PaperlayerView exact revision on arXiv

Source-linked AI summary

Secure Communication Via an Untrusted Non-Regenerative Relay in Fading Channels

Jing Huang, Amitav Mukherjee, A. Lee Swindlehurst

arXiv:1302.1216v1cs.ITmath.IT

TL;DR

The paper studies how to protect confidential messages from an untrusted relay while using it to improve transmission reliability. It derives exact secrecy outage probabilities for three schemes and analyzes their asymptotic behavior, finding that all schemes approach outage probability one with many relay antennas, while antenna selection improves secrecy performance and cooperative jamming can provide receiver diversity.

  • Problem

    The study examines the tradeoff between diversity for the legitimate receiver and inadvertent diversity of information leaked to the untrusted relay.

    Source: §I · p. 3

  • Method

    The paper derives exact secrecy outage probabilities for direct transmission, conventional non-regenerative relaying, and destination cooperative jamming, then analyzes their asymptotic behavior across operating regimes.

    Source: §VI · p. 28

  • Results

    With many relay antennas, all three schemes’ SOPs converge to unity; with antenna selection, all schemes improve, and cooperative jamming can provide receiver diversity.

    Source: §VI · p. 28

  • Takeaways & Limitations

    The policy comparison identifies cooperative jamming with antenna selection as a scheme that can provide a receiver diversity advantage.

    Source: §VI · p. 28

Abstract

from arXiv · show

We investigate a relay network where the source can potentially utilize an untrusted non-regenerative relay to augment its direct transmission of a confidential message to the destination. Since the relay is untrusted, it is desirable to protect the confidential data from it while simultaneously making use of it to increase the reliability of the transmission. We first examine the secrecy outage probability (SOP) of the network assuming a single antenna relay, and calculate the exact SOP for three different schemes: direct transmission without using the relay, conventional non-regenerative relaying, and cooperative jamming by the destination. Subsequently, we conduct an asymptotic analysis of the SOPs to determine the optimal policies in different operating regimes. We then generalize to the multi-antenna relay case and investigate the impact of the number of relay antennas on the secrecy performance. Finally, we study a scenario where the relay has only a single RF chain which necessitates an antenna selection scheme, and we show that unlike the case where all antennas are used, under certain conditions the cooperative jamming scheme with antenna selection provides a diversity advantage for the receiver. Numerical results are presented to verify the theoretical predictions of the preferred transmission policies.

I. INTRODUCTION · II. MATHEMATICAL MODEL

The paper analyzes secrecy outage probability in an untrusted-relay network, comparing direct transmission, AF relaying, and cooperative jamming. It also examines antenna-count effects and a relay model limited to one RF chain.

I. INTRODUCTION

  • The work focuses on AF-relaying SOP because it suits realistic fading channels with finite-delay constraints and offers greater security than decode-and-forward with less complexity than compress-and-forward.Source: §I · pp. 2–3
  • The open problem is balancing diversity for the legitimate receiver against the inadvertent diversity of information leaked to the untrusted relay in the first hop.Source: §I · p. 3
  • The analysis derives exact SOPs for direct transmission, conventional AF relaying, and destination cooperative jamming, which degrades the relay’s eavesdropping capability in the first hop.Source: §I · p. 3
  • Asymptotic analysis identifies optimal policies for a single-antenna relay across power budgets and channel gains, while the broader analysis determines when each scheme has an advantage.Source: §I · p. 3
  • Antenna selection improves secrecy performance over beamforming, but SOP still grows with relay antenna count in nearly all cases; it is non-increasing only with cooperative jamming when second-hop CSI is hidden from the relay.Source: §I · p. 3

II. MATHEMATICAL MODEL

  • The mathematical model is a half-duplex, two-hop network with single-antenna Alice and Bob and an untrusted AF relay equipped with K antennas.Source: §II · p. 4
  • The model assumes quasi-static Rayleigh fading, an available direct Alice–Bob link, and the same power budget P at every node.Source: §II · p. 4

A. Relay Protocol · B. Cooperative Jamming

The relay uses MRC for reception and MRT for forwarding, while the cooperative-jamming scheme has Bob transmit interference in the first phase and remove its contribution from the forwarded signal. Both schemes are modeled as equivalent one-stage SIMO wiretap channels, with secrecy rate determined by the difference between legitimate-link and eavesdropper-link mutual information.

