Construction of 1-Resilient Boolean Functions with Optimal Algebraic Immunity and Good Nonlinearity

This paper presents a construction for a class of 1-resilient Boolean functions with optimal algebraic immunity on an even number of variables by dividing them into two correlation classes, i.e. equivalence classes. From which, a nontrivial pair of functions has been found by applying the generating matrix. For $n$ is small (e.g. $n=6$), a part of these functions achieve almost optimal nonlinearity. Apart from their good nonlinearity, the functions reach Siegenthaler\u27s \cite{Siegenthaler} upper bound of algebraic degree. Furthermore, a class of 1-resilient functions on any number $n>2$ of variables with at least sub-optimal algebraic immunity is provided