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Cooperation with an Untrusted Relay: A Secrecy Perspective

Xiang He, Aylin Yener

arXiv:0910.1511v1cs.IT

TL;DR

The paper asks whether untrusted relay nodes should participate despite secrecy constraints. It analyzes two relay models and finds that cooperation is unhelpful in the first but beneficial in the second.

  • Problem

    The paper examines whether untrusted network nodes should participate while information remains secret from them.

  • Method

    The paper provides an achievable secrecy rate and a channel transformation that separates the relay from the eavesdropper.

  • Results

    In the first model, the relay-destination link does not increase secrecy rate, whereas in the second model relay cooperation improves secrecy rate over treating the relay as an eavesdropper.

  • Takeaways & Limitations

    The untrusted relay should not be deployed for perfect secrecy in the first model, but cooperation is beneficial in the second model.

Abstract

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We consider the communication scenario where a source-destination pair wishes to keep the information secret from a relay node despite wanting to enlist its help. For this scenario, an interesting question is whether the relay node should be deployed at all. That is, whether cooperation with an untrusted relay node can ever be beneficial. We first provide an achievable secrecy rate for the general untrusted relay channel, and proceed to investigate this question for two types of relay networks with orthogonal components. For the first model, there is an orthogonal link from the source to the relay. For the second model, there is an orthogonal link from the relay to the destination. For the first model, we find the equivocation capacity region and show that answer is negative. In contrast, for the second model, we find that the answer is positive. Specifically, we show by means of the achievable secrecy rate based on compress-and-forward, that, by asking the untrusted relay node to relay information, we can achieve a higher secrecy rate than just treating the relay as an eavesdropper. For a special class of the second model, where the relay is not interfering itself, we derive an upper bound for the secrecy rate using an argument whose net effect is to separate the eavesdropper from the relay. The merit of the new upper bound is demonstrated on two channels that belong to this special class. The Gaussian case of the second model mentioned above benefits from this approach in that the new upper bound improves the previously known bounds. For the Cover-Kim deterministic relay channel, the new upper bound finds the secrecy capacity when the source-destination link is not worse than the source-relay link, by matching with the achievable rate we present.

I. INTRODUCTION

The paper asks whether an untrusted relay with lower security clearance should participate in secret communication. It finds that cooperation is useless in one orthogonal model but beneficial in another, where compress-and-forward can increase secrecy.

  • Motivation: The central problem is whether an untrusted relay should participate when source-destination communication must remain secret from it.The relay has lower security clearance than the destination, creating a conflict between cooperation and secrecy.
  • Model 1: In Model 1, the relay is useless for secrecy, so it should not be deployed when perfect secrecy is desired.The paper finds the capacity-equivocation region and shows that the relay-destination link does not increase secrecy rate.
  • Model 2: In Model 2, compress-and-forward lets the relay increase an otherwise zero secrecy rate without knowing the secret message.The resulting achievable secrecy rate exceeds the rate obtained by treating the relay merely as an eavesdropper.
  • Upper bound: For a special Model 2 class without relay self-interference, the paper derives a tighter upper bound by separating the relay and eavesdropper.Introducing a second eavesdropper does not reduce secrecy capacity for this class.
  • Results: The new upper bound improves previous Gaussian Model 2 bounds and matches compress-and-forward for the Gaussian Cover-Kim deterministic relay channel when the source-destination link is not worse than the source-relay link.In that condition, the bound establishes secrecy capacity.

II. ACHIEVABLE SECRECY RATE FOR THE GENERAL RELAY CHANNEL WITH CO-LOCATED EAVESDROPPER

The paper formulates secrecy against a co-located relay eavesdropper and derives an achievable equivocation region using compress-and-forward with Wyner-Ziv coding.

  • The source sends message W to the destination while keeping it secret from the relay over a memoryless three-node relay channel.
  • The relay’s transmitted signal can reveal additional information about W, so secrecy must account for both its received and transmitted signals.
  • The achievable region uses compress-and-forward with a relay compression variable and the constraint I(Xr;Y) > I(Ŷr;Yr|Y,Xr).
  • Wyner-Ziv coding is enabled because the co-located eavesdropper has perfect knowledge of the relay’s transmitted signal, allowing block equivocations to be combined.
  • Introducing an auxiliary variable U before X can potentially enlarge the achievable equivocation region.

III. TWO SPECIAL CASES OF THE GENERAL MODEL: RELAY NETWORKS WITH ORTHOGONAL COMPONENTS

The paper studies two orthogonal-component relay models: Model 1 separates the source-to-relay link, while Model 2 separates the relay-to-destination link.

