A novel technique of image encryption through projective coordinates of elliptic curve

Efficient multiple pseudo-random number sequences (PRNS) and substitution boxes (S-boxes) are two of the most consequential construction blocks jointly assumed commonly for secure data encryption. Multiple aspects pave the way to address large-scale multimedia data. However, the computational efforts on multiple constructions may limit the required ciphering. Therefore, reducing the computational cost of various patterns, such as PRNS and S-boxes, is the core requirement for an efficient cryptosystem. For this achievement, this article addresses the challenge of constructing secure S-boxes with enhanced nonlinearity (NL) and keyspace in image encryption. Our technique aims to bolster security in digital image encryption methods by prioritizing robustness over conventional complexity. We present a novel and efficient cryptosystem that utilizes Projective Coordinates (PCs) of Elliptic Curves (ECs) for encrypting digital images. Initially, we leverage the power of PCs of ECs over the set of integers modulo pr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}^{r}$$\end{document}, where p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document} is prime and r=9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r = 9$$\end{document}. Using ECs and applying trace mappings, we create optimal 8×8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$8\times 8$$\end{document} S-boxes for pixel substitution in digital images. Moreover, the proposed scheme generates 2108\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2}^{108}$$\end{document} S-boxes in a single case of the proposed scheme. In addition, Pseudo-Random Numbers (PRNs) are generated from ECs over modulo pr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}^{r}$$\end{document} to enhance security. Further, computational experiments demonstrate that our proposed cryptosystem offers superior protection against linear, differential, and statistical attacks compared to existing cryptosystems.