具有最优代数免疫度的2~m个变量的对称Boolean函数

On 2~m-variable symmetric Boolean functions with maximum algebraic immunity 2～（m-1）

本文给出了2m个变量的对称Boolean函数f具有最优代数免疫度AI2m(f)=2m-1的一个充分必要条件.由此得到一个递归公式,从而构造出全部具有最优代数免疫度的2m个变量的对称Boolean函数(m2).最后证明了这样的Boolean函数的个数为3·2m.

In this paper we prove a necessary and sufficient condition for a 2m-variable symmetric Boolean function f having maximum algebraic immunity AI2m（f） = 2m-1. As its consequences, we present a simple recursive procedure to construct all 2m-variable symmetric Boolean functions f with maximum algebraic immunity for all m 2, and prove that the number of such functions is 3·2m.