具有最小平方和指标的平衡布尔函数性质刻画

Characterization of a Balanced Boolean Function with the Minimum of the Sum-of-squares Indicator

全局雪崩准则是布尔函数的重要密码学性质,具有最小平衡和指标的平衡布尔函数是一个重要的研究方向,本文利用两个不同布尔函数的互相关全局雪崩准则得到了n元平衡布尔函数f达到最小平方和指标时对应的n-1元分解函数f1和f2满足的密码学性质:(1)f1和f2的自相关和为3-值分布;(2)f1和f2的互相关为2-值分布;(3)乘积函数的汉明重量大小由变元n决定;(4)f1、f2的平方和指标之和减去f的平方和指标的大小由汉明重量和变元n决定.同时证明了当f1和f2的汉明重量乘积达到一定条件时该n元平衡布尔函数的平方和指标严格大于最小的平方和指标.最后给出了达到最大非线性度的偶数元平衡布尔函数的一些密码学性质.

The global avalanche characteristic is an important cryptographic property of Boolean functions, and balanced Boolean functions with the minimum of the sum-of-square indicators is an important research direction. Some cryptographic properties of decomposition Boolean functions(f1, f2) are obtained by the global avalanche characteristics of a balanced Boolean function f:(1) the addition of autocorrelation functions of f1 and f2 has 3 different values;(2) the cross-correlation distribution of f1 and f2 has 2 different values;(3) the hamming weight of product function between f1 and f2 is determined by n;(4)the sum of the sum-of-square indicator between f1 and f2 minus the sum-of-square indicators of f is totally determined by n and the hamming weight of f1 and f2. We also proved that under certain conditions, an n-variable balanced Boolean function with the sum-of-square indicators greater than the minimum of the sum-of-square indicators is obtained by the hamming weight of the decomposition Boolean functions. Finally, some properties of even variable balanced Boolean function with the maximum nonlinearity are given.