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

摘要

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 2(n- 1) - 2(4r- 1) or 2(n- 1) - 2(5r- 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.