A Lower Bound of the Second-order Nonlinearities of Boolean Bent Functions

In this paper we find a lower bound of the second-order nonlinearities of Boolean bent functions of the form f(x) = Tr-1(n) (alpha(1)x(d1) + alpha(2)x(d2)), where d(1) and d(2) are Niho exponents. A lower bound of the second-order nonlinearities of these Boolean functions can also be obtained by using a recent result of Li, Hu and Gao (eprint.iacr.org/2010/009.pdf). It is shown in Section 3, by a direct computation, that for large values of n, the lower bound obtained in this paper are better than the lower bound obtained by Li, Hu and Gao.