A characterisation of S-box fitness landscapes in cryptography

Abstract

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. CCS CONCEPTS• Software and its engineering → Software notations and tools; Search-based software engineering;

Artificial Intelligence for the Design of Symmetric Cryptographic Primitives

“…Along the same direction, it would also be useful to study more in detail the difficulty of the combinatorial optimization problems related to Boolean functions and S-boxes, for example, by employing fitness landscape analysis techniques on the underlying search spaces. As far as the authors know, only Jakobovic et al [25] performed an analysis of the fitness landscape associated with S-boxes, and there are still several avenues for further research that could be explored in this domain.…”

A Survey of Advances in Landscape Analysis for Optimisation

“…Landscape analysis has also been used to understand real-world problems. Examples include: the design of wind turbines [87]; university course timetabling [88]; genetic improvement of software [89][90][91]; automated test case generation for software testing [92,93]; computational protein design [94]; design of substitution boxes in cryptography [95]; hyperparameter optimisation for metaheuristics [96]; and building energy optimisation [97].…”

Toward more efficient heuristic construction of Boolean functions

“…This assumption is based on previous basin-edges models, which also do not consider weighted edges [29,31,32].…”

“…Using a similar methodology, the first landscape analysis in the greater field of security investigated cryptographic S-Boxes [32]. For the chosen fitness functions and two neighborhood operators (considered in isolation), it was observed that the number of local optima is substantial, and a conjecture has been made that links S-Boxes of odd dimensions to their problem difficulty.…”