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Wireless Covert Communications Aided by Distributed Cooperative Jamming over Slow Fading Channels

Tong-Xing Zheng, Ziteng Yang, Chao Wang, Zan Li, Jinhong Yuan, Xiaohong Guan

arXiv:2105.05485v1cs.IT

TL;DR

The paper addresses how to communicate covertly over slow fading channels while limiting detection by a warden. It selects friendly jammers using receiver-channel thresholds and jointly optimizes that threshold with message rate under a detection-error constraint. Numerical results report higher covert throughput for multi-jammer assistance than for a single jammer, with gains increasing and eventually saturating as helper count grows.

  • Problem

    The paper studies covert communication between a legitimate transmitter and receiver under surveillance by a warden over slow fading channels.

  • Method

    It activates helpers whose instantaneous receiver-channel gains fall below a threshold and jointly optimizes the selection threshold and transmission rate for covert throughput.

  • Results

    Multi-jammer assistance significantly outperforms the single-jammer scheme in covert throughput, while throughput improves with helper count before saturation.

  • Takeaways & Limitations

    A moderate number of cooperative nodes can effectively improve covert throughput, but very stringent covertness requirements can favor the single-jammer scheme.

Abstract

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In this paper, we study covert communications between {a pair of} legitimate transmitter-receiver against a watchful warden over slow fading channels. There coexist multiple friendly helper nodes who are willing to protect the covert communication from being detected by the warden. We propose an uncoordinated jammer selection scheme where those helpers whose instantaneous channel gains to the legitimate receiver fall below a pre-established selection threshold will be chosen as jammers radiating jamming signals to defeat the warden. By doing so, the detection accuracy of the warden is expected to be severely degraded while the desired covert communication is rarely affected. We then jointly design the optimal selection threshold and message transmission rate for maximizing covert throughput under the premise that the detection error of the warden exceeds a certain level. Numerical results are presented to validate our theoretical analyses. It is shown that the multi-jammer assisted covert communication outperforms the conventional single-jammer method in terms of covert throughput, and the maximal covert throughput improves significantly as the total number of helpers increases, which demonstrates the validity and superiority of our proposed scheme.

I. INTRODUCTION

The paper motivates covert communications as hiding communication existence from wardens and proposes multi-jammer assistance with threshold-based selection to balance covertness and reliability. It jointly optimizes jammer selection and transmission rate, with numerical results showing improved covert throughput and saturation as helpers increase.

  • Motivation: Covert communications protect the existence of message delivery, complementing mechanisms that primarily protect message content from interception.Exposure of communication activity can trigger suspicion and further signal analysis.
  • Research gap: Prior work studied covert communication under channel, noise, relay, and interference uncertainty, but multiple friendly jammers over slow fading channels remained insufficiently explored.The paper identifies multi-jammer cooperative jamming and intelligent jammer selection as its focus.
  • Contributions: The proposed uncoordinated scheme activates each helper when its instantaneous channel gain to the receiver is below a threshold, reducing receiver interference while confusing the warden.The helpers use fixed jamming power, while their selection creates uncertainty about aggregate interference at the warden.
  • Contributions: The paper derives detection-error and outage expressions under a worst-case warden and jointly optimizes the selection threshold and transmission rate subject to a covertness constraint.The optimization targets maximum covert throughput while requiring average detection error to exceed a prescribed level.
  • Results: Numerical results show that covert throughput first improves and then saturates as the number of cooperative nodes increases, with a minimum deployment size required for a given covertness level.The minimum required number of nodes increases when jamming power becomes smaller.

II. SYSTEM MODEL

The system comprises a transmitter, receiver, warden, and friendly nodes whose channel-dependent activation provides cooperative jamming. The threshold scheme uses local receiver-channel information to create warden uncertainty while exposing a covertness–reliability trade-off.

  • Communication scenario: A transmitter covertly sends to a receiver under warden surveillance, assisted by N friendly nodes that can be selected as jammers.The network includes channel path loss, Rayleigh fading, and publicly known topology.
  • Channel estimation: The receiver broadcasts a public pilot so helpers obtain their instantaneous CSI to the receiver, while transmitter and receiver use a secret pilot for channel estimation.The secret pilot is unknown to the warden and enables receiver-side estimation of the transmitter–receiver channel.
  • Jammer selection: Each helper independently transmits jamming when its channel power gain to the receiver is below threshold τ; otherwise it remains silent.τ = 0 gives no jammer assistance, while τ →∞ selects all helpers.
  • Design trade-off: Increasing τ activates more jammers and improves covertness but can severely interfere with the receiver, so τ must balance covertness and reliability.The scheme requires low-level collaboration rather than globally coordinated CSI and beamformer optimization.
  • Jammer selection: The scheme creates uncertainty at the warden about aggregate interference even though all helpers use the same fixed jamming power.The warden lacks the instantaneous helper-to-receiver channel gains needed to determine the exact jammer count or aggregate interference.

