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On Covert Communication with Noise Uncertainty

Biao He, Shihao Yan, Xiangyun Zhou, Vincent K. N. Lau

arXiv:1612.09027v1cs.IT

TL;DR

Prior work assessed covertness under noise uncertainty using Willie’s worst-case detection performance, which the paper identifies as overly optimistic. The paper instead incorporates the noise-uncertainty distribution, defines average covert probability and covert outage probability, and derives covert rates for bounded and unbounded models. Its results show that positive covert rates are achievable at low detection probability, while the covert rate approaches zero as detection probability approaches zero.

  • Problem

    Prior noise-uncertainty studies used a worst-case warden perspective that can overestimate the true covertness of a system.

  • Method

    The paper incorporates the distribution of noise uncertainty and uses average covert probability and covert outage probability to analyze bounded and unbounded models.

  • Results

    Positive covert rates are achievable with a low probability of detection, while the covert rate approaches zero as detection probability approaches zero.

  • Takeaways & Limitations

    Statistical covertness metrics provide a basis for deriving rates subject to covertness requirements under both noise-uncertainty models.

  • Takeaways & Limitations

    The existing worst-case requirement cannot guarantee any true level of covertness for systems satisfying ξup > 1 − ǫ.

Abstract

from arXiv · show

Prior studies on covert communication with noise uncertainty adopted a worst-case approach from the warden's perspective. That is, the worst-case detection performance of the warden is used to assess covertness, which is overly optimistic. Instead of simply considering the worst limit, in this work, we take the distribution of noise uncertainty into account to evaluate the overall covertness in a statistical sense. Specifically, we define new metrics for measuring the covertness, which are then adopted to analyze the maximum achievable rate for a given covertness requirement under both bounded and unbounded noise uncertainty models.

I. INTRODUCTION

Covert communication aims to hide transmissions from a warden, but prior noise-uncertainty analyses used a worst-case perspective that can overestimate covertness. This work instead accounts for noise-uncertainty distributions and analyzes bounded and unbounded models.

  • Covert communication protects privacy and helps prevent attacks or conceal military operations from detection.
  • The AWGN square root law limits reliable covert transmission to O(√n) bits over n channel uses, yielding zero asymptotic covert rate.
  • Noise uncertainty can break the square root law, but prior analyses evaluated Willie’s detection performance at a worst-case noise power.
  • The paper proposes distribution-aware covertness evaluation for both bounded and unbounded noise-uncertainty models.

A. Hypothesis Testing Problem at Willie

Willie performs binary hypothesis testing between noise alone and signal plus noise using a radiometer. Covertness requires the sum of false-alarm and misdetection probabilities to remain close to one.

  • Willie distinguishes H0, noise only, from H1, signal plus noise, using the received signal vector.
  • Willie’s radiometer uses the average received energy T(yw) as its detection statistic and compares it with a threshold γ.
  • False-alarm and misdetection probabilities are defined as PFA = P(D1 | H0) and PMD = P(D0 | H1), respectively.
  • The covert requirement is ξ = PFA + PMD ≥ 1 − ǫ, while Willie seeks to minimize ξ.

B. Noise Uncertainty

Noise uncertainty reflects Willie’s unknown exact receiver-noise power. The paper models it as either bounded over a finite range or unbounded over the full dB line, with known statistics but unknown realization.

  • Noise uncertainty is the lack of knowledge of the exact noise power, arising from sources such as thermal, quantization, environmental, and calibration effects.
  • The paper considers bounded and unbounded noise-uncertainty models at Willie.
  • 1) Bounded Uncertainty Model:: In the bounded model, the exact noise power lies within a finite range around the nominal noise power.
  • 1) Bounded Uncertainty Model:: The bounded model uses a log-uniform noise-power distribution, with ρdB quantifying the uncertainty size.
  • 2) Unbounded Uncertainty Model:: The unbounded model allows noise power across [−∞, +∞] in dB, while its statistics are specified even though the exact power remains unknown.

2) Unbounded Uncertainty Model:

For unbounded noise uncertainty, the analysis uses the known statistical distribution of Willie’s unknown noise power to assess covertness rather than relying on a single worst-case value.

  • In the unbounded model, the exact noise power spans [−∞, +∞] in dB and its deviation from nominal power follows a normal distribution.
  • The noise-uncertainty statistics are known to the analysis, but Willie’s exact noise power is unknown; Bob’s noise uncertainty is omitted.
  • Covertness is examined through ξ = PFA + PMD, and the paper introduces approaches beyond the existing analysis of noise uncertainty.

A. Limitation of Existing Approach

The existing robust-statistics approach measures covertness using Willie’s worst-case detection performance, which can report apparent covertness without guaranteeing true covertness.

  • The existing work measures covertness using the upper limit of ξ under noise uncertainty.
  • The requirement under this approach is ξup ≥1−ǫ.
  • This measure characterizes Willie’s worst-case performance rather than overall performance across noise uncertainty.
  • A system satisfying ξup > 1−ǫ cannot guarantee any true level of covertness.
  • The worst-case metric can meet the covert requirement when Willie’s exact noise power is not at the worst-case limit.

