Positive-rate Covert Communication

• Positive-rate covert communication is defined by achieving a reliable, constant data rate while ensuring adversaries’ detection remains at random-guessing levels.
• Techniques such as leveraging transmitter-side channel state information, jamming, noise uncertainty, and multi-antenna nulling overcome the traditional square-root law limitations.
• System designs use optimization methods, helper nodes, and even quantum protocols to practically implement covert networking with measurable positive capacity.

Positive-rate ^^^^1^^^^t communication refers to scenarios in which a transmitter can reliably send information to a legitimate receiver at a strictly positive rate, while ensuring that an adversary (warden or eavesdropper) is fundamentally unable to detect the presence of communication, except at the trivial error probability associated with random guessing. This field is motivated by applications requiring not only secrecy but also undetectability of transmission, such as anti-surveillance, resistance to RF-based attack, low-observability networking, and next-generation quantum communications.

1. Fundamental Principles and Covertness Metrics

The canonical model involves three parties—transmitter (Alice), legitimate receiver (Bob), and warden (Willie)—over a noisy channel. "Covertness" typically means that the probability for the warden to correctly detect transmission (aggregate of false-alarm and missed-detection probabilities) remains arbitrarily close to that of pure guessing. Two formal metrics dominate:

• Detection error sum: $\alpha + \beta \geq 1 - \epsilon$, where $\alpha$ and $\beta$ are false-alarm and missed-detection probabilities, and $\epsilon$ is a small parameter.
• Kullback-Leibler (KL) divergence: $D(P_{Z^n} \| Q_0^{\otimes n}) \to 0$, where $P_{Z^n}$ is the distribution of the warden's observation under communication, $Q_0$ when Alice is silent. Pinsker's inequality links KL divergence to total variation and error rate.

In typical discrete memoryless channel (DMC) and AWGN settings with no shared randomness or channel state knowledge, the square-root law prevails: only $O(\sqrt n)$ covert bits are achievable in $n$ uses ($C_{\mathrm{covert}} = 0$ per channel use) (Bash et al., 2015).

Positive-rate covert communication—achieving $C_{\mathrm{covert}} > 0$—requires leveraging structural advantages: channel state information (CSI) at the transmitter, noise-uncertainty, jamming or helper nodes, multi-antenna systems with transmit nullspaces, action-dependent states, or hybrid network-layer techniques (Lee et al., 2017, Bendary et al., 2019, Huang et al., 2021, ZivariFard et al., 22 Jan 2025).

2. Channel Models Supporting Positive Covert Rates

2.1 Noisy Channels with Noise Uncertainty and Jamming

When the warden's noise statistics are uncertain (e.g., $\sigma_w^2$ only known to lie in a range), Alice can exploit this by operating at a power that remains hidden within the warden's uncertainty, achieving positive rate covert communication (Bash et al., 2015). Independently, friendly jammers can transmit additional noise unknown to Willie, permitting Alice to transmit at constant power and positive rate (Huang et al., 2021).

2.2 State-dependent Channels and Action-dependent State Information

If channels are endowed with (possibly transmitter-controllable) random states—either random, as in random fading/interference, or action-dependent as in channels with injecting "cover" noise—transmitter-side CSI or action-dependent state information (ADSI) can be leveraged to mask the covert transmission completely (Lee et al., 2017, ZivariFard et al., 22 Jan 2025). The transmitter codes to induce at the warden exactly the output distribution seen during silence, via soft covering or resolvability, which permits strictly positive, single-letter covert capacity.

2.3 Multi-antenna and Spatial Techniques

MIMO channels admit positive-rate covert schemes when the transmitter can steer nulls in the direction of Willie's antennas (i.e., the transmit covariance can be restricted to the null-space of $H_w$). In the massive-MIMO regime ($N_a \to \infty$), the covert capacity approaches the ordinary MIMO capacity as isotropic leakage vanishes (Bendary et al., 2019).

2.4 Helper and Cooperative Mechanisms

Jamming-aided and helper-assisted schemes (including quantum) allow the legitimate parties to arrange for the warden's observations to be rendered uninformative, by correlating signals or employing codebook designs that simulate innocent (no-communication) distributions at Willie's receiver. Specifically, helpers can coordinate their codebooks to zero out the warden's mutual information, leading to positive-rate capacity regions for both classical and quantum MACs (Huang et al., 2021, ZivariFard et al., 26 Apr 2025).

2.5 Robust/Hybrid and Network-layer Approaches

Realistic systems deploy a variety of methods including artificial noise emission (full-duplex receivers) (Hu et al., 2017), moving/flexible antennas (Wang et al., 1 Dec 2025), and transmission in hybrid mode (adaptive frequency/time/mode selection) (Zhang et al., 2023, Wang et al., 2023). Additionally, methods based on amplitude or bit-rate modulation over real networks show experimentally that positive covert throughput is achievable in practice, with observed rates up to 5 bps and spectral efficiencies near 0.92 bps/Hz (Soderi et al., 2024).

3. Capacity Formulas and Achievability Conditions

Capacity for positive-rate covert communication depends on the ability to match or mask the warden's marginal statistics:

• DMC with state/ADSI: Covert capacity is given by

$$C_{\mathrm{covert, nc}} = \max_{P_{U|S}, x(u,s): P_Z = Q_0} [I(U;Y) - I(U;S)],$$

under the constraint that the warden's output law is indistinguishable from silence, and with sufficiently large key if needed (Lee et al., 2017, ZivariFard et al., 22 Jan 2025).

