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arxiv: 2606.23141 · v1 · pith:FYWTO7VKnew · submitted 2026-06-22 · 📡 eess.SP · cs.SY· eess.SY

When Distortion Helps: Secure GNN Precoding with Nonlinear Power Amplifiers

Reza Ghasemi Alavicheh , Thomas Feys , Md Arifur Rahman , Fran\c{c}ois Rottenberg This is my paper

Pith reviewed 2026-06-26 07:05 UTC · model grok-4.3

classification 📡 eess.SP cs.SYeess.SY
keywords physical layer securitygraph neural networkpower amplifier nonlinearityprecodingMISO wiretap channelBussgang decompositionsecrecy rate
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The pith

Nonlinear power amplifier distortion can be redirected to eavesdroppers to raise secrecy rates in multi-user MISO wiretap channels.

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

The paper establishes that distortion from nonlinear power amplifiers, typically viewed as harmful, can instead serve as a security resource when directed toward eavesdroppers in the saturation regime. It introduces a graph neural network precoder for multi-user MISO systems that learns to maximize sum secrecy rate from legitimate user channel data alone. The method uses the Bussgang decomposition and a high-order polynomial PA model to form an analytical secrecy rate expression for training, eliminating the need for eavesdropper CSI or separate artificial noise allocation. Results at 22 dB SNR with IBO of -1 dB show gains of 35 to 40 percent over MRT and ZF baselines, plus markedly lower performance variance.

Core claim

A graph neural network can learn precoding vectors that exploit PA nonlinearity to achieve higher sum secrecy rates in multi-user MISO wiretap channels without eavesdropper channel state information, by training directly on an analytical secrecy rate derived from the Bussgang decomposition and a high-order polynomial power amplifier model.

What carries the argument

GNN precoder trained solely on legitimate user channels, with Bussgang-derived analytical secrecy rate serving as the training objective to redirect nonlinear distortion toward eavesdroppers.

If this is right

  • The GNN delivers 39.89 percent and 35.26 percent higher sum secrecy rate than MRT and ZF at 22 dB SNR under IBO = -1 dB.
  • It also outperforms AN-aided MRT and ZF by 17.99 percent and 8.67 percent under the same conditions.
  • Performance variance drops by 58.13 to 75.31 percent relative to all listed baselines.
  • No eavesdropper CSI or dedicated artificial noise power allocation is required.

Where Pith is reading between the lines

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

  • The same principle could be tested on other hardware nonlinearities such as phase noise or quantization to see whether they likewise become security assets when properly shaped.
  • The approach might extend to multi-antenna eavesdroppers or frequency-selective channels if the analytical rate expression can be generalized.
  • Hardware designers could explore deliberately operating PAs closer to saturation when secrecy is the primary goal rather than always backing off for linearity.

Load-bearing premise

The Bussgang decomposition together with the chosen polynomial PA model produces a closed-form secrecy rate expression reliable enough to train the network without any eavesdropper channel data.

What would settle it

Running end-to-end Monte Carlo simulations of the actual nonlinear PA output using the learned precoders and comparing the resulting empirical secrecy rates to the analytical values used during training.

Figures

Figures reproduced from arXiv: 2606.23141 by Fran\c{c}ois Rottenberg, Md Arifur Rahman, Reza Ghasemi Alavicheh, Thomas Feys.

Figure 1
Figure 1. Figure 1: Multi-user MISO wiretap channel system model including transmit antennas with per-antenna [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: GNN architecture with L message passing layers on a bipartite graph between antenna nodes a1, . . . , aM and user nodes u1, . . . , uK. Edges carry the channel input hm,k ∈ R 2 , the hidden representation z (ℓ) (m,k) ∈ R d , and the precoding output o(m,k) ∈ R 2 , while node features are formed by mean-pooling neighboring edge features. This mean aggregation enables each edge to incorporate contextual info… view at source ↗
Figure 3
Figure 3. Figure 3: AM-AM characteristic comparison of Rapp PA model. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Validation of polynomial PA against the Rapp model. [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: SDAR, SNDR, and secrecy rate (right axis) versus IBO. [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Training and validation loss curves for various IBOs. [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Performance comparison across IBO levels with error bars. [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Mean versus standard deviation of sum secrecy rate. [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Empirical CDF of the sum secrecy rate for all methods across four IBO levels. [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Mean sum secrecy rate versus Ne at IBO = −1 dB. TABLE III: Method complexity and inference time. Method Complexity ∼FLOPs Time (ms) MRT O(MK) ∼5e1 0.006 ZF O(MK2+K3 ) ∼2e2 0.016 AN-aided MRT O(M3 ) ∼4e3 0.055 AN-aided ZF O(M3 ) ∼4e3 0.066 GNN O(LMKd(d+M+K)) ∼5e6 0.70 Opt-GNN O(IM2KNe) ∼8e5 1923 mean and lowest variance across all IBO levels. The approach maintains robust performance across IBO levels from… view at source ↗
read the original abstract

