Experimental Results on Higher-Order Differential Spectra of 6 and 8-bit Invertible S-Boxes

Designing S-box with good cryptographic properties still remains as one of the most important areas of research in symmetric key cryptography. For quite sometime, inverse function (x↦ x- 1, i.e., x2n-2 ) over F2n has been the most popular choice for S-boxes due to good resistance against differential and linear cryptanalysis. Very recently Tang et. al. (2020) proved that inverse function admits a bias (error) of 12n-2 when considered in its second-order differential spectrum. In this paper we present experimental results related to higher-order differential spectrum of multiplicative inverse functions for n= 6 and 8 and compare the result with APN permutation for n= 6. In particular, we observe that APN permutation over F26 has larger bias in its second-order differential spectrum with probability 18 (12n-2 ). This fact admits the possibility of higher-order differential attacks against block ciphers which employ APN permutations as a nonlinear layer.