Square Root Law for Covert Communications

• The square root law defines the maximum covert bits that can be reliably transmitted as scaling with Θ(√n) over n channel uses under strict low probability of detection constraints.
• It is rigorously proven across classical AWGN, discrete memoryless, quantum, and bosonic channels through sparse signaling strategies and hypothesis-testing techniques.
• Experimental validations and theoretical bounds confirm its efficiency while highlighting challenges in key sharing, synchronization, and hardware limitations.

The square root law for covert communications is a fundamental principle in information and quantum theory that characterizes the maximum information that can be reliably and covertly transmitted over $n$ independent uses of a noisy channel, under stringent low probability of detection (LPD) constraints. In both classical and quantum settings, the law asserts that covert throughput fundamentally scales as $\Theta(\sqrt n)$, a sharp departure from the $O(n)$ scaling of unconstrained capacity. This law has been rigorously proven across a broad variety of models, including classical AWGN channels, classical-quantum and quantum channels, wireless networks (with various forms of interference and adversary models), and compound/uncertainty channels.

1. Statement of the Square Root Law

The square root law (SRL) stipulates that the maximal number of information bits (or qubits, or entangled pairs) that can be sent both reliably and covertly over $n$ independent channel uses scales at most as $\Theta(\sqrt n)$. For classical AWGN and DMCs, as well as quantum and bosonic channels, this bound is tight under canonical model assumptions.

Classical AWGN Channel Example:

• In the presence of a passive warden ("Willie") observing an independent AWGN channel with noise variance $\sigma_w^2$, for blocklength $n$, the largest reliably and covertly transmittable message $M$ satisfies

$$\log_2 M = \kappa(\delta) \sqrt n + o(\sqrt n), \;\; \kappa(\delta) = \frac{\sigma_w^2 Q^{-1}(1-\delta/2)}{ \sigma_b^2 \ln 2 \sqrt{2} },$$

where the sum of adversary's false-alarm and missed-detection probabilities is bounded below by $1-\delta$ (Bash et al., 2012, Bash et al., 2015).

Discrete Memoryless Channels and Compound Channels:

• For DMCs with an "off" input symbol and covertness measured in total variation or KL divergence, the law again dictates

$\max \log |\mathcal{M}| \approx O(\sqrt n)$

for message set $\mathcal M$ (Salehkalaibar et al., 2019, Zhang et al., 2020).

Bosonic Channels and Quantum Generalizations:

• Over lossy thermal-noise bosonic channels, covert bits or qubits are limited by

$$M = L\sqrt n + o(\sqrt n), \quad L = c_{\rm cov} c_{\rm rel}$$

with explicit constants in terms of photon number constraints and channel parameters (Bullock et al., 2019, Anderson et al., 11 Jun 2025, Anderson et al., 2024).

Universality:

• The law holds for point-to-point classical, classical-quantum, compound, multiuser, and interference channels under nontrivial LPD constraints (Cho et al., 2020, Bullock et al., 2016).

2. Mathematical Formulation and Information-Theoretic Constants

The SRL is not merely a qualitative scaling: the precise leading constants are derived via second-order asymptotics and depend on the relative entropy and $\chi^2$-divergence (classical or quantum) between the "active" and "inactive" channel-induced output distributions.

General DMC/Compound Model (Salehkalaibar et al., 2019):

For a compound channel with two states $W_s$, the covert capacity in the large-key regime is: $$L^* = \frac{2\,\Phi^{-1}\left(\tfrac{1+\delta}{2}\right)}{\sqrt{\Delta}\,\mathbb{D}},$$ where

$$\Delta = \mathbb E_{Q_0}\left[\left(\frac{\tilde Q_1 - \tilde Q_2}{Q_0}\right)^2\right], \; \mathbb{D} = D(\tilde Q_1 \| Q_0) = D(\tilde Q_2 \| Q_0).$$

Bosonic (Quantum) Channel (Bullock et al., 2019): $$c_{\mathrm{cov}} = \frac{\sqrt{2\eta \bar n_B (1 + \eta \bar n_B)}}{1 - \eta}, \;\; M = \sqrt{n} \delta\, c_{\mathrm{cov}}\, c_{\mathrm{rel}} + o(\sqrt{n}),$$ where $\eta$ is the transmissivity, $\bar n_B$ is background photon number.

Multiuser and Interference Channels (Cho et al., 2020): $$R_k \leq \alpha_k D(W_k^{(k)} \| W_0^{(k)}) / \sqrt{\chi^2(\alpha)/2},$$ where $R_k$ is the normalized covert rate for user $k,$ and $\chi^2(\alpha)$ is a channel- and user-split-dependent cross-divergence term.

