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Cartesian Genetic Programming Approach for Generating Substitution Boxes of Different Sizes

Substitution Boxes (S-boxes) play an important role in many modern-day cryptography algorithms. Indeed, without carefully chosen S-boxes many ciphers would be easy to break. The design of suitable S-boxes attracts a lot of attention in cryptography community. The evolutionary algorithms (EAs) community also had several attempts to evolve S-boxes with good cryptographic properties. When using EAs one usually uses permutation representation in order to preserve the bijectivity of the resulting S-boxes. In this paper we experiment with Cartesian Genetic Programming (CGP) and Genetic Programming (GP) in order to evolve bijective S-boxes of various sizes that have good cryptographic properties. Besides the standard CGP representation, we use an approach that allows CGP and GP to be mapped to the permutation encoding.