A characterisation of S-box fitness landscapes in cryptography

Substitution Boxes (S-boxes) are nonlinear objects often used in the design of cryptographic algorithms. The design of high quality S-boxes is an interesting problem that attracts a lot of attention. Many attempts have been made in recent years to use heuristics to design S-boxes, but the results were often far from the previously known best obtained ones. Unfortunately, most of the effort went into exploring different algorithms and fitness functions while little attention has been given to the understanding why this problem is so difficult for heuristics. In this paper, we conduct a fitness landscape analysis to better understand why this problem can be difficult. Among other, we find that almost each initial starting point has its own local optimum, even though the networks are highly interconnected.