A relation between self-reciprocal 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) is 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_qm and the partial conjugates given as follows are linearly independent over F_q, {γ - γ^-1, (γ - γ^-1)^q, · · · , (γ - γ^-1)qm-1}, (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.