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

This paper investigates the optimal input distributions for maximizing secrecy metrics—specifically Secrecy Outage Probability (SOP), Ergodic Secrecy Rate (ESR), and Ergodic Positive Secrecy Rate (EPSR)—over random wireless channels by introducing new channel ordering concepts and providing closed-form solutions for Rayleigh fading scenarios.

Original authors: Clement Leroy, Tarak Arbi, Benoit Geller, Olivier Rioul

Published 2026-04-27

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Clement Leroy, Tarak Arbi, Benoit Geller, Olivier Rioul

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to have a private conversation in a crowded, noisy room. You have a friend (Bob) you want to talk to, but there is also a spy (Eve) lurking nearby, trying to eavesdrop on your secrets.

This paper is a mathematical study of how to talk to your friend as securely as possible when you don't know exactly where the spy is or how loud the background noise is—you only know the general patterns of the room (the "statistics").

Here is the breakdown of the paper using a few metaphors.

1. The Three Ways to Measure "Success"

The researchers looked at three different ways to define if your conversation was "secure." Think of these as different ways to grade a spy mission:

The Secrecy Outage Probability (SOP): The "Oops" Factor. This measures how often you accidentally spill a secret. If you decide to speak at a certain speed, how often does the spy manage to catch enough words to understand you?
The Ergodic Secrecy Rate (ESR): The "Average Speed." If you had this conversation 1,000 times, what would be your average speed of sharing secrets without the spy knowing?
The Ergodic Positive Secrecy Rate (EPSR): The "Productive Moments." This ignores the times when the spy is too loud and you can't talk at all. It only asks: "When we are successfully talking privately, how much information are we actually moving?"

2. The "Golden Rule" of Talking (The Gaussian Input)

In many wireless communication problems, mathematicians suggest using a "Gaussian" signal.

The Analogy: Imagine you are sending a message by tapping on a table. A "Gaussian" signal is like tapping in a very specific, natural, rhythmic way that sounds like random white noise to an outsider. It’s the "standard" way to talk in the math world.

The paper discovers that if your friend’s ear is "better" than the spy’s ear (meaning your friend's channel is "less noisy"), then the "Golden Rule" holds: Just use that standard, rhythmic Gaussian tapping. It is the absolute best way to minimize your "Oops" factor and maximize your "Productive Moments."

3. When the Rule Breaks (The Counterexamples)

This is the most interesting part of the paper. The researchers found that the "Golden Rule" isn't a law of nature; it’s a conditional suggestion.

The "Masking" Metaphor:
Imagine the spy is very good at hearing your voice, but very bad at hearing a heavy drumbeat. If you just use the "standard" rhythmic tapping (Gaussian), the spy might still catch your words.

However, if you use a "Structured" approach—where you send your message (the secret) and then simultaneously hit a loud drum (Artificial Noise) to drown out your own voice—you might actually be more secure. The drumbeat makes it impossible for the spy to distinguish your taps from the noise, even though your friend (who knows to listen for the rhythm) can still hear you.

The paper proves that if the spy's channel is "weird" or "unpredictable" in certain ways, the standard Gaussian way is actually not the best. In those cases, you need to be more clever, perhaps by using "Artificial Noise" to mask your secrets.

Summary in Plain English

If your friend has a clear connection and the spy is struggling: Just use the standard, mathematically "smooth" way of sending data (Gaussian). It’s simple and optimal.
If the spy is unpredictable or has a strange advantage: The standard way fails. You might need to get creative, like adding "fake noise" to your signal to confuse the spy, even if it makes the signal look less "smooth."
The Big Takeaway: You can't use a "one size fits all" strategy for security. To be truly safe, you have to understand the pattern of the noise in the room before you decide how to speak.

This paper, titled "On the Optimum Secrecy Outage Probability and Ergodic Secrecy Rate over Wireless Channels," investigates the optimization of secrecy performance in wireless wiretap channels where the transmitter (Alice) lacks instantaneous channel state information (CSI) and only possesses knowledge of the channel statistics.

1. The Problem

In a standard MIMO wiretap model, secrecy is achieved by transmitting a signal that Bob can decode but Eve cannot. While it is well-established that Gaussian inputs are optimal when instantaneous CSI is known, the problem becomes significantly more complex when the channels are random.

The authors address the optimization of three specific secrecy metrics under an average power constraint:

Secrecy Outage Probability (SOP): The probability that the instantaneous secrecy rate falls below a target rate $r$ .
Ergodic Positive Secrecy Rate (EPSR): The expected value of the secrecy rate, considering only positive realizations (relevant for on-off power control).
Ergodic Secrecy Rate (ESR): The expected value of the secrecy rate over all channel realizations.

The central challenge is determining the optimal input distribution $p(u, x)$ that maximizes these metrics when the channel matrices $H$ (Bob) and $G$ (Eve) are random variables.

2. Methodology

The authors employ information-theoretic tools, including the Pinsker decomposition and the I-MMSE relationship, to analyze the concavity of secrecy functions. To handle the randomness of the channels, they introduce two new partial orderings for ergodic channels:

Uniformly Less Noisy: The legitimate channel is better than the eavesdropper channel for every possible instantaneous channel realization.
Less Noisy on Average: The legitimate channel is better than the eavesdropper channel in terms of expected mutual information.

The paper categorizes the analysis into two main scenarios:

MIMOME (Multiple-Input, Multiple-Output, Multiple-Eavesdropper): General case with multiple antennas.
SIMOME (Single-Input, Multiple-Output, Multiple-Eavesdropper): A special case where Alice has only one antenna ( $N_A = 1$ ).

3. Key Contributions and Results

A. Optimality of Gaussian Inputs

The paper provides rigorous proofs for when a non-precoded Gaussian input is the optimal strategy:

For SOP and EPSR: If the legitimate channel is uniformly less noisy than the eavesdropper channel, a non-precoded Gaussian input is optimal for both metrics in the MIMOME case.
For ESR: If the legitimate channel is less noisy on average than the eavesdropper channel, a non-precoded Gaussian input is optimal.
SIMOME Scenario: For the SIMOME case ( $N_A = 1$ ), the authors prove that Gaussian inputs are optimal for SOP and EPSR without requiring any channel ordering assumptions.

B. Non-Optimality and Counterexamples

A significant contribution is the demonstration that Gaussian signaling is not universally optimal:

ESR in SIMOME: The authors prove (Theorem 6) that for the ESR, the optimal distribution depends heavily on the specific channel statistics. They provide examples where BPSK modulation outperforms Gaussian signaling.
Importance of Ordering: They provide a counterexample showing that if the "uniformly less noisy" assumption is violated, a structured input (combining information-bearing signals with artificial noise) can strictly outperform a non-precoded Gaussian input in both SOP and ESR.

C. Closed-Form Expressions

The authors derive closed-form expressions for Rayleigh fading channels, specifically for the SOP and EPSR in SIMOME settings, which are useful for practical system design.

4. Significance

This paper is significant for several reasons:

Bridging Theory and Practice: It moves beyond the idealized "perfect CSI" assumption to a more realistic "statistical CSI" model, which is the standard in fast-fading wireless environments.
Design Guidelines: It provides clear mathematical conditions (the two partial orderings) under which engineers can safely assume Gaussian signaling is optimal, saving computational complexity.
Complexity Awareness: By proving that Gaussian signaling is not always optimal for ESR or in non-ordered channels, it highlights the need for more sophisticated signaling strategies (like artificial noise or non-Gaussian modulation) in certain security-critical environments.