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 F2n over F2 for n < 1200 whenever the complete factorization of 2" 1 is known. Our results show that a subclass of optimal normal basis generators always have high multiplicative orders, at least 0((2" \)/n), and are very often primitive. For a given optimal normal basis generator a in 7> and an arbitrary integer e , we show that ae can be computed in 0(n ■ v(e)) bit operations, where v(e) is the number of Ps in the binary representation of e . For a prime power q and a positive integer « , let Fq and iy be the finite fields of q and q" elements, respectively. A normal basis TV for Fqn over Fq is a basis of the form (a, aq , ... , aq" ), where a e Fqn . In this case a is said to be a normal element or normal basis generator. The complexity of TV, denoted by Cn , is defined to be the number of nonzero entries fy in the « expressions