Generalization of statistical criteria for Sboxes

Linear cryptanalysis is one of the most important tools in the analysis of symmetric ciphers. This attack makes use of linear properties of Sboxes. In this paper, we investigate the linear properties of Sboxes, mathematically, and generalize the criteria used in linear attacks in two aspects: the first aspect introduces nonlinear criteria for Sboxes; the second aspect is generalizing linear criteria in finite fields, and also for balanced linear transformations. The first aspect, generalizes the works of Harpes, Kramer and Massey in EUROCRYPT'95 and the second aspect is a generalization of the works of Baigneres in his dissertation in 2008. Combining these two generalizations, we study generalized (nonlinear) criteria for distinguishing Sboxes from random ones. At last, we prove a theorem which we could name “Generalized Piling up Lemma”.