Evolution of Walsh Transforms with Genetic Programming

The design of Boolean functions which exhibit high-quality cryptography properties is a crucial aspect when implementing secure stream ciphers. To this end, several methods have been proposed to search for secure Boolean functions, and, among those, evolutionary algorithms play a prominent role. In this paper, Genetic Programming (GP) is applied for the evolution of Boolean functions in order to maximize one essential property for strong cryptography functions, namely non-linearity. Differently from other approaches, the evolution happens in the space of Walsh Transforms, instead of using a direct representation of the Boolean functions. Specifically, we evolve coefficients of the Walsh Transform to obtain a generic Walsh spectrum, from which it is possible, through spectral inversion, to obtain a pseudo-Boolean function that, consequently, can be mapped to (one of) the nearest Boolean one. Since that function might not be unique, we propose a strategy in which balancedness, another important cryptography property, is preserved as much as possible. To show that the evolutionary search is actually effective in this task, we evolved Boolean functions from 6 to 16 variables. The results show that not only GP is effective in evolving Boolean functions with high non-linearity, but also that balanced functions are discovered.