布尔函数的二阶非线性度的下界

Lower Bounds of Second-Order Nonlinearity of Boolean Functions

对形如f(x)=tr(∑﹂(n-1)/2」i,j=1bijxd)的n元布尔函数的二阶非线性度进行了研究,其中d=2i+2j+1,bij GF(2),1≤i<j≤L(n-1)/2」.当n为奇数时,找出了函数f(x)达到最大非线性度的导数;当n为偶数时,找出了函数f(x)的半Bent函数的导数.基于这些具有高非线性度的导数,给出了f(x)二阶非线性度的紧下界.结果表明f(x)具有较高的二阶非线性度,可以抵抗二次函数逼近和仿射逼近攻击.

This paper deals with the second-order nonlinearities of the Boolean functions f（x）=tr（∑（n-1）/2」i,j=1bijxd） with n variables,where d=2i＋2j＋1,bij GF（2） and 1≤ij≤L（n-1）/2」.The derivatives with the maximal nonlinearity of f（x） are determined for odd n,and,for even n,the derivatives which are semi-Bent functions are obtained.Based on these derivatives with high nonlinerity,the tight lower bounds of the second-order nonlinearity of f（x） are given.The results show that f（x） with high second-order nonlinearity,can resist the quadratic and affine approximation attacks.