如何对基于 RLWE 的同态密文进行水印处理
Ciphertext-Native Watermarking for RLWE-Based Homomorphic Encryption
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中文总结 AI 辅助
针对 RLWE 同态加密方案忽视密文版权保护等实际需求的问题，提出水印技术，利用 RLWE 多项式代数结构嵌入水印，给出 ARWMark 和 MRWMark 两种方案，经理论分析和实验验证其有效性且不损原安全性。
中文摘要 AI 辅助
近年来，基于环上误差学习（RLWE）问题的同态加密（HE）方案迅速发展并广泛应用于安全计算任务。但现有大多数 HE 方案主要关注同态计算的可行性和效率，常忽略诸如密文版权保护、源认证和计算监督等实际需求。为解决这些问题，我们提出一种针对基于 RLWE 的 HE 密文的水印技术。RLWE 多项式的代数结构使我们能在不影响明文值的情况下将小噪声作为水印信息嵌入密文多项式。然而，HE 密文通常会经历多次同态运算，这可能会扭曲甚至去除嵌入的水印信息。为此我们提出两种实用解决方案。第一种是基于噪声分层的水印方案 ARWMark，对同态加法运算具有鲁棒性。第二种方案 MRWMark 利用线性方程的根构建，对同态加法和乘法运算均具有弹性，同时支持零比特水印。我们进行了详细的理论分析，证明我们的方案在不损害 HE 原有安全性的同时，确保了所提出水印技术的正确性和鲁棒性。此外，我们进行了大量实验来证明两种水印方案的有效性。
英文摘要
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.
