Global Avalanche Characteristics and Nonlinearity of Boolean Function with the Hamming Weight k

Some properties of autocorrelation coefficient and cross-correlation coefficient are given. The restricted relationship among n(n variables, wt(f) (the Hamming weight of Boolean function f (x) and t (t-th propagation criteria) was derived, then a lower bound on the sum-of-squares of any Boolean functions with Hamming weight k is concluded. Finally, the results generalized a upper bound on nonlinearity of Boolean function only depending on Hamming weight. This paper improved known results.