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arxiv: 2604.19890 · v1 · submitted 2026-04-21 · 💻 cs.CR

Efficient Arithmetic-and-Comparison Homomorphic Encryption with Space Switching

Erwin Eko Wahyudi , Yan Solihin , Qian Lou This is my paper

Pith reviewed 2026-05-10 01:55 UTC · model grok-4.3

classification 💻 cs.CR
keywords homomorphic encryptionspace switchingarithmetic operationscomparison operationsFV schemeprivacy-preserving computationdatabase workloadsfully homomorphic encryption
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The pith

Space switching lets a single homomorphic encryption scheme handle both arithmetic and comparison operations continuously.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to enable both arithmetic calculations and comparisons on encrypted data inside one FV-style scheme instead of switching schemes or approximating comparisons with polynomials. It does this by moving data between a large number space suited to arithmetic and a digit space suited to comparisons using two transition procedures. A sympathetic reader would care because prior methods either cost too much time for large data or introduce errors that break accuracy-sensitive tasks such as database queries. Experiments on real workloads show the approach runs up to 17 times faster than scheme switching and 15 times faster than direct comparison, supporting practical privacy-preserving computation.

Core claim

The paper claims that space switching integrates arithmetic and comparison computation seamlessly within FV-style schemes. The method works by noting that the two operation types need different plaintext spaces and by introducing a reduction step to move from the number space to the digit space plus a modulus-raising step to map results back, allowing continuous evaluation of both operation types inside the same scheme.

What carries the argument

Space switching, which consists of a reduction step from the number space to the digit space followed by a modulus-raising step to return to the original space.

If this is right

  • Arithmetic and comparison operations can be evaluated continuously inside one scheme without changing encryption parameters mid-computation.
  • Mixed workloads such as database queries run up to 17 times faster than when using scheme switching.
  • The same workloads run up to 15 times faster than when using direct comparison methods.
  • Accuracy-sensitive tasks avoid errors near critical points that appear in polynomial approximation approaches.
  • The design supports practical privacy-preserving computation on real-world data sets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same space-transition idea could be tested in other homomorphic schemes that rely on modular arithmetic.
  • It may simplify building larger privacy-preserving systems that combine calculations with conditional decisions.
  • Further tuning of the transition steps could produce even larger speed gains when data sizes grow.

Load-bearing premise

The steps that move encrypted values between a large number space and a small digit space can be carried out efficiently and correctly without weakening security or adding too much extra work.

What would settle it

Run the reduction and modulus-raising steps on actual encrypted data for mixed arithmetic-plus-comparison tasks, then measure total runtime, result correctness, and security level against the claimed speedups over scheme switching.

Figures

Figures reproduced from arXiv: 2604.19890 by Erwin Eko Wahyudi, Qian Lou, Yan Solihin.

Figure 1
Figure 1. Figure 1: Database queries inherently combine comparison (e.g., filtering with predicates) and arithmetic (e.g., multiplication [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview of continuous evaluation of arithmetic [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Example of digit decomposition in Zp4 . The Chen-Han method [12] uses a similar approach to Halevi-Shoup, with the main difference being in how the last number of each row is computed. In Chen-Han, they introduced another polynomial Gp,e based on the following lemma: Lemma 3 (Section 3.2 in [12]). For every prime p and e ≥ 1, there exists a polynomial Gp,e of degree at most (e − 1)(p − 1) + 1 such that for… view at source ↗
Figure 4
Figure 4. Figure 4: Example of proposed digit decomposition in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Runtime comparison for the TPC-H Bench￾mark [56] Query 6 across scheme switching, direct com￾parison, and space switching. minutes and 3 hours 19 minutes for 2 12 and 2 13 rows, re￾spectively. However, at 2 14 rows, scheme switching becomes significantly worse, taking more than 6 hours, nearly twice as slow as direct comparison. This inefficiency arises from the way comparison operations are handled. Initi… view at source ↗
Figure 5
Figure 5. Figure 5: Amortized runtime comparison for the LT operator [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims to introduce a 'space switching' method within FV/BGV-style homomorphic encryption that integrates arithmetic operations over Z_{p^r} and comparison operations over Z_p via two new homomorphic procedures (reduction from Z_{p^r} to Z_p followed by modulus raising back to Z_{p^r}), enabling continuous mixed computation in a single scheme without scheme switching or polynomial approximations, and reports up to 17x speedups over scheme switching and 15x over direct comparison on database workloads, with code released.

Significance. If the reduction and modulus-raising steps can be realized homomorphically with controlled noise growth and without compromising semantic security, the result would be significant for practical FHE deployments requiring mixed arithmetic and comparisons (e.g., secure databases or decision trees), as it avoids the overhead of scheme switching and the accuracy issues of approximations. The public code and artifacts are a positive contribution that supports reproducibility.

major comments (3)
  1. [Space switching method] The description of the space switching method (reduction Z_{p^r} to Z_p and modulus-raising): the manuscript provides no explicit noise-growth analysis, circuit depth, or homomorphic implementation details for these steps. This is load-bearing for the central claim of 'seamless continuous evaluation' and the reported speedups, as non-trivial circuits would consume the noise budget and require bootstrapping.
  2. [Experimental evaluation] Experimental results section: the 17x and 15x speedup claims are reported without accompanying noise budgets, bootstrapping frequency, exact parameter sets (e.g., p, r, security level), or circuit sizes for the new procedures. This prevents verification that the speedups are achieved under standard FV noise constraints rather than idealized conditions.
  3. [Security discussion] Security analysis: no reduction or argument is given showing that the new reduction and raising maps preserve semantic security of the underlying FV scheme. Since these maps are custom and not native operations, this is required to support the claim that the approach does not weaken security.
minor comments (2)
  1. [Introduction] Notation for plaintext spaces (Z_{p^r} vs. Z_p) is introduced without a small concrete example early in the paper; adding one would improve readability for readers unfamiliar with digit decomposition.
  2. [Abstract] The abstract states 'continuous evaluation of arithmetic and comparison within the same scheme' but does not define what 'continuous' precisely means in terms of circuit depth or number of operations before noise exhaustion.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and valuable comments, which will help strengthen the manuscript. We address each major comment below and will revise the paper accordingly to provide the requested details and analysis.

read point-by-point responses

  1. Referee: [Space switching method] The description of the space switching method (reduction Z_{p^r} to Z_p and modulus-raising): the manuscript provides no explicit noise-growth analysis, circuit depth, or homomorphic implementation details for these steps. This is load-bearing for the central claim of 'seamless continuous evaluation' and the reported speedups, as non-trivial circuits would consume the noise budget and require bootstrapping.

    Authors: We agree that the current manuscript would benefit from a more explicit treatment of these aspects. In the revision, we will add a dedicated subsection detailing the homomorphic implementations of the reduction and modulus-raising procedures, their circuit depths, and a noise-growth analysis under the standard FV noise model. This will confirm that the operations support the claimed seamless continuous evaluation within the available noise budget without requiring bootstrapping for the evaluated workloads. revision: yes

  2. Referee: [Experimental evaluation] Experimental results section: the 17x and 15x speedup claims are reported without accompanying noise budgets, bootstrapping frequency, exact parameter sets (e.g., p, r, security level), or circuit sizes for the new procedures. This prevents verification that the speedups are achieved under standard FV noise constraints rather than idealized conditions.

    Authors: We acknowledge that additional parameters and metrics are needed for full verification. In the revised experimental section, we will report the exact parameter sets (including p, r, and security level), noise budgets, bootstrapping frequency, and circuit sizes for the space switching procedures. The reported speedups were obtained under standard FV constraints with these settings, and we will include summary tables to facilitate independent checks. revision: yes

  3. Referee: [Security discussion] Security analysis: no reduction or argument is given showing that the new reduction and raising maps preserve semantic security of the underlying FV scheme. Since these maps are custom and not native operations, this is required to support the claim that the approach does not weaken security.

    Authors: We thank the referee for highlighting this point. In the revision, we will include a security argument showing that the reduction and raising maps are deterministic functions on the plaintext space that are evaluated using the native homomorphic operations of the FV scheme. Semantic security is therefore preserved by reduction to the semantic security of FV (based on ring-LWE), with no additional assumptions or leakage introduced. revision: yes

Circularity Check

0 steps flagged

No circularity detected in space switching construction

full rationale

The paper's core contribution is a novel construction that defines reduction from Z_{p^r} to Z_p and the corresponding modulus-raising map as explicit procedures inside an FV/BGV scheme. These maps are presented as the authors' design choice, not derived from or equivalent to any fitted parameter or prior result by construction. Performance claims rest on experimental measurements rather than statistical predictions or self-referential equations. No load-bearing step invokes a self-citation chain, uniqueness theorem from the same authors, or renaming of an existing empirical pattern. The derivation chain therefore remains independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim relies on standard properties of homomorphic encryption schemes and the new space switching procedures.

axioms (1)
  • domain assumption The plaintext spaces for arithmetic and comparison operations in FV-style schemes are distinct (Z_{p^r} vs Z_p)
    This is the key identification made by the paper to motivate the switching.
invented entities (1)
  • Space switching no independent evidence
    purpose: To enable seamless transition between arithmetic and comparison computations in the same FHE scheme
    Introduced as a new procedure consisting of reduction and modulus-raising steps.

pith-pipeline@v0.9.0 · 5505 in / 1296 out tokens · 44166 ms · 2026-05-10T01:55:44.700949+00:00 · methodology

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