A. Relay Protocol

  • With K RF chains, the relay uses MRC on reception and MRT on transmission when forwarding the message.The relay forwards the output of its MRC beamformer to Bob.Source: A. Relay Protocol · p. 4, B. Cooperative Jamming · p. 6
  • AF and cooperative-jamming schemes are equivalent to a one-stage 1 × 2 SIMO wiretap channel,, with achievable secrecy rate Rs = [IB − IE]+.Here, IB and IE are the mutual information of the legitimate and eavesdropper links, respectively.Source: A. Relay Protocol · p. 5

B. Cooperative Jamming

  • In half-duplex cooperative jamming, Bob forgoes receiving Alice’s information in the first phase and transmits a jamming signal instead.The jamming signal is known to Bob.Source: B. Cooperative Jamming · p. 5
  • In the second phase, the relay scales its received signal and forwards it to Bob using amplify-and-forward.The relay–Bob channel is assumed reciprocal.Source: B. Cooperative Jamming · pp. 5–6
  • Bob removes the intentional interference from the received signal because he knows the jamming signal he transmitted.He can subtract the jamming contribution forwarded by the relay regardless of the relay’s receive beamformer.Source: B. Cooperative Jamming · p. 6
  • When evaluating the relay’s mutual information under cooperative jamming, the relay is assumed to use MMSE (maximum-SINR) beamforming to counteract Bob’s jamming.The relay forwards the output of its MRC beamformer to Bob.Source: B. Cooperative Jamming · p. 6

III. TRANSMISSION WITH AN UNTRUSTED RELAY · A. Single-antenna Relay

For the single-antenna relay, the section derives exact secrecy outage probabilities for direct transmission (DT), amplify-and-forward (AF), and cooperative jamming (CJ), and examines their asymptotic behavior. It identifies conditions favoring DT or limiting AF and CJ, while focusing its theoretical analysis on full-power transmission.

A. Single-antenna Relay

  • Exact secrecy outage probabilities are derived for DT, AF, and CJ with a single-antenna relay.The section gives exact SOP expressions for AF and CJ and frames the analysis as covering all three schemes.Source: A. Single-antenna Relay · pp. 6, 8, 9
  • DT can provide better secrecy performance than AF and CJ in some cases.Under DT, the relay is treated as an eavesdropper; fading can still permit a positive secrecy rate when its average channel is stronger, though that event has probability below 1/2 [2].Source: A. Single-antenna Relay · pp. 6–7
  • As the average Alice-to-relay channel gain tends to infinity, Alice cannot reliably transmit secret messages under DT.The passage gives proximity of the untrusted relay to Alice as an example of this limiting condition.Source: A. Single-antenna Relay · p. 7
  • At high SNR, AF's SOP approaches a value independent of transmit SNR rather than zero.Increasing transmit power raises SNR at both the legitimate user and the eavesdropper.Source: A. Single-antenna Relay · p. 8
  • CJ is unsuitable when the relay-to-Bob channel is weak because Bob ignores the direct link and relies on the second hop.The probability of a positive secrecy rate for CJ increases monotonically with the average relay-to-Bob channel gain when the average Alice-to-relay gain is fixed.Source: A. Single-antenna Relay · p. 9
  • The theoretical analysis assumes all nodes transmit at maximum power; closed-form solutions for outage minimization under other power constraints appear intractable.Later simulations show examples of performance gains from optimal power allocation.Source: A. Single-antenna Relay · p. 9

B. Asymptotic Behavior · 2) Case of · 3) Case of ¯γR,B →∞or 0:

Asymptotic scheme preferences depend on power budgets and channel conditions: CJ can be preferable at high SNR or with a weak direct link, while DT or AF can prevail in other limiting regimes.

B. Asymptotic Behavior

2) Case of

  • As ¯γA,B →∞, the outage probability tends to zero.Source: 2) Case of · p. 10
  • When the direct link is strong, DT and AF outperform CJ; when it is weak, CJ performs better.The comparison is made as the average direct-link SNR varies.Source: 2) Case of · p. 10

3) Case of ¯γR,B →∞or 0:

C. Multi-antenna Relay · 1) Direct Transmission (DT): · 2) Amplify-and-Forward (AF):

For a multi-antenna relay, the paper characterizes secrecy outage probability (SOP) and its dependence on relay antenna count. Both direct transmission and amplify-and-forward relaying have SOPs that approach unity as the relevant antenna count grows.

