Substitution boxes (S-boxes) are often used as the most important nonlinear components in many symmetric encryption algorithms. The cryptographic properties of an S-box directly affect the security of the whole cipher system. Recently, generalized global avalanche characteristics (GGAC) were introduced to measure the correlation between two arbitrary Boolean functions. In this paper, to better evaluate the security of an S-box, we present two cross-correlation indicators for it. In addition, by studying the related properties of the cross-correlation between two balanced Boolean functions, we propose the lower bounds on the sum-of-squares indicator related to GGAC for two balanced functions and also for an S-box.
The relationship among cross-correlation of arbitrary four Boolean functions is presented. Several known cross-correlation properties of Boolean functions are generalized. Based on them, a lower bound for the maximal cross-correlation (in absolute value) of two Boolean functions (if one of the functions is bent) is obtained.
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