Maximal nonlinearity in balanced boolean functions with even number of inputs, revisited

The problem of obtaining maximal nonlinearity in Boolean functions is well researched, both from the cryptographic and the evolutionary computation side. However, the results are still not conclusive enough to be able to show how good a heuristic approach is when tackling this problem. In this paper, we investigate how to obtain the maximal possible nonlinearity in balanced Boolean functions, but we also analyze how difficult is the problem itself. In order to do so, we conduct experiments with Estimation of distribution algorithms as well as the fitness landscape analysis and the deception analysis. Our results indicate that the first difficulties arise from the inappropriate fitness function and representation of solutions coupled with a huge search space. The fitness landscape analysis does not reveal any significant differences that could justify the assumed jump in problem difficulty when going from Boolean functions with 6 inputs to those with 8 inputs. Finally, we show that this problem is not order-1 deceptive.