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arxiv: 2604.22587 · v1 · submitted 2026-04-24 · 💻 cs.IT · math.IT

On the Optimum Secrecy Outage Probability and Ergodic Secrecy Rate over Wireless Channels

Clement Leroy , Tarak Arbi , Benoit Geller , Olivier Rioul This is my paper

Pith reviewed 2026-05-08 09:38 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords secrecy outage probabilityergodic secrecy rateGaussian inputfading channelspartial orderingswireless secrecyRayleigh fading
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The pith

When the main channel is uniformly less noisy than the eavesdropper channel, non-precoded Gaussian inputs optimize the secrecy outage probability and ergodic positive secrecy rate.

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

The paper studies secrecy in wireless channels with an eavesdropper when the transmitter knows only channel statistics. It defines two partial orderings for random channels: uniformly less noisy and less noisy on average. It establishes that under the uniformly less noisy ordering, a non-precoded Gaussian input is optimal for the secrecy outage probability and the ergodic positive secrecy rate. The same holds for the ergodic secrecy rate under the less noisy on average ordering. For single-antenna channels, the optimality for secrecy outage and positive rate holds without any ordering assumptions. Closed forms are given for Rayleigh fading.

Core claim

When the main channel is uniformly less noisy than the eavesdropper channel, the optimal input distribution is a non-precoded Gaussian input for both the SOP and the EPSR. Furthermore, the same input distribution is optimal for the ESR when the less noisy on average order holds. Similar optimality results for the SOP and the EPSR are obtained for single-transmit-antenna channels without requiring any channel ordering assumptions.

What carries the argument

The uniformly less noisy partial ordering of random channels that proves optimality of non-precoded Gaussian inputs for secrecy metrics.

If this is right

  • Closed-form expressions for the secrecy metrics are obtained for Rayleigh fading channels.
  • The optimality of non-precoded Gaussian inputs holds for SOP and EPSR under the uniformly less noisy condition.
  • Optimality for ESR holds under the less noisy on average condition.
  • For single-antenna channels, SOP and EPSR optimality holds without channel ordering assumptions.

Where Pith is reading between the lines

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

  • In systems with statistical channel knowledge, precoding may not be needed for optimal secrecy if the ordering condition is satisfied.
  • The partial orderings introduced can be applied to compare secrecy performance in additional channel models.
  • These findings suggest Gaussian signaling is sufficient for secrecy in many fading environments.

Load-bearing premise

The main and eavesdropper channels satisfy the newly defined uniformly less noisy or less noisy on average partial orderings and the transmitter has only statistical knowledge of the channels.

What would settle it

A calculation showing that for channels satisfying the uniformly less noisy order, some non-Gaussian input distribution achieves a lower secrecy outage probability than the Gaussian input would falsify the optimality result.

Figures

Figures reproduced from arXiv: 2604.22587 by Benoit Geller, Clement Leroy, Olivier Rioul, Tarak Arbi.

Figure 1
Figure 1. Figure 1: Illusration of the general channel model view at source ↗
Figure 2
Figure 2. Figure 2 view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the construction of γ1, γ2 and h0. We now construct h and g with finite support and whose square norm lies in J . We define the channel distribution ph to have a single mass point, while pg has two mass points of equal weights. This models a non-varying main channel and an eavesdropper channel that can take only two distinct states. We choose the two mass points of pg to have square norms γ… view at source ↗
Figure 3
Figure 3. Figure 3: SOP⋆ as a function of r with NA = 1, NE = 2, PT = 3, khk = 5 and different values of σ. Both examples are based on the same model: a SISOSE wiretap channel in which the main channel fading coefficient h is constant and equal to a positive number h0, while the eavesdropper channel fading coefficient g is 0 with probability 1/2 and equal to a positive number g0 with probability 1/2. The total input power and… view at source ↗
Figure 4
Figure 4. Figure 4: shows the ESR for a Gaussian input and a BPSK input as functions of g0, with h0 = 3 (since IBPSK(3) ' 0.97 > 0.75). In this case,  1+h0 2 1 4 2 − 1 ' 10.3 and (1 + h0) 2 − 1 = 15 view at source ↗
Figure 5
Figure 5. Figure 5: Counterexample illustration The eavesdropper channel g consists of one fixed component and one component with a random sign. Specifically, g takes two possible values, g1 and g2, each with probability 0.5: g1 = view at source ↗
read the original abstract

