A new method of constructing $$(k+s)$$ -variable bent functions based on a family of s-plateaued functions on k variables

Abstract

It is important to study the new construction methods of bent functions. In this paper, we first propose a secondary construction method of \((k+s)\)-variable bent function g through a family of s-plateaued functions \(f_0,f_1,\ldots ,f_{2^s-1}\) on k variables with disjoint Walsh supports, which can be obtained through any given \((k-s)\)-variable bent function f by selecting \(2^s\) disjoint affine subspaces \(S_0,S_1,\ldots ,S_{2^s-1}\) of \({\mathbb {F}}_2^k\) with dimension \(k-s\) to specify the Walsh support of these s-plateaued functions respectively, where s is a positive integer and \(k-s\) is a positive even integer. The dual functions of these newly constructed bent functions are determined. This secondary construction method of bent functions has a great improvement in counting. As a generalization, we find that the one initial \((k-s)\)-variable bent function f can be replaced by several different \((k-s)\)-variable bent functions. Compared to the first construction method, the latter one gives much more bent functions. It is worth mentioning that it can give all the 896 bent functions on 4 variables.