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China Communications

Total variation based asymptotics of covert communication in AWGN channels

Wang Jianquan, Wanbin Tang, Xiaoping Li

Published May 1, 2025• 0 citations
Publisher

Abstract

Covert communication guarantees the security of wireless communications via hiding the existence of the transmission. This paper focuses on the first and second order asymptotics of covert communication in the AWGN channels. The covertness is measured by the total variation distance between the channel output distributions induced with and without the transmission. We provide the exact expressions of the maximum amount of information that can be transmitted with the maximum error probability and the total variation less than any small numbers. The energy detection and the random coding are employed to prove our results. We further compare our results with those under relative entropy. The results show how many additional amounts of information can be transmitted covertly when changing the covertness constraint to total variation.

DOI
10.23919/jcc.fa.2021-0804.202505

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Covert communication guarantees the security of wireless communications via hiding the existence of the transmission. This paper focuses on the first and second order asymptotics of covert communication in the AWGN channels. The covertness is measured by the total variation distance between the channel output distributions induced with and without the transmission. We provide the exact expressions of the maximum amount of information that can be transmitted with the maximum error probability and the total variation less than any small numbers. The energy detection and the random coding are employed to prove our results. We further compare our results with those under relative entropy. The results show how many additional amounts of information can be transmitted covertly when changing the covertness constraint to total variation.
We study the first- and second-order asymptotics of covert communication over binary-input discrete memoryless channels for three different covertness metrics and under maximum probability of error constraint. When covertness is measured in terms of the relative entropy between the channel output distributions induced with and without communication, we characterize the exact first- and second-order asymptotics of the number of bits that can be reliably transmitted with a maximum probability of error less than E and a relative entropy less than δ. When covertness is measured in terms of the variational distance between the channel output distributions or in terms of the probability of missed detection for fixed probability of false alarm, we establish the exact first-order asymptotics and bound the second-order asymptotics. Pulse position modulation achieves the optimal first-order asymptotics for all three metrics, as well as the optimal second-order asymptotics for relative entropy. The main conceptual contribution of this paper is to clarify how the choice of a covertness metric impacts the information-theoretic limits of covert communications. The main technical contribution underlying our results is a detailed expurgation argument to show the existence of a code satisfying the reliability and covertness criteria.
We study the second-order asymptotics of covert communication over binary-input DMC for three different metrics of covertness. When covertness is measured in terms of the relative entropy between the channel output distributions induced with and without communication, we characterize the exact second-order asymptotics of the number of bits that can be reliably transmitted with a probability of error less than ϵ\epsilonϵ and a relative entropy less than δ\deltaδ. When covertness is measured in terms of the variational distance between the channel output distributions or in terms of the probability of missed detection for fixed probability of false alarm, we establish the exact first-order asymptotics and bounds on the second-order asymptotics. The main conceptual contribution of this paper is to clarify how the choice of a covertness metric impacts the information-theoretic limits of the number of covert and reliable bits. The main technical contribution of the underlying results is a detailed analysis of probability of existence of a code satisfying the reliability and covert criteria.
Covert communication is crucial in privacy protection for users of wireless networks, and efforts have been made to improve the system throughput subject to a constraint on a warden's detection error probability. However, most existing research is based on an unclear relation between system throughput and covertness, due to the use of the Kullback-Leibler divergence. To solve this problem, in this paper, we explore the exact throughput-covertness relation by investigating the likelihood variation distance. An algorithm is established to find the minimum channel uses required for the transmission of a given amount of information and a certain covertness constraint. The rigorous proof of the convergence of the algorithm is also given. Our results show that an overly strict constraint on covertness would lead to a large sacrifice of system throughput, and it is not cost-effective to increase the number of channel uses to improve covertness if only a small amount of information is to be transmitted. These results could be very helpful for system designers in achieving a desirable tradeoff between system throughput and covertness.
It is well known that finite blocklength analysis plays an important role in evaluating performances of communication systems in practical settings. This paper considers the achievability and converse bounds on the maximal channel coding rate (throughput) at a given blocklength and error probability in covert communication over AWGN channels. The covert constraint is given in terms of an upper bound on total variation distance (TVD) between the distributions of eavesdropped signals at an adversary with and without presence of active and legitimate communication, respectively. For the achievability, Gaussian random coding scheme is adopted for convenience in the analysis of TVD. The classical results of finite blocklength regime are not applicable in this case. By exploiting and extending canonical approaches, we first present new and more general achievability bounds for random coding schemes under maximal or average probability of error requirements. The general bounds are then applied to covert communication in AWGN channels where codewords are generated from Gaussian distribution while meeting the maximal power constraint. We further show an interesting connection between attaining tight achievability and converse bounds and solving two total variation distance based minimax and maxmin problems. The TVD constraint is analyzed under the given random coding scheme, which induces bounds on the transmission power through divergence inequalities. Further comparison is made between the new achievability bounds and existing ones derived under deterministic codebooks. Our thorough analysis thus leads us to a comprehensive characterization of the attainable throughput in covert communication over AWGN channels.
Covert communication over an additive white Gaussian noise (AWGN) channel with finite block length is investigated in this paper. The attention is on the covert criterion, which has not been considered in finite block length circumstance. As an accurate quantity metric of discrimination, the variation distance with given finite block length n and signal-noise ratio (snr) is obtained. We give both its analytic solution and expansions which can be easily evaluated. It is shown that K-L distance, which is frequently adopted as the metric of discrimination at the adversary in asymptotic regime, is not convincing in finite block length regime compared with the total variation distance. Moreover, the convergence rate of the total variation with different snr is analyzed when the block length tends to infinity. The results will be very helpful for understanding the behavior of the total variation distance and practical covert communication.
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.