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arXiv:2604.22587

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

Clément Leroy, Tarak Arbi, Benoît Geller, Olivier Rioul

Published Apr 24, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2604.22587

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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 rrr 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.
The achievable secrecy of time-invariant wireless channels is typically characterized by the secrecy capacity of the communication system, whose mathematical expression is well established. However, in most practical scenarios, channels cannot be assumed to remain constant or to be perfectly known at both the transmitter and the legitimate receiver. In such cases, no closed-form expression for the secrecy capacity is available. To address this, several metrics are studied, that characterize secrecy in random channel environments: Ergodic Secrecy Rate (ESR), Ergodic Positive Secrecy Rate (EPSR), and Secrecy Outage Probability (SOP). We derive the optimal values of these metrics for SIMOME channels (Single-Input, Multiple-Output, Multiple-Eavesdropper) and various MIMOME (Multiple-Input, Multiple-Output, Multiple-Eavesdropper) scenarios.
In this paper, we consider the problem of secret communication through Rayleigh fading single-input multiple-output (SIMO) broadcast channel in the presence of an eavesdropper. At first, we define the ergodic secrecy capacity and find the optimal power allocation at the transmitter that achieves the secrecy capacity for the full channel state information (CSI) case, where transmitter has access to both the main channel and eavesdropper's channel gains, and for the case of only main channel CSI at the transmitter. Then, we define the secrecy capacity in case of open-loop transmission scheme and present the analytical expression for the lower bound of ergodic secrecy capacity. We also find the analytical expression for secure outage probability to study the secure outage performance of the proposed model.
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This thesis examines the challenges of information-theoretic secret communication that exploits the temporal and spatial variations of the wireless medium to improve secret communication rates. We first examine the secrecy capacity of a system consisting of independent parallel channels with one transmitter, one intended receiver and one eavesdropper. We show that the secrecy capacity of the system is the sum of the secrecy capacities of the individual subchannels. We further derive the optimal power allocation strategy for a system of parallel AWGN channels subject to a total power constraint, and also extend the results to random fading channels with additive Gaussian noise. We then study the achievable secrecy rate with Gaussian random codes for the situation where the channel of the intended receiver is a constant AWGN channel, while the eavesdropper's channel is fast Rayleigh fading with unknown realizations but known statistics to the transmitter. The proposed method with artificial noise and bursting provides ways to achieve positive secrecy rate even when Bob's channel is worse than Eve's channel on average. We also examine the achievable secrecy rate for a multiple antenna system, and the optimal input structure needed to achieve this rate. For the multiple input single output case, an analytical solution is derived. Multiple antenna systems provide extra degrees of freedom to the transmitter so that a beamforming-like approach can be used to provide advantage to the intended receiver against the eavesdropper. Next we derive a secrecy capacity outer bound region for a class of one-sided interference channels. The outer bound is shown to be tight for a class of binary deterministic one-sided interference channels, and can be achieved within one bit for some Gaussian one-sided interference channels.Finally, as Gaussian random codes are impractical, we evaluate achievable secrecy rates with discrete signaling. We observe that with discrete signaling, there exists an optimal power that maximizes the achievable secrecy rate. For the AWGN channel, larger constellation is always better. While for fading channel, the optimal constellation size varies with the power constraint, and discrete signaling can perform better than random Gaussian coding.