Basis Translation Matrix between Two Isomorphic Extension Fields via Optimal Normal Basis

  • Nogami, Yasuyuki (Department of Communication Network Engineering, Okayama University) ;
  • Namba, Ryo (Department of Communication Network Engineering, Okayama University) ;
  • Morikawa, Yoshitaka (Department of Communication Network Engineering, Okayama University)
  • Received : 2007.06.18
  • Published : 2008.04.30

Abstract

This paper proposes a method for generating a basis translation matrix between isomorphic extension fields. To generate a basis translation matrix, we need the equality correspondence of a basis between the isomorphic extension fields. Consider an extension field $F_{p^m}$ where p is characteristic. As a brute force method, when $p^m$ is small, we can check the equality correspondence by using the minimal polynomial of a basis element; however, when $p^m$ is large, it becomes too difficult. The proposed methods are based on the fact that Type I and Type II optimal normal bases (ONBs) can be easily identified in each isomorphic extension field. The proposed methods efficiently use Type I and Type II ONBs and can generate a pair of basis translation matrices within 15 ms on Pentium 4 (3.6 GHz) when $mlog_2p$ = 160.

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