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IEEE International Symposium on Information Theory (ISIT)

Second-Order Asymptotics for Covert Communication over MIMO AWGN Channels

Changhong Liu, Jingjing Wang, Lin Zhou

Published Jun 1, 2026• 0 citations
Publisher

Abstract

As claimed in International Telecommunications Union (ITU) Recommendation ITU-R M.2160, 6G communication systems require extremely high-security and low-latency communication, necessitating the study of covert communication operating at finite blocklengths. In the point-to-point (P2P) setting, covert communication enables a transmitter to send messages reliably over a noisy channel to a legitimate receiver without being detected by any third party. Under the covertness metric of Kullback–Leibler divergence (KLD), we derive exact second-order asymptotics for P2P covert communication over a multiple-input multiple-output (MIMO) additive white Gaussian noise (AWGN) channel. In particular, our theoretical benchmarks refine the first-order asymptotics of Wang and Bloch (TIFS 2021), which is known as the square root law by showing that the non-asymptotic maximal number of transmitted messages has a back off that scales in the order \(\Theta(n^\frac{1}{4})\) beyond the first-order term scaling in the order \(\Theta(n^\frac{1}{2})\) when the blocklength is \(n\). Thus, our second-order asymptotic bound provides a better approximation to the finite blocklength performance of optimal codes. Furthermore, compared with the single antenna result of Yu et al. (arXiv:2305.17924v3), we demonstrate the impact of the number of antennas \(m\) and reveal spatial diversity gains of MIMO, advocating the use of MIMO for covert communication to achieve a high transmission rate. To prove our results, we extend the quasi-\(\eta\)-neighborhood framework from single-antenna real value channels to multi-antenna complex value MIMO channels. To ensure covertness, the transmission power vanishes as the blocklength \(n\) increases. Thus, we judiciously analyze the finite blocklength performance of MIMO communication by modifying critical steps concerning the Berry–Esseen Theorem to deal with vanishing second and third absolute moments of information densities that rely on blocklength, which is in stark contrast with the non-covert case.

