有限域上的弱自对偶正规基

On Weakly Self-dual Normal Bases over Finite Fields

对有限域上的弱自对偶正规基的乘法表的特征进行了刻画,并对其复杂度进行了研究,得到了在几种不同类型的有限域扩张时此类正规基的下界描述.例如,若q为素数幂,E=F_q^n为q元域F=F_q的n次扩张,N={a_i=a^q~i|i=0,1,…,n-1}为E在F上的一组弱自对偶正规基,其对偶基由β=ca_r生成,其中c∈F~*,0≤r≤n-1,则当r≠0,n/2时,N的复杂度C_N为偶数且C_N≥4n-2.

This paper studies the complexity of weakly self-dual normal bases over finite fields.The authors characterize when a normal bases is weakly self-dual through simple criteria on its the multiplication table.As a consequences of this result,the authors prove various lower bounds for the complexity of weakly self-dual normal bases.For example, suppose that a weakly self-dual normal basis N of Fq^n over Fq is generated by a and its dual basis generated byαr=α^q^r.It is proved that the complexity of N is even and is at least 4n-2,provided that r≠0 or n/2.