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Proactive Eavesdropping via Cognitive Jamming in Fading Channels

Jie Xu, Lingjie Duan, Rui Zhang

arXiv:1512.02754v2cs.ITmath.IT

TL;DR

Infrastructure-free suspicious links may evade conventional monitoring, while a distant legitimate monitor may be unable to decode them. The paper uses cognitive jamming and fading-state power allocation for event-based and content-based surveillance; simulations report better performance than passive and constant-power jamming.

  • Problem

    Infrastructure-free suspicious links can be difficult to monitor through conventional Internet-backbone or cellular interception, and a distant monitor may not decode the link efficiently.

    Source: §I · pp. 1–2

  • Method

    The full-duplex monitor deliberately jams the suspicious receiver and optimizes its jamming power across fading states for event-based and content-based monitoring objectives.

    Source: §I · p. 2

  • Results

    The proposed approach greatly outperforms passive eavesdropping, constant-power jamming, and on-off jamming in numerical results.

    Source: §I · p. 3

  • Takeaways & Limitations

    The paper presents cognitive jamming as a wireless surveillance approach for monitoring suspicious communication links.

    Source: §I · pp. 1–2

    Extending the approach to massive multi-antenna suspicious users that adapt beamformers may require many coordinated legitimate monitors, a mode-selection and coordination problem left for future work.

    Source: §VIII · p. 13

Abstract

from arXiv · show

To enhance the national security, there is a growing need for government agencies to legitimately monitor suspicious communication links for preventing intended crimes and terror attacks. In this paper, we propose a new wireless information surveillance paradigm by investigating a scenario where a legitimate monitor aims to intercept a suspicious wireless communication link over fading channels. The legitimate monitor can successfully eavesdrop (decode) the information of the suspicious link at each fading state only when its achievable data rate is no smaller than that at the suspicious receiver. In practice, such legitimate eavesdropping is particularly challenging, since the legitimate monitor may be far away from the suspicious transmitter and cannot eavesdrop efficiently. To overcome this issue, we propose a new approach, namely proactive eavesdropping via cognitive jamming, in which the legitimate monitor purposely jams the receiver so as to change the suspicious communication (e.g., to a smaller data rate) for overhearing more efficiently. In particular, we consider delay-sensitive and delay-tolerant applications for the suspicious data communications, under which the legitimate monitor maximizes the eavesdropping non-outage probability for event-based monitoring and the relative eavesdropping rate for content analysis, respectively, by optimizing its jamming power allocation over different fading states subject to an average power constraint. Numerical results show that the proposed proactive eavesdropping via cognitive jamming approach greatly outperforms the conventional passive eavesdropping without jamming and the proactive eavesdropping with constant-power jamming.

I. INTRODUCTION

The paper reframes wireless security as legitimate surveillance of suspicious links, including infrastructure-free communications that conventional monitoring may miss. It proposes cognitive jamming to improve decoding and formulates power-allocation objectives for event-based and content-based monitoring.

I. INTRODUCTION

  • Infrastructure-free links can be difficult to monitor through conventional Internet-backbone or cellular-office interception, motivating authorized wireless surveillance.Source: §I · p. 1

II. SYSTEM MODEL AND PROBLEM FORMULATION

  • Successful decoding requires the monitor’s received SNR to be no smaller than the suspicious receiver’s; the analysis assumes perfect SIC and global CSI for its fundamental limits.Source: §I · p. 2, §II · pp. 3–4

I. INTRODUCTION

  • The proposed approach deliberately jams the suspicious receiver to lower its achievable rate, helping a full-duplex legitimate monitor decode the suspicious link.The monitor adapts jamming power across fading states under an average power constraint.Source: §I · p. 2
  • For delay-sensitive monitoring, the objective is eavesdropping non-outage probability; for delay-tolerant content analysis, it is relative eavesdropping rate.The monitor optimizes its jamming allocation over fading states for both objectives.Source: §I · p. 2
  • Numerical results report that cognitive jamming greatly outperforms passive eavesdropping, constant-power jamming, and on-off jamming.Source: §I · p. 3
  • The paper also develops online cognitive jamming under residual self-interference and local CSI, reporting performance similar to the ideal scheme especially with separate antennas.Source: §I · p. 3

III. OPTIMAL COGNITIVE JAMMING IN DELAY-SENSITIVE SUSPICIOUS APPLICATIONS

For delay-sensitive applications, the paper maximizes the probability of successful eavesdropping subject to an average jamming-power constraint. Its optimal solution follows from time-sharing and Lagrangian duality, and allocates jamming to fading states where it can secure decoding.

