Homomorphic-based Encryption using Weil Pairing: A Foundation for Fully Homomorphic Encryption on Elliptic Curves

We present a novel cryptographic scheme that integrates homomorphic encryption with elliptic curve cryptography through the application of Weil pairing theory. Our approach, termed Elliptic Curve Homomorphic Cryptography (ECHC), leverages the isomorphic relationship between elliptic curves and external direct products of Zₙ established by the Weil theorem. This construction provides a foundation for achieving fully homomorphic encryption on elliptic curves while maintaining computational efficiency. We provide comprehensive mathematical proofs, security analysis, and complexity evaluations demonstrating that ECHC offers improved performance over traditional homomorphic encryption schemes with security levels comparable to established elliptic curve cryptosystems. The proposed scheme supports both additive and multiplicative homomorphic operations, making it a viable stepping stone toward practical fully homomorphic encryption implementations.