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Secrecy Metrics over Random Gaussian SIMOME and MIMOME Channels

Clément Leroy, Tarak Arbi, Benoît Geller, Olivier Rioul

Published Nov 1, 2026• 0 citations
OpenAlex

Abstract

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.

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1 On Secrecy Capacity of Fast Fading MIMOME Wiretap Channels With Statistical CSIT

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This paper studies the secrecy performance of multiple-input multiple-output (MIMO) wiretap channels, also termed as multiple-input multiple-output multiple-eavesdropper (MIMOME) channels, under transmit antenna selection (TAS) and BPSK/QPSK modulations. In the main channel between the transmitter and the legitimate receiver, a single transmit antenna is selected to maximize the instantaneous Signal to Noise Ratio (SNR) at the receiver. At the receiver and the eavesdropper, selection combining (SC) is utilized. Additionally, suppose that the transmitted message is modulated by BPSK/QPSK modes. We first derive the closed-form approximated expression for the ergodic secrecy rate under Rayleigh fading, assuming that the channel state information of the eavesdropper (CSIE) is available at the transmitter. Next, analytical formulas for the approximated and asymptotic secrecy outage probability are also developed when CSIE is unavailable (NCSIE). Simulation results are provided to demonstrate the approximation precision of the derived results above. Furthermore, the asymptotic results reveal that the secrecy diversity order degrades into 0 due to the finite-alphabet inputs, which is totally different from that driven by the Gaussian inputs.

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MIMOME channel secrecy capacity

Mohammad Rakibul Islam, Jinsang Kim, Md Shamsul Arefin

Although conventional cryptographic security mechanisms are essential to the overall problem of security, the openness of wireless medium poses both threats and opportunities for securing transmission. Information theoretic security is attracting researchers for its robust nature. In this paper, a Gaussian multiple input multiple output multiple eavesdropper (MIMOME) channel is considered where a transmitter is communicating to a receiver in the presence of an eavesdropper. The transmitter is equipped with multiple antennas, while the receiver and the eavesdropper also contains multiple antennas. We present a technique for determining the secrecy capacity of the MIMO channel under Gaussian noise. To do so, we transform the channel into multiple single input multiple output (SIMO) Gaussian wiretap channel and then use scalar approach using standard techniques of communications theory. Then we consider the effect of fading and found out the limiting values of the secrecy capacity.

International Conference on Computer and Information Technology (ICCIT)•Dec 1, 2008•4 citations
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.
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.
In this paper, we consider secure transmissions in ergodic Rayleigh fast-faded multiple-input multiple-output multiple-antenna-eavesdropper (MIMOME) wiretap channels with only statistical channel state information at the transmitter (CSIT). When the antennas at the legitimate receiver are more than (or equal to) those at the eavesdropper, we prove the first MIMOME secrecy capacity with partial CSIT by establishing a new secrecy capacity upper-bound. The key step is to form an MIMOME degraded channel by dividing the legitimate receiver's channel matrix into two submatrices, and setting one of the submatrices to be the same as the eavesdropper's channel matrix. Next, under the total power constraint over all transmit antennas, we analytically solve the channel-input covariance matrix optimization problem to fully characterize the MIMOME secrecy capacity. Typically, the MIMOME optimization problems are non-concave. However, thanks to the proposed degraded channel, we can transform the stochastic MIMOME optimization problem to be a Schur-concave one and then find its solution. Besides total power constraint, we also investigate the secrecy capacity when the transmitter is subject to the practical per-antenna power constraint. The corresponding optimization problem is even more difficult since it is not Schur-concave. Under the two power constraints considered, the corresponding MIMOME secrecy capacities can both scale with the signal-to-noise ratios (SNR) when the difference between numbers of antennas at legitimate receiver and eavesdropper are large enough. However, when the legitimate receiver and eavesdropper have a single antenna each, such SNR scalings do not exist for both cases.
In this paper, we consider secure transmissions in ergodic Rayleigh fast-faded multiple-input multiple-output multiple-antenna-eavesdropper (MIMOME) wiretap channels with only statistical channel state information at the transmitter (CSIT). When the antennas at the legitimate receiver are more than (or equal to) those at the eavesdropper, we prove the first MIMOME secrecy capacity with partial CSIT by establishing a new secrecy capacity upper-bound. The key step is to form an MIMOME degraded channel by dividing the legitimate receiver's channel matrix into two submatrices, and setting one of the submatrices to be the same as the eavesdropper's channel matrix. Next, under the total power constraint over all transmit antennas, we analytically solve the channel-input covariance matrix optimization problem to fully characterize the MIMOME secrecy capacity. Typically, the MIMOME optimization problems are non-concave. However, thanks to the proposed degraded channel, we can transform the stochastic MIMOME optimization problem to be a Schur-concave one and then find its solution. Besides total power constraint, we also investigate the secrecy capacity when the transmitter is subject to the practical per-antenna power constraint. The corresponding optimization problem is even more difficult since it is not Schur-concave. Under the two power constraints considered, the corresponding MIMOME secrecy capacities can both scale with the signal-to-noise ratios (SNR) when the difference between numbers of antennas at legitimate receiver and eavesdropper are large enough. However, when the legitimate receiver and eavesdropper have a single antenna each, such SNR scalings do not exist for both cases.
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.
This paper studies the secrecy performance of multiple-input multiple-output (MIMO) wiretap channels, also termed as multiple-input multiple-output multiple-eavesdropper (MIMOME) channels, under transmit antenna selection (TAS) and BPSK/QPSK modulations. In the main channel between the transmitter and the legitimate receiver, a single transmit antenna is selected to maximize the instantaneous Signal to Noise Ratio (SNR) at the receiver. At the receiver and the eavesdropper, selection combining (SC) is utilized. Additionally, suppose that the transmitted message is modulated by BPSK/QPSK modes. We first derive the closed-form approximated expression for the ergodic secrecy rate under Rayleigh fading, assuming that the channel state information of the eavesdropper (CSIE) is available at the transmitter. Next, analytical formulas for the approximated and asymptotic secrecy outage probability are also developed when CSIE is unavailable (NCSIE). Simulation results are provided to demonstrate the approximation precision of the derived results above. Furthermore, the asymptotic results reveal that the secrecy diversity order degrades into 0 due to the finite-alphabet inputs, which is totally different from that driven by the Gaussian inputs.
Although conventional cryptographic security mechanisms are essential to the overall problem of security, the openness of wireless medium poses both threats and opportunities for securing transmission. Information theoretic security is attracting researchers for its robust nature. In this paper, a Gaussian multiple input multiple output multiple eavesdropper (MIMOME) channel is considered where a transmitter is communicating to a receiver in the presence of an eavesdropper. The transmitter is equipped with multiple antennas, while the receiver and the eavesdropper also contains multiple antennas. We present a technique for determining the secrecy capacity of the MIMO channel under Gaussian noise. To do so, we transform the channel into multiple single input multiple output (SIMO) Gaussian wiretap channel and then use scalar approach using standard techniques of communications theory. Then we consider the effect of fading and found out the limiting values of the secrecy capacity.