Homomorphic encryption (HE) schemes based on the Ring Learning with Errors (RLWE) problem have been rapidly developed and widely applied to secure computation tasks, such as privacy-preserving deep learning inference and database queries. However, existing HE schemes mainly focus on the feasibility and efficiency of homomorphic computation, while practical requirements including ciphertext copyright protection, provenance tracking, and computation supervision remain largely unexplored. In this work, we propose a watermarking technique for RLWE-based HE ciphertexts. By exploiting the algebraic structure of RLWE polynomials, we embed watermark information into ciphertext noise without affecting plaintext correctness. To address watermark degradation caused by homomorphic operations, we introduce two practical schemes. The first scheme, ARWMark, leverages noise stratification to achieve robustness against homomorphic additive operations. The second scheme, MRWMark, is based on the roots of a linear equation and supports zero-bit watermarking while remaining robust against both homomorphic addition and multiplication. We provide rigorous theoretical analysis demonstrating that the proposed schemes preserve the original security of HE while maintaining correctness and watermark robustness. Extensive experiments further validate the effectiveness and practicality of the proposed watermarking schemes.

翻译：暂无翻译