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Two-hop Secure Communication Using an Untrusted Relay

Xiang He, Aylin Yener

arXiv:0910.2718v1cs.IT

TL;DR

The paper addresses secure communication when a source and destination must rely on an untrusted relay and cannot use a direct link. It uses cooperative jamming from the destination or an external helper, derives an upper bound, and shows that the achievable-rate gap vanishes as relay power grows.

  • Problem

    The paper studies how to keep a confidential message secret from a relay that is essential for communication between a source and destination without a direct link.

  • Method

    The paper uses cooperative jamming by the transmitting destination or an external helper and derives an upper bound through a second-eavesdropper and genie-based wiretap-channel transformation.

  • Results

    A positive secrecy rate is achievable, and the gap between achievable rates and the upper bound converges to zero as relay power goes to ∞.

  • Takeaways & Limitations

    Cooperative jamming can enable secure two-hop communication through an untrusted relay when the destination has transmission capability or an external helper is available.

  • Takeaways & Limitations

    The upper-bound analysis assumes that no feedback is used for encoding at the source or destination.

Abstract

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We consider a source-destination pair that can only communicate through an untrusted intermediate relay node. The intermediate node is willing to employ a designated relaying scheme to facilitate reliable communication between the source and the destination. Yet, the information it relays needs to be kept secret from it. In this two-hop communication scenario, where the use of the untrusted relay node is essential, we find that a positive secrecy rate is achievable. The center piece of the achievability scheme is the help provided by either the destination node with transmission capability, or an external "good samaritan" node. In either case, the helper performs cooperative jamming that confuses the eavesdropping relay and disables it from being able to decipher what it is relaying. We next derive an upper bound on the secrecy rate for this system. We observe that the gap between the upper bound and the achievable rate vanishes as the power of the relay node goes to infinity. Overall, the paper presents a case for intentional interference, i.e., cooperative jamming, as an enabler for secure communication.

I. INTRODUCTION

The paper studies two-hop networks where an untrusted relay is essential because the source and destination lack a viable direct link. It shows that destination transmission or external cooperative jamming can enable positive secrecy rates, and derives a tighter upper bound whose gap vanishes as relay power grows.

  • Problem: The source and destination communicate only through an intermediate relay because direct communication cannot sustain a non-trivial reliable rate under power constraints.The relay is therefore necessary, despite not being trusted with the confidential message.
  • Problem: Without destination transmission capability, the no-direct-link network has zero secrecy capacity because the physically degraded relay knows everything the destination knows.The destination receives signals only from the relay, making confidential transmission impossible in that setting.
  • Achievability: Destination transmission enables cooperative jamming that confuses the relay and achieves a positive secrecy rate; an external cooperative jammer can provide the same type of help.The model assumes half-duplex nodes and a two-phase communication protocol.
  • Model and novelty: The paper focuses on an untrusted but legitimate relay, unlike prior cooperative-jamming work centered on an external eavesdropper.The relay faithfully follows the designated relaying scheme but lacks sufficient security clearance for the confidential message.
  • Upper bound: A computable secrecy-rate upper bound is obtained by introducing a second eavesdropper, applying genie arguments, and transforming the channel into a wiretap channel with a helpful jammer.The resulting bound is strictly tighter than the corresponding bound without secrecy constraints and than a generalized entropy-power-inequality bound when relay received SNR exceeds 0dB.
  • Results: The gap between the upper bound and achievable rates converges to zero as the relay power goes to ∞.This establishes asymptotic agreement between the achievable scheme and the derived bound in the high-relay-power regime.

II. CHANNEL MODEL

The model uses half-duplex nodes and alternates communication between phase one, when the source transmits and the destination jams, and phase two, when the relay transmits. Reliable delivery is required while the relay’s available observations reveal only limited information about the source message.

  • Communication phases: Communication alternates between phase one and phase two under a random or deterministic schedule.The schedule contains n phase-one uses and m phase-two uses, which need not be consecutive.
  • Node transmissions: During phase one, the source transmits X1 while the destination transmits jamming signal X2 to confuse the relay.In phase two, the relay transmits Xr, computed from its prior observations and local randomness.
  • Reliability and secrecy: The destination decodes the source message from its phase-one transmissions and phase-two receptions, with decoding error required to be small.The relay is treated as an eavesdropper whose local randomness and transmitted signals are available for inferring the message.
  • Reliability and secrecy: The secrecy constraint limits the information about W extractable by the relay from its available observations.The secrecy rate is defined over schedules with phase-one time-sharing factor α and optimized over α for achievability.
  • Cooperative-jamming models: An external cooperative jammer is a more general model than destination jamming when the destination receives only a noisy copy of the jamming signal.Giving the noise sequence to the destination reveals the jamming signal, so an upper bound for the destination-jamming model also bounds the external-jammer model.
  • Protocol boundary: Proper protocol initialization is necessary because otherwise the destination may miss transmission initiation and the relay may receive the source message without jamming protection.The paper identifies this as an apparent vulnerability of the two-phase protocol.

