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

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.