Minimum Distance between Bent and 1-Resilient Boolean Functions

In this paper we study the minimum distance between the set of bent functions and the set of 1-resilient Boolean functions and present a lower bound on that. The bound is proved to be tight for functions up to 10 input variables. As a consequence, we present a strategy to modify the bent functions, by toggling some of its outputs, in getting a large class of 1-resilient functions with very good nonlinearity and autocorrelation. In particular, the technique is applied upto 12-variable functions and we show that the construction provides a large class of 1-resilient functions reaching currently best known nonlinearity and achieving very low autocorrelation values which were not known earlier. The technique is sound enough to theoretically solve some of the mysteries of 8-variable, 1-resilient functions with maximum possible nonlinearity. However, the situation becomes complicated from 10 variables and above, where we need to go for complicated combinatorial analysis with trial and error using computational facility.