III. ACHIEVABLE RATE

The paper constructs achievable secrecy rates using deterministic periodic schedules and a quantized compress-and-forward relay description. The resulting rate depends on schedule timing and power allocation, with relay power always maximized but source power potentially reduced.

  • Schedule construction: The achievable-rate construction alternates n′ phase-one uses and m′ phase-two uses periodically, repeated M times.The schedule sequence lets M, n′, and m′ grow while preserving the selected phase-one time-sharing factor.
  • Achievable secrecy rate: Theorem 1 states that a secrecy rate is achievable for the model with the specified schedule and quantization design.The relay’s compress-and-forward description uses an auxiliary quantized observation, with quantization-noise variance c determined by a stated condition.
  • Power allocation: For any fixed time-sharing factor α, the relay should always transmit at maximum power Pr.
  • Power allocation: The achievable rate is not monotonic in source power P′1 for a fixed jamming power P2.As P′1 approaches 0 or infinity, Re approaches 0, so the optimal source power may be below the available budget P1.
  • Asymptotic behavior: As the relay power constraint P̄r tends to infinity, the achievable secrecy-rate expression has the asymptotic behavior stated in Remark 5.

IV. UPPER BOUND

The paper derives an upper bound by optimizing the channel-use schedule and transforming the model through added eavesdroppers, genie information, and independent-noise arguments. The resulting bound is asymptotically close to the achievable rate as relay power grows, under the stated power-scaling conditions.

  • Schedule optimization: The upper-bound derivation first fixes an optimal schedule with phase-one channel uses preceding phase two.Any schedule can be rearranged this way without changing the achievable secrecy rate.
  • Channel transformations: Adding a second eavesdropper preserves the secrecy rate because the message is already secret from that additional observation.The construction is shown as a two-eavesdropper channel.
  • Channel transformations: Removing the relay’s eavesdropper constraint and revealing selected signals through a genie produces a more tractable upper-bound channel.The genie reveals the relay signal to the destination and the phase-two source signal to both relay and destination, removing the jamming signal’s influence from the relay link.
  • Channel transformations: Independent link noises justify the transformed channel in which the revealed information makes the phase-two relay signal useless for the relay.The resulting equivalent model is shown in Figure 5.
  • Upper bound: Theorem 2 upper bounds secrecy rate using the stated logarithmic expression, with ρ selected optimally and powers normalized by the time-sharing factor α.P1, P2, and Pr are the average power constraints for the two nodes and relay.
  • Asymptotic behavior: When source/helper powers grow with fixed helper power, the upper-bound gap depends only on helper power; when source and helper powers scale together, the gap converges to 0.The latter case makes the upper bound asymptotically tight.

A. Comparison with the bound derived with generalized Entropy Power Inequality

This section compares the preceding upper bound with another computable bound obtained using a generalized entropy power inequality. The alternative bound is within 0.5 bit/channel use of achievable rates as relay power grows, under a stated source-plus-helper power condition.

  • Derivation: The comparison bound is derived using a generalized entropy power inequality previously applied to Gaussian multiple-access channels with secrecy constraints.The derivation uses Fano’s inequality, independence of the two input sequences, and an inequality from the cited work.
  • Bound comparison: The alternative bound is nontrivial when P1 > P2 because it is tighter than the bound obtained by removing secrecy constraints.The paper also compares which bound is tighter across power regimes.
  • Asymptotic comparison: When relay power tends to infinity, the gap between achievable rates and the alternative bound is smaller than 0.5 bit/channel use.In this limit, α tends to 1 and Pi tends to the corresponding average power for i = 1, 2.
  • Bound comparison: For P1 + P2 > 1, the gap to the first upper bound is also bounded by 0.5 bit/channel use as relay power tends to infinity.Under this condition, the alternative bound is always looser than the bound from the earlier derivation.
  • Scope boundary: When P1 + P2 < 1, the paper does not determine which of the two bounds is tighter.The paper notes that secrecy capacity is very small in these cases.