C. Multi-antenna Relay

  • The multi-antenna analysis characterizes SOP and examines how the number of relay antennas affects secrecy performance.This section generalizes the analysis to a multi-antenna relay.Source: C. Multi-antenna Relay · p. 12

1) Direct Transmission (DT):

  • For DT with multiple relay antennas and MRC beamforming, the outage analysis uses a central chi-square distribution with 2K degrees of freedom.The cited derivation describes the relevant random variable as having this distribution.Source: 1) Direct Transmission (DT) · p. 12
  • The DT scheme’s SOP approaches unity as K grows, reflecting worse secrecy with more eavesdropper antennas.The passage attributes this deterioration to the eavesdropper’s increased antenna count.Source: 1) Direct Transmission (DT) · p. 12

2) Amplify-and-Forward (AF):

3) Cooperative Jamming (CJ):

The cooperative-jamming scheme has unfavorable asymptotic secrecy when the relay uses multiple antennas: outage remains nonzero at high SNR and approaches unity as antenna count grows. With one relay antenna, its outage instead converges to zero.

3) Cooperative Jamming (CJ):

  • Because the CJ secrecy-outage expression is more complicated than for AF and lacks a clear closed form, the analysis focuses on asymptotics in transmit SNR and relay antenna count.Source: 3) Cooperative Jamming (CJ) · p. 15
  • For K > 1, the secrecy outage probability approaches a nonzero constant as transmit SNR grows; for K = 1, it converges to zero.The relay can suppress Bob’s jamming signal when it has multiple antennas, explaining the less favorable secrecy outcome.Source: 3) Cooperative Jamming (CJ) · p. 15
  • As the number of relay antennas grows, the cooperative-jamming secrecy outage probability approaches unity.The derivation notes that a term increasing with antenna count drives the outage probability toward unity.Source: 3) Cooperative Jamming (CJ) · pp. 15–16
  • Under certain conditions, antenna selection can reverse the multi-antenna secrecy trend: increasing relay antennas improves secrecy when the relay has a single RF chain.Source: 3) Cooperative Jamming (CJ) · p. 16
  • These conclusions assume Alice sends no information to Bob in the second phase and may not hold for protocols that allow it, such as Protocol III.Source: 3) Cooperative Jamming (CJ) · p. 16

IV. SECRECY WITH RELAY ANTENNA SELECTION

The section studies secrecy when an untrusted, non-malicious relay selects antennas instead of beamforming. It characterizes SOP for AF and cooperative jamming (CJ) and examines how relay antenna count affects secrecy.

IV. SECRECY WITH RELAY ANTENNA SELECTION

  • The relay selects its strongest receive antenna to maximize wiretapping ability, then its best transmit antenna to forward the message to Bob.The relay is assumed untrusted but not malicious.Source: §IV · p. 16
  • Because antenna selection limits the relay’s ability to counter artificial jamming, while Bob retains diversity, CJ is expected to improve secrecy performance.The relay loses beamforming flexibility to cope with artificial jamming signals, whereas Bob can still benefit from antenna-selection diversity.Source: §IV · p. 16
  • The section characterizes secrecy outage probability for AF and CJ and studies the impact of the number of relay antennas.Source: §IV · p. 16

A. Direct Transmission (DT) · B. Amplify-and-Forward (AF) · 1) Relaying with second-hop CSI:

The sections describe direct transmission with relay antenna selection and AF under different CSI assumptions; for AF with second-hop CSI, secrecy outage approaches unity as the relay antenna count grows.

