THC-DL: Three-Party Homomorphic Computing Delegation Learning for Secure Optimization

Delegation learning (DL) flourishes data sharing, enabling agents to delegate data to the cloud for model training. To preserve privacy, homomorphic encryption (HE) offers an effective solution for privacy-preserving machine learning (PPML) in DL, yet faces critical challenges in functionality (nonlinear activation support), practicality (ciphertext blow-up from iterative computations), and security (data leakage risks caused by public knowledge of the mathematical principles applied in model training). To tackle these challenges, we propose THC-DL, a three-party HE framework addressing these challenges holistically for the first time. We elaborately design the ciphertext secure comparison (DL-CSC) protocol to satisfy secure comparison with private inputs, enabling efficient nonlinear operations with $\mathcal {O}(1)$ communication complexity that reduces runtime to 12.5% of the DGK (Dolev-Greensmith-Kent protocol, the well-known comparison protocol). Second, we construct a Truncation-Mapping (Tru-Map) scheme, a mechanism that transforms input data by truncating and mapping it into a domain that facilitates more efficient processing, and the addition of truncated mapped data to resolve ciphertext blow-up by adaptively scaling ciphertexts during iterative training, ensuring correctness. Third, we formalize data leakage risks in HE-based quadratic convex optimization (standard in ML) and apply THC-DL to construct a secure optimization scheme. Theoretical analysis confirms THC-DL’s resilience against input recovery attacks, even when adversaries exploit public model parameters. Experiments on a real-world platform validate DL-CSC’s efficiency and scalability while reducing the computational complexity and communication complexity from $\mathcal {O}(n)$ to $\mathcal {O}(1)$ where n denotes the length of the input bits.