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.
KeywordsGowers uniformity norms Second-order nonlinearity Maiorana–McFarland bent functions Differentially \(\delta \)-uniform functions APN functions
Mathematics Subject Classification06E30 94C10
The authors thank Palash Sarkar for suggesting the problem and for several long discussions. We are also thankful for the useful comments of the reviewers that has immensely helped us to significantly improve both technical and editorial quality of the manuscript. Bimal Mandal acknowledges IIT Roorkee for supporting his research.
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