The autocorrelation distribution of balanced Boolean function

Abstract

The global avalanche characteristics (the sumof- squares indicator and the absolute indicator) measure the overall avalanche characteristics of a cryptographic Boolean function. Sung et al. (1999) gave the lower bound on the sumof- squares indicator for a balanced Boolean function satisfying the propagation criterion with respect to some vectors. In this paper, if balanced Boolean functions satisfy the propagation criterion with respect to some vectors, we give three necessary and sufficient conditions on the auto-correlation distribution of these functions reaching the minimum the bound on the sum-of-squares indicator. And we also find all Boolean functions with 3-variable, 4-variable, and 5-variable reaching the minimum the bound on the sum-of-squares indicator.

Keywords

Boolean functions / auto-correlation distribution / global avalanche characteristics / balanced / propagation criterion

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