---
title: Square Root Law for Covert Communications
url: https://www.emergentmind.com/topics/square-root-law-for-covert-communications
type: topic
---

# Square Root Law for Covert Communications

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 $1-\delta$ [1202.6423, 1506.00066].

**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 $\mathcal M$ [1906.06675, 2007.13333].

**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 [1907.04228, 2506.09474, 2401.06764].

**Universality**:

- The law holds for point-to-point classical, classical-quantum, compound, multiuser, and interference channels under nontrivial LPD constraints [2003.04531, 1601.06826].

## 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** [1906.06675]:

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** [1907.04228]:
\[
c_{\mathrm{cov}} = \frac{\sqrt{2\eta \bar n_B (1 + \eta \bar n_B)}}{1 - \eta}, \;\;
M = \sqrt{n} \delta\, c_{\mathrm{cov}}\, c_{\mathrm{rel}} + o(\sqrt{n}),
\]
where $\eta$ is the transmissivity, $\bar n_B$ is background photon number.

**Multiuser and Interference Channels** [2003.04531]:
\[
R_k \leq \alpha_k D(W_k^{(k)} \| W_0^{(k)}) / \sqrt{\chi^2(\alpha)/2},
\]
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 [1906.06675, 1601.06826].
- **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 [2501.13103, 2506.09474].

### 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 [1202.6423, 1506.00066, 1404.7347].
- 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 [1906.06675].

**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 [2007.13333].

**Classical-Quantum and Quantum Channels**:

- The SRL holds in full generality for memoryless classical-quantum channels with arbitrary finite input [1601.06826, 1603.05823]. 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 [2506.09474, 2503.21002].

**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 [1712.05099, 1909.12752, 1610.00384].
- With friendly jamming, the prefactor in the $O(\sqrt n)$ bound can be improved by scaling with the jammer density [1610.00384].

**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 [1705.02303].

## 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 [2506.02297].
- Time and frequency synchronization, particularly for sparse signaling, usually require some non-covert (public) preamble or reference [2506.02297].

**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 [2506.02297, 1404.7347].

**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 [2007.13333].

**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 [2401.06764, 2506.09474].

## 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 [1601.06826, 2003.04531].
- 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 [1403.1013].

**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 [1805.06182].

**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 [1712.05099, 1909.12752].

## 7. Summary Table: SRL Across Communication Models

| Channel Class                   | Covert Scaling | Key Dependency | Comments/References             |
|----------------------------------|---------------|---------------|---------------------------------|
| AWGN, classical                  | $O(\sqrt n)$  | $O(n)$        | [1202.6423, 1506.00066, 1610.00384] |
| Compound DMC                     | $O(\sqrt n)$  | $O(\sqrt n)$  | [1906.06675]                    |
| Classical-quantum, c-q, quantum  | $O(\sqrt n)$  | $O(\sqrt n)$  | [1601.06826, 1603.05823, 2501.13103, 2401.06764] |
| Bosonic (optical) channels       | $O(\sqrt n)$  | $O(\sqrt n)$  | [1907.04228, 2506.09474, 1404.7347] |
| Multiuser/Interference channel   | $O(\sqrt n)$  | $O(\sqrt n)$  | [2003.04531]                    |
| Identification codes             | $\log\log M = O(\sqrt n)$ | None | [2007.13333] |
| Interference-uncertainty networks| $O(\log \sqrt n)$| N/A        | [1712.05099, 1909.12752]        |
| MIMO AWGN                        | $O(\sqrt n)$, exponentially better in $N$ | $O(n)$ | [1705.02303] |

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.

## References

- "Covert Communication Over a Compound Channel" [1906.06675]
- "Limits of Reliable Communication with Low Probability of Detection on AWGN Channels" [1202.6423]
- "Fundamental limits of quantum-secure covert communication over bosonic channels" [1907.04228]
- "Covert Entanglement Generation over Bosonic Channels" [2506.09474]
- "Experimental Covert Communication Using Software-Defined Radio" [2506.02297]
- "Fundamental Limits of Covert Communication over Classical-Quantum Channels" [1601.06826]
- "Covert Identification over Binary-Input Discrete Memoryless Channels" [2007.13333]
- "Covert Single-hop Communication in a Wireless Network with Distributed Artificial Noise Generation" [1610.00384]
- "Treating Interference as Noise is Optimal for Covert Communication over Interference Channels" [2003.04531]
- "Covert Optical Communication" [1404.7347]
- "Covert Communication Gains from Adversary's Ignorance of Transmission Time" [1403.1013]
- "Hiding Communications in AWGN Channels and THz Band with Interference Uncertainty" [1909.12752]
- "Achievability of Covert Quantum Communication" [2501.13103]
- "Covert Quantum Communication Over Optical Channels" [2401.06764]
- "Covert Entanglement Generation and Secrecy" [2503.21002]
- "Covert Wireless Communications with Active Eavesdropper on AWGN Channels" [1805.06182]
- "Fundamental Limits of Covert Communication over MIMO AWGN Channel" [1705.02303]
- "Hiding Information in Noise: Fundamental Limits of Covert Wireless Communication" [1506.00066]

Source: https://www.emergentmind.com/topics/square-root-law-for-covert-communications