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

Abstract

In this paper we find a lower bound of the second-order nonlinearities of Boolean bent functions of the form ${\rm f}({\rm x}) = {\rm Tr}_{1}^{{\rm n}}(\rmalpha_{1}{\rm x}^{{\rm d}_{1}} + \rmalpha_{2}{\rm x}^{{\rm d}_{2}})$, where d₁ and d₂ 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.