3. Achievability and Converse Mechanisms

Achievability

Code constructions universally employ sparse signaling: only $O(\sqrt n)$ out of $n$ channel uses are "active" (i.e., nonzero, or non-innocent), with which information is encoded and transmitted. Achievability leverages random coding, soft-covering (resolvability), and hypothesis-testing bounds.

• Sparse codebooks: Codewords with $\Theta(\sqrt n)$ active symbols ensure covertness by keeping the induced output distribution statistically close (in total variation or quantum relative entropy) to the null distribution (Salehkalaibar et al., 2019, Bullock et al., 2016).
• ML or square-root decoding at the receiver ensures reliable recovery with vanishing error probability when the code size matches the SRL rate.
• Quantum protocols: Sparse transmission rounds, Pauli-twirling, and quantum error-correcting codes achieve the same scaling for qubits and ebits (Anderson et al., 22 Jan 2025, Anderson et al., 11 Jun 2025).

Converse

Converses are established via statistical hypothesis testing at the adversary.

• Any attempt to transmit more than $O(\sqrt n)$ bits increases the adversary’s detection probability (i.e., adversary’s total variation or relative entropy between noise and signal-plus-noise laws scales to $O(1)$ or higher), allowing sub-unity error probability (Bash et al., 2012, Bash et al., 2015, Bash et al., 2014).
• Or, to evade detection, Alice must spread power so broadly that Bob’s channel becomes too noisy for reliable decoding.

4. Extensions: Multiuser, Quantum, Networks

Compound Channels:

• In channels with unknown state known to Bob but not Willie, the square-root law is retained; the exponent's constant is determined by both states’ divergences and their cross-moments (Salehkalaibar et al., 2019).

Identification Codes:

• For covert identification, the iterated log-size of the message set scales as $\Theta(\sqrt n)$; the per-message rate is again governed by the same constants as in standard covert transmission, and no key is needed (Zhang et al., 2020).

Classical-Quantum and Quantum Channels:

• The SRL holds in full generality for memoryless classical-quantum channels with arbitrary finite input (Bullock et al., 2016, Wang, 2016). The precise scaling constant is determined via second derivative (quantum $\chi^2$-divergence) of the output relative entropy.

Entanglement Generation:

• The maximum entangled dimension that can be generated covertly follows the SRL: $\log \dim \mathcal H_M = O(\sqrt n)$, with identical constants to covert classical information (Anderson et al., 11 Jun 2025, Kimelfeld et al., 26 Mar 2025).

Networks with Interference and Friendly Jamming:

• In wireless networks, aggregate interference uncertainty improves or alters the SRL scaling: while per-link throughput may drop to $O(\log \sqrt n)$, network-wide spatial throughput can remain positive (Liu et al., 2017, Liu et al., 2019, Soltani et al., 2016).
• With friendly jamming, the prefactor in the $O(\sqrt n)$ bound can be improved by scaling with the jammer density (Soltani et al., 2016).

MIMO AWGN and Massive-MIMO Regimes:

• In MIMO AWGN channels, the SRL holds with a throughput scaling exponentially in the number of antennas. In massive-MIMO with randomly oriented warden channels, Alice can asymptotically achieve unconstrained capacity with covertness (Abdelaziz et al., 2017).

5. Practical Considerations and Experimental Validation

Finite Blocklengths and Hardware Limitations:

• Sparse transmission imposes significant constraints on hardware, especially for ADC/DAC dynamic ranges, synchronization, and noise floor estimation (Bali et al., 2 Jun 2025).
• Time and frequency synchronization, particularly for sparse signaling, usually require some non-covert (public) preamble or reference (Bali et al., 2 Jun 2025).

Experimental Validation

• Recent experiments with SDR-based RF systems and free-space optical links have confirmed the square-root scaling and detection-error constraints, matching theoretical predictions (Bali et al., 2 Jun 2025, Bash et al., 2014).

Key Requirements and Shared Secret:

• Achieving the SRL typically necessitates a pre-shared secret (codebook or key) of $O(\sqrt n)$ to $O(n)$ bits, but recent results in identification settings show that keyless covert identification is possible (Zhang et al., 2020).

Lower Bound Tightness and Open Problems:

• Constant-factor multiplicative gaps remain between known achievability and converse bounds in certain quantum and bosonic scenarios, due to limitations in quantum code design (e.g., efficient photonic QECCs) and upper bounds on quantum channel capacity (Anderson et al., 2024, Anderson et al., 11 Jun 2025).

