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arXiv:1710.09754

Time-Division is Optimal for Covert Communication over Some Broadcast Channels

Vincent Y. F. Tan, Si-Hyeon Lee

Published Oct 26, 2017• 1 citation
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Abstract

We consider a covert communication scenario where a transmitter wishes to communicate simultaneously to two legitimate receivers while ensuring that the communication is not detected by an adversary, the warden. The legitimate receivers and the adversary observe the transmission from the transmitter via a three-user discrete or Gaussian memoryless broadcast channel. We focus on the case where the "no-input" symbol is not redundant, i.e., the output distribution at the warden induced by the no-input symbol is not a mixture of the output distributions induced by other input symbols, so that the covert communication is governed by the square root law, i.e., at most $Θ(\sqrt{n})$ bits can be transmitted over $n$ channel uses. We show that for such a setting, a simple time-division strategy achieves the optimal throughputs for a non-trivial class of broadcast channels; this is not true for communicating over broadcast channels without the covert communication constraint. Our result implies that a code that uses two separate optimal point-to-point codes each designed for the constituent channels and each used for a fraction of the time is optimal in the sense that it achieves the best constants of the $\sqrt{n}$-scaling for the throughputs. Our proof strategy combines several elements in the network information theory literature, including concave envelope representations of the capacity regions of broadcast channels and El Gamal's outer bound for more capable broadcast channels.

DOI
10.48550/arxiv.1710.09754

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We consider a covert communication scenario where a transmitter wishes to communicate simultaneously to two legitimate receivers while ensuring that the communication is not detected by an adversary, the warden. The legitimate receivers and the adversary observe the transmission from the transmitter via a three-user discrete or Gaussian memoryless broadcast channel. We focus on the case where the “no-input” symbol is not redundant, i.e., the output distribution at the warden induced by the no-input symbol is not a mixture of the output distributions induced by other input symbols, so that the covert communication is governed by the square root law, i.e., at most Θ(√n) bits can be transmitted over n channel uses. We show that for such a setting, a simple time-division strategy achieves the optimal throughputs for a non-trivial class of broadcast channels; this is not true for communicating over broadcast channels without the covert communication constraint. Our result implies that a code that uses two separate optimal point-to-point codes each designed for the constituent channels and each used for a fraction of the time is optimal in the sense that it achieves the best constants of the √n-scaling for the throughputs. Our proof strategy combines several elements in the network information theory literature, including concave envelope representations of the capacity regions of broadcast channels and El Gamal's outer bound for more capable broadcast channels.
We consider a covert communication scenario where a legitimate transmitter wishes to communicate simultaneously to two legitimate receivers while ensuring that the communication is not detected by an adversary, also called the warden. The legitimate receivers and the adversary observe the transmission from the legitimate transmitter via a three-user discrete or Gaussian memoryless broadcast channel. We focus on the case where the symbol is not redundant, i.e., the output distribution at the warden induced by the no-input symbol is not a mixture of the output distributions induced by other input symbols, so that the covert communication is governed by the square root law, i.e., at most Θ(n)\Theta(\sqrt{n})Θ(n​) bits can be transmitted over nnn channel uses. We show that for such a setting of covert communication over broadcast channels, a simple time-division strategy achieves the optimal throughputs. Our result implies that a code that uses two separate optimal point-to-point codes each designed for the constituent channels and each used for a fraction of the time is optimal in the sense that it achieves the best constants of the n\sqrt{n}n​-scaling for the throughputs. Our proof strategy combines several elements in the network information theory literature, including concave envelope representations of the capacity regions of broadcast channels and El Gamal's outer bound for more capable broadcast channels.
We consider a scenario in which K transmitters attempt to communicate covert messages reliably to a legitimate receiver over a discrete memoryless multiple-access channel (MAC) while simultaneously escaping detection from an adversary who observes their communication through another discrete memoryless MAC. We assume that each transmitter may use a secret key that is shared only between itself and the legitimate receiver. We show that each of the K transmitters can transmit on the order of √n reliable and covert bits per n channel uses, exceeding which, the warden will be able to detect the communication. We identify the optimal pre-constants of the scaling, which leads to a complete characterization of the covert capacity region of the K-user binary-input MAC. We show that, asymptotically, all sum-rate constraints are inactive unlike the traditional MAC capacity region. We also characterize the channel conditions that have to be satisfied for the transmitters to operate without a secret key.
We study the covert communication over K-user discrete memoryless interference channels (DM-ICs) with a warden. It is assumed that the warden's channel output distribution induced by K "off" input symbols, which are sent when no communication occurs, is not a convex combination of those induced by any other combination of input symbols (otherwise, the square-root law does not hold). We derive the exact covert capacity region and show that a simple point-to-point based scheme with treating interference as noise is optimal. In addition, we analyze the secret key length required for the reliable and covert communication with the desired rates, and present a channel condition where a secret key between each user pair is unnecessary. The results are extended to the Gaussian case and the case with multiple wardens.
We study the covert communication over K-user-pair discrete memoryless interference channels (DM-ICs) with a warden. It is assumed that the warden's channel output distribution induced by K “off” input symbols, which are sent when no communication occurs, is not a convex combination of those induced by any other combination of input symbols (otherwise, the square-root law does not hold). We derive the exact covert capacity region and show that a simple point-to-point based scheme with treating interference as noise is optimal. In addition, we analyze the secret key length required for the reliable and covert communication with the desired rates, and present a channel condition where a secret key between each user pair is unnecessary. The results are extended to the Gaussian case and the case with multiple wardens.
We consider the covert communication over the two-user Gaussian multiple access channel (GMAC) with causal feedback. It is shown that any causal feedback does not enlarge the covert capacity region and the time-division scheme is optimal for a class of channels, which includes the case where the warden observes what the legitimate receiver receives. For the proof, we derive an outer bound on the covert capacity region of general GMACs with noiseless causal feedback, which implies that the square-root law holds for arbitrary channel gains. Furthermore, the extension to multi-user case is presented.