B. Newly-Adopted Approach

The paper replaces worst-case evaluation with statistical approaches that account for the prior distribution of noise power, covering both bounded and unbounded uncertainty models.

  • The paper introduces two approaches that evaluate overall performance at Willie instead of focusing on the worst-case scenario.
  • The Bayesian statistics approach measures covertness by averaging ξ over the a priori distribution of noise power.
  • The average covert probability ¯ξ captures average covertness over multiple communications or experiments, with requirement ¯ξ ≥1−ǫ.
  • The outage-based approach measures the probability that ξ < 1−ǫ.
  • The covert outage probability pout is the probability that covert communication fails, with requirement pout ≤δ.
  • Both metrics depend on the a priori noise-power distribution, and with N→∞, pout = 1−¯ξ makes their requirements equivalent.

IV. COVERT RATE

The covert rate is derived by first finding the received-power threshold that satisfies the average covert-probability constraint, under a log-uniform noise-power model. The analysis also shows that the prior upper-limit measure can overstate covertness.

  • The covert-rate problem maximizes Alice’s reliable communication rate subject to an average covert-probability constraint.The covert rate is denoted by R, and the constraint is expressed through the average covert probability.
  • For each noise-uncertainty model, the analysis first derives Willie’s received-power threshold and then obtains the corresponding covert rate.The threshold is defined by the required covertness condition ξ̄ ≥ 1 − ǫ.
  • Under log-uniform noise uncertainty, Proposition 1 gives the received-power threshold below which the required covertness condition holds.The noise power at Willie is assumed to follow a log-uniform distribution.
  • As ǫ → 0, the log-uniform threshold P_LU approaches zero, requiring sufficiently small received power for arbitrarily small ǫ.As ǫ → 1, P_LU approaches (ρ − 1/ρ)σ_n^2.
  • The prior ξ_up-based design cannot guarantee any true level of covertness, because its limiting threshold matches the threshold associated with arbitrarily small ǫ.This conclusion is stated for the bounded uncertainty analysis.

B. Unbounded Uncertainty Model

For unbounded noise uncertainty modeled by a log-normal distribution, the paper derives an approximated received-power threshold because the exact closed form is intractable. It then uses that threshold to obtain the corresponding covert rate.

  • The log-normal noise-power model makes the closed-form power threshold mathematically intractable, so an approximated threshold is provided.The approximation is introduced in the proposition that follows the model assumption.
  • Proposition 2 gives the received-power threshold below which the required covertness condition holds for log-normally distributed noise power.The resulting threshold is an approximation rather than a closed-form exact expression.
  • When noise uncertainty is small, the noise-power density is approximated by a Gaussian function to support the threshold derivation.The Gaussian approximation is used in the proof steps for Proposition 2.
  • The derivation uses Willie’s optimal detection threshold γ* = max {φ1 + P_w/2, P_w} before solving the covertness inequality for P_w.Solving ξ̄ ≥ 1 − ǫ for P_w completes the proof.
  • As ǫ → 0, P_LN approaches zero, whereas the ξ_up-based threshold diverges as ǫ → 1 under unbounded uncertainty.The paper uses this behavior to reject ξ_up as an appropriate covertness measure.

V. NUMERICAL RESULTS

The numerical results validate the log-normal threshold approximation and examine covert rate under bounded and unbounded noise uncertainty. Covert rate increases with uncertainty and the required ǫ, while the prior ξ_up-based result substantially overestimates achievable covertness.

  • The approximated P_LN matches Monte Carlo simulations precisely, with almost unnoticeable error when σ_Δ,dB = 0.5.The approximation becomes more accurate as the noise variance decreases; the nominal noise power is σ_n,dB^2 = −100.
  • Figure 1 illustrates the accuracy of the approximated P_LN against numerically obtained values.The comparison uses Monte Carlo simulations.
  • Figure 2(a) plots R_LU against ρ_dB for bounded log-uniform uncertainty, while Figure 2(b) plots R_LN against σ_Δ,dB for unbounded log-normal uncertainty.The parameter setting includes r_b = r_w.
  • Covert rate increases as Willie’s noise uncertainty and/or the required ǫ increases, but approaches zero when noise-power uncertainty vanishes.The zero-uncertainty limit is ρ_dB, σ_Δ,dB → 0.
  • The prior result is much larger than the achievable covert rate under the proposed analysis, so ξ_up overestimates covertness and is inappropriate for this study.The prior result is based on an arbitrarily small detection probability and the ξ_up measure.

VI. CONCLUSION

The paper introduces average covert probability and covert outage probability for noise-uncertain wardens, analyzing both bounded and unbounded uncertainty. It derives covertness-constrained rates and finds positive cover rates at low detection probabilities, while the rate vanishes as detection probability approaches zero.

  • Average covert probability and covert outage probability are proposed as covertness metrics under warden noise uncertainty.
  • Both bounded and unbounded noise uncertainty models are considered.
  • For each uncertainty model, the paper derives the rate below which communication satisfies a given covertness requirement.
  • A positive cover rate is achievable with a low probability of detection, but the covert rate approaches zero as detection probability approaches zero.
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