A new construction of highly nonlinear S-boxes

In this paper we give a new construction of highly nonlinear vectorial Boolean functions. This construction is based on coding theory, more precisely we use concatenation to construct Boolean functions from codes over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}_q$\end{document} containing a first-order generalized Reed–Muller code. As it turns out this construction has a very compact description in terms of Boolean functions, which is of independent interest. The construction allows one to design functions with better nonlinearities than known before.