Fractional Duals of the Poisson Process on Time Scales with Applications in Cryptography

Abstract

A super-structure system for probability densities, covering not just typical types but also fractional ones, was developed using the time scale theory. From a mathematical point of view, we discover duals of the Poisson process on the time scale T=R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {T}=\mathbb {R}$$\end{document} for the time scales T=Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {T}=\mathbb {Z}$$\end{document} and T=qZ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb {T}=q<^>{\mathbb {Z}},$$\end{document} evaluating del-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla -$$\end{document}calculus and Delta-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta -$$\end{document}calculus. Also, we search the fractional extensions of the Poisson process on these time scales and detect duals of them. A simulation allows for comparing the nabla and delta types of the observed distributions, not just typical types but also fractional ones. As an application, we also propose new substitution boxes (S-boxes) using the proposed stochastic models and compare the performance of S-boxes created in this way. Given that the S-box is the core for confusion in Advanced Encryption Standard (AES), the formation of these new S-boxes represents an interesting application of these stochastic models.