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This paper introduces a space switching method to efficiently integrate arithmetic and comparison operations within FV-style homomorphic encryption schemes. The approach uses reduction and modulus-raising steps to transition between different plaintext spaces ($\mathbb{Z}_{p^r}$ and $\mathbb{Z}_{p}$), enabling continuous evaluation of both operation types. Empirical results on database workloads demonstrate a significant performance improvement of up to $17\times$ compared to scheme switching and $15\times$ compared to direct comparison.
Forget slow scheme switching: this new method speeds up homomorphic encryption for mixed arithmetic and comparison tasks by up to 17x.
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.
