Permutations satisfying $(P_1)$ and $(P_2)$ properties and $\ell$-optimal bent functions

TL;DR

This paper explores special permutation properties $(P_1)$ and $(P_2)$, introduces $ ext{ell}$-optimal bent functions with minimal linearity, and develops methods to generate diverse bent functions outside the Maiorana-McFarland class, enhancing understanding of their structure.

Contribution

It provides a detailed analysis of permutation properties, constructs $ ext{ell}$-optimal bent functions, and generalizes the swapping variables method to produce many bent functions outside the known classes.

Findings

Permutations with properties $(P_1)$ and $(P_2)$ can generate bent functions outside the Maiorana-McFarland class.

Explicit constructions of $ ext{ell}$-optimal bent functions within the $D_0$ class are provided.

A subclass of $D_0$ has 5-valued spectra decompositions, and the generalized swapping variables method produces large families of bent functions outside $M^lat$.

Abstract

An important classification of permutations over $\mathbb{F}_2^m$, suitable for constructing Maiorana-McFarland bent functions on $\mathbb{F}_2^m \times \mathbb{F}_2^m$ with the unique $M$-subspace of maximal dimension, was recently considered in Pasalic et al. (IEEE Trans. Inf. Theory 70(6): 4464-4477, 2024). More precisely, two properties called $(P_1)$ and $(P_2)$ were introduced and a generic method of constructing permutations having the property $(P_1)$ was presented, whereas no such results were provided related to the $(P_2)$ property. In this article, we provide a deeper insight on these properties, their mutual relationship, and specify some explicit classes of permutations having these properties. Such permutations are then employed to generate a large variety of bent functions outside the completed Maiorana-McFarland class $M^\#$. We also introduce $\ell$-optimal bent functions as bent functions with the lowest possible linearity index; such functions can be considered as opposite to Maiorana-McFarland bent functions. We give explicit constructions of $\ell$-optimal bent functions within the $D_0$ class, which in turn can be employed in certain secondary constructions of bent functions for providing even more classes of bent functions that are provably outside $M^\#$. Moreover, we demonstrate that a certain subclass of $D_0$ has an additional property of having only 5-valued spectra decompositions. Finally, we generalize the so-called "swapping variables" method which then allows us to specify large families of bent functions outside $M^\#$. In this way, we give a better explanation of the origin of bent functions in dimension eight, since the vast majority of them is outside ${M}^\#$.