Improving the lower bound on the maximum nonlinearity of 1-resilient Boolean functions and designing functions satisfying all cryptographic criteria

In this article we improve the lower bound on the maximum nonlinearity of 1-resilient Boolean functions, for n even, by proposing a method of constructing this class of functions attaining the best nonlinearity currently known. Thus for the first time, at least for small values of n, the upper bound on nonlinearity can be reached in a deterministic manner in difference to some heuristic search methods proposed previously. The nonlinearity of these functions is extremely close to the maximum nonlinearity attained by bent functions and it might be the case that this is the highest possible nonlinearity of 1-resilient functions. Apart from this theoretical contribution, it turns out that the cryptographic properties of these functions are overall good apart from their moderate resistance to fast algebraic attacks (FAA). This weakness is repaired by a suitable modification of the original functions giving a class of balanced functions with almost optimal resistance to FAA whose nonlinearity is better than the nonlinearity of other methods.