关于多输出布尔函数的第二类非线性度

On Nonlinearity of the Second Type of Multi-output Boolean Functions

本文对多输出布尔函数的第二类非线性度进行研究,该定义是衡量抵抗最佳多输出仿射逼近攻击性能的一项重要准则.利用多输出布尔函数的Walsh变换,我们给出第二类非线性度的一种表达式,并在此基础上得到第二类非线性度的一个上界.进一步地,我们给出了当第一类非线性度达到最优时,其第二类非线性度的一个界.此外,本文还给出任意多输出布尔函数与所有多输出线性函数之间距离的均值.

In this paper, nonlinearity of the second type of the multi-output Boolean functions is studied, which is an important cryptographic criterion to measure the ability on the resistance of the best multi-output affine approximation attack. By using the Walsh transform of the multi-output Boolean functions, we present an explicit representation for nonlinearity of the second type, and based on this nonlinearity, we obtain an upper bound of this type of nonlinearity. Furthermore, we give the bounds for the nonlinearity of the second type when the nonlinearity of the first type is optimal. Additionally, the average value of distances between a multi-output Boolean function and all other multi-output linear functions is provided.