A relation between self-re ciprocal transformation and normal basis over odd characteristic field

Let q and f (x) be an odd characteristic and an irreducible polynomial of degree m over F_q , respectively. Then, suppose that F(x) = x ^m f (x + x-1) becomes irreducible over F_q . This paper shows that the conjugate zeros of F(x) with respect to F_q form a normal basis in F_q2m if and only if those of f (x) form a normal basis in F_q^m and the part of conjugates given as follows are linearly independent over F_q , {γ - γ^-1, (γ - γ^-1)q, ⋯ , (γ - γ^-1) ^qm-1 }, where γ is a zero of F(x) and thus a proper element in F_q2m . In addition, from the viewpoint of q-polynomial, this paper proposes an efficient method for checking whether or not the conjugate zeros of F(x) satisfy the condition.