Improved Rijndael-Like S-Box and Its Transform Domain Analysis

In this paper, we propose a simple scheme which produces a new S-box from a given S-box. We use the well-known conversion technique between the polynomial functions over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{F}_{2^n}}$\end{document} and the boolean functions from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{F}_2^n}$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{F}_2}$\end{document}. We have applied the scheme to Rijndael S-box and obtained 29 new S-boxes, of which only one is a bijection with better algebraic expression than the original Rijndael S-box and has the same spectral properties as the original Rijndael S-box. All others turned out to be non-bijective, and have different spectral properties, and hence, they all are inequivalent to the original as boolean functions.