On orders of optimal normal basis generators

In this paper we give some experimental results on the multiplicative orders of optimal normal basis generators in F 2 n over F 2 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 high multiplicative orders, at least O((2 n - 1)/n), and are very often primitive. For a given optimal normal basis generator a in F 2 n 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.