On the global avalanche characteristics between two Boolean functions and the higher order nonlinearity

The criterion for the global avalanche characteristics (GAC) of cryptographic functions is an important property. To measure the correlation between two arbitrary Boolean functions, we propose two new criteria called the sum-of-squares indicator and the image indicator of the cross-correlation between two Boolean functions. The two indicators generalize the GAC criterion. Based on the properties of the cross-correlation function, we deduce the rough lower and the rough upper bounds on the two indicators by hamming weights of two Boolean functions, and generalize some properties between the Walsh spectrum and the cross-correlation function. Furthermore, we give the tight upper and the tight lower bounds on the two indicators. Finally, we show some relationships between the upper bounds on the two indicators and the higher order nonlinearity.