Normal Bases and Their Dual-Bases over Finite Fields

In this paper,we prove the following results:1)A normal basis N over a finite field isequivalent to its dual basis if and only if the multiplication table of N is symmetric;2)The normal basisN is self-dual if and only if its multiplication table is symmetric and Tr(α2)=1,where α generatesN; 3)An optimal normal basis N is self-dual if and only if N is a type-Ⅰ optimal normal basis withq=n=2 or N is a type-Ⅱ optimal normal basis.