Extending Elliptic-Curve Cryptography to Multi-Dimensional Elliptical Surfaces

Abstract

This research introduces a novel cryptographic framework that extends classical elliptic curve cryptography (ECC) to higher-dimensional elliptic surfaces using matrix-based transformations. Traditional ECC operates over two-dimensional curves with scalar multiplication as the core operation. In contrast, our proposed methodology utilizes matrix embeddings of elliptic curve points and matrix-based key transformations to enable secure key exchange. By employing 2×2 matrices with 64-bit entries, the scheme preserves cryptographic strength equivalent to standard 256-bit ECC while enhancing structural efficiency and scalability. This matrix approach leads to simultaneous, multidimensional instances of the Elliptic Curve Discrete Logarithm Problem (ECDLP), thereby increasing resistance to known attacks such as Pollard’s rho and the MOV reduction. Furthermore, we integrate bilinear pairings into the multi-dimensional context to support advanced cryptographic constructs, including identity-based encryption and multi-party agreements. Numerical examples validate the feasibility of our scheme, and comparative analysis highlights improved security, scalability, and quantum resistance with manageable computational overhead. Additionally, we propose a class of nonlinear differential equations where elliptic curves emerge as special cases, opening avenues for future research in mathematical cryptography. The proposed model demonstrates strong potential for next-generation cryptographic applications, particularly in lightweight, secure, and post-quantum environments.