A construction of bent functions with optimal algebraic degree and large symmetric group

Abstract

As maximal, nonlinear Boolean functions, bent functions have many theoretical and practical applications in combinatorics, coding theory, and cryptography. In this paper, we present a construction of bent function fa,S with n = 2m variables for any nonzero vector (Formula Presented.) and subset S of Fm2 satisfying a + S = S. We give a simple expression of the dual bent function of fa,S and prove that fa,S has optimal algebraic degree m if and only if |S| ≡ 2(mod4). This construction provides a series of bent functions with optimal algebraic degree and large symmetric group if a and S are chosen properly. We also give some examples of those bent functions fa,S and their dual bent functions.