Designs, Codes and Cryptography

, Volume 2, Issue 4, pp 315–323

Optimal normal bases

  • Shuhong Gao
  • Hendrik W. LenstraJr.

DOI: 10.1007/BF00125200

Cite this article as:
Gao, S. & Lenstra, H.W. Des Codes Crypt (1992) 2: 315. doi:10.1007/BF00125200


Let KL be a finite Galois extension of fields, of degree n. Let G be the Galois group, and let (<α)<∈G be a normal basis for L over K. An argument due to Mullin, Onyszchuk, Vanstone and Wilson (Discrete Appl. Math. 22 (1988/89), 149–161) shows that the matrix that describes the map xαx on this basis has at least 2n - 1 nonzero entries. If it contains exactly 2n - 1 nonzero entries, then the normal basis is said to be optimal. In the present paper we determine all optimal normal bases. In the case that K is finite our result confirms a conjecture that was made by Mullin et al. on the basis of a computer search.

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • Shuhong Gao
    • 1
  • Hendrik W. LenstraJr.
    • 2
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA