三类布尔函数的相关函数研究

Research on Correlation Function for Three Classes of Boolean Functions

布尔函数的相关函数能刻画其扩散特征和线性结构特征,所以研究相关函数的性质对于布尔函数理论具有重要作用。为此,根据自相关和互相关函数的定义,分析通过迹表示的二次布尔函数f(x)=Tr_1~n(x^(2^i+1)+x(2~′+1))的自相关函数值,给出互相关函数平方的一个表达式C_(f,g)~2(α)=(?)(-1)^(D_(f,g)(a)+D_(f,g)(a+ω)),利用该表达式给出任意三次布尔函数的自相关函数平方和的上界,并借助该上界进一步研究两类迹表示的三次布尔函数的绝对值指标上界问题。

The correlation function of Boolean function can depict the diffusion characteristics and linear structure characteristics, and the properties of correlation function plays an important role in Boolean function theory. According to the definition of auto-correlation functionand cross-correlation function, the auto-correlation function of a special form quadratic Boolean function f（x）=Tr1n（x2i＋1＋x2j＋1）presented in this paper and the expressionis given. Based on it, it gives the upper bound of auto-correlation function of three times Boolean function square, and the upper bounds of absolute indicators of two classes trace Boolean functions are investigated.