Abstract
Let K ⊂ L 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 x → αx 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.C. Mullin, A characterization of th extremal distributions of optimal normal bases, Proc. Marshall Hall Memorial Conference, Burlington, Vermont, 1990, to appear.
R.C. Mullin, I.M. Onyszchuk, S.A. Vanstone, and R.M. Wilson, Optimal normal bases in GF(pn), Discrete Appl. Math. Vol. 22 (1988/89), pp. 149–161.
Author information
Authors and Affiliations
Additional information
Communicated by S.A. Vanstone
Rights and permissions
About this article
Cite this article
Gao, S., Lenstra, H.W. Optimal normal bases. Des Codes Crypt 2, 315–323 (1992). https://doi.org/10.1007/BF00125200
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00125200