The Type k-Gaussian Normal Bases over Finite Fields and Their Dual Bases
Qunying Liao · Journal of Sichuan Normal University · 2011
It is well-known that normal bases are widely used in applications of finite fields in areas such as coding theory,cryptography,signal processing,and so on.Z.X.Wan et al(Finite Fields and Their Applications,2007,13(4):411-417.) computed the complexity of the dual basis of a type Ⅰ optimal normal basis of Fqn over Fq which is equal to 3n-2 or 3n-3 according to q is odd or even,respectively.This is a special class of type k-Gaussian normal bases.Recently,Q.Y.Liao et al(J.Sichuan University:Science Nautural,2010,47(6):1221-1224.) gave the dual basis and the complexity of a type 2-Gaussian normal basis.In this paper,for a general type k-Gaussian normal basis N,we obtain the dual basis and a upper bound for the complexity of N when n≥ k≥ 1.Furthermore,we prove that the upper bound can be achieved for k=3,and then determine all(weakly) self-dual type k-Gaussian normal bases.