III. OPTIMAL COGNITIVE JAMMING IN DELAY-SENSITIVE SUSPICIOUS APPLICATIONS

  • The optimization maximizes eavesdropping non-outage probability under an average jamming-power constraint.Source: §III · p. 5
  • Time-sharing ensures strong duality for the nonconvex problem, enabling the paper to obtain an optimal solution using Lagrangian duality.Source: §III · p. 5
  • The optimal policy jams only fading states where jamming is needed to achieve successful eavesdropping, while already-decodable states require no jamming.In the states requiring jamming, the required power makes the monitor’s and receiver’s SINRs equal; the policy selects states under the average-power constraint.Source: §III · p. 5

IV. OPTIMAL COGNITIVE JAMMING IN DELAY-TOLERANT SUSPICIOUS APPLICATIONS WITH FIXED POWER TRANSMISSION

For delay-tolerant suspicious applications with fixed transmitter power, the monitor maximizes relative eavesdropping rate by allocating jamming power across fading states. The solution turns the optimization into feasibility tests and uses duality and search to obtain the optimal allocation.

IV. OPTIMAL COGNITIVE JAMMING IN DELAY-TOLERANT SUSPICIOUS APPLICATIONS WITH FIXED POWER TRANSMISSION

  • The monitor rewrites the relative-rate objective using an auxiliary threshold t and finds its optimum by testing feasibility at fixed t and applying bisection.A feasible test implies the optimal t is at least the tested value; an infeasible test means it is lower.Source: §IV · pp. 5–6
  • For each fixed t, strong duality permits solving the non-convex feasibility problem through its Lagrange dual.The paper attributes strong duality to the time-sharing condition and gives a dual-function test for infeasibility.Source: §IV · p. 6
  • The dual optimization decomposes into subproblems for individual fading states, yielding state-dependent candidate jamming powers.The per-state solution distinguishes successful from unsuccessful eavesdropping states.Source: §IV · p. 6
  • The complete procedure combines the feasibility solver with bisection over t to recover the optimal jamming allocation.The feasibility algorithm is summarized in Table I.Source: §IV · p. 6, Table I · p. 6

V. OPTIMAL COGNITIVE JAMMING IN DELAY-TOLERANT SUSPICIOUS APPLICATIONS WITH WATER-FILLING POWER

For delay-tolerant applications with water-filling at the suspicious transmitter, the monitor maximizes relative eavesdropping rate while accounting for how its jamming changes the transmitter’s power allocation. The solution optimizes jamming jointly with the water-filling level.

V. OPTIMAL COGNITIVE JAMMING IN DELAY-TOLERANT SUSPICIOUS APPLICATIONS WITH WATER-FILLING POWER

  • The transmitter’s water-filling allocation depends on the monitor’s jamming profile, so the monitor optimizes relative eavesdropping rate over jamming while accounting for that adaptation.The water level is determined by the transmitter’s average-power constraint.Source: §V · p. 7
  • The water-filling transmitter must know the interference power at its receiver, which can be measured there and fed back to the transmitter.The interference power is g2(ν)q(ν).Source: §V · p. 7
  • The monitor’s optimization is equivalent to a bilevel problem: the transmitter maximizes its average rate below, and the monitor maximizes relative eavesdropping rate above.The lower-level transmitter solution is water-filling for a given auxiliary level.Source: §V · p. 7

A. Finding the Feasible Regime of β

For water-filling transmission, the paper first identifies feasible values of β, then optimizes the monitor’s jamming allocation for each feasible value. The resulting relative eavesdropping rate versus β rises and then falls in the reported example, suggesting a simpler search may suffice.

A. Finding the Feasible Regime of β

  • The feasible β interval is bounded above by the no-jamming case and below by full jamming, with the lower bound found through feasibility tests and bisection.The paper establishes strong duality for the feasibility problem despite its nonlinear equality constraint.Source: A. Finding the Feasible Regime of β · p. 7

B. Optimizing {q(ν)} for Problem (P3) Under Any Given Feasible β

  • For each feasible β, the monitor solves a sequence of feasibility problems using Lagrange duality, then searches over β to obtain the optimal jamming profile.The fixed-β solution uses bisection over an auxiliary threshold and an ellipsoid method for the dual problem.Source: B. Optimizing {q(ν)} for Problem (P3)… · pp. 8–9
  • The per-state jamming solution selects among candidate powers according to their associated objective values and conditions on successful eavesdropping and transmitter water-filling.One candidate corresponds to jamming enough to make the transmitter allocate no power in that fading state.Source: B. Optimizing {q(ν)} for Problem (P3)… · p. 8
  • The reported relative eavesdropping rate first increases and then decreases with β, suggesting bisection may replace exhaustive search and reduce complexity.The plotted example uses average jamming power Q = 20 dB; the authors report the trend in other simulations but note it is difficult to prove rigorously.Source: Figure 2 · p. 9, B. Optimizing {q(ν)} for Problem (P3)… · p. 9

C. Comparison Among Different Optimal Cognitive Jamming Solutions

Optimal jamming differs across application objectives and transmitter power strategies. The comparisons show that water-filling can require more average jamming than fixed-power transmission to attain the same relative eavesdropping rate.

