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

Abstract

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 ($x\mapsto x^{-1}$, i.e., $x^{2^n-2}$) over $\mathbb {F}_{2^n}$ 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 $\frac{1}{2^{n-2}}$ 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 $\mathbb {F}_{2^6}$ has larger bias in its second-order differential spectrum with probability $\frac{1}{8}$ ( $\frac{1}{2^{n-2}}$). This fact admits the possibility of higher-order differential attacks against block ciphers which employ APN permutations as a nonlinear layer.