# A characterisation of S-box fitness landscapes in cryptography

## TL;DR

This paper analyzes the fitness landscapes of S-boxes in cryptography to understand why heuristic methods struggle, revealing that most starting points lead to local optima despite interconnected search spaces.

## Contribution

It provides the first detailed fitness landscape analysis of S-box design problems, highlighting the landscape's complexity and the prevalence of local optima.

## Key findings

- Most initial points lead to local optima.
- S-box fitness landscapes are highly interconnected.
- Difficulty in heuristic design is due to landscape complexity.

## 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.