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IEEE International Symposium on Information Theory (ISIT)

The Continuous-Time Poisson Channel Has Infinite Covert Communication Capacity

Ligong Wang

Published Jun 1, 2018• 13 citations
Publisher

Abstract

We consider the problem of communication over a continuous-time, infinite-bandwidth Poisson channel without peak-power constraint, but subject to a covertness constraint: the relative entropy between the output distributions when a codeword is transmitted and when no input is provided to the channel must tend to zero as total communication time grows large. We show that, under such a constraint, the capacity of this channel, in nats per second, is infinity.

DOI
10.1109/isit.2018.8437350

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We consider the problem of communication over a continuous-time, infinite-bandwidth Poisson channel without peak-power constraint, but subject to a covertness constraint: the relative entropy between the output distributions when a codeword is transmitted and when no input is provided to the channel must tend to zero as total communication time grows large. We show that, under such a constraint, the capacity of this channel, in nats per second, is infinity.
We consider the problem of communication over a continuous-time Poisson channel subject to a covertness constraint: The relative entropy between the output distributions when a message is transmitted and when no input is provided must be small. In the absence of both bandwidth and peak-power constraints, we show the covert communication capacity of this channel, in nats per second, to be infinity. When a peak-power constraint is imposed on the input, the covert communication capacity becomes zero, and the “square-root scaling law” applies. When a bandwidth constraint but no peak-power constraint is imposed, the covert communication capacity is again shown to be zero, but we have not determined whether the square-root law holds or not.
We consider a problem of communication over a continuous-time additive Gaussian noise channel. A covertness constraint is imposed on the communication protocol, which says that the channel output must statistically resemble pure noise. When there is no bandwidth constraint on the input, we argue that the covert communication capacity of this channel is positive, in contrast to the band-limited case where covertness requires that the amount of transmission grow at most like the square root of total communication time. This claim is formally proven in the case where the Gaussian noise is white with respect to the bandwidth used by the transmitter.
This paper considers the problem of communication over a discrete memoryless channel subject to the constraint that the probability that an adversary who observes the channel outputs can detect the communication is low. Specifically, the relative entropy between the output distributions when a codeword is transmitted and when no input is provided to the channel must be sufficiently small. For a channel whose output distribution induced by the zero input symbol is not a mixture of the output distributions induced by other input symbols, it is shown that the maximum number of bits that can be transmitted under this criterion scales like the square root of the blocklength. Exact expressions for the scaling constant are also derived.
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.
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.
This paper considers the problem of communication over a discrete memoryless channel (DMC) or an additive white Gaussian noise (AWGN) channel subject to the constraint that the probability that an adversary who observes the channel outputs can detect the communication is low. In particular, the relative entropy between the output distributions when a codeword is transmitted and when no input is provided to the channel must be sufficiently small. For a DMC whose output distribution induced by the “off” input symbol is not a mixture of the output distributions induced by other input symbols, it is shown that the maximum amount of information that can be transmitted under this criterion scales like the square root of the blocklength. The same is true for the AWGN channel. Exact expressions for the scaling constant are also derived.