New Bent Functions from Permutations and Linear Translators

Abstract

Starting from the secondary construction originally introduced by Carlet [“On Bent and Highly Nonlinear Balanced/Resilient Functions and Their Algebraic Immunities”, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 2006], that we shall call “Carlet‘s secondary construction”, Mesnager has showed how one can construct several new primary constructions of bent functions. In particular, she has showed that three tuples of permutations over the finite field $\mathbb {F}_{2^m}$ such that the inverse of their sum equals the sum of their inverses give rise to a construction of a bent function given with its dual. It is not quite easy to find permutations satisfying such a strong condition $(\mathcal {A}_m)$. Nevertheless, Mesnager has derived several candidates of such permutations in 2015, and showed in 2016 that in the case of involutions, the problem of construction of bent functions amounts to solve arithmetical and algebraic problems over finite fields.

This paper is in the line of those previous works. We present new families of permutations satisfying $(\mathcal {A}_m)$ as well as new infinite families of permutations constructed from permutations in both lower and higher dimensions. Our results involve linear translators and give rise to new primary constructions of bent functions given with their dual. And also, we show that our new families are not in the class of Maiorana-McFarland in general.