Efficient Construction of a Substitution Box Based on a Mordell Elliptic Curve Over a Finite Field

TL;DR

This paper introduces a fast, efficient method for constructing cryptographically strong S-boxes using Mordell elliptic curves over prime fields, enhancing security and reducing computational resources.

Contribution

It presents a novel linear-time, constant-space S-box construction method based on Mordell elliptic curves, improving efficiency over existing elliptic curve methods.

Findings

The method generates secure S-boxes with comparable cryptographic strength.

It operates in linear time and constant space, outperforming previous approaches.

Computational results confirm the security and efficiency of the proposed scheme.

Abstract

Elliptic curve cryptography (ECC) is used in many security systems due to its small key size and high security as compared to the other cryptosystems. In many well-known security systems substitution box (S-box) is the only non-linear component. Recently, it is shown that the security of a cryptosystem can be improved by using dynamic S-boxes instead of a static S-box. This fact necessitates the construction of new secure S-boxes. In this paper, we propose an efficient method for the generation of S-boxes based on a class of Mordell elliptic curves (MECs) over prime fields by defining different total orders. The proposed scheme is developed in such a way that for each input it outputs an S-box in linear time and constant space. Due to this property, our method takes less time and space as compared to all existing S-box construction methods over elliptic curve. Furthermore, it is shown by the computational results that the proposed method is capable of generating cryptographically strong S-boxes with comparable security to some of the existing S-boxes constructed over different mathematical structures.