Constructions of 2-resilient rotation symmetric Boolean functions with odd number of variables

Abstract

In this paper, a new method for constructing 2-resilient rotation symmetric Boolean functions with odd number of variables is presented. Based on an equivalent characterization of this class of functions and the relation between the orbit matrices and their complements, a system of equations about 2-tuples distribution matrix is established. Then the constructions of some 2-resilient rotation symmetric Boolean functions with odd number of variables can be converted into the solutions of the system of equations. We also give a sufficient condition for the constructed functions to have the maximum algebraic degree. Moreover, we provide a lower bound on the number of nonlinear 2-resilient rotation symmetric Boolean functions with odd number of variables. Particularly, the lower bound is 152 for seven number of variables.