Paper 2015/077

On the Primary Constructions of Vectorial Boolean Bent Functions

Yuwei Xu and Chuankun Wu

Abstract

Vectorial Boolean bent functions, which possess the maximal nonlinearity and the minimum differential uniformity, contribute to optimum resistance against linear cryptanalysis and differential cryptanalysis for the cryptographic algorithms that adopt them as nonlinear components. This paper is devoted to the new primary constructions of vectorial Boolean bent functions, including four types: vectorial monomial bent functions, vectorial Boolean bent functions with multiple trace terms, $\mathcal{H}$ vectorial functions and $\mathcal{H}$-like vectorial functions. For vectorial monomial bent functions, this paper answers one open problem proposed by E. Pasalic et al. and characterizes the vectorial monomial bent functions corresponding to the five known classes of bent exponents. For the vectorial Boolean bent functions with multiple trace terms, this paper answers one open problem proposed by A. Muratović-Ribić et al., presents six new infinite classes of explicit constructions and shows the nonexistence of the vectorial Boolean bent functions from $\mathbb{F}_{2^{n}}$ to $\mathbb{F}_{2^{k}}$ of the form $\sum_{i=1}^{2^{k-2}}Tr^{n}_{k}(ax^{(2i-1)(2^{k}-1)})$ with $n=2k$ and $a\in\mathbb{F}_{2^{k}}^{*}$. Moreover, $\mathcal{H}$ vectorial functions are further characterized. In addition, a new infinite class of vectorial Boolean bent function named as $\mathcal{H}$-like vectorial functions are derived, which includes $\mathcal{H}$ vectorial functions as a subclass.