有限域上的2-型高斯正规基及其对偶基(英文)

The type 2 Gaussian normal basis and its dual basis over finite fields

设q为素数p的幂,F_q^n为有限域F_q的n(n≥2)次扩域.熟知k-型高斯正规基当k=1时为Ⅰ型最优正规基,当q=k=2时为Ⅱ型最优正规基.本文证明了k-型高斯正规基生成元的迹函数为-1,确定了2-型高斯正规基的复杂度及其对偶基的生成元与复杂度.

Let q be a power of the prime p and Fq- the extension of the finite field Fq with degree n（n≥2）. It＇s well-known that the type k Gaussian normal basis of Fq^n over Fq is a type Ⅰ or type Ⅱoptimal normal basis depends on k= 1 or q= k= 2 correspondingly. In the present paper, the authors prove that the trace of a generator of the type k Gaussian normal basis of Fq. over Fq equals to - 1. For the case k= 2, they determine the dual basis B of N and the complexities for N and B.