Abstract: Global avalanche characteristics matrix representation method starting from the expression of global avalanche characteristics was proposed and a new tool for research global avalanche characteristics was provided. According to properties of sum-of-square indicator, an odd almost optimal Boolean functions which meet savariety of properties of cryptography was constructed and hold slower sum-of-square indicator via M-M Boolean functions. It is a problem that how to construct a kind of functions satisfying GAC. There is a effecint way that modifying M-M functions with designing new mapping.