On orders of optimal normal basis generators

Abstract

In this paper we give some experimental results on the multiplicative orders of optimal normal basis generators in F 2 n {F_{{2^n}}} over F 2 {F_2} for n ≤ 1200 n \leq 1200 whenever the complete factorization of 2 n − 1 {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 ) O(({2^n} - 1)/n) , and are very often primitive. For a given optimal normal basis generator α \alpha in F 2 n {F_{{2^n}}} and an arbitrary integer e, we show that α e {\alpha ^e} can be computed in O ( n ⋅ v ( e ) ) O(n \cdot v(e)) bit operations, where v ( e ) v(e) is the number of 1’s in the binary representation of e.