关键词: 布尔函数, 密码学, 非线性度, Walsh变换, 导数

Abstract:

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.

Key words: Boolean functions, cryptography, nonlinearities, Walsh transforms, derivatives