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.
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.
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 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 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.
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.
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 arenδlogeand (2)21(nδ)41(loge)43⋅Q−1(ϵ), respectively bynchannel uses, whereδandϵare constraints imposed on covertness and channel decoding error probabilities, respectively. The technique we use in the achievability is quasi-ε-neighborhood notion from information geometry. For finite blocklengthn, 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 dimensionn, which further leads to the direct converse bound in terms of covert metric.
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 Θ(n41) beyond the first-order term scaling in the order Θ(n21) 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-η-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.