplication algorithm and existence probability of gauss period normal basis

Kenta Nekado, Yasuyuki Nogami, Hidehiro Kato, Yoshitaka Morikawa

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


Recently, pairing-based cryptographic application sch emes have attracted much attentions. In order to make the schemes more efficient, not only pairing algorithm but also arithmetic operations in exten sion field need to be efficient. For this purpose, the authors have proposed a series of cyclic vector multiplication algorithms (CVMAs) correspond ing to the adopted bases such as type-I optimal normal basis (ONB). Note here that every basis adapted for the conventional CVMAs are just special classes of Gauss period normal bases (GNBs). In general, GNB is charac terized with a certain positive integer h in addition to characteristic p and extension degree m, namely type-(h.m) GNB in extension field F The parameter h needs to satisfy some conditions and such a positive integer h infinitely exists. From the viewpoint of the calculation cost of CVMA, it is preferred to be small. Thus, the minimal one denoted by hmjn will be adapted. This paper focuses on two remaining problems: 1) CVMA has not been expanded for general GNBs yet and 2) the minimal hmjn sometimes becomes large and it causes an inefficient case. First, this paper expands CVMA for general GNBs. It will improve some critical cases with large hmin reported in the conventional works. After that, this paper shows a theorem that, for a fixed prime number r, other prime numbers modulo r uniformly distribute between ito r - 1. Then, based on this theorem, the existence probability of type-(hmjn,m) GNB in Fpm and also the expected value of hmin are explicitly given.

Original languageEnglish
Pages (from-to)172-179
Number of pages8
JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Issue number1
Publication statusPublished - Jan 2011


  • All one polynomial field
  • Cyclic vector multiplication algorithm
  • Extension field
  • Gauss period normal basis

ASJC Scopus subject areas

  • Signal Processing
  • Computer Graphics and Computer-Aided Design
  • Electrical and Electronic Engineering
  • Applied Mathematics


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