Evolving Constructions for Balanced, Highly Nonlinear Boolean Functions
This work addresses a specific challenge in cryptography for designing secure Boolean functions, but it is incremental as it builds upon known constructions like indirect sum.
The paper tackled the problem of evolving constructions for balanced, highly nonlinear Boolean functions, which are difficult to scale for larger sizes, and found that genetic programming can discover constructions that generalize across multiple sizes and identify equivalent syntactic representations.
Finding balanced, highly nonlinear Boolean functions is a difficult problem where it is not known what nonlinearity values are possible to be reached in general. At the same time, evolutionary computation is successfully used to evolve specific Boolean function instances, but the approach cannot easily scale for larger Boolean function sizes. Indeed, while evolving smaller Boolean functions is almost trivial, larger sizes become increasingly difficult, and evolutionary algorithms perform suboptimally. In this work, we ask whether genetic programming (GP) can evolve constructions resulting in balanced Boolean functions with high nonlinearity. This question is especially interesting as there are only a few known such constructions. Our results show that GP can find constructions that generalize well, i.e., result in the required functions for multiple tested sizes. Further, we show that GP evolves many equivalent constructions under different syntactic representations. Interestingly, the simplest solution found by GP is a particular case of the well-known indirect sum construction.