Efficient Arithmetic-and-Comparison Homomorphic Encryption with Space Switching

2026-04-21

AI Summary

Existing fully homomorphic encryption (FHE) schemes struggle to efficiently and accurately support mixed arithmetic and comparison operations within a unified framework, often resorting to costly scheme switching or error-prone polynomial approximations. This work proposes a novel space-switching technique that enables seamless integration of these two operation types in FV-like schemes by leveraging plaintext space reduction from ℤ_{p^r} to ℤ_p, modulus lifting, and digit decomposition. For the first time, this approach achieves error-free, low-overhead hybrid computation within a single homomorphic encryption framework. Experimental results on representative database workloads demonstrate a 17× speedup over conventional scheme-switching methods and a 15× improvement compared to direct comparison approaches, substantially enhancing the practicality of privacy-preserving computation.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Privacy
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Privacy-enhancing technologies
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Fully homomorphic encryption (FHE) enables computation on encrypted data without decryption, making it central to privacy-preserving applications. However, no existing scheme efficiently supports both arithmetic and comparison operations in a unified framework. Prior approaches such as scheme switching and polynomial approximation face serious limitations: switching incurs prohibitive overhead for large inputs, while approximation methods introduce errors near critical points, restricting use in accuracy-sensitive tasks. We propose space switching method to integrate arithmetic and comparison computation seamlessly within FV-style schemes. Our approach identifies that the two types of operations require different plaintext spaces and introduces two procedures: a reduction step to transition from the number space $\mathbb{Z}_{p^r}$ to the digit space $\mathbb{Z}_{p}$, and a modulus-raising step to map results back to $\mathbb{Z}_{p^r}$. This design enables continuous evaluation of arithmetic and comparison within the same scheme. Experiments show that our method achieves up to $17\times$ faster performance than scheme switching and $15\times$ faster than direct comparison on database workloads, demonstrating its practicality for real-world privacy-preserving computation. Code and artifacts are available at https://github.com/UCF-Lou-Lab-PET/Universal-BGV.

Problem

• Fully Homomorphic Encryption
• Arithmetic Operations
• Comparison Operations
• Privacy-Preserving Computation

Innovation

• space switching
• fully homomorphic encryption
• arithmetic-comparison integration
• plaintext space transformation
• privacy-preserving computation