6. Implications, Limitations, and Generalizations

Fundamental Limitation:

• The square root law marks a strict boundary for covert throughput under LPD constraints: regardless of physical-layer sophistication, increasing the message size faster than $O(\sqrt n)$ induces detection risk or reliability breakdown.

Known Exceptions:

• The law can be circumvented only if the adversary's noise/distribution is not independent of transmission, is uncertain, or if the null hypothesis is in the convex hull of active distributions, permitting $\Theta(n)$ covert bits (Bullock et al., 2016, Cho et al., 2020).
• In adversarial settings with timing uncertainty (i.e., adversary does not know the interval when communication occurs), the throughput can scale as $\Theta(\sqrt{n \log T(n)})$ where $T(n)$ is the number of possible slots (Bash et al., 2014).

Active Adversaries:

• When the adversary can adaptively move or sample (active Willie), simple trend tests can defeat naive covert schemes. Countermeasures include randomized on/off scheduling or leveraging network density to create "shadow" networks (Liu et al., 2018).

Network and Spatial Throughput:

• While the $O(\sqrt n)$ bound holds per Alice–Bob pair, spatial throughput in dense interference-limited networks remains nonzero, as aggregate interference masks covert signals (Liu et al., 2017, Liu et al., 2019).

7. Summary Table: SRL Across Communication Models

Channel Class	Covert Scaling	Key Dependency	Comments/References
AWGN, classical	$O(\sqrt n)$	$O(n)$	(Bash et al., 2012, Bash et al., 2015, Soltani et al., 2016)
Compound DMC	$O(\sqrt n)$	$O(\sqrt n)$	(Salehkalaibar et al., 2019)
Classical-quantum, c-q, quantum	$O(\sqrt n)$	$O(\sqrt n)$	(Bullock et al., 2016, Wang, 2016, Anderson et al., 22 Jan 2025, Anderson et al., 2024)
Bosonic (optical) channels	$O(\sqrt n)$	$O(\sqrt n)$	(Bullock et al., 2019, Anderson et al., 11 Jun 2025, Bash et al., 2014)
Multiuser/Interference channel	$O(\sqrt n)$	$O(\sqrt n)$	(Cho et al., 2020)
Identification codes	$\log\log M = O(\sqrt n)$	None	(Zhang et al., 2020)
Interference-uncertainty networks	$O(\log \sqrt n)$	N/A	(Liu et al., 2017, Liu et al., 2019)
MIMO AWGN	$O(\sqrt n)$, exponentially better in $N$	$O(n)$	(Abdelaziz et al., 2017)

In all cases, violation of the SRL scaling is only possible by relaxing the covertness constraint, assuming information-theoretically undetectable signals (e.g., time uncertainty, convex-hull conditions), or non-standard adversarial models.

• "Covert Communication Over a Compound Channel" (Salehkalaibar et al., 2019)
• "Limits of Reliable Communication with Low Probability of Detection on AWGN Channels" (Bash et al., 2012)
• "Fundamental limits of quantum-secure covert communication over bosonic channels" (Bullock et al., 2019)
• "Covert Entanglement Generation over Bosonic Channels" (Anderson et al., 11 Jun 2025)
• "Experimental Covert Communication Using Software-Defined Radio" (Bali et al., 2 Jun 2025)
• "Fundamental Limits of Covert Communication over Classical-Quantum Channels" (Bullock et al., 2016)
• "Covert Identification over Binary-Input Discrete Memoryless Channels" (Zhang et al., 2020)
• "Covert Single-hop Communication in a Wireless Network with Distributed Artificial Noise Generation" (Soltani et al., 2016)
• "Treating Interference as Noise is Optimal for Covert Communication over Interference Channels" (Cho et al., 2020)
• "Covert Optical Communication" (Bash et al., 2014)
• "Covert Communication Gains from Adversary's Ignorance of Transmission Time" (Bash et al., 2014)
• "Hiding Communications in AWGN Channels and THz Band with Interference Uncertainty" (Liu et al., 2019)
• "Achievability of Covert Quantum Communication" (Anderson et al., 22 Jan 2025)
• "Covert Quantum Communication Over Optical Channels" (Anderson et al., 2024)
• "Covert Entanglement Generation and Secrecy" (Kimelfeld et al., 26 Mar 2025)
• "Covert Wireless Communications with Active Eavesdropper on AWGN Channels" (Liu et al., 2018)
• "Fundamental Limits of Covert Communication over MIMO AWGN Channel" (Abdelaziz et al., 2017)
• "Hiding Information in Noise: Fundamental Limits of Covert Wireless Communication" (Bash et al., 2015)