This paper considers the problem of communication over a memoryless classical-quantum wiretap channel subject to the constraint that the eavesdropper on the channel should not be able to learn with high confidence whether the legitimate parties are using the channel to communicate or not. Specifically, the relative entropy between the output quantum states at the eavesdropper when a codeword is transmitted and when no input is provided must be sufficiently small. Extending earlier works, this paper proves the “square-root law” for a broad class of classical-quantum channels: the maximum amount of information that can be reliably and covertly transmitted over n uses of such a channel scales like √n. The scaling constant is also determined.
This paper considers the problem of communication over a memoryless classical-quantum wiretap channel subject to the constraint that the eavesdropper on the channel should not be able to learn with high confidence whether the legitimate parties are using the channel to communicate or not. Specifically, the relative entropy between the output quantum states at the eavesdropper when a codeword is transmitted and when no input is provided must be sufficiently small. Extending earlier works, this paper proves the “square-root law” for a broad class of classical-quantum channels: the maximum amount of information that can be reliably and covertly transmitted over n uses of such a channel scales like √n. The scaling constant is also determined.
We investigate covert communication over general memoryless classical-quantum channels with fixed finite-size input alphabets. We show that the square root law (SRL) governs covert communication in this setting when product a ofninput states is used:LSRLn+o(n)covert bits (but no more) can be reliably transmitted innuses of classical-quantum channel, whereLSRL>0is a channel-dependent constant that we callcovert capacity. We also show that ensuring covertness requiresJSRLn+o(n)bits secret key shared by the communicating parties prior to transmission, whereJSRL≥0is a channel-dependent constant. We assume a quantum-powerful adversary that can perform an arbitrary joint (entangling) measurement on allnchannel uses. We determine the single-letter expressions forLSRLandJSRL, and establish conditions whenJSRL=0(i.e., no pre-shared secret key is needed). Finally, we evaluate scenarios where covert communication is not governed by the SRL.
Recently, the fundamental limits of covert, i.e., reliable-yet-undetectable, communication have been established for general memoryless channels and for lossy-noisy bosonic (quantum) channels with a quantum-limited adversary. The key import of these results was the square-root law (SRL) for covert communication, which states that O(√n) covert bits, but no more, can be reliably transmitted over n channel uses with O(√n) bits of secret pre-shared between communicating parties. Here we prove the achievability of the SRL for a general memoryless classical-quantum channel, showing that SRL covert communication is achievable over any quantum communication channel with a product-state transmission strategy. We leave open the converse, which, if proven, would show that even using entangled transmissions and entangling measurements, the SRL for covert communication cannot be surpassed over an arbitrary quantum channel.
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 paper, we study the conditions under which covert communication at positive rates is feasible over discrete memoryless classical, classical-quantum, and quantum channels. For classical point-to-point channels, we show that if the dimension of the channel input probability simplex exceeds that of the channel output probability simplex, equivalently, if the input alphabet has larger cardinality than the output alphabet, then positive covert rates are achievable for certain classes of Discrete Memoryless Channels (DMCs). We further show that allowing the innocent symbol (i.e., the symbol transmitted in the no-communication mode) to be chosen appropriately can enlarge the class of DMCs for which positive covert rates are achievable. We also study covert communication over classical Multiple-Access Channels (MACs), where the additional transmitter effectively enlarges the set of available channel input pairs, and we show that the conditions required to satisfy the covertness constraint are less restrictive for MACs than for point-to-point channels. We extend these results to classical-quantum DMCs by showing that if the dimension of the input probability simplex exceeds the affine dimension of the set of output states that can be induced at the channel output, then positive covert rates are achievable for certain classes of classical-quantum DMCs. Finally, for quantum channels, we show that if the innocent state (i.e., the state transmitted in the no-communication mode) is mixed, then the covertness constraint can always be satisfied by a non-trivial input ensemble. Consequently, positive covert rates are achievable whenever the legitimate receiver can distinguish at least two states in a suitable such ensemble.
We establish the fundamental limits of covert communication over quantum channels, focusing on both finite-dimensional classical-quantum (CQ) channels and the infinite-dimen\-sional lossy thermal-noise bosonic channel. For general memoryless CQ channels, we characterize when covert communication is possible and prove a square root law (SRL): at most Ln+o(n) bits can be reliably transmitted covertly over n uses of the channel where L is the channel-dependent covert capacity. We derive a single-letter expression for L and show that the required pre-shared secret key also scales as n under mild conditions. We also identify and characterize corner cases where either constant-rate or zero-rate covert communication is possible. For the bosonic channel, we prove that the SRL holds and derive exact covert capacities under different resource assumptions. In particular, we show that the covert secrecy and covert entanglement generation capacities both equal the covert classical capacity without entanglement-assistance, Lno-EA, and show that the required pre-shared classical secret key scales as n. We also find that quadrature phase-shift keying (QPSK) achieves the optimal covert capacity, outperforming binary modulation in contrast to the classical channel setting. When Alice and Bob instead share entanglement, the covert classical capacity improves to LEAnlogn+o(nlogn), though this scaling is sensitive to noise in the entangled resource. Finally, we propose a practical covert transceiver and simulate its performance compared to recently experimentally-tested transceivers under realistic conditions.
We study the fundamental limits of covert communications over general memoryless additive-noise channels. We assume that the legitimate receiver and the eavesdropper share the same channel and therefore see the same outputs. Under mild integrability assumptions, we find a general upper bound on the square-root scaling constant, which only involves the variance of the logarithm of the probability density function of the noise. Furthermore, we show that, under some additional assumptions, this upper bound is tight. We also provide upper bounds on the length of the secret key required to achieve the optimal scaling.