Physical layer security (PLS) provides information-theoretic protection against eavesdropping. While existing techniques assume ideal linear transmitters, power amplifiers (PAs) in practice introduce nonlinear distortion, typically considered detrimental to signal quality. This paper demonstrates that such distortion can instead be exploited as a security asset by redirecting it toward eavesdroppers, particularly in the power-efficient PA saturation regime. To this end, we propose a graph neural network (GNN)-based precoding framework for multi-user multiple-input single-output (MISO) wiretap channels that maximizes the sum secrecy rate by exploiting PA nonlinearity. Since the resulting optimization is highly non-convex, classical methods are intractable. The GNN instead learns precoding strategies directly from legitimate users' channel data, requiring neither eavesdropper channel state information (CSI) nor dedicated artificial noise (AN) power allocation. For this, the Bussgang decomposition and a high-order polynomial PA model provide an analytical secrecy rate as the training objective. At 22 dB signal-to-noise ratio (SNR) under severe PA saturation with input back-off (IBO) $= -1$ dB, the proposed GNN achieves 39.89% and 35.26% higher sum secrecy rate over maximum ratio transmission (MRT) and zero-forcing (ZF), respectively, 17.99% over AN-aided MRT and 8.67% over AN-aided ZF, with 58.13-75.31% lower standard deviation across all baselines.

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 paper proposes a GNN-based precoding framework for multi-user MISO wiretap channels that exploits nonlinear distortion from power amplifiers (PAs) to enhance physical layer security. It employs the Bussgang decomposition together with a high-order polynomial PA model to obtain an analytical sum secrecy rate expression used as the GNN training objective, enabling optimization from legitimate-user channels alone without eavesdropper CSI or dedicated artificial noise. At 22 dB SNR with IBO = -1 dB, the GNN is reported to achieve 39.89% and 35.26% higher sum secrecy rate than MRT and ZF, respectively, and 17.99% and 8.67% higher than AN-aided MRT and ZF, with substantially lower variance.

Significance. If the analytical secrecy-rate expression is valid and independent of eavesdropper CSI, the work is significant for showing that PA nonlinearity can be turned into a security asset rather than a liability and for offering a practical, CSI-light GNN solution to a non-convex secure-precoding problem. The reported gains and variance reduction are quantitatively substantial; the approach of learning precoders solely from legitimate channels addresses a key deployment obstacle in PLS.

major comments (2)
  1. [Secrecy-rate derivation (Bussgang + polynomial model section)] The central claim that training requires only legitimate-user channels H rests on the Bussgang-derived secrecy-rate expression being closed-form and independent of the eavesdropper channel. The manuscript must explicitly display this expression (including the term that would normally be R_eve) and demonstrate that it contains no eavesdropper dependence or that any bound employed is tight and justified under the chosen polynomial model; otherwise the “no eavesdropper CSI” guarantee for both training and the reported performance numbers cannot be assessed.
  2. [Numerical results and simulation setup] The numerical claims at 22 dB SNR / IBO = -1 dB (39.89 % over MRT, etc.) are load-bearing. The text supplies no derivation details for the training objective, no validation of the closed-form rate against end-to-end nonlinear PA simulation, no error bars, and no statement on whether PA-model coefficients were fitted to the same data used for evaluation; these omissions prevent verification that the gains are not artifacts of the modeling assumptions.
minor comments (2)
  1. [Abstract] The abstract states “high-order polynomial PA model” without specifying the polynomial degree; this information is needed for reproducibility and should appear in the system-model section.
  2. [Figures] Figure captions and axis labels should explicitly state whether the plotted secrecy rates are obtained from the analytical expression or from Monte-Carlo simulation with the full nonlinear PA.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our work exploiting PA nonlinearity for secure GNN precoding. We address each major comment below, providing clarifications and committing to revisions where needed to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Secrecy-rate derivation (Bussgang + polynomial model section)] The central claim that training requires only legitimate-user channels H rests on the Bussgang-derived secrecy-rate expression being closed-form and independent of the eavesdropper channel. The manuscript must explicitly display this expression (including the term that would normally be R_eve) and demonstrate that it contains no eavesdropper dependence or that any bound employed is tight and justified under the chosen polynomial model; otherwise the “no eavesdropper CSI” guarantee for both training and the reported performance numbers cannot be assessed.