DOI
10.1109/isit62367.2026.11653958

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As claimed in International Telecommunications Union (ITU) Recommendation ITU-R M.2160, 6G communication systems require extremely high-security and low-latency communication, necessitating the study of covert communication operating at finite blocklengths. In the point-to-point (P2P) setting, covert communication enables a transmitter to send messages reliably over a noisy channel to a legitimate receiver without being detected by any third party. Under the covertness metric of Kullback–Leibler divergence (KLD), we derive exact second-order asymptotics for P2P covert communication over a multiple-input multiple-output (MIMO) additive white Gaussian noise (AWGN) channel. In particular, our theoretical benchmarks refine the first-order asymptotics of Wang and Bloch (TIFS 2021), which is known as the square root law by showing that the non-asymptotic maximal number of transmitted messages has a back off that scales in the order (\Theta(n^\frac{1}{4})) beyond the first-order term scaling in the order (\Theta(n^\frac{1}{2})) when the blocklength is (n). Thus, our second-order asymptotic bound provides a better approximation to the finite blocklength performance of optimal codes. Furthermore, compared with the single antenna result of Yu et al. (arXiv:2305.17924v3), we demonstrate the impact of the number of antennas (m) and reveal spatial diversity gains of MIMO, advocating the use of MIMO for covert communication to achieve a high transmission rate. To prove our results, we extend the quasi-(\eta)-neighborhood framework from single-antenna real value channels to multi-antenna complex value MIMO channels. To ensure covertness, the transmission power vanishes as the blocklength (n) increases. Thus, we judiciously analyze the finite blocklength performance of MIMO communication by modifying critical steps concerning the Berry–Esseen Theorem to deal with vanishing second and third absolute moments of information densities that rely on blocklength, which is in stark contrast with the non-covert case.
We study the second-order asymptotics of optimal codes for covert communication over quasi-static multi-antenna fading channels, under the covertness metric of Kullback--Leibler (KL) divergence. In particular, we study all four cases regarding the availability of channel state information (CSI) for the legitimate transmitter and receiver, assume that the warden knows perfect CSI for the channel from the legitimate transmitter to itself, whereas the legitimate transmitter knows only that the warden's channel matrix belongs to a bounded deterministic uncertainty set. Specifically, we show that, when the blocklength is nnn, the first-order covert rate satisfies the square root law, scaling as Θ(n−12)Θ(n^{-\frac{1}{2}})Θ(n−21​) with the coefficient determined by the traces of the channel matrices of the legitimate users and the warden, and the second-order rate vanishes. We also show that the availability of CSI at the transmitter enables optimal covert power allocation across the spatial sub-channels, which can increase the covert rate, relatively to equal power allocation, without changing the square root law. Furthermore, we reveal the significant spatial diversity gain provided by multiple-antenna systems for covert communication and demonstrate the critical role of the number of antennas to achieve high throughput covert communication. For the covertness analysis, we extend the quasi-ηηη-neighborhood framework to quasi-static fading channels. For the reliability analysis, due to the vanishing power imposed by the covertness constraint, we refine the non-covert analysis by Yang et al. (TIT, 2014), by carefully controlling higher-order terms and exploiting the properties of covert outage probability.
This work tackles the asymptotics of the maximal throughput of covert communications over AWGN channels when the covert metric is Kullback-Leibler divergence (KL divergence). It is shown that the first and second order asymptotics of the maximal throughput arenδlog⁡e\sqrt{n\delta\log e}nδloge​and (2)12(nδ)14(log⁡e)34⋅Q−1(ϵ){ }^{\frac{1}{2}}(n \delta)^{\frac{1}{4}}(\log e)^{\frac{3}{4}} \cdot Q^{-1}(\epsilon)21​(nδ)41​(loge)43​⋅Q−1(ϵ), respectively bynnnchannel uses, whereδ\deltaδandϵ\epsilonϵare constraints imposed on covertness and channel decoding error probabilities, respectively. The technique we use in the achievability is quasi-ε\varepsilonε-neighborhood notion from information geometry. For finite blocklengthnnn, the generating distributions are chosen to be a family of truncated Gaussian distributions with decreasing variances. The law of decreasing is carefully designed so that it maximizes the throughput at the main channel in the asymptotic sense under the condition that the output distributions satisfy the covert constraint. For the converse, the optimality of Gaussian distribution for minimizing KL divergence under second order moment constraint is extended from dimension 1 to dimensionnnn, which further leads to the direct converse bound in terms of covert metric.
Physical-layer security aims to exploit the randomness inherent in communication channels in order to guarantee confidentiality even against computationally unlimited attackers. In this thesis, we focus on covert communication, also known as “communication with low probability of detection”. This is a scenario where a transmitter and receiver try to prevent an eavesdropper from making a good guess on whether a communication is ongoing or not. We assume that the receiver and the eavesdropper share the same channel and therefore see the same outputs. In this scenario, the transmitter shares a key with the legitimate receiver to provide the necessary advantage to detect the communication and reliably decode the message. Asymptotically, the amount of information that can be sent reliably and covertly scales like the square root of the number of channel uses; this is known as the square root law of covert communication.We study the corresponding scaling constant of the square root law for general memoryless channels including non-Gaussian additive noise, as well as Gaussian channels with memory. In the latter case, we show that the scaling constant is the same as over the memoryless Gaussian channel. For continuous memoryless channels, under mild integrability conditions, we show that the scaling constant is upper bounded by a simple expression which only involves the variance of the logarithm of the probability density function of the noise. Moreover, we show that under some additional assumptions, this upper bound is tight. Furthermore, we provide upper bounds on the length of the secret key required to achieve covertness.The second objective of this work is to investigate the limits of covert communication over continuous memoryless channels in the finite blocklength regime. We provide bounds for the first and second-order asymptotics for covert communication over an AWGN channel under a maximal error probability criterion and for the first-order asymptotics under an average error probability criterion.
The problem of covert communication over Multiple-Input Multiple-Output (MIMO) Additive White Gaussian Noise (AWGN) channels is investigated, in which a transmitter attempts to reliably communicate with a legitimate receiver while avoiding detection by a passive adversary. The covert capacity of the MIMO AWGN channel is characterized under a variational distance covertness constraint when the MIMO channel matrices are static and known. The characterization of the covert capacity is also extended to a class of channels in which the legitimate channel matrix is known but the adversary’s channel matrix is only known up to a rank and a spectral norm constraint.
Covert communication conceals transmission of messages between Alice and Bob from an adversary, Willie, who tries to determine if a transmission took place or not. While covert communication in a basic, standard setting where all variables are known to Willie, results in the well-known square-root law, when a jammer is present and assists Alice by creating uncertainty in Willie's decoder, a strictly positive transmission rate is possible. In this work, we analyze the case where the jammer is equipped with multiple antennas. Specifically, we analyze the effect of multiple antennas at the jammer on Alice's transmission power and consequently on the transmission rate. We consider both the case where the channel knowledge of Willie is known as well as the case where it is unknown. We formulate several optimization problems for the transmission strategies of the jammer to maximize his assistance to Alice, in terms of maximizing Bob's received SNR and consequently the covert rate. When the channel information is known to the jammer, we show that under an achievable covertness scheme, the optimal strategy of the jammer is to perform beamforming towards a single direction with all his available power. This direction though, is not trivial, since it reflects a tradeoff point between minimizing the interference at Bob and maximizing the interference at Willie. When the channel knowledge is unknown, we show that the optimal strategy of the jammer is either to transmit isotropically to all directions or to the null-space of Bob, where this choice depends on certain channel conditions. This is in contrast to current schemes in the literature. Furthermore, we extend the optimization problems to the case where Bob is also equipped with multiple antennas, and provide insightful results, shown to be asymptotically optimal, accompanied by simulations.
This paper investigates the achievable message transmission rate of covert communication over slow fading channels where the channel coefficient remains unchanged over a finite transmission block. Such communication is often implemented with an extremely low transmission power aiming to achieve low probability of detection by the eavesdropper and under tolerable probability of decoding error for the legitimate receiver. The exact expression of achievability and asymptotic covert bounds are derived, which are shown in accordance with the square root law in AWGN (Additive White Gaussian Noise) channels with large transmission blocks. The case that the eavesdropper has unknown channel state information is also studied in the paper.