Fundamental Limits of Covert Communication

Traditional security (e.g., encryption) prevents unauthorized access to message content; however, detection of the mere presence of a message can have significant negative impact on the privacy of the communicating parties. Unlike these standard methods, covert or low probability of detection (LPD) communication not only protects the information contained in a transmission from unauthorized decoding, but also prevents the detection of a transmission in the first place. In this thesis we investigate the fundamental laws of covert communication. We first study covert communication over additive white Gaussian noise (AWGN) channels, a standard model for radio-frequency (RF) communication. We present a square root limit on the amount of information transmitted covertly and reliably over such channels. Specifically, we prove that if the transmitter has the channels to the intended receiver and the warden that are both AWGN, then O(\sqrt{n}) covert bits can be reliably transmitted to the receiver in n uses of the channel. Conversely, attempting to transmit more than O(\sqrt{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--\u3e\infty. Next we study the impact of warden\u27s ignorance of the communication attempt time. We prove that if the channels from the transmitter to the intended receiver and the warden are both AWGN, and if a single n-symbol period slot out of T(n) such slots is selected secretly (forcing the warden to monitor all T(n) slots), then O(\min{\sqrt{n\log T(n)},n}) covert bits can be transmitted reliably using this slot. Conversely, attempting to transmit more than O(\sqrt{n\log T(n)}) bits either results in detection with probability one or a non-zero probability of decoding error at the receiver. We then study covert optical communication and characterize the ultimate limit of covert communication that is secure against the most powerful physically-permissible adversary. We show that, although covert communication is impossible when a channel injects the minimum noise allowed by quantum mechanics, it is attainable in the presence of any noise excess of this minimum (such as the thermal background). In this case, O(\sqrt{n}) covert bits can be transmitted reliably in n optical channel uses using standard optical communication equipment. The all-powerful adversary may intercept all transmitted photons not received by the intended receiver, and employ arbitrary quantum memory and measurements. Conversely, we show that this square root scaling cannot be circumvented. Finally, we corroborate our theory in a proof-of-concept experiment on an optical testbed

Fundamental limits of quantum-secure covert optical sensing

We present a square root law for active sensing of phase $\theta$ of a single pixel using optical probes that pass through a single-mode lossy thermal-noise bosonic channel. Specifically, we show that, when the sensor uses an $n$-mode covert optical probe, the mean squared error (MSE) of the resulting estimator $\hat{\theta}_n$ scales as $\langle (\theta-\hat{\theta}_n)^2\rangle=\mathcal{O}(1/\sqrt{n})$; improving the scaling necessarily leads to detection by the adversary with high probability. We fully characterize this limit and show that it is achievable using laser light illumination and a heterodyne receiver, even when the adversary captures every photon that does not return to the sensor and performs arbitrarily complex measurement as permitted by the laws of quantum mechanics.Comment: 13 pages, 1 figure, submitted to ISIT 201

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

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 $o(\sqrt{n})$ bits can be sent from the transmitter to the receiver in $n$ channel uses while lower-bounding $\alpha+\beta\geq1-\epsilon$ for any $\epsilon>0$, where $\alpha$ and $\beta$ 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 $O(\sqrt{n})$ bits can be sent in $n$ LPD channel uses. Conversely, attempting to transmit more than $O(\sqrt{n})$ bits either results in detection by the warden with probability one or a non-zero probability of decoding error at the receiver as $n\rightarrow\infty$.Comment: Major revision in v2. Context, esp. the relationship to steganography updated. Also, added discussion on secret key length. Results are unchanged from previous version. Minor revision in v3. Major revision in v4, Clarified derivations (adding appendix), also context, esp. relationship to previous work in communication updated. Results are unchanged from previous revision

Bounding the quantum limits of precision for phase estimation with loss and thermal noise

We consider the problem of estimating an unknown but constant carrier phase modulation $\theta$ using a general -- possibly entangled -- $n$-mode optical probe through $n$ independent and identical uses of a lossy bosonic channel with additive thermal noise. We find an upper bound to the quantum Fisher information (QFI) of estimating $\theta$ as a function of $n$, the mean and variance of the total number of photons $N_{\rm S}$ in the $n$-mode probe, the transmissivity $\eta$ and mean thermal photon number per mode ${\bar n}_{\rm B}$ of the bosonic channel. Since the inverse of QFI provides a lower bound to the mean-squared error (MSE) of an unbiased estimator $\tilde{\theta}$ of $\theta$, our upper bound to the QFI provides a lower bound to the MSE. It already has found use in proving fundamental limits of covert sensing, and could find other applications requiring bounding the fundamental limits of sensing an unknown parameter embedded in a correlated field.Comment: No major changes to previous version. Change in the title and abstract, change in the presentation and structure, an example of the bound is now included, and some references were added. Comments are welcom

Fundamental Limits of Thermal-noise Lossy Bosonic Multiple Access Channel

Bosonic channels describe quantum-mechanically many practical communication links such as optical, microwave, and radiofrequency. We investigate the maximum rates for the bosonic multiple access channel (MAC) in the presence of thermal noise added by the environment and when the transmitters utilize Gaussian state inputs. We develop an outer bound for the capacity region for the thermal-noise lossy bosonic MAC. We additionally find that the use of coherent states at the transmitters is capacity-achieving in the limits of high and low mean input photon numbers. Furthermore, we verify that coherent states are capacity-achieving for the sum rate of the channel. In the non-asymptotic regime, when a global mean photon-number constraint is imposed on the transmitters, coherent states are the optimal Gaussian state. Surprisingly however, the use of single-mode squeezed states can increase the capacity over that afforded by coherent state encoding when each transmitter is photon number constrained individually.Comment: 8 pages, 3 figure

Covert Communication over Classical-Quantum Channels

The square root law (SRL) is the fundamental limit of covert communication over classical memoryless channels (with a classical adversary) and quantum lossy-noisy bosonic channels (with a quantum-powerful adversary). The SRL states that $\mathcal{O}(\sqrt{n})$ covert bits, but no more, can be reliably transmitted in $n$ channel uses with $\mathcal{O}(\sqrt{n})$ bits of secret pre-shared between the communicating parties. Here we investigate covert communication over general memoryless classical-quantum (cq) channels with fixed finite-size input alphabets, and show that the SRL governs covert communications in typical scenarios. %This demonstrates that the SRL is achievable over any quantum communications channel using a product-state transmission strategy, where the transmitted symbols in every channel use are drawn from a fixed finite-size alphabet. We characterize the optimal constants in front of $\sqrt{n}$ for the reliably communicated covert bits, as well as for the number of the pre-shared secret bits consumed. We assume a quantum-powerful adversary that can perform an arbitrary joint (entangling) measurement on all $n$ channel uses. However, we analyze the legitimate receiver that is able to employ a joint measurement as well as one that is restricted to performing a sequence of measurements on each of $n$ channel uses (product measurement). We also evaluate the scenarios where covert communication is not governed by the SRL