A. Direct Transmission (DT)

  • In DT, the untrusted relay selects the antenna with the largest Alice-to-relay channel gain for wiretapping, and this maximum gain increases with relay antenna count.Source: A. Direct Transmission (DT) · pp. 16–17
  • The DT secrecy outage probability is derived using the channel distributions and order statistics.The passages define the direct-link gain and relay-channel gains and refer to order-statistics theory.Source: A. Direct Transmission (DT) · pp. 16–17

B. Amplify-and-Forward (AF)

  • AF is considered both when the relay has Bob’s second-hop CSI and when it lacks that CSI, as with a passive receiver or absent training.Source: B. Amplify-and-Forward (AF) · p. 17

1) Relaying with second-hop CSI:

  • Proposition 4 gives an exact secrecy outage probability for AF relaying with antenna selection at a given target secrecy rate.Source: 1) Relaying with second-hop CSI · pp. 17–18
  • With second-hop CSI, AF antenna selection chooses receive and transmit antennas by first maximizing the relay’s wiretap SNR, then considering assistance to Bob.Source: 1) Relaying with second-hop CSI · p. 17
  • As the relay antenna count grows, AF secrecy outage approaches unity because the relay gains diversity proportionally more than Bob, harming confidentiality.Both the relay and Bob gain diversity, but the relay’s greater benefit comes at the expense of information confidentiality.Source: 1) Relaying with second-hop CSI · pp. 18–19

2) Relaying without second-hop CSI: … 2) Relaying without second-hop CSI:

Without second-hop CSI, conventional relaying performs worse, but cooperative jamming can improve secrecy: the legitimate user retains first-hop diversity while the relay loses cooperative-jamming diversity gain.

2) Relaying without second-hop CSI:

C. Cooperative Jamming (CJ)

1) Relaying with second-hop CSI:

  • With second-hop CSI, the cooperative-jamming SOP is difficult to calculate exactly, and its analysis notes that the denominator grows with K while the numerator is upper-bounded by 1 + ρ|hA,m∗|2.The numerator is not necessarily increasing as K grows.Source: 1) Relaying with second-hop CSI · p. 20

2) Relaying without second-hop CSI:

  • In the no-CSI case, the relay has no information about hR,B when Bob sends no training data and jams only while Alice transmits; it selects the first-hop receive antenna with the largest channel gain.Discriminatory channel-estimation training methods can also prevent the relay from acquiring CSI from Bob,.Source: 2) Relaying without second-hop CSI · p. 20
  • Since |h_{m*,B}|^2 is independent of |h_{A,m*}|^2, transmitting on the first-hop-selected antenna is equivalent to random second-hop antenna selection; the exact SOP is a special case of (54).Source: 2) Relaying without second-hop CSI · pp. 19–20
  • Without second-hop CSI, cooperative jamming with antenna selection improves secrecy despite losing transmit-selection diversity, because the relay loses diversity gain while the legitimate user retains first-hop diversity.The corollary states this secrecy improvement when the relay’s second-hop CSI is hidden.Source: 2) Relaying without second-hop CSI · p. 21

V. NUMERICAL RESULTS · A. Single-Antenna Case

Numerical examples compare DT, AF, and CJ across single- and multi-antenna scenarios; the single-antenna results show how preferred schemes vary with SNR and channel gains.

V. NUMERICAL RESULTS

  • The numerical study evaluates outage for DT, AF, and CJ across transmit powers, average channel gains, and antenna counts, using a target secrecy rate of R = 0.1 bits per channel use.The target rate is as assumed in,.Source: §V · p. 21

A. Single-Antenna Case

  • AF and CJ are evaluated through Eqs. (8), (14), and (20), agree well with simulations, and exactly predict the performance cross-over points.Source: A. Single-Antenna Case · p. 21
  • At high SNR CJ outperforms AF, while AF is better at low SNR; as SNR grows, SOP tends to constants for DT and AF but to zero for CJ.DT is suboptimal at medium to high SNR compared with using the relay.Source: A. Single-Antenna Case · p. 21, Figure 1 · p. 22
  • The fixed-to-optimal power-allocation performance gap is more pronounced for AF than CJ, and optimal allocation follows the asymptotic analysis.AF uses the direct link, making its secrecy performance more sensitive to power allocation.Source: A. Single-Antenna Case · p. 22
  • As the relay-to-destination channel gain approaches zero, CJ outage approaches one, while DT and AF approach nonzero constants; DT retains an advantage over AF.As the relay-to-destination gain grows large, CJ has the best performance in the plotted setting, but relative performance may vary with the other channel gains.Source: A. