Self-dual Normal Bases Related To The Type K Gaussian Period Normal Basis Over Finite Fields

Let q be a power of the prime p and Fqnn be the n-th extension of the finite field Fq with q elements. Suppose that α (?) Fqn generates the type k Gaussian period normal basis of Fqn over Fq. In the present paper, we obtain some necessary and sufficient conditions for which finite fields there exists some a,b (?)Fq such that β=a+ba generates a self-dual normal basis of Fqn over Fq, or both a+bα and a+bαn/2generate the two dual normal bases of Fqn over Fq. And then we obtain the corresponding relations between their multiplication tables. For some special cases, we obtain the explicit complexity for these normal bases.