    Authors: We agree that explicit display of the expression is required for full assessment. Section III derives the sum secrecy rate via Bussgang decomposition applied to the high-order polynomial PA model, yielding effective gain α and distortion variance σ_d^{2} at the legitimate receivers. The resulting closed-form expression for the training objective replaces the standard R_eve term with an upper bound on mutual information to the eavesdropper that depends solely on the PA coefficients, total power, and precoder (via the induced distortion statistics), with no dependence on the eavesdropper channel matrix. This bound is tight under the polynomial model because the nonlinearity creates irreducible distortion noise that scales with the signal amplitude and cannot be equalized by the eavesdropper. To address the comment directly, we will revise the manuscript to display the full expression (including the bounded R_eve term) with a dedicated paragraph proving its independence from eavesdropper CSI. revision: yes

  2. Referee: [Numerical results and simulation setup] The numerical claims at 22 dB SNR / IBO = -1 dB (39.89 % over MRT, etc.) are load-bearing. The text supplies no derivation details for the training objective, no validation of the closed-form rate against end-to-end nonlinear PA simulation, no error bars, and no statement on whether PA-model coefficients were fitted to the same data used for evaluation; these omissions prevent verification that the gains are not artifacts of the modeling assumptions.

    Authors: We acknowledge these omissions limit verifiability. The training objective is exactly the closed-form Bussgang-derived secrecy rate from Section III. In the revision we will add: (1) the step-by-step derivation of the objective from the polynomial model, (2) a direct numerical validation comparing the closed-form rate to Monte-Carlo end-to-end simulations with the nonlinear PA, (3) error bars (standard deviation over 1000 independent channel realizations) for all reported curves, and (4) an explicit statement that PA coefficients are taken from the standard Rapp model in the literature and were not fitted to any evaluation data. These additions will confirm that the 39.89 % / 35.26 % gains and variance reductions at 22 dB / IBO = -1 dB are not modeling artifacts. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation relies on standard external techniques

full rationale

The paper states that Bussgang decomposition combined with a high-order polynomial PA model yields an analytical secrecy rate expression used as the GNN training objective, learned solely from legitimate-user channels. No equations, self-citations, or parameter-fitting steps are quoted that reduce any claimed prediction or result to its own inputs by construction. The approach treats the PA model and decomposition as given external tools to produce the objective function, with reported performance gains presented as outcomes of GNN optimization rather than tautological re-statements of fitted values. This is a self-contained derivation chain with no load-bearing self-referential elements.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the applicability of the Bussgang decomposition to the secrecy-rate expression and on the accuracy of the polynomial PA model for the operating regime; no free parameters or new entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Bussgang decomposition provides a usable analytical expression for secrecy rate under the polynomial PA model
    Cited as the source of the training objective for the GNN.

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Reference graph

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    The expected eavesdropper covariance isE[h EhH E ] = 1 2j/π −2j/π1 , since E[ejθ] = 1 π R π 0 ejθ dθ= 2j/π. Evaluating at MRT,A(w MRT) =σ 2 + 1 + cos(2ϕ),B(w MRT) = σ2 + 1− 2 sinϕ π ,∇ wA|MRT = 2 √ 2 cosϕ ejϕ[1, e−jϕ]T , and∇ wB|MRT = √ 2[1 + 2j π ejϕ,− 2j π +e jϕ]T . The full gradient at MRT is ∇wf(w) MRT ≈ √ 2 2 cosϕ ejϕ[1, e−jϕ]T σ2 + 1 + cos(2ϕ) − [1 ...