两类三次旋转对称布尔函数的汉明重量和非线性度

The Hamming Weight and Nonlinearity of Two Classes of Cubic Rotation-symmetric Boolean Functions

旋转对称布尔函数是一类具有良好密码学性质的布尔函数,自被提出来就得到了学者们的广泛关注.本文研究了形如f（x）=∑i=0^n-1 xix1＋ixm＋1＋i和ft（x）=∑i=0^n-1 xixt＋ixm＋i的两类三次旋转对称布尔函数的汉明重量及非线性度.通过对F2^n进行分解,可将函数转化为特殊形式,使得求取函数的傅里叶变换变得相对容易.再利用汉明重量及非线性度与傅里叶变换之间的关系,求出了这两类函数的汉明重量和非线性度的计算公式.

Rotation-symmetric Boolean functions are a class of Boolean functions with good cryptographic properties, which have been extensively studied since they were proposed. In this paper, we study the Hamming weight and nonlinearity of two classes of Rotation-symmetric functions, which are of the forms： f（x）=∑i=0^n-1 xix1＋ixm＋1＋i and f（x）=∑i=0^n-1 xixt＋ixm＋i. We transform the function into a special form by decomposing the F2^n, so that it is easier to obtain the Fourier transform of the function. Then we use the relationship between the Hamming weight and nonlinearity with the Fourier transform, and give the formula by which one can calculate the Hamming weight and the nonlinearitv of these two classes of functions.