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arxiv: 2510.13661 · v2 · pith:ZR6W7NJEnew · submitted 2025-10-15 · 💻 cs.IT · cs.CR· math.IT

Local Information-Theoretic Security via Euclidean Geometry

Emmanouil M.Athanasakos , Nicholas Kalouptsidis , Hariprasad Manjunath This is my paper

Pith reviewed 2026-05-18 06:12 UTC · model grok-4.3

classification 💻 cs.IT cs.CRmath.IT
keywords information theoretic securitywiretap channelEuclidean information theorylocal secrecy capacitycontraction coefficientsgeneralized eigenvaluesquadratic programmingKKT conditions
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The pith

Local Euclidean approximations convert wiretap secrecy optimization into a quadratic program solved via generalized eigenvalues to yield an analytical formula for approximate local secrecy capacity.

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

This paper develops a geometric method using Euclidean information theory to examine local properties of secure communication over discrete memoryless wiretap channels. It formulates an optimization that maximizes the legitimate rate while constraining eavesdropper leakage and encoding cost. Local geometric approximations recast this non-convex problem as a quadratic program. The optimal Lagrange multipliers are recovered by solving a linear program whose constraints are expressed through generalized eigenvalues of channel matrices. The approach supplies an analytical expression for approximate local secrecy capacity and defines secret local contraction coefficients as the largest generalized eigenvalues of a matrix pencil, which measure the local ratio of utility to leakage.

Core claim

By applying local Euclidean geometric approximations to the constrained optimization of mutual information quantities in wiretap channels, the inherently non-convex problem is transformed into a tractable quadratic program; the optimal Lagrange multipliers for this program are obtained by solving a linear program whose constraints derive from the Karush-Kuhn-Tucker conditions and involve generalized eigenvalues of channel-derived matrices, thereby producing an analytical formula for the approximate local secrecy capacity together with a new class of secret local contraction coefficients characterized as the largest generalized eigenvalues of a matrix pencil that quantify the maximum ratio of

What carries the argument

Local Euclidean geometric approximations that reduce the mutual-information secrecy optimization to a quadratic program whose Lagrange multipliers are recovered from a linear program on generalized eigenvalues of channel matrices.

If this is right

  • The non-convex secrecy optimization is converted into a tractable quadratic program.
  • Optimal Lagrange multipliers are recovered by solving a linear program whose constraints come from KKT conditions expressed via generalized eigenvalues.
  • An analytical formula for the approximate local secrecy capacity is obtained.
  • Secret local contraction coefficients are defined as the largest generalized eigenvalues of a matrix pencil.
  • Bounds are established relating these local coefficients to their global counterparts based on true mutual information.

Where Pith is reading between the lines

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

  • If the approximations remain accurate for larger alphabets, the method could reduce the need for exhaustive numerical search when estimating local secrecy performance in multi-user settings.
  • The eigenvalue characterization of contraction coefficients might identify channel parameters that locally maximize security efficiency without requiring full mutual-information computation.
  • The same local geometric reduction could be tested on related problems such as secure key agreement or authentication to derive analogous analytical expressions.

Load-bearing premise

The local geometric approximations are accurate enough that the quadratic program and its linear-program solution for the Lagrange multipliers faithfully represent the original constrained optimization over mutual information quantities.

What would settle it

For the binary symmetric wiretap channel, compute the true local maximum of the secrecy objective by exhaustive search over input distributions near a candidate point and compare the value to the result of the quadratic-program approximation.

Figures

Figures reproduced from arXiv: 2510.13661 by Emmanouil M.Athanasakos, Hariprasad Manjunath, Nicholas Kalouptsidis.

Figure 1
Figure 1. Figure 1: IB Comparison for a BSC(𝑝bob = 0.1) with uniform 𝑃𝑋 and binary 𝑈. Solid blue represents the true IB curve (Blahut-Arimoto solution) and red circles dashed the analytical EIT-IB. As observed in [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The LP solution for optimal multipliers for a numerically generated channel with [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Normalized approximate local secrecy capacity as a function of the constraint ratio. [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: 𝐶SIC as a function of Eve’s channel quality for different output quantization levels |Z|, with |𝑋| = 8, Bob’s 𝐸𝑏/𝑁0 = 8.0dB, 𝑅 = 0.5, Θ = 0.1 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the EIT-approximated local secrecy capacity (dashed red line) for [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Operational regimes of 𝐶SIC for the BSWC, as characterized by Theorem 10. The solid black line shows 𝐶SIC as a function of the leakage allowance Θ, for fixed 𝑝bob = 0.1, 𝑞eve = 0.25, and 𝑅 = 0.5. The curve shows the clear transition from the leakage-dominant regime to the rate-dominant regime [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Verification of KKT condition for BSWC, with [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Utility vs. Leakage for general DM-WTC with [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Distribution of Utility/Leakage Ratios with [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Illustration of the achieved utility-to-leakage ratio from the SIC solution (solid blue) versus [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Bounds on the global secret contraction coefficient ( [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The axes represent the total expected squared magnitudes ( [PITH_FULL_IMAGE:figures/full_fig_p041_12.png] view at source ↗
read the original abstract