C. Detection Strategy at Warden

The warden detects whether transmission occurs by applying a radiometer to average received power under two hypotheses. The paper evaluates false alarms and missed detections under a worst-case optimized detection threshold and uses covert throughput as the design objective.

  • Detection model: The warden performs binary hypothesis testing between no transmitter activity H0 and covert transmission H1.Under each hypothesis, the received signal contains noise and potentially aggregate jamming, with H1 additionally containing the covert signal.
  • Detection model: The warden uses a radiometer that compares average received power over n channel uses with a detection threshold µ.The analysis adopts n →∞, corresponding to an infinite number of signal observations.
  • Detection metric: Detection error combines the false alarm probability PF A = P(D1|H0) and miss detection probability PMD = P(D0|H1).The paper uses the combined detection error rate to measure warden performance.
  • Worst-case analysis: The design considers a worst-case warden that selects the detection threshold minimizing its detection error.This minimum error is denoted ξ∗, while the transmitter lacks instantaneous CSI of the transmitter-to-warden channel.
  • Optimization objective: Covert throughput depends on transmission rate R and outage probability δ, which measures whether reliability at rate R fails over fading channels.The optimization maximizes throughput subject to minimum average detection error ¯ξ∗ ≥ 1 − ε.

III. ANALYSIS OF COVERTNESS PERFORMANCE

The analysis derives detection error by combining false-alarm and miss-detection probabilities for a given transmitter–warden channel, then designs the warden’s worst-case threshold to minimize average detection error.

  • Detection-error analysis: For a given ht,w, the paper derives detection error from the false alarm rate and miss detection rate.The analysis then considers a warden that adapts its detection threshold to fading channels and network geometry.

A. Detection Error Probability

The paper derives false-alarm and miss-detection rates by modeling the aggregate power from independently selected jammers, then characterizes how detection thresholds, selection thresholds, and jamming power affect detection error.

  • The aggregate jamming power is difficult to characterize because its terms are independent but non-identically distributed across helper-to-warden distances.
  • The selected-jammer set is represented by subsets of each cardinality, with probabilities obtained from independent jammer-selection probabilities.
  • A gamma random-variable approximation replaces the exact aggregate-power distribution to obtain mathematically tractable false-alarm and miss-detection expressions.
  • Theorem 1 gives the false-alarm rate PF A and miss-detection rate PMD as functions of the detection threshold and aggregate jamming-power distribution.
  • Detection thresholds at or below the noise power cause complete false alarms, while thresholds below the signal-plus-noise power create miss-detection behavior and other cases depend on aggregate jamming power.
  • The analyses support the superiority of the threshold-based jammer-selection scheme for degrading warden detection.
  • When τ →∞, all cooperative nodes jam and detection error becomes independent of τ for fixed detection threshold and warden channel.
  • As τ, Pj → 0, detection error approaches zero for σ2 w < µ ≤ ρ1, whereas sufficiently large τ and Pj can make detection error approach one.

B. Minimum Average Detection Error Probability

Because the transmitter lacks instantaneous warden-channel CSI, the paper minimizes average detection error over that channel under a worst-case warden threshold, while analyzing both CSI-information cases.

  • The transmitter cannot calculate exact detection error without instantaneous warden-channel CSI, so performance is averaged over the warden channel under a worst-case threshold.
  • Case 1: W knows instantaneous CSI: When W knows instantaneous ht,w, it can adapt µ in real time, and the minimum detection error is obtained by minimizing ξ over µ for each channel realization.
  • Case 2: W does not know instantaneous CSI: When W lacks instantaneous ht,w, it selects µ from channel statistics to minimize average detection error, whose expression is derived in Theorem 2.
  • The average detection-error metric is treated as a Bayesian covertness measure and is stated to be equivalent to covert outage probability under the considered system.
  • The average detection error contains one component from opportunistic jammer selection and another from missing instantaneous warden-channel CSI.
  • As τ →∞ or τ →0, average detection error converges to a τ-independent constant because uncertainty in the number of selected jammers vanishes.
  • Very large τ and Pj can severely interfere with R, so τ must balance covertness against communication reliability.

IV. COVERT THROUGHPUT MAXIMIZATION

The paper formulates covert-throughput maximization by first deriving transmission outage probability, then jointly optimizing jammer-selection threshold τ and transmission rate R subject to a covertness constraint.

  • The optimization first calculates transmission outage probability δ, then designs τ and R to maximize covert throughput under a detection-error constraint.

A. Transmission Outage Probability

Transmission outage occurs when the channel capacity from T to R falls below the transmission rate, and its probability is derived using the selected-jammer channel distribution and total probability.

  • An outage occurs when C = log2(1 + Λr) is below R because the destination channel is random to T.
  • Lemma 3 gives the transmission outage probability from T to R by averaging over possible jammer selections and the fading channel.
  • The selected-jammer channel distribution is derived conditionally from the selection rule |hjk,r|2 d−α r < τ.
  • The outage expression separates the no-jammer outage contribution from the additional outage risk caused by introducing jammers.
  • As τ →0 or τ →∞, the transmission outage probability approaches a constant value.

