A new lower bound on the second-order nonlinearity of a class of monomial bent functions

Abstract

The second-order nonlinearity can provide knowledge on classes of Boolean functions used in symmetric-key cryptosystems, coding theory, and Gowers norm. It is well-known that bent functions possess the highest nonlinearity on even number of variables and so it will be of great interest to investigate the lower bound on the second-order nonlinearity of such functions. In 2008, Canteaut et al. (Finite Fields Appl. 14(1), 221–241, 2) found a class of monomial bent functions on n = 6r variables and proved that their derivatives have nonlinearities either 2n− 1 − 24r− 1 or 2n− 1 − 25r− 1. In this paper, we completely determine the distributions of the nonlinearities of the derivatives of this class of bent functions. Further, we present a new lower bound on the second-order nonlinearity of this class of bent functions, which is better than the previous one.