C. Comparison Among Different Optimal Cognitive Jamming Solutions

  • When eavesdropping is already successful without jamming, all three optimal jamming policies allocate zero power in that fading state.This is the case where the eavesdropping link is better than the suspicious communication link.Source: C. Comparison Among Different Optimal… · p. 9
  • For delay-sensitive applications, the monitor jams only states where it can eavesdrop successfully after jamming; delay-tolerant monitoring may also jam unsuccessful states to reduce the suspicious link’s rate.The latter can increase the relative eavesdropping rate in the states where eavesdropping succeeds.Source: C. Comparison Among Different Optimal… · p. 9
  • The relative eavesdropping rate under fixed-power transmission is higher than under water-filling, so water-filling generally requires more average jamming for the same rate.The paper notes that this comparison is difficult to establish analytically and points to its numerical results in Fig. 8.Source: C. Comparison Among Different Optimal… · p. 9

VI. NUMERICAL RESULTS

The simulations compare proactive cognitive jamming with constant-power and passive eavesdropping across delay-sensitive and delay-tolerant applications. Optimal cognitive power control improves eavesdropping outcomes, while water-filling at the suspicious transmitter can reduce relative eavesdropping performance.

VI. NUMERICAL RESULTS

  • The simulations use Rayleigh fading, independent channel coefficients, normalized noise powers, and a suspicious-transmitter power of P = 20 dB unless otherwise stated.The legitimate monitor is assumed far from both suspicious endpoints relative to their separation.Source: §VI · pp. 9–10
  • Cognitive jamming with optimal power control beats constant-power jamming for delay-sensitive non-outage probability and delay-tolerant average eavesdropping rate.For delay-sensitive applications, both proactive schemes also outperform passive eavesdropping.Source: §VI · p. 10
  • The delay-tolerant fixed-power results show that cognitive jamming achieves a higher average eavesdropping rate than constant-power jamming.The authors describe gains in both the percentage of data bits overheard and the total volume of data.Source: §VI · p. 10
  • At Q = 20 dB, fixed-power transmission yields a higher relative eavesdropping rate than water-filling, especially at low suspicious-transmitter power P.The paper reports that water-filling dynamics degrade proactive eavesdropping performance and require higher average jamming power to match fixed-power performance.Source: §VI · p. 10

VII. PRACTICAL IMPLEMENTATION OF PROACTIVE EAVESDROPPING

The practical implementation section considers residual self-interference and local channel-state information, then develops an online cognitive-jamming scheme for delay-sensitive applications. With residual interference that is not too strong, the optimal non-outage solution retains a threshold-based jamming structure.

VII. PRACTICAL IMPLEMENTATION OF PROACTIVE EAVESDROPPING

  • An online cognitive-jamming scheme addresses residual self-interference and local CSI for delay-sensitive non-outage maximization.The discussion focuses on that case; analogous delay-tolerant analysis is omitted.Source: §VII · p. 10

A. Eavesdropping Non-Outage Probability Maximization in the Case with Residual SI

  • After cancellation, residual self-interference scales with jamming power through the effective loop-back channel gain.The text notes that combined analog and digital cancellation can achieve up to 110 dB of self-interference reduction.Source: A. Eavesdropping Non-Outage… · p. 11
  • Threshold-based jamming remains optimal with residual self-interference only when the interference is not too strong.The threshold is the minimum jamming power needed for successful eavesdropping when the eavesdropping link is weaker than the suspicious link.Source: A. Eavesdropping Non-Outage… · p. 11

B. Online Cognitive Jamming Under Practical Assumptions

The practical online scheme learns a required jamming power from observed signals and adapts a threshold under local-CSI and residual-self-interference assumptions. Simulations report performance close to optimal jamming, with outcomes depending on antenna placement and residual self-interference.

B. Online Cognitive Jamming Under Practical Assumptions

  • The online scheme is designed for finite time blocks with local knowledge of the eavesdropping and effective loop-back channels, while the suspicious and jamming channels are unknown.The monitor can infer the suspicious communication rate under a given jamming power from received signals.Source: B. Online Cognitive Jamming Under… · p. 11

1) Learning the Required Jamming Power:

2) Online Threshold-Based Jamming Design:

  • After learning the required power, the scheme jams only when that power is at or below an adaptively updated threshold, adjusting the threshold according to average power use.The threshold increases when average jamming power remains below the constraint and decreases otherwise.Source: 2) Online Threshold-Based Jamming Design · p. 12
  • With sufficiently small step size and enough time blocks, the online threshold converges to the optimal threshold.The paper relates the update to a subgradient approximation for the dual problem.Source: 2) Online Threshold-Based Jamming Design · p. 12

