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arxiv: 2605.01654 · v1 · submitted 2026-05-03 · 💻 cs.CR · math.FA

Limit Properties at Critical Indices of Linear Canonical Riesz Potentials and Their Applications to Security of Multi-Image Encryption

Zunwei Fu , Dachun Yang , Shuhui Yang This is my paper

Pith reviewed 2026-05-10 16:32 UTC · model grok-4.3

classification 💻 cs.CR math.FA
keywords linear canonical Riesz potentialLCRPimage encryptionconvergencechirp functionsmulti-imagecryptosystemRiesz potential
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The pith

Linear canonical Riesz potentials converge for grating functions where classical ones diverge and match polygon indicators away from boundaries, enabling secure multi-image encryption.

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

This paper introduces the linear canonical Riesz potential and examines its limit properties at critical indices for grating functions and characteristic functions of convex polygons. It shows that these potentials converge for grating functions thanks to chirp functions, unlike the diverging classical Riesz potentials, and that their limits match the indicator function inside and outside the polygon but not on the boundary. These mathematical properties are then used to construct an asymmetric cascaded encryption method for multiple images, with the inverse operator being the linear canonical Laplacian. Security tests indicate the system resists various attacks and performs efficiently. A sympathetic reader would care because this links specific convergence behaviors in potential theory to practical improvements in image cryptosystems.

Core claim

We introduce the linear canonical Riesz potential and give its symbol in terms of linear canonical transforms. For grating functions, classical Riesz potentials diverge while LCRPs converge due to chirp functions. For the characteristic function of a convex polygon, the limit at non-boundary points equals the function, but differs at boundaries, contrasting with Schwartz functions where limits always match. Based on these and the inverse linear canonical Laplacian, we propose an asymmetric cascaded LCRP method for multi-image encryption whose security evaluations show robustness, and which is more efficient than fractional Riesz potential approaches.

What carries the argument

The linear canonical Riesz potential, whose convergence at critical indices for grating and polygon indicator functions, combined with the linear canonical Laplacian as its inverse, carries the argument for the encryption application.

Load-bearing premise

The inverse of the LCRP is precisely the linear canonical Laplacian operator and that the observed convergence properties suffice to guarantee cryptographic security in the proposed system.

What would settle it

Finding a grating function where the LCRP diverges at the critical index, or an attack that breaks the encryption scheme despite the convergence properties, would falsify the central claims.

Figures

Figures reproduced from arXiv: 2605.01654 by Dachun Yang, Shuhui Yang, Zunwei Fu.

Figure 1
Figure 1. Figure 1: Gaussian Function in the Spatial Domain and Di [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: LCRP and LCLO with A and β = 1.1. (a) (b) (c) (d) (e) (f) [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: LCRP and LCLO with B and β = 1.1 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: LCRP with C and, respectively, β = 0.5, β = 1, and β = 1.5 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: LCRP with D and, respectively, β = 0.5, β = 1, and β = 1.5 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Encryption Process of MIE-AC-LCRP Method. [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Decryption Process of MIE-AC-LCRP Method. [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Correct Encryption and Decryption Simulation. [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Error Encryption and Decryption Simulation (i). [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Error Encryption and Decryption Simulation (ii). [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Error Encryption and Decryption Simulation (iii). [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: MSEs Between Decrypted Image of Error Key [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: MSEs Between Decrypted Image of Error Key [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: MSEs of Erroneously Decrypted Images in Single-Image Encryption. [PITH_FULL_IMAGE:figures/full_fig_p031_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: New Original Images to be Encrypted [PITH_FULL_IMAGE:figures/full_fig_p033_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Histograms of Encrypted Images [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Noise Attack Analysis [PITH_FULL_IMAGE:figures/full_fig_p034_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Occlusion Attack Analysis. The robustness of the MIE-AC-LCRP method against occlusion attacks mathematically stems from the non-local nature of the LCRP and the destructive interference mechanism of its oscillatory factors [PITH_FULL_IMAGE:figures/full_fig_p035_18.png] view at source ↗
read the original abstract

In this article we introduce the linear canonical Riesz potential (for short, LCRP) and give its symbol in terms of linear canonical transforms. Driven by image processing, we establish the convergence/divergence of these LCRPs for different kinds of functions. Concretely, for grating functions, we prove that their classical Riesz potentials diverge, whereas their LCRP converge due to the key role of chirp functions. For the characteristic function ${\mathbf 1}_P$ of a convex polygon $P$, we show that the limit of its Riesz potential at any non-boundary point $\boldsymbol{x}$ equals ${\mathbf 1}_P(\boldsymbol{x})$, but its limit at the boundaries differ from ${\mathbf 1}_P$, while it is known that, for any Schwartz function $f$, the limit of its Riesz potential at any point $\boldsymbol{x}$ always equals $f(\boldsymbol{x})$. Based on these and the inverse operator of the LCRP (namely the linear canonical Laplacian operator), we propose an asymmetric cascaded LCRP method for the multi-image encryption and create an efficient and secure cryptosystem. Systematic security evaluations, including sensitivity, statistical, noise attack, and occlusion attack analyses, demonstrate its robustness and its security. Even for a single image, the proposed method is more efficient than the known encryption approach based on the fractional Riesz potential. The novelty of these results lies in that the convergence and the divergence of LCRTs at the critical indices, respectively, for ``good" Schwartz functions and for ``bad" discrete image functions essentially affect the security of image encryption and decryption.

