The Study of Data Security Algorithms Based on Elliptic Curves

Abstract

Many data security algorithms have been developed based on elliptic curves (ECs) to address data encryption scenarios. However, these studies contain some imperfections. This work aims to remove the existing limitations and construct robust encryption techniques to tackle unauthorized access to data adequately. We have developed a hash-sharing mechanism using Neal Koblitz’s technique to create a secure and efficient key-sharing algorithm. Additionally, the hash-based key has been utilized as the initial parameters of a chaotic map to produce chaotic data, which is then used to construct TetraVex tiles for the image encryption algorithm. Furthermore, we have improved the proposed hash-sharing approach by utilizing Diffie Hellman (DH) to provide an efficient sharing technique. Quadratic Residue (QR)/Non- Quadratic Residue (NQR) elements of the EC y− coordinates have been used to perform pixel swapping of images, with diffusion performed using DNA, providing a robust encryption algorithm. Moreover, in the era of quantum computing, the DH key exchange protocol is vulnerable to brute force attacks if the parameters are not chosen carefully. We have added a security layer using a secret generator for the DH key exchange protocol, which is used in multiple RGB image encryption algorithms. In addition, a predominant approach for S-box generation involves utilizing Mordell ECs, chosen for their high security with small key space attributes. However, prevalent S-box algorithms derived from these ECs exhibit structural and algorithmic limitations, rendering them less adept for deployment in small devices. We present a novel approach to overcome these challenges. We have proposed an efficient lattice ordering-based S-box algorithm by employing a Mordell EC as its foundation. This endeavor seeks to rectify existing deficiencies, resulting in an alternative S-box characterized by improved algorithmic complexity and relatively diminished computation time compared to existing works centered around the Mordell EC. Additionally, discrepancies were identified in the structure used to generate the S-box, including the failure to generate a zero element and the inability to generate the S-box when the cardinality of the codomain set is less than prime p. We have addressed the identified algorithmic and mathematical limitations, offering a promising S-box generation solution for cryptographic applications. 2 The reported approach exhibits the solution to the overwhelming hash-sharing issue, enhanced efficiency, and the capability of the encryption algorithm to withstand various attacks from the community of cryptanalysts. It provides better alternative S-box generators to address the identified limitations.