This paper introduces a methodology based on Euclidean information theory to investigate local properties of secure communication over discrete memoryless wiretap channels. We formulate a constrained optimization problem that maximizes a legitimate user's information rate while imposing explicit upper bounds on both the information leakage to an eavesdropper and the informational cost of encoding the secret message. By leveraging local geometric approximations, this inherently non-convex problem is transformed into a tractable quadratic programming structure. It is demonstrated that the optimal Lagrange multipliers governing this approximated problem can be found by solving a linear program. The constraints of this linear program are derived from Karush-Kuhn-Tucker conditions and are expressed in terms of the generalized eigenvalues of channel-derived matrices. This framework facilitates the derivation of an analytical formula for an approximate local secrecy capacity. Furthermore, we define and analyze a new class of secret local contraction coefficients. These coefficients, characterized as the largest generalized eigenvalues of a matrix pencil, quantify the maximum achievable ratio of approximate utility to approximate leakage, thus measuring the intrinsic local leakage efficiency of the channel. We establish bounds connecting these local coefficients to their global counterparts defined over true mutual information measures. The efficacy of the proposed framework is demonstrated through detailed analysis and numerical illustrations for both general multi-mode channels and the canonical binary symmetric wiretap channel.

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 presents a framework using Euclidean information theory to analyze local aspects of secure communications in discrete memoryless wiretap channels. It converts a non-convex optimization problem maximizing the legitimate rate under leakage and encoding cost constraints into a quadratic program through local geometric approximations of mutual informations. Optimal Lagrange multipliers are obtained by solving a linear program based on KKT conditions involving generalized eigenvalues of matrices derived from the channel. This leads to an analytical expression for an approximate local secrecy capacity and the introduction of secret local contraction coefficients, defined as the largest generalized eigenvalues of a matrix pencil, which measure the local leakage efficiency. Bounds relating these to global contraction coefficients are provided, along with numerical results for general multi-mode channels and the binary symmetric wiretap channel.

Significance. Assuming the local approximations are accurate within a suitable neighborhood, this work offers a novel analytical tool for approximating local secrecy capacities and defines new coefficients that capture the channel's local secrecy properties. The connection between local and global measures, along with the numerical validation on the binary symmetric channel, adds value to the information-theoretic security literature by providing a geometric perspective that may simplify analysis near operating points.

major comments (3)
  1. [Section deriving the quadratic program from local geometric approximations] The transformation of the inherently non-convex secrecy-rate maximization problem into a quadratic programming structure via local geometric approximations of mutual information quantities is load-bearing for the central claims. The manuscript does not provide explicit error bounds on the approximation or characterize the neighborhood size around the operating point where the quadratic model remains valid for the discrete memoryless wiretap channel.
  2. [Section on the linear program and KKT conditions] The recovery of optimal Lagrange multipliers by solving a linear program whose constraints come from KKT conditions and are expressed via generalized eigenvalues of channel-derived matrices requires a complete, verifiable derivation. Without this, the analytical formula for the approximate local secrecy capacity cannot be confirmed to faithfully represent the approximated problem.
  3. [Section defining and analyzing secret local contraction coefficients] The secret local contraction coefficients are defined as the largest generalized eigenvalues of a matrix pencil and bounds to global counterparts over true mutual information are claimed. The exact form of the matrix pencil and the proof of these bounds should be stated explicitly in a theorem to allow assessment of when the local coefficients meaningfully quantify leakage efficiency.
minor comments (2)
  1. The abstract states that efficacy is demonstrated through 'detailed analysis and numerical illustrations' for the binary symmetric wiretap channel, but a summary table of key numerical outcomes (e.g., approximated capacity values versus exact ones) would improve readability.
  2. [Preliminaries or notation section] Notation for the channel matrices and the matrix pencil should be introduced with explicit definitions early in the manuscript to avoid ambiguity when discussing generalized eigenvalues.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed report, which highlights both the novelty of the Euclidean-geometry approach to local secrecy and the areas where additional rigor would strengthen the presentation. We address each major comment below and will incorporate the suggested clarifications and formal statements into the revised manuscript.

read point-by-point responses

  1. Referee: [Section deriving the quadratic program from local geometric approximations] The transformation of the inherently non-convex secrecy-rate maximization problem into a quadratic programming structure via local geometric approximations of mutual information quantities is load-bearing for the central claims. The manuscript does not provide explicit error bounds on the approximation or characterize the neighborhood size around the operating point where the quadratic model remains valid for the discrete memoryless wiretap channel.