We study the secrecy of wireless channels in the presence of an eavesdropper, where the channels are random and the transmitter only has knowledge of the channel statistics. We investigate the optimal input distribution with respect to several secrecy metrics: the Secrecy Outage Probability (SOP), defined as the probability that the coding rate $r$ exceeds the instantaneous secrecy rate; the Ergodic Secrecy Rate (ESR), defined as the expected secrecy rate over channel realizations; and the Ergodic Positive Secrecy Rate (EPSR), defined as the expected value of the positive part of the secrecy rate. We introduce two partial orderings for random channels: the uniformly less noisy order and the less noisy on average order. We show that when the main channel is uniformly less noisy than the eavesdropper channel, the optimal input distribution is a non-precoded Gaussian input for both the SOP and the EPSR. Furthermore, we show that the same input distribution is optimal for the ESR when the less noisy on average order holds. In addition, similar optimality results for the SOP and the EPSR are obtained for single-transmit-antenna channels without requiring any channel ordering assumptions. Closed-form expressions of the secrecy metrics are derived for special cases of Rayleigh fading channels.

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

0 major / 3 minor

Summary. The manuscript studies optimal input distributions for secrecy metrics (secrecy outage probability SOP, ergodic secrecy rate ESR, and ergodic positive secrecy rate EPSR) over random wireless channels when the transmitter has only statistical CSI. It introduces two new partial orderings on random channels—the uniformly less noisy order and the less noisy on average order—and proves that a non-precoded Gaussian input is optimal for SOP and EPSR when the main channel is uniformly less noisy than the eavesdropper channel. The same input is optimal for ESR under the less noisy on average ordering. Analogous optimality results hold for single-transmit-antenna channels without any ordering assumptions. Closed-form expressions for the metrics are derived in special cases of Rayleigh fading.

Significance. If the optimality claims hold, the work extends classical Gaussian optimality results for wiretap channels to the statistical-CSI regime via new channel orderings that generalize less-noisy concepts. This provides a principled way to select inputs for MIMO secrecy systems under partial CSI and yields practical closed-forms for Rayleigh fading. The approach is consistent with information-theoretic ordering techniques and could enable further analysis of secrecy metrics in more general settings.

minor comments (3)
  1. Section 2: The definitions of the uniformly less noisy and less noisy on average orderings would benefit from explicit comparison to classical notions (e.g., the standard less-noisy order or stochastic dominance) to clarify their novelty and implications.
  2. The numerical results section: Parameter values (SNR, fading variances, target rate r) used in the plots should be stated explicitly in the captions or text for reproducibility.
  3. Appendix derivations: A few intermediate steps in the closed-form expressions for Rayleigh fading could be expanded to improve readability without altering the results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and the recommendation of minor revision. The referee's summary correctly captures our main results on the optimality of non-precoded Gaussian inputs for SOP and EPSR under the uniformly less noisy ordering, and for ESR under the less noisy on average ordering, along with the closed-form expressions for Rayleigh fading.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper introduces two new partial orderings (uniformly less noisy and less noisy on average) as independent definitions, then derives optimality of non-precoded Gaussian inputs for SOP, EPSR, and ESR from those orderings plus standard information-theoretic secrecy rate expressions. SISO results hold without the orderings. No load-bearing self-citations, no fitted parameters renamed as predictions, and no equations that reduce by construction to their own inputs. The central claims rest on the newly defined orderings and classical wiretap analysis rather than circular re-labeling or self-referential fitting.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the newly introduced definitions of channel partial orderings as domain assumptions, plus the standard assumption of statistical CSI only; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption Uniformly less noisy order between main and eavesdropper channels
    Newly introduced ordering used to prove optimality for SOP and EPSR.
  • domain assumption Less noisy on average order between main and eavesdropper channels
    Newly introduced ordering used to prove optimality for ESR.

pith-pipeline@v0.9.0 · 5536 in / 1450 out tokens · 92537 ms · 2026-05-08T09:38:47.573649+00:00 · methodology

discussion (0)

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Reference graph

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