A Feasible Method for Computing the Greatest Common Divisor and Least Common Multiple Under Ciphertext Based on Fully Homomorphic Encryption

With the rapid development of cloud computing and privacy-preserving computation, how to perform secure computation while ensuring data privacy has become an important research topic. As fundamental mathematical operations, the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) are widely used in cryptography, data mining, and other fields. However, traditional GCD algorithms cannot be directly applied to encrypted data. This paper proposes an efficient method for computing GCD and LCM on ciphertexts based on the BFV homomorphic encryption scheme. By leveraging the homomorphic properties of the BFV scheme, the method designs a secure and efficient protocol for ciphertext-based GCD/LCM computation. The protocol, through steps such as preprocessing, size comparison, difference calculation, and iterative optimization, enables direct computation of GCD/LCM on encrypted data without decryption, outputting encrypted results and effectively protecting data privacy. Theoretical analysis and experimental results show that the method can accurately compute GCD and LCM under ciphertext, and the ciphertext depth of the scheme is optimized to reduce the communication overhead, which provides a new perspective for secure computation under privacy protection.