  • Model 1: Model 1 has an orthogonal source-to-relay channel, with overall transition law p(Y,Yr|XR,XD,Xr)=p(Y|XD,Xr)p(Yr|XR,Xr).
  • Gaussian models: The Gaussian Model 1 description uses independent Gaussian noises with variance N, channel gains a and b, and source and relay power constraints.
  • Model 2: Model 2 uses a broadcast channel from source to relay and destination, plus a separate orthogonal relay-to-destination link.
  • The upper-bound class includes the Gaussian Model 2 and the Gaussian Cover-Kim deterministic relay channel.
  • The paper derives Model 1’s equivocation capacity region, Model 2’s achievable region, and an upper bound for a special Model 2 class.

IV. EQUIVOCATION CAPACITY REGION FOR MODEL 1

For Model 1, the paper characterizes the equivocation capacity region and concludes that the untrusted relay does not improve secrecy; the secret message can avoid the relay.

  • Theorem 2 gives the equivocation capacity region for Model 1.
  • The total decoded rate is bounded by I(XD,Xr;Y), while the source-to-relay and direct components contribute through I(XR;Yr|Xr) and I(XD;Y|Xr).
  • Coding scheme: The coding scheme splits each block’s message into secret WD(k) and relay-assisted WR(k) parts, using block Markov coding and separate codebooks.
  • Decoding: The destination decodes the relay-assisted part before using the resulting relay codeword to decode the secret part.
  • Secrecy consequence: The relay-to-destination link does not improve secrecy, and secret information is mapped only through XD rather than through the relay.
  • Secrecy consequence: Thus, the paper concludes that the untrusted relay should not be deployed in Model 1.

V. AN ACHIEVABLE REGION FOR MODEL 2

For Model 2, the paper presents an achievable equivocation-rate region based on compress-and-forward and analyzes Gaussian compress-and-forward and amplify-and-forward schemes. The relay-to-destination link can enable secrecy, while source power control may improve secrecy rates.

  • Theorem 3 gives an achievable equivocation-rate region for Model 2.
  • Compress-and-forward can achieve a non-zero secrecy rate when the relay-to-destination gain b is sufficiently large, even when a > 1.
  • A sufficiently strong relay-to-destination link makes the untrusted relay useful, despite the relay and eavesdropper being co-located.
  • Amplify-and-forward can also achieve a non-zero secrecy rate for sufficiently large b, but its rate is strictly smaller than compress-and-forward.
  • Under secrecy constraints, the secrecy rate need not be maximized at maximum source power, and power control can improve both schemes.The paper attributes this effect particularly to small b and demonstrates it numerically for a = 1.2.

A. The Enhanced Channel

The enhanced-channel argument adds a statistically equivalent second eavesdropper and removes the original relay eavesdropper. Under specified channel conditions, this preserves secrecy capacity and yields an upper-bounding enhanced channel.

  • The construction relies on a relay model without feedback from the relay output X_r to its input Y_r.
  • The relay-eavesdropper separation argument replaces the co-located eavesdropper with an external one by adding a statistically equivalent second eavesdropper and removing the first.
  • The second eavesdropper cannot be added arbitrarily, because it may hear X_r and cooperate with the first eavesdropper.
  • Theorem 4 states that conditions (43)–(44) preserve secrecy capacity after introducing the second eavesdropper.
  • When the relay is not self-interfering, adding the second eavesdropper causes no secrecy-rate loss and produces an enhanced channel upper-bounding the original secrecy rate.

B. Upper Bound for a Special Class of Model 2

For a special class of Model 2 without relay self-interference, the paper derives an upper bound using the enhanced-channel construction and characterizes it through a theorem-defined distribution set.

  • The bound is derived for channels whose conditional distribution factors according to the specified orthogonal, non-self-interfering model.
  • The admissible set P contains joint distributions satisfying the channel factorizations in (55) and (56).
  • Theorem 5 upper-bounds the secrecy rate for relay channels in which the relay is the eavesdropper.
  • The upper bound combines a relay-link term I(X_r; Y_R) with a minimum involving destination information conditioned on the relay or separated eavesdropper observation.
  • The alternative upper bound can be tightened by choosing Y_e, but it does not improve the first term in the earlier bound.

C. The Gaussian Case of Model 2

For the Gaussian Model 2, the paper specializes the special-class upper bound and compares it with achievable rates as the relay-to-destination gain varies. The bound becomes tight for a strong relay-to-destination link and decreases as that gain vanishes.

  • Corollary 3 gives an upper bound on secrecy rate for the Gaussian Model 2 with independent noise components.
  • Figure 9 compares the Gaussian upper bound with achievable rates while fixing a = 1 and varying the relay-to-destination gain b.
  • As b →∞, the Gaussian upper bound becomes tight; as b →0, the upper bound decreases.
  • The decrease as b →0 is attributed to the first term in Corollary 3.

VII. THE COVER-KIM DETERMINISTIC RELAY CHANNEL

The section analyzes the Gaussian Cover–Kim deterministic relay channel, deriving an upper bound and an achievable secrecy rate. The two coincide for α ≤ 1, establishing secrecy capacity in that regime.

