Trace-orthogonal normal bases

Abstract

In this paper, we shall first obtain some basic theoretical results on trace-orthogonal normal bases of GF( q n ) over GF( q ): We show that such a basis exists if and only if a self-dual normal basis exists (in fact, any such basis is equivalent to a self-dual one) and we give several characterizations of the trace-orthogonal normal bases in terms of two matrices M and T (associated with every normal basis N ) describing the multiplicative structure of GF( q n ). The matrix T is actually the matrix used in investigating the complexity of the normal basis N . Using this fact, we also completely determine the trace-orthogonal optimal normal bases. In the special case q = 2, n even, we then give a simple construction associating with every self-dual normal basis N another such basis N ∗ and relate the complexities of these two bases. This allows us to obtain an upper bound on the complexity of self-dual normal bases in this case which turns out to explain several entries in the available tables on computer searches regarding the complexity of normal bases. Finally, we give a product construction for (trace-orthogonal) normal bases.