    Authors: We agree that explicit error bounds and a precise characterization of the neighborhood would improve the manuscript. The local quadratic model follows from the standard second-order Taylor expansion of mutual information in the Euclidean information geometry framework. In the revision we will add a new subsection that derives an explicit remainder bound using the integral form of Taylor's theorem applied to the mutual-information functional and characterizes the valid neighborhood in terms of the operator norm of the third derivative tensor, thereby quantifying when the quadratic approximation remains accurate for discrete memoryless wiretap channels. revision: yes

  2. Referee: [Section on the linear program and KKT conditions] The recovery of optimal Lagrange multipliers by solving a linear program whose constraints come from KKT conditions and are expressed via generalized eigenvalues of channel-derived matrices requires a complete, verifiable derivation. Without this, the analytical formula for the approximate local secrecy capacity cannot be confirmed to faithfully represent the approximated problem.

    Authors: The mapping from the KKT stationarity conditions to the linear program over generalized eigenvalues is outlined in the main text and supporting appendix. To make the derivation fully verifiable, we will expand the appendix with a self-contained, step-by-step proof that starts from the Lagrangian of the quadratic program, applies the KKT conditions, and shows how the resulting stationarity equations reduce to the generalized eigenvalue problem whose solutions yield the optimal multipliers and the closed-form expression for the approximate local secrecy capacity. revision: yes

  3. Referee: [Section defining and analyzing secret local contraction coefficients] The secret local contraction coefficients are defined as the largest generalized eigenvalues of a matrix pencil and bounds to global counterparts over true mutual information are claimed. The exact form of the matrix pencil and the proof of these bounds should be stated explicitly in a theorem to allow assessment of when the local coefficients meaningfully quantify leakage efficiency.

    Authors: We will add a formal theorem statement that explicitly defines the matrix pencil in terms of the Hessians of the locally approximated mutual informations (i.e., the pair of matrices constructed from the channel transition probabilities and the second-order expansions) and proves the claimed bounds by invoking the variational characterization of generalized eigenvalues together with the monotonicity properties preserved by the local Euclidean approximation near the operating point. This will clarify the regime in which the local coefficients serve as meaningful measures of leakage efficiency. revision: yes

Circularity Check

0 steps flagged

No circularity: approximate secrecy capacity and contraction coefficients derived from channel matrices via standard optimization on local approximations

full rationale

The derivation converts the constrained mutual-information maximization into a quadratic program using local Euclidean geometric approximations of the rate and leakage functions. Lagrange multipliers for the resulting QP are recovered by solving an LP whose constraints come from KKT stationarity conditions expressed via generalized eigenvalues of matrices constructed directly from the channel transition probabilities. The secret local contraction coefficients are explicitly defined as the largest generalized eigenvalues of the associated matrix pencil; this is a direct algebraic characterization of the approximated utility-to-leakage ratio, not a redefinition of the original mutual-information quantities. Bounds relating the local coefficients to their global mutual-information counterparts are stated separately and do not rely on the local quantities being identical to the global ones. No fitted parameters are renamed as predictions, no self-citations supply uniqueness theorems, and no ansatz is imported via prior work. The central results therefore remain independent of the target quantities they approximate.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The approach rests on standard convex optimization and linear-algebra facts plus the modeling assumption that local Euclidean approximations preserve the essential secrecy-leakage trade-off; no free parameters or new physical entities are introduced in the abstract.

axioms (2)
  • domain assumption Local geometric approximations transform the original non-convex mutual-information optimization into a tractable quadratic program whose KKT conditions yield a linear program solvable via generalized eigenvalues.
    Invoked in the abstract when stating that the inherently non-convex problem is transformed into quadratic programming and that optimal Lagrange multipliers are found by solving a linear program derived from KKT conditions.
  • domain assumption Generalized eigenvalues of channel-derived matrices correctly characterize the maximum ratio of approximate utility to approximate leakage.
    Used to define the secret local contraction coefficients and to obtain the analytical formula for approximate local secrecy capacity.
invented entities (1)
  • secret local contraction coefficients no independent evidence
    purpose: Quantify the maximum achievable ratio of approximate utility to approximate leakage as the largest generalized eigenvalue of a matrix pencil
    Newly defined class of coefficients introduced to measure intrinsic local leakage efficiency of the channel.

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    Relation between the paper passage and the cited Recognition theorem.

    By leveraging local geometric approximations, this inherently non-convex problem is transformed into a tractable quadratic programming structure... secret local contraction coefficients... largest generalized eigenvalues of a matrix pencil

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Reference graph

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