Gowers $U_3$ norm of some classes of bent Boolean functions

Abstract

The Gowers $U_3$ norm of a Boolean function is a measure of its resistance to quadratic approximations. It is known that smaller the Gowers $U_3$ norm for a Boolean function larger is its resistance to quadratic approximations. Here, we compute Gowers $U_3$ norms for some classes of Maiorana–McFarland bent functions. In particular, we explicitly determine the value of the Gowers $U_3$ norm of Maiorana–McFarland bent functions obtained by using APN permutations. We prove that this value is always smaller than the Gowers $U_3$ norms of Maiorana–McFarland bent functions obtained by using differentially $\delta$-uniform permutations, for all $\delta \ge 4$. We also compute the Gowers $U_3$ norms for a class of cubic monomial functions, not necessarily bent, and show that for $n=6$, these norm values are less than that of Maiorana–McFarland bent functions. Further, we computationally show that there exist 6-variable functions in this class which are not bent but achieve the maximum second-order nonlinearity for 6 variables.