Second-Order Nonlinearity of a Boolean Function Class with Low Spectra

Abstract

In 2015, Cao and Hu (Cao, X., Hu, L.: Two boolean functions with five-valued walsh spectra and high nonlinearity. International Journal of Foundations of Computer Science, pp. 537–556 (2015)) introduced a certain class of Boolean functions, possessing low Walsh spectra, high nonlinearity, and high algebraic degree. For this class of Boolean functions, computation of higher-order nonlinearities (even second-order) is a tedious task. Therefore, in this article, we study the lower bound on the second-order nonlinearity of the above-mentioned class of Boolean functions for $$n=4.$$ Also, we deduce that the bound, thus obtained is the maximum possible bound. We also demonstrated that our lower bound is greater than the lower bound on the second-order nonlinearity of other classes of cubic Boolean functions.