V. NUMERICAL RESULTS

The numerical results examine scenarios varying power budgets, relay and jammer power, and whether the time-sharing factor α is fixed or optimized. Across these settings, the upper bound is often close to the achievable rate, while power control and optimization of α affect the gap.

  • V. NUMERICAL RESULTS: The scenarios vary power budgets, relay and jammer power, and whether the time-sharing factor α is fixed or dynamically optimized.The numerical cases also include an external cooperative jammer, for which the upper bound holds but the achievable-rate gap is wider.
  • V. NUMERICAL RESULTS: When the relay has more power than the source and jammer, the achievable rate typically increases linearly with the source SNR.For Pr →∞, the optimal time-sharing factor α converges to 1.
  • V. NUMERICAL RESULTS: As P1 →∞, the gap between the upper bound and achievable rate goes to zero; for P2 = 30dB, the gap is 9.98 × 10^-4 bits/channel use.In the P2 = 30dB case, the upper bound almost coincides with the achievable rate.
  • V. NUMERICAL RESULTS: The improvement of the new bounds over the cut-set bound without secrecy constraints is generally small when jammer power is large.The numerical comparison includes the non-secrecy cut-set bound in each figure.
  • V. NUMERICAL RESULTS: Without source power control, the achievable rate eventually decreases to zero when the relay’s power is limited.This behavior appears only in the limited-relay-power setting.
  • V. NUMERICAL RESULTS: Optimizing both rates over α widens the gap because the second term in upper bound (41) becomes significant, although the new bound still improves substantially over the non-secrecy bound.The second term tends to balance the two terms in the bound when α is optimized.

VI. CONCLUSION

The paper shows that secrecy is achievable in a two-hop relay network even when the relay is untrusted and essential for communication. Cooperative jamming yields a positive secrecy rate, while the derived upper bound approaches the achievable rate under high-power conditions.

  • VI. CONCLUSION: A nonzero secrecy rate is achievable despite imposing secrecy constraints at the untrusted relay.The source and destination communicate through the relay without a direct link.
  • VI. CONCLUSION: The destination or a dedicated helper jams the relay and uses the jamming signal as side information.This cooperative-jamming mechanism confuses the relay while supporting decoding at the destination.
  • VI. CONCLUSION: The paper derives an upper bound that is strictly tighter than the corresponding bound without secrecy constraints and tighter than a generalized entropy-power-inequality bound.The comparison is stated for the no-feedback encoding setting.
  • VI. CONCLUSION: The gap between the upper bound and achievable rate converges to 0 as the transmitter, relay, and jammer powers go to ∞.Numerical results also show the upper bound is generally close to the achievable rate.
  • VI. CONCLUSION: With a relay whose power is in abundance, the upper bound is indistinguishable from the achievable rate for a fixed time-sharing factor.The paper describes its bound as asymptotically tighter in this case than a comparison bound whose gap does not vanish.
  • VI. CONCLUSION: Whether cooperative jamming achieves the secrecy capacity for various multiuser channels remains an open problem.The conclusion motivates further study in larger networks and more general settings.

APPENDIX A

The appendix develops the achievable secrecy-rate result for a relay channel with a jammer by applying a compress-and-forward construction to an equivalent memoryless relay channel. It then specifies the channel representation, independent input distributions, and limiting argument used to establish the achievable-rate range.

  • APPENDIX A: The proof introduces supporting results and applies an achievable secrecy-rate result for a general relay channel.The appendix uses the compress-and-forward framework as the basis for the construction.
  • APPENDIX A: A stochastic source encoder randomly selects a codeword from a message-determined bin to confuse the relay.The bin size is 2^N I(X;Y_r|X_r), and the resulting secrecy-rate expression includes −I(X;Y_r|X_r).
  • APPENDIX A: The jammer extension uses an i.i.d. jamming signal X_2 independent of the source and relay inputs.Under p(X,X_2,X_r)=p(X)p(X_2)p(X_r), the induced channel is memoryless.
  • APPENDIX A: Figure 12 reformulates the original channel so the jammer and receiver can be treated separately.This separation is valid because the jammer does not use past received signals to compute its jamming signal.
  • APPENDIX A: The achievable-rate range follows by substituting the mutual-information terms, dividing by m′+n′, and taking m′+n′→∞.The appendix states the resulting condition as I(X_r;Y)>I(Ŷ_r;Y_r|Y,X_r).
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