C. Numerical Examples

  • Online jamming with residual self-interference performs similarly to optimal jamming and significantly outperforms constant-power, on-off, and passive benchmarks in the co-located case.The optimal scheme with residual self-interference performs worse than without it, especially above 30 dBm, because residual self-interference is significant at the co-located monitor.Source: C. Numerical Examples · p. 12, Figure 10 · p. 13
  • With separately located antennas, online jamming with residual self-interference performs similarly to optimal jamming and much better than the other benchmark schemes.Effective self-interference cancellation makes optimal jamming with and without residual self-interference perform the same in this setup.Source: C. Numerical Examples · p. 12, Figure 11 · p. 13

VIII. CONCLUDING REMARKS

  • The paper formulates delay-sensitive and delay-tolerant jamming-power allocation problems under an average-power constraint and obtains optimal solutions using Lagrange duality despite non-convexity.The conclusion also identifies intelligent suspicious users, multi-antenna defenses, and acquiring global CSI as open issues for future study.Source: §VIII · pp. 12–13

APPENDIX

The appendix proves Proposition 3.1 by solving the primal problem through its Lagrange dual. It decomposes the per-state optimization and finds the dual variable by bisection.

A. Proof of Proposition 3.1

  • The proof uses Lagrange duality, with a dual variable for the average jamming-power constraint, to solve problem (P1) through its dual problem.The proof first obtains the dual function for a given dual variable, then minimizes it to find the optimal value.Source: A. Proof of Proposition 3.1 · p. 13
  • For each given dual variable, the Lagrangian problem decomposes into subproblems, one for each fading state.The proof handles cases according to the sign of g2(ν), comparing candidate choices when it is positive.Source: A. Proof of Proposition 3.1 · pp. 13–14
  • The proof finds the optimal dual variable by bisection using the subgradient, then substitutes it to obtain the optimal jamming allocation.When Q is below the expected unconstrained allocation, the optimal dual variable tends to zero; otherwise the average-power constraint is met with equality.Source: A. Proof of Proposition 3.1 · p. 14

B. Proof of Proposition 4.1

The appendix establishes an infeasibility test for problem (P2.2) and derives the per-fading-state solution used in Proposition 4.2. The arguments rely on duality and a comparison of cases.

B. Proof of Proposition 4.1

  • Problem (P2.2) is infeasible if and only if nonnegative dual variables exist for which its dual function is negative.The proof establishes the two directions using a feasible auxiliary problem and strong duality.Source: B. Proof of Proposition 4.1 · p. 14

C. Proof of Proposition 4.2

  • For Proposition 4.2, the per-state solution treats the g2(ν)≤0 case separately from the g2(ν)>0 case.In the positive case, the proof compares the optimal values from the two subcases.Source: C. Proof of Proposition 4.2 · p. 14
  • The positive-gain case includes a convex subproblem whose solution is compared with the alternative case to select the optimum.The proof notes that a strict power inequality may prevent exact attainment of one candidate, but its value is lower than the alternative in that condition.Source: C. Proof of Proposition 4.2 · p. 14

D. Checking the Feasibility of Problem (28)

The section tests feasibility of problem (28) through its dual and gives an optimal solution to an associated problem. The latter decomposes across fading states.

D. Checking the Feasibility of Problem (28)

E. Proof of Proposition 5.2

The proof decomposes problem (31) into fading-state subproblems and solves problem (55) by considering three cases. It selects the optimal solution by comparing the resulting values, subject to the feasibility condition for the second case.

E. Proof of Proposition 5.2

  • When q(ν) ≥ ˆq1(ν), the optimum is q(ν) = ˆq1(ν), with ˆr0(ν) = 0 and ˆp(ν) = 0.The resulting value is ˆv1(ν).Source: E. Proof of Proposition 5.2 · p. 15
  • For q(ν) < ˆq1(ν) and X(ν) = 1, convexity makes the optimum one of the endpoints ˆq2(ν) or ˆq1(ν).This case is feasible only when ˆq2(ν) < ˆq1(ν); the endpoint values are ˆv2(ν) and ˆv1(ν), respectively.Source: E. Proof of Proposition 5.2 · p. 15
  • For q(ν) < ˆq1(ν) and X(ν) = 0, the convex problem has an optimum at ˆq3(ν) = [min(ˆq1(ν), ˆq2(ν), ˆq4(ν))]+.The objective's first-order derivative is zero at ˆq4(ν), and the resulting value is ˆv3(ν).Source: E. Proof of Proposition 5.2 · p. 15
  • Comparing the three case values yields the optimal solution to problem (55), with the second case feasible only when ˆq2(ν) < ˆq1(ν).The cases produce ˆv1(ν), ˆv2(ν), and ˆv3(ν), respectively.Source: E. Proof of Proposition 5.2 · p. 15

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