On generalizing the power function exponent constructions with genetic programming

Claude Carlet, Domagoj Jakobovic, Stjepan Picek

Abstract: Many works are investigating Almost Perfect Nonlinear (APN) functions and, in particular, APN power functions. Such functions are of the form F(x) = xd, and they have practical relevance as they reach in characteristic 2 the best possible differential uniformity. This work investigates whether genetic programming (GP) can "reinvent" the known expressions used to obtain exponent values d resulting in APN functions. The ultimate goal is to find classes of exponents that would be "transversal" to the known infinite classes of APN exponents, and would contain new APN exponents (for values of n necessarily larger than those for which an exhaustive search could be made so far). This would be already a breakthrough, and our hope is to find this way new infinite classes of APN exponents. Our results show this is possible but difficult, and a careful trade-off between finding new values and skipping known values is required.