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arXiv:1909.11324 · IEEE Transactions on Information Forensics and Security

Finite Blocklength Analysis of Gaussian Random Coding in AWGN Channels Under Covert Constraint

Xinchun Yu, Shuangqin Wei, Yuan Luo

Published Oct 21, 2020• 22 citations
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Abstract

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.

DOI
10.1109/tifs.2020.3032292

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Finite Blocklength Analysis of Gaussian Random coding in AWGN Channels under Covert constraints II: A Viewpoint of Total Variation Distance

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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.

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On the Second Order Asymptotics of Covert Communications over AWGN Channels

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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 are$\sqrt{n\delta\log e}$and (2)${ }^{\frac{1}{2}}(n \delta)^{\frac{1}{4}}(\log e)^{\frac{3}{4}} \cdot Q^{-1}(\epsilon)$, respectively by$n$channel uses, where$\delta$and$\epsilon$are constraints imposed on covertness and channel decoding error probabilities, respectively. The technique we use in the achievability is quasi-$\varepsilon$-neighborhood notion from information geometry. For finite blocklength$n$, 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 dimension$n$, which further leads to the direct converse bound in terms of covert metric.

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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.

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We consider the problem of covert communication with random slot selection over binary-input Discrete Memoryless Channels (DMCs) and Additive White Gaussian Noise (AWGN) channels, in which a transmitter attempts to reliably communicate with a legitimate receiver while simultaneously maintaining covertness with respect to (w.r.t.) an eavesdropper. Covertness refers to the inability of the eavesdropper to distinguish the transmission of a message from the absence of communication, modeled by the transmission of a fixed channel input. Random slot selection refers to the transmitter’s ability to send a codeword in a time slot with known boundaries selected uniformly at random among a predetermined number of slots. Our main contribution is to develop bounds for the information-theoretic limit of communication in this model, called the covert capacity, when the number of time slots scales sub-exponentially with the codeword length. Our upper and lower bounds for the covert capacity are within a multiplicative factor of$\sqrt {2}$independent of the channel. This result partially fills a characterization gap between the covert capacity without random slot selection and the covert capacity with random selection among an exponential number of slots in the codeword length. Our key technical contributions consist of 1) a tight upper bound for the relative entropy characterizing the effect of random slot selection on the covertness constraint in our achievability proof; 2) a careful converse analysis to characterize the maximum allowable weight or power of codewords to meet the covertness constraint. Our results suggest that, unlike the case without random slot selection, the choice of covertness metric does not change the covert capacity in the presence of random slot selection.

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In this work, the covert communication assisted by a probabilistic jammer is investigated in the finite blocklength regime, in which the jammer emits jamming signals with a certain probability to alleviate its negative effect on legitimate link but still enhance the covertness performance. Specifically, we first derive a tractable expression of covertness constraint based on the average KL divergence. Then, a closed-form expression of average decoding error probability (DEP) involving the effect of probabilistic jamming is presented. Moreover, joint optimization of transmit power and blocklength at the transmitter for maximizing the average effective covert throughput (AECT) is formulated and tackled, which demonstrates the trade-off between the transmit power and blocklength to improve covert communication performance. Our results show that the probabilistic jamming can indeed enhance the covert communication performance, and the jamming probability can be optimized to further improve the AECT.

International Conference on Wireless Communications and Signal Processing (WCSP)•Nov 1, 2022•5 citations
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.
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.
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 consider the problem of covert communication with random slot selection over binary-input Discrete Memoryless Channels (DMCs) and Additive White Gaussian Noise (AWGN) channels, in which a transmitter attempts to reliably communicate with a legitimate receiver while simultaneously maintaining covertness with respect to (w.r.t.) an eavesdropper. Covertness refers to the inability of the eavesdropper to distinguish the transmission of a message from the absence of communication, modeled by the transmission of a fixed channel input. Random slot selection refers to the transmitter’s ability to send a codeword in a time slot with known boundaries selected uniformly at random among a predetermined number of slots. Our main contribution is to develop bounds for the information-theoretic limit of communication in this model, called the covert capacity, when the number of time slots scales sub-exponentially with the codeword length. Our upper and lower bounds for the covert capacity are within a multiplicative factor of2\sqrt {2}2​independent of the channel. This result partially fills a characterization gap between the covert capacity without random slot selection and the covert capacity with random selection among an exponential number of slots in the codeword length. Our key technical contributions consist of 1) a tight upper bound for the relative entropy characterizing the effect of random slot selection on the covertness constraint in our achievability proof; 2) a careful converse analysis to characterize the maximum allowable weight or power of codewords to meet the covertness constraint. Our results suggest that, unlike the case without random slot selection, the choice of covertness metric does not change the covert capacity in the presence of random slot selection.
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.
In this work, the covert communication assisted by a probabilistic jammer is investigated in the finite blocklength regime, in which the jammer emits jamming signals with a certain probability to alleviate its negative effect on legitimate link but still enhance the covertness performance. Specifically, we first derive a tractable expression of covertness constraint based on the average KL divergence. Then, a closed-form expression of average decoding error probability (DEP) involving the effect of probabilistic jamming is presented. Moreover, joint optimization of transmit power and blocklength at the transmitter for maximizing the average effective covert throughput (AECT) is formulated and tackled, which demonstrates the trade-off between the transmit power and blocklength to improve covert communication performance. Our results show that the probabilistic jamming can indeed enhance the covert communication performance, and the jamming probability can be optimized to further improve the AECT.