B. Optimization of Covert Throughput

The optimization jointly selects the jammer threshold and covert transmission rate: first minimize outage for a fixed rate under covertness, then maximize throughput over the rate.

  • Selection-threshold optimization: The optimal selection threshold τ* minimizes transmission outage probability while satisfying the covertness constraint.For fixed transmit and jamming powers, node count, and tolerable detection probability, Proposition 1 characterizes τ* as the minimum feasible threshold.
  • Selection-threshold optimization: Because larger τ activates more jammers and increases interference at the receiver, the outage-minimizing threshold is the smallest one meeting covertness.
  • Selection-threshold optimization: The covertness-constrained threshold is obtained by solving ¯ξ*ι = 1 − ε and can be calculated by inverting ¯ξ*ι with bisection.The constraint applies for ι = 1, 2.
  • Rate optimization: The covert throughput Ω(R) increases and then decreases with transmission rate because outage δ(R) increases with R and approaches one at high rates.Thus, both very small and very large rates yield low throughput.
  • Rate optimization: The optimal transmission rate R* can be efficiently found by bisection, and substituting τ* and R* into the throughput expression gives maximum covert throughput Ω*.

V. NUMERICAL RESULTS

Numerical results validate the theoretical analyses and show how jammer selection, channel knowledge, jamming power, covert rate, and helper count affect detection error, outage probability, and covert throughput. The multi-jammer scheme generally improves covert throughput over the single-jammer benchmark, although stringent covertness requirements and excessive jamming expose reliability trade-offs.

  • Detection performance: Detection error increases with helper count and jamming power, while acquiring the detection-channel CSI reduces covertness, especially for smaller helper counts.The CSI-aware minimum detection error is lower than the CSI-unaware value, and the gap enlarges for smaller N.
  • Validation: Monte-Carlo results match theoretical values for detection error and transmission outage probability, validating the analyses and approximation method.The comparison is reported for Fig. 4.
  • Threshold effects: Both minimum detection error and transmission outage probability increase with the selection threshold, reflecting more active jammers and stronger interference at the legitimate receiver.For outage probability, the increase also occurs with covert rate and jamming power.
  • Threshold optimization: The optimal selection threshold decreases with helper count and jamming power because greater aggregate-jamming uncertainty permits a smaller threshold under a fixed covertness constraint.This behavior is attributed to increased uncertainty in distinguishing desired signal and jamming power.
  • Covert throughput: Covert throughput first increases and then decreases with covert rate, approaches zero at both rate extremes, and increases as the number of cooperative nodes grows.The helper-count gain is attributed to greater jamming-power randomness without a significant outage increase from the selection criterion.
  • Scheme comparison and scaling: The multi-jammer scheme significantly outperforms the single-jammer benchmark over a wide covertness range, while throughput gains from additional helpers eventually saturate.For extremely small ε, the single-jammer scheme can be superior; N_min is 5 at Pj = 10dBm and 8 at Pj = 5dBm.

VI. CONCLUSION

The paper proposes multi-jammer covert communication over slow fading channels and jointly optimizes jammer selection and covert rate under a covertness constraint. Numerical comparisons show higher covert throughput than the single-jammer scheme, while the optimal threshold decreases with more helpers or greater jamming power.

  • System and approach: The scheme selects cooperative nodes as jammers to confuse the warden while supporting covert transmission from T to R.The paper analyzes detection error and transmission outage probability for this multi-jammer setting.
  • Optimization: The optimal selection threshold and covert rate are jointly designed to maximize covert throughput under a covertness constraint.The paper reports that an optimal jammer selection threshold exists.
  • Numerical findings: The multi-jammer scheme achieves higher covert throughput than the single-jammer case in numerical comparisons.The comparison is made against a conventional single-jammer scheme.
  • Numerical findings: The optimal selection threshold decreases as either the number of cooperative nodes or the jamming power increases.The paper also reports that a minimum number of cooperative nodes is needed to meet a specified covertness level.

APPENDIX

The appendix establishes the statistical basis for the warden’s detector and derives cumulant quantities for the aggregate received-signal model. It shows that the energy test statistic is sufficient for detecting whether the transmitter is active, regardless of the aggregate jamming-power distribution.

  • Detector sufficiency: The appendix proves that the energy statistic T_w=(1/n)Σ_i|y_w[i]|^2 is sufficient for detecting whether T is transmitting.The proof applies the Fisher–Neyman factorization theorem.
  • Signal model: The received signal model uses θ=1 when T transmits and θ=0 otherwise.The signal and noise variables are modeled as complex Gaussian random variables.
  • Detector sufficiency: The sufficiency result holds regardless of the form of the aggregate jamming-power distribution f_σ²_j(x).The conditional distribution depends on the observations only through T_w.
  • Cumulant derivation: The appendix derives the first and second cumulants of X_m,s from the moments and mutual independence of the relevant jammer-to-warden channel gains.The parameters v_m,s and ω_m,s are obtained from these cumulant calculations.
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