Comparative Analysis of Homomorphic Encryption Based on Elliptic Curves over Rings

Data security in the cloud can be achieved through encryption. However, if processing encrypted messages still requires decryption, there is a risk that decryption keys could be obtained by attackers. Homomorphic encryption offers a solution by allowing the processing of encrypted data without the need for decryption. There are three main categories of homomorphic encryption: partial homomorphic encryption (PHE), somewhat homomorphic encryption (SWHE), and fully homomorphic encryption (FHE). Among these, PHE is preferred when cost efficiency, lower storage requirements, and better data processing speed are needed. Homomorphic encryption can be implemented using various algorithms, including symmetric key algorithms and public key algorithms. Elliptic curve algorithms, a type of public key algorithm, provide efficiency with smaller key sizes while maintaining security levels comparable to integer-based algorithms like RSA. In this study, elliptic curves operating over algebraic ring structures are used to enhance the security of the PHE algorithm. Three quotient rings were successfully implemented in the PHE algorithm, and performance analysis was conducted. The performance analysis indicates that encryption execution time increases with larger generated random values and the number of messages. Decryption execution time also increases with larger private key sizes and the number of messages. Furthermore, the analysis of time differences between the three elliptic curves in encryption and decryption processes shows that the differences are not significant.