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Fundamental Limits of Communication With Low Probability of Detection

Ligong Wang, Gregory W. Wornell, Lizhong Zheng

Published Jun 1, 2016• 208 citations
OpenAlex

Abstract

© 2016 IEEE. 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.

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Fundamental Limits of Communication With Low Probability of Detection

Ligong Wang, Gregory W. Wornell, Lizhong Zheng

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.

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Limits of low-probability-of-detection communication over a discrete memoryless channel

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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.

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Information-theoretic limits of covert communication over additive-noise channels

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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.

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Limits of Reliable Communication with Low Probability of Detection on AWGN Channels

Boulat A. Bash, Dennis Goeckel, Don Towsley

We present a square root limit on the amount of information transmitted reliably and with low probability of detection (LPD) over additive white Gaussian noise (AWGN) channels. Specifically, if the transmitter has AWGN channels to an intended receiver and a warden, both with non-zero noise power, we prove that o(√n) bits can be sent from the transmitter to the receiver in n channel uses while lower-bounding α + β ≥ 1-ε for any ε > 0, where α and β respectively denote the warden's probabilities of a false alarm when the sender is not transmitting and a missed detection when the sender is transmitting. Moreover, in most practical scenarios, a lower bound on the noise power on the channel between the transmitter and the warden is known and O(√n) bits can be sent in n LPD channel uses. Conversely, attempting to transmit more than O(√n) bits either results in detection by the warden with probability one or a non-zero probability of decoding error at the receiver as n→∞.

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Bounds on Covert Capacity With Sub-Exponential Random Slot Selection

Shi-Yuan Wang, Keerthi Suria Kumar Arumugam, Matthieu R. Bloch

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.

arXiv:2409.07777•IEEE Transactions on Information Theory•Jul 16, 2025•1 citation

Covert Capacity-Key Tradeoff Over Discrete Memoryless Networks

Abdelaziz Bounhar, Mireille Sarkiss, Michèle Wigger

This paper characterizes the covert capacity-key tradeoffs for discrete memoryless channels (DMC) and discrete memoryless multiple-access channels (DMMAC). The focus is on channels where communication rates are measured as the number of transmitted bits divided by the square-root of the blocklength and by the square-root of the covertness constraint. Previous results had determined the largest achievable covert rates for these channels, as well as the minimum key rates required to achieve these largest covert rates. In this work, we additionally determine the minimum key rates required at reduced covert rates. Stated differently, we determine the set of all covert rates that are achievable for a given set of key rates. This is termed thecovert capacity-key tradeoff. Our results on the covert capacity-key tradeoffs over DMCs and DMMACs show that for small key rates and when the adversary observes the inputs through a better channel than the legitimate receiver, binary signalling at all transmitters is optimal even for larger input alphabets and the capacity-key tradeoff grows linearly. Moreover, in this regime the covert capacity-key tradeoff of DMMACs is square implying that each of the transmitters can simultaneously achieve its own largest covert rate depending only on its own available key rate, irrespective of the other transmitter. For larger key rates or when the adversary is not uniformly stronger than the legitimate receiver over all inputs, the covert capacity-key tradeoff region of DMMACs is non-square and a tradeoff arises between the rates of the various transmitters.

IEEE Journal on Selected Areas in Information Theory•Jan 1, 2026•0 citations
© 2016 IEEE. 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.
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.
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.
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.
We present a square root limit on the amount of information transmitted reliably and with low probability of detection (LPD) over additive white Gaussian noise (AWGN) channels. Specifically, if the transmitter has AWGN channels to an intended receiver and a warden, both with non-zero noise power, we prove that o(√n) bits can be sent from the transmitter to the receiver in n channel uses while lower-bounding α + β ≥ 1-ε for any ε > 0, where α and β respectively denote the warden's probabilities of a false alarm when the sender is not transmitting and a missed detection when the sender is transmitting. Moreover, in most practical scenarios, a lower bound on the noise power on the channel between the transmitter and the warden is known and O(√n) bits can be sent in n LPD channel uses. Conversely, attempting to transmit more than O(√n) bits either results in detection by the warden with probability one or a non-zero probability of decoding error at the receiver as n→∞.
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.
This paper characterizes the covert capacity-key tradeoffs for discrete memoryless channels (DMC) and discrete memoryless multiple-access channels (DMMAC). The focus is on channels where communication rates are measured as the number of transmitted bits divided by the square-root of the blocklength and by the square-root of the covertness constraint. Previous results had determined the largest achievable covert rates for these channels, as well as the minimum key rates required to achieve these largest covert rates. In this work, we additionally determine the minimum key rates required at reduced covert rates. Stated differently, we determine the set of all covert rates that are achievable for a given set of key rates. This is termed thecovert capacity-key tradeoff. Our results on the covert capacity-key tradeoffs over DMCs and DMMACs show that for small key rates and when the adversary observes the inputs through a better channel than the legitimate receiver, binary signalling at all transmitters is optimal even for larger input alphabets and the capacity-key tradeoff grows linearly. Moreover, in this regime the covert capacity-key tradeoff of DMMACs is square implying that each of the transmitters can simultaneously achieve its own largest covert rate depending only on its own available key rate, irrespective of the other transmitter. For larger key rates or when the adversary is not uniformly stronger than the legitimate receiver over all inputs, the covert capacity-key tradeoff region of DMMACs is non-square and a tradeoff arises between the rates of the various transmitters.