Avalanche and bit independence properties for the ensembles of randomly chosen n x n S-boxes

Abstract

Cryptographic test methods such as avalanche, strict avalanche and bit independence criteria, which measure the degree of security of the s-boxes of substitution-permutation networks, are applied to randomly generated ensembles of n x n s-boxes. Statistical analysis of experimental data directs the work towards defining "relative errors'' and examining the avalanche and strict avalanche criteria within "relative error ranges''. Histograms of relative errors in each ensemble are evaluated, and combining the results of different ensembles corresponding to different values of the s-box size, variations of maximum relative errors versus the size of the s-box are depicted. Some predictions follow that the larger the s-box size, the more probable that these criteria are satisfied; thus it is possible to form more secure substitution-permutation networks. Correlations among the test criteria are also evaluated in random ensembles to find out to what extent those criteria measure different cryptographic aspects of s-boxes.