On orders of optimal normal basis generators

In this paper we give some computational results on the multiplicative orders of optimal normal basis generators in F2n over F2 for n ≤ 1200 whenever the complete factorization of 2 n − 1 is known. Our results show that a subclass of optimal normal basis generators always have very high multiplicative orders and are very often primitive. For a given optimal normal basis generator α in F2n and an arbitrary integer e, we show that α e can be computed in O(n · v(e)) bit operations, where v(e) is the number of 1’s in the binary representation of e. For a prime power q and a positive integer n, let Fq and Fqn be the finite fields of q and q n elements, respectively. A normal basis N for Fqn over Fq is a basis of the form (α, α , . . . , α n−1 ) where α ∈ Fqn . In such case α is said to be a normal element or normal basis generator. The complexity of N , denoted by cN , is defined to be the number of nonzero entries tij in the n expressions