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arXiv:1703.01362 · IEEE Transactions on Information Theory

First- and Second-Order Asymptotics in Covert Communication

Mehrdad Tahmasbi, Matthieu R. Bloch

Published Oct 29, 2018• 99 citations
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Abstract

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.

DOI
10.1109/tit.2018.2878526

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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.
We determine the exact second-order asymptotics of covert communication over binary-input discrete memoryless channels when covertness is measured by variational distance. Previous work by Tahmasbi and Bloch [IEEE Trans. Inf. Theory, Apr. 2019] characterized the first-order asymptotics and derived achievability and converse bounds on the second-order term, but these bounds do not match. The gap arises from an additional penalty of order n^{1/4} in the achievability bound. We show that this penalty can be removed through a sharper analysis of the distribution of the warden's output induced by pulse-position modulation. Specifically, we express the variational distance through the Bhattacharyya coefficient of two distributions and the expectation of a continuous function of the log-likelihood ratio. Because the resulting expectation involves a continuous function rather than the probability of a likelihood-ratio event, an analysis of the characteristic function combined with a Gaussian smoothing argument reduces the approximation error from O(n^{-1/4}) (derived from the Berry--Esseen bound in prior work) to O(n^{-1/2}). With this better controlled approximation error, we manage to derive a matching achievability result to the existing converse result, thus establishing the exact second-order asymptotics.
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.
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.
In this paper, we study covert communication over a compound discrete memoryless channel (DMC). There are two channel states in which one of them is arbitrarily chosen and remains fixed during the transmission. The objective is to reliably send a message from the transmitter to the receiver. An adversary who is observing the channel output should not be able to infer the channel state. Two covertness metrics are considered. In the first metric, covertness is measured using the KL-divergence of the channel output marginal of each state with a fixed distribution. Different cases where such a distribution can be specified, are studied. The optimal transmission rate of each case is established. In the second metric, the covertness is measured by using the total variation distance of the channel output marginals of the two states. Upper and lower bounds on the optimal transmission covert rate are derived. The bounds match for a special case and characterize the optimal throughput.
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.