On Primitive Optimal Normal Elements of Finite Fields

Abstract:

      Let $q$ be a prime or prime power and $F_{q^{n}}$ the extension of $q$ elements finite field $F_{q}$ with degree $n~(n>1)$. Davenport, Lenstra and Schoof proved that there exists a primitive element $\alpha\in F_{q^{n}}$ such that $\alpha$ generates a normal basis of $F_{q^{n}}$ over $F_{q}$. Later, Mullin, Gao and Lenstra, etc., raised the definition of optimal normal bases and constructed such bases. In this paper, we determine all primitive type I optimal normal bases and all finite fields in which there exists a pair of reciprocal elements $\alpha$ and $\alpha^{-1}$ such that both of them generate optimal normal bases of $F_{q^{n}}$ over $F_{q}$. Furthermore, we obtain a sufficient condition for the existence of primitive type II optimal normal bases over finite fields and prove that all primitive optimal normal elements are conjugate to each other.