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 / 0 minor

Summary. The manuscript introduces the linear canonical Riesz potential (LCRP) and its symbol in terms of linear canonical transforms. It establishes that classical Riesz potentials diverge on grating functions while LCRPs converge due to chirp functions, and that the limit of the Riesz potential of the characteristic function 1_P of a convex polygon recovers 1_P(x) at non-boundary points but differs at boundaries (contrasting with Schwartz functions where the limit always equals f(x)). Using these limit properties at critical indices together with the claim that the inverse of the LCRP is the linear canonical Laplacian, the authors construct an asymmetric cascaded LCRP scheme for multi-image encryption and report empirical security results from key sensitivity, statistical, noise, and occlusion tests, asserting that the analytic convergence/divergence behaviors essentially affect encryption security and that the method is more efficient than prior fractional Riesz potential approaches.

Significance. If the translation from the proven limit properties to cryptographic security can be made rigorous, the work would supply a novel analytic foundation for image encryption that exploits specific divergence at boundaries and chirp-induced convergence, potentially yielding efficiency gains and robustness. The introduction of the LCRP operator and its explicit symbol also contributes to the theory of linear canonical transforms and potential operators, with possible extensions to image processing.

major comments (2)
  1. Abstract and security evaluations: The central claim that 'the convergence and the divergence of LCRTs at the critical indices, respectively, for 'good' Schwartz functions and for 'bad' discrete image functions essentially affect the security' is load-bearing for the application but unsupported by a formal argument. The security analysis consists solely of standard empirical tests (key sensitivity, histogram, correlation, noise, occlusion) without a security game, reduction showing that the cited limit discrepancies (boundary mismatch for 1_P or chirp convergence for gratings) prevent plaintext/key recovery, or analysis of how these properties survive discretization to pixel arrays. This gap prevents verification that the analytic results confer the claimed security advantage.
  2. Inverse operator and decryption: The construction of the cascaded encryption scheme relies on the statement that the inverse of the LCRP is exactly the linear canonical Laplacian operator. The manuscript should provide or cite an explicit derivation of this inversion, including verification that it holds at the critical indices where the convergence/divergence results are established, as any mismatch would undermine the decryption step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive suggestions. We address each major comment below with clarifications and proposed revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: Abstract and security evaluations: The central claim that 'the convergence and the divergence of LCRTs at the critical indices, respectively, for 'good' Schwartz functions and for 'bad' discrete image functions essentially affect the security' is load-bearing for the application but unsupported by a formal argument. The security analysis consists solely of standard empirical tests (key sensitivity, histogram, correlation, noise, occlusion) without a security game, reduction showing that the cited limit discrepancies (boundary mismatch for 1_P or chirp convergence for gratings) prevent plaintext/key recovery, or analysis of how these properties survive discretization to pixel arrays. This gap prevents verification that the analytic results confer the claimed security advantage.

    Authors: We acknowledge that the manuscript relies on empirical security evaluations rather than a formal cryptographic reduction or game-based proof linking the limit properties directly to attack resistance. The analytic results on convergence for grating functions (via chirps) and boundary behavior for polygon indicators are presented as the motivation for selecting LCRP over classical Riesz potentials, ensuring the operator is well-defined on the discrete image data used in encryption. The empirical tests (key sensitivity, statistical, noise, and occlusion) then validate practical robustness, with efficiency gains over fractional Riesz methods also shown. We agree a more explicit discussion of how the properties survive pixel discretization would help. In revision we will add a dedicated paragraph in the introduction and security section clarifying the role of the limit properties in guaranteeing decryption invertibility for image-like functions, while noting that security claims remain empirically supported rather than formally reduced. revision: partial

  2. Referee: Inverse operator and decryption: The construction of the cascaded encryption scheme relies on the statement that the inverse of the LCRP is exactly the linear canonical Laplacian operator. The manuscript should provide or cite an explicit derivation of this inversion, including verification that it holds at the critical indices where the convergence/divergence results are established, as any mismatch would undermine the decryption step.

    Authors: The inverse relation is asserted based on the symbol of the LCRP in the linear canonical transform domain, where the multiplier is the reciprocal of that for the linear canonical Laplacian. We will insert an explicit derivation (including the symbol computation and inversion formula) into the section defining the LCRP operator. This derivation will be checked to hold at the specific critical indices used in the grating and polygon limit theorems, confirming that decryption recovers the plaintext without mismatch at those values. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper first proves convergence/divergence properties of LCRPs for grating functions (via chirp modulation) and for the indicator 1_P of convex polygons (limit equals 1_P away from boundaries) by direct comparison to classical Riesz potentials and Schwartz functions. It then invokes the inverse operator (linear canonical Laplacian) to define the encryption scheme and reports standard empirical security tests. No quoted step reduces by construction to its own inputs, no fitted parameter is relabeled as prediction, and no load-bearing self-citation or ansatz smuggling appears; the analytic results and cryptographic evaluations remain independent of each other.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the existence and invertibility of the linear canonical Riesz potential defined via linear canonical transforms, plus standard properties of Riesz potentials on Schwartz space and the assumption that security metrics on test images generalize to real attacks.

axioms (2)
  • domain assumption The linear canonical transform is invertible and its symbol defines a valid potential operator
    Invoked when the LCRP is introduced and its inverse is identified with the linear canonical Laplacian.
  • domain assumption Convergence at critical indices for grating and polygon functions follows from the chirp modulation
    Stated as the key reason classical divergence is avoided.
invented entities (1)
  • Linear canonical Riesz potential (LCRP) no independent evidence
    purpose: A modified potential operator that converges for discrete image-like functions where the classical Riesz potential diverges
    Introduced in the paper as the central new object; no independent physical evidence is given beyond the mathematical definition.

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