  • Upper bound: Theorem 7 upper bounds the secrecy rate of the Gaussian Cover–Kim deterministic relay channel.The proof uses a transformation that separates the eavesdropper and relay.
  • Capacity characterization: The upper bound and achievable rate coincide when α ≤ 1.This condition means the source–destination link is not worse than the source–relay link.
  • Capacity characterization: For α ≤ 1, compress-and-forward achieves the secrecy capacity.The conclusion follows by matching the achievable rate with the upper bound.
  • Noise correlation: The secrecy capacity can exceed the direct-link capacity if R0 > C(P).The paper attributes this benefit to correlation between the noises on the source links.
  • Noise correlation: With independent noises, the secrecy capacity cannot exceed C(P).This provides a contrasting bound for the same channel family.
  • Numerical illustration: For R0 = 0.5 bits/channel use and P = 1, the upper bound and achievable rate meet for α ≤ 1.Figure 10 illustrates the resulting secrecy-capacity match.

VIII. CONCLUSION

The paper studies whether untrusted relays can preserve or improve secrecy. Its two orthogonal relay models behave differently: the first offers no secrecy benefit, while the second can benefit from cooperation.

  • Model 1: For the first orthogonal model, the relay–destination link does not increase the secrecy rate.The authors therefore conclude that the untrusted relay should not be deployed when perfect secrecy is desired.
  • Model 2: For the second model, an achievable secrecy rate using relay cooperation improves on treating the relay as an eavesdropper.This establishes a positive role for the untrusted relay under that model.
  • Model 2: The second model can let the source and destination communicate beneficially despite the secrecy constraint.The conclusion is specifically tied to the model with an orthogonal relay–destination component.
  • Upper bounds: A channel transformation separates the relay and eavesdropper to produce an upper bound for a special class of untrusted-relay channels.The approach tightens previously known bounds for the Gaussian relay channel with an orthogonal relay–destination link.
  • Cover–Kim channel: For the Gaussian Cover–Kim deterministic relay channel, the approach finds secrecy capacity when the source–destination link is not worse than the source–relay link.The result follows from matching the new upper bound with the achievable rate.
  • Conclusion: The paper concludes that cooperation and secrecy can coexist in communication with untrusted partners.This conclusion is presented as a broader implication of the model-specific results.

D. Decoder at the Destination

The destination decodes the relay’s transmitted codeword and compression information before recovering the source codeword and message. Reliable decoding follows from the stated rate conditions.

  • Backward decoding: The destination first decodes X^n(N−1), then performs block-by-block backward decoding.Each block recovers the relay codeword, compression labels, and preceding source codeword.
  • Relay decoding: The relay codeword is decoded from the destination output when D < I(X_r; Y).The decoded relay index identifies the bin containing the previous compression label.
  • Compression decoding: The destination uses the relay codeword and side information to recover the compression label through bin decoding.It then reconstructs the compressed relay observation for source-codeword decoding.
  • Rate conditions: The source-codebook rate must satisfy the mutual-information condition in (122), alongside the relay-compression condition in (123).These conditions support the decoding analysis for the proposed scheme.
  • Reliability: The decoded message is a deterministic function of the decoded source codeword.Therefore, the message-decoding error probability is bounded by the source-codeword decoding error probability.

E. Equivocation Computation

The section derives the achievable equivocation region and converse bounds for the coding scheme, then specializes the analysis to channels where the relay does not interfere with itself.

  • Equivocation analysis: The equivocation analysis uses stochastic source encoding, independent source blocks, and the relay’s compress-and-forward structure.Fano-type bounds and Markov relations control the information revealed to the relay.
  • Achievable region: The achievable region is obtained by combining the two cases R1 ≤ lim C and R1 > lim C.The union of the resulting regions yields the final rate region under the compression constraint.
  • Achievable region: The region is constrained by I(X_r; Y) > I(Ŷ_r; Y_r|Y, X_r).This condition ensures that the destination can recover the relay’s compressed observation.
  • Special channel class: For the special non-self-interfering class, the relay input can be chosen as X_r,i = Y_r,i−1 without reducing the achievable secrecy rate.This converts the destination observation into a pair consisting of the previous relay output and current destination output.
  • Special channel class: The resulting channel is equivalent to a 1 × 2 MIMO wiretap channel.The eavesdropper observes Y_r while the destination receives the relay output together with Y_D.
  • Examples: In the constructed Gaussian example, the destination can cancel the noise completely, whereas the eavesdropper has finite AWGN capacity.The secrecy capacity is therefore unbounded in that example.
  • Examples: Adding a second eavesdropper that receives the destination’s signal reduces the secrecy capacity of the constructed system to 0.The new eavesdropper has the same marginal distribution as the relay eavesdropper but observes the destination signal.
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