• AWGN with CSI: With noncausal CSI, the capacity becomes

$C_{\mathrm{covert, nc}} = \frac{1}{2} \log(1+P^*)$

where $P^*$ is the "pre-subtractable" part of transmitter power after masking state (Lee et al., 2017, ZivariFard et al., 22 Jan 2025).

• Multi-antenna/MIMO: A strictly positive covert rate is achievable if and only if the transmit covariance can satisfy $H_w Q H_w^\dagger = 0$, i.e., there is a nontrivial null-space at the warden (Bendary et al., 2019).
• Quantum settings: If a helper can align the induced warden state with the innocent marginal, the multiuser covert region matches the reliability region, i.e., positive rates up to the mutual informations available through the legitimate channel (ZivariFard et al., 26 Apr 2025). In Minkowski vacuum mining protocols, choosing the energy gap, windowing, and coupling strength can obtain nonzero covert quantum capacity by extracting timelike entanglement (Bradler et al., 2016).
• Relay and full-duplex receiver setups: Relay injection, opportunistic forwarding, and artificial noise emission create scenarios where the effective covert rate can be strictly positive under realistic detection constraints at the warden (Hu et al., 2017, Hu et al., 2017, Hu et al., 2017).

4. Optimization and Design Methodologies

Solving for positive-rate covert communication typically entails:

• Joint power/rate/allocation optimization constrained by covertness and reliability (Wang et al., 1 Dec 2025, Bendary et al., 2019, Zhang et al., 19 Aug 2025).
• Alternating optimization—decomposing into power control, antenna placement, or rate selection subproblems, solved iteratively (Wang et al., 1 Dec 2025).
• Lagrangian and KKT tools for block-fading channels; e.g., generalized water-filling with caps under per-block covert constraints (Zhang et al., 19 Aug 2025).
• Block-Markov encoding and key generation for channels with action/state dependencies and low-rate shared keys (ZivariFard et al., 22 Jan 2025).
• DRL-based (deep reinforcement learning) methods (e.g., DDQN) to tackle causal-CSI allocation under non-Markovian objectives (Zhang et al., 19 Aug 2025).
• Robust and semidefinite programming (SDP) to handle uncertainty in the warden's channel estimates and achieve covertness under bounded CSI error (Wang et al., 2023).

5. Prototypical System Examples

Scenario	Conditions for $C_\text{covert}>0$	Key Reference
AWGN, perfect noise knowledge	Not achievable: only $O(\sqrt n)$ bits ($C_\mathrm{covert}=0$)	(Bash et al., 2015)
AWGN, warden noise uncertainty	$C_\mathrm{covert}>0$ if uncertainty band covers Alice's power	(Bash et al., 2015)
DMC, transmitter CSI	$C_\mathrm{covert}>0$ if state known at transmitter, masking warden's output	(Lee et al., 2017)
MIMO nullspace	$C_\mathrm{covert}>0$ if transmit nullspace to warden exists	(Bendary et al., 2019)
Block-fading, Tx/Rx have CSI	$C_\mathrm{covert}>0$ if at least one block has $h_\ell \geq g_\ell$	(Zhang et al., 19 Aug 2025)
Hybrid backscatter with PA	$C_\mathrm{covert}>0$ by optimizing transmit power and antenna placements	(Wang et al., 1 Dec 2025)
Quantum MAC with helper	$C_\mathrm{covert}>0$ if helper camouflages warden's statistics	(ZivariFard et al., 26 Apr 2025)

6. Practical Implications and System Design Insights

Positive-rate covert communication fundamentally hinges on transmitter-side control, channel knowledge, and system structure to manipulate the distribution of the warden's observations. For physical-layer and physical network-layer designs, these insights imply:

• Exploiting advantageous side-information or flexibility (CSI, active jamming, reconfigurable antennas, radar cooperation) is essential.
• Noise-uncertainty and helper mechanisms—both in theory and practical networked systems—enable surmounting the square-root law and support covert communication at rates competitive with ordinary channels.
• Hybrid systems (e.g., combining hardware-level action dependence, network-layer modulation, and adaptive spatial transmission) open further avenues for covert throughput under stringent detection constraints.
• Quantum protocols (both optical and quantum field theoretic) admit fundamentally new positive-rate covert paradigms (timelike entanglement mining, covertness via channel resolvability), further motivating ultrasecure quantum networking (Arrazola et al., 2016, Bradler et al., 2016, ZivariFard et al., 26 Apr 2025).
• Mechanisms such as pilot attacks in hardware-trojan scenarios demonstrate the ease with which practical systems can be pushed from square-root to linear positive-rate covert regimes in the absence of robust defense (Bakirtas et al., 2024).

Rigorous engineering of such systems must jointly consider channel uncertainties, adversarial capabilities, and information-theoretic trade-offs to achieve reliable and undetectable communication in both classical and quantum domains.

7. Future Directions and Open Problems

Emerging research continues to address:

• Achievability under limited or partial CSI, dynamic environments, and adversaries with adaptive detection capability.
• Universal coding and practical schemes that approach information-theoretic covert capacity in nonasymptotic regimes.
• Multilayered covert networking, integrated sensing-communication designs, and application to distributed and ad hoc scenarios.
• Robust covert quantum communication over noisy, large-scale optical or superconducting systems.
• Automated detection and countermeasures—both practical IDS design and theoretical minimax detection—against positive-rate covert channels.

The field remains rich for investigation at the intersection of information theory, wireless systems, and quantum communication (Zhang et al., 19 Aug 2025, Wang et al., 2023, Hu et al., 2017).