Generic constructions of $\mathbb {Z}$-bent functions

Abstract

$\mathbb {Z}$-bent functions, mappings from $\mathbb {F}_2^n$ to a subset of $\mathbb {Z}$, were introduced by Dobbertin and Leander (Des Codes Cryptogr 49:3–22, 2008) as an attempt to capture the origin of standard bent functions and in particular to understand better a recursive construction framework of bent functions. Nevertheless, many questions have been left open in Dobbertin and Leander (2008) such as efficient construction methods of $\mathbb {Z}$-bent functions of different levels, where these levels specify precisely a subset of $\mathbb {Z}$ containing both the image values of f and its normalized Fourier coefficients. In this article, using different design techniques, we provide several generic construction methods of $\mathbb {Z}$-bent functions of arbitrary levels, thereby solving an open problem posed in Dobbertin and Leander (2008). On the other hand, apart from an independent theoretical interest in these objects, our rigor treatment of the so-called gluing technique reveals that this approach is equivalent to a classical concept of concatenation. More precisely, gluing four suitable n-variables $\mathbb {Z}$-bent functions of level one to obtain an $(n+2)$-variable bent function directly corresponds to a concatenation of four suitable n-variable Boolean functions. Nevertheless, the recursive framework based on $\mathbb {Z}$-bent functions remains to be investigated further in this context.