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arxiv: 2606.03513 · v1 · pith:A3VFPS45new · submitted 2026-06-02 · 💻 cs.CR

Privacy-Preserving High-Resolution Image Gradient Computation Based on Fully Homomorphic Encryption

Yufei Zhou This is my paper

Pith reviewed 2026-06-28 09:36 UTC · model grok-4.3

classification 💻 cs.CR
keywords homomorphic encryptionprivacy-preserving image processingSobel operatorgradient computationpolynomial approximationhigh-resolution imagesmulti-ciphertext frameworksemi-honest model
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The pith

Dividing high-resolution images into sub-images keeps homomorphic encryption parameters small for privacy-preserving Sobel gradient computation.

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

The paper presents a framework that splits large images into multiple sub-images so homomorphic encryption can use smaller parameters and smaller keys. Parallel processing of the resulting ciphertexts, combined with a repeated packing method for convolution, reduces both encryption and server-side computation costs under the semi-honest model. For the Sobel operator the authors replace the usual reciprocal calculation with a new polynomial approximation derived from the sign function, which improves gradient-direction accuracy. The overall approach targets high-resolution images where direct application of prior HE methods becomes impractical.

Core claim

A multi-ciphertext privacy-preserving framework is obtained by partitioning a large image into independent sub-images; this permits smaller HE parameters while the sub-image ciphertexts are processed in parallel and later recombined. Convolution is optimized by repeated packing, and the Sobel gradient direction is computed via a new polynomial approximation of the reciprocal that is constructed from the sign function. The resulting protocol computes encrypted image gradients without revealing the plaintext image.

What carries the argument

Multi-ciphertext framework that partitions the input image into sub-images together with a sign-function-based polynomial approximation of the reciprocal.

If this is right

  • Smaller HE parameters reduce both key size and encryption time for the client.
  • Parallel sub-image processing and the new bootstrapping placement lower overall latency.
  • Repeated packing reduces the number of homomorphic multiplications needed for convolution.
  • The sign-based reciprocal approximation can be reused in other HE protocols that require division or normalization.

Where Pith is reading between the lines

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

  • The same sub-image partitioning may apply to other local operators such as edge detection or filtering kernels beyond Sobel.
  • Error introduced at sub-image boundaries could be measured by comparing against a non-partitioned baseline on images of varying sizes.
  • The approximation technique might be tuned further for different noise budgets or security levels without changing the overall framework.
  • Integration with existing HE libraries could be tested by measuring wall-clock time on standard hardware for megapixel-scale inputs.

Load-bearing premise

Splitting the image into sub-images and recombining the encrypted gradient results produces the same output as running the Sobel operator on the intact image.

What would settle it

Encrypt a known high-resolution test image, run the full pipeline, decrypt the gradient fields, and check whether every pixel matches (within the scheme's noise bound) the gradients obtained by applying the plaintext Sobel operator directly to the original image.

Figures

Figures reproduced from arXiv: 2606.03513 by Yufei Zhou.

Figure 1
Figure 1. Figure 1: Comparison between single-ciphertext encryption and multi-ciphertext [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sobel operator computation process based on multi-ciphertext encryp [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of Sobel magnitude without boundary handling. The [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example of SIMD convolution construction. The image size is [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Examples of different image partitioning methods. [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of multi-ciphertext encryption and parallel convolution [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Example of gradient angle computation. ‘Select’ refers to the [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Convolution overhead under LHE for different image sizes. [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of user online overhead for different image sizes. [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of gradient angle visualization for real images. [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
read the original abstract

With growing emphasis on privacy protection, homomorphic encryption (HE) has emerged as a core method for privacy-preserving image processing, as it enables operations directly on encrypted data. However, existing research predominantly focuses on low-resolution image processing, and techniques for privacy-preserving high-resolution image processing remain underexplored. As the image size increases, the HE parameters must be adjusted accordingly, and directly applying existing methods can lead to significant computational overhead. In this work, we propose a multi-ciphertext privacy-preserving framework for large images, enabling efficient image encryption and computation under the semi-honest model. Specifically, we divide the large image into multiple sub-images, which allows us to maintain smaller HE parameters and reduce key size. By parallel processing the sub-image ciphertexts and introducing a new bootstrapping placement strategy, we significantly reduce encryption overhead and enhance user experience. On the server side, we optimize the large image convolution operation through a repeated packing technique and implement the Sobel operator computation based on HE. To improve gradient direction calculation for the Sobel operator, we introduce a new polynomial approximation method for the reciprocal function based on the sign function, which can be applied to other HE-based protocols.

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

2 major / 2 minor

Summary. The manuscript proposes a multi-ciphertext privacy-preserving framework for Sobel gradient computation on high-resolution images under fully homomorphic encryption in the semi-honest model. Large images are partitioned into sub-images to permit smaller HE parameters and reduced key sizes; sub-image ciphertexts are processed in parallel with a new bootstrapping placement strategy, repeated packing for convolution, and a polynomial approximation to the reciprocal function (derived from the sign function) to compute gradient directions.

Significance. If the sub-image recombination is shown to be equivalent to monolithic Sobel evaluation and the approximation error is bounded with concrete accuracy measurements, the framework would provide a concrete route to scaling FHE-based image processing beyond low-resolution regimes. The repeated-packing optimization and sign-based reciprocal approximation are technically interesting contributions that could be reused in other HE protocols.

major comments (2)
  1. [Abstract / Framework Description] Abstract and framework description: the central efficiency claim rests on dividing the image into independent sub-images, encrypting and computing each with smaller HE parameters, then recombining. Sobel operators are 3x3 and therefore require adjacent-pixel neighborhoods; without an explicit overlap, padding, or stitching protocol, gradients along sub-image seams cannot be guaranteed to match full-image results. No equivalence argument, error bound, or experimental verification (e.g., pixel-wise difference maps or PSNR between sub-image and monolithic outputs) is supplied, directly undermining the claim that the multi-ciphertext method preserves mathematical correctness.
  2. [Polynomial Approximation Method] Polynomial approximation section: the new sign-based polynomial for the reciprocal is presented as an improvement for gradient-direction calculation, yet the manuscript supplies neither the explicit polynomial coefficients, the approximation interval and degree, nor any error analysis (maximum absolute or relative error) or comparison against prior HE-friendly approximations. Without these, it is impossible to assess whether the method meets the accuracy requirements of the Sobel pipeline.
minor comments (2)
  1. [Abstract] The abstract states that parallel processing and the new bootstrapping placement “significantly reduce encryption overhead,” but no concrete runtime figures, parameter sets, or comparison against a single-ciphertext baseline are given even at the level of order-of-magnitude estimates.
  2. [Convolution Optimization] Notation for ciphertext packing and the exact form of the repeated-packing convolution should be introduced with a small worked example or diagram to clarify how the 3x3 Sobel kernels are realized under the packing scheme.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which identify important gaps in the presentation of our multi-ciphertext framework. We address each major comment below and will revise the manuscript to incorporate the requested details and verifications.

read point-by-point responses

  1. Referee: [Abstract / Framework Description] Abstract and framework description: the central efficiency claim rests on dividing the image into independent sub-images, encrypting and computing each with smaller HE parameters, then recombining. Sobel operators are 3x3 and therefore require adjacent-pixel neighborhoods; without an explicit overlap, padding, or stitching protocol, gradients along sub-image seams cannot be guaranteed to match full-image results. No equivalence argument, error bound, or experimental verification (e.g., pixel-wise difference maps or PSNR between sub-image and monolithic outputs) is supplied, directly undermining the claim that the multi-ciphertext method preserves mathematical correctness.

    Authors: We agree that the manuscript does not explicitly describe an overlap or stitching protocol, which is necessary for correct 3x3 Sobel neighborhoods at sub-image boundaries. In the revised version we will add a dedicated subsection detailing the sub-image division with a one-pixel overlap on each side, provide a mathematical argument establishing equivalence to monolithic Sobel evaluation under this protocol, and include experimental verification consisting of pixel-wise difference maps and PSNR values between the recombined sub-image outputs and a monolithic baseline. revision: yes

  2. Referee: [Polynomial Approximation Method] Polynomial approximation section: the new sign-based polynomial for the reciprocal is presented as an improvement for gradient-direction calculation, yet the manuscript supplies neither the explicit polynomial coefficients, the approximation interval and degree, nor any error analysis (maximum absolute or relative error) or comparison against prior HE-friendly approximations. Without these, it is impossible to assess whether the method meets the accuracy requirements of the Sobel pipeline.

    Authors: We acknowledge that the current manuscript does not supply the explicit coefficients, degree, interval, error bounds, or comparisons for the sign-based reciprocal approximation. The revised manuscript will expand the polynomial approximation section to include the concrete polynomial, its degree and approximation interval, a full error analysis reporting maximum absolute and relative errors, and direct comparisons against prior HE-friendly reciprocal approximations, allowing readers to evaluate its accuracy within the Sobel pipeline. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a methodological proposal for a multi-ciphertext HE framework that splits large images into sub-images for smaller parameters, uses repeated packing for convolution, and introduces a sign-function-based polynomial approximation for reciprocal in Sobel gradient direction. No load-bearing steps reduce by the paper's own equations or self-citations to fitted parameters, self-definitions, or renamed inputs; the central claims rest on independent engineering choices and a new approximation technique whose correctness is asserted separately from any fitted quantities. The sub-image recombination assumption is an unverified premise rather than a circular reduction, and no derivation chain collapses to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields minimal ledger entries; the semi-honest model is the primary domain assumption invoked.

axioms (1)
  • domain assumption The server follows the protocol but does not deviate maliciously (semi-honest model)
    Explicitly stated as the threat model under which the framework operates.

pith-pipeline@v0.9.1-grok · 5734 in / 1331 out tokens · 24276 ms · 2026-06-28T09:36:00.162754+00:00 · methodology

discussion (0)

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