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 \times 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.
VERGİLİ, IŞIL and YÜCEL, MELEK D. (2001) "Avalanche and Bit Independence Properties for the Ensembles of Randomly Chosen n \times n S-Boxes," Turkish Journal of Electrical Engineering and Computer Sciences: Vol. 9: No. 2, Article 3. Available at: https://journals.tubitak.gov.tr/elektrik/vol9/iss2/3