This paper investigates the influence of genotype size on evolutionary algorithms' performance. We consider genotype compression (where genotype is smaller than phenotype) and expansion (genotype is larger than phenotype) and define different strategies to reconstruct the original variables of the phenotype from both the compressed and expanded genotypes. We test our approach with several evolutionary algorithms over three sets of optimization problems: COCO benchmark functions, modeling of Physical Unclonable Functions, and neural network weight optimization. Our results show that genotype expansion works significantly better than compression, and in many scenarios, outperforms the original genotype encoding. This could be attributed to the change in the genotype-phenotype mapping introduced with the expansion methods: this modification beneficially transforms the domain landscape and alleviates the search space traversal.

We investigate whether it is possible to evolve cryptographically strong S-boxes that have additional constraints on their structure. We investigate two scenarios: where S-boxes additionally have a specific sum of values in rows, columns, or diagonals and the scenario where we check that the difference between the Hamming weights of inputs and outputs is minimal. The first case represents an interesting benchmark problem, while the second one has practical ramifications as such S-boxes could offer better resilience against side-channel attacks. We explore three solution representations by using the permutation, integer, and cellular automata-based encoding. Our results show that it is possible to find S-boxes with excellent cryptographic properties (even optimal ones) and reach the required sums when representing S-box as a square matrix. On the other hand, for the most promising S-box representation based on trees and cellular automata rules, we did not succeed in finding S-boxes with small differences in the Hamming weights between the inputs and outputs, which opens an interesting future research direction. Our results for this scenario and different encodings inspired a mathematical proof that the values reached by evolutionary algorithms are the best possible ones.

Substitution boxes (S-boxes) are nonlinear mappings that represent one of the core parts of many cryptographic algorithms (ciphers). If S-box does not possess good properties, a cipher would be susceptible to attacks. To design suitable S-boxes, we can use heuristics as it allows significant freedom in the selection of required cryptographic properties. Unfortunately, with heuristics, one is seldom sure how good a trade-off between cryptographic properties is reached or if optimizing for one property optimizes implicitly for another property. In this paper, we consider what is to the best of our knowledge, the most detailed analysis of trade-offs among S-box cryptographic properties. More precisely, we ask questions if one property is optimized, what is the worst possible value for some other property, and what happens if all properties are optimized. Our results show that while it is possible to reach a large variety of possible solutions, optimizing for a certain property would commonly result in good values for other properties. In turn, this suggests that a single-objective approach should be a method of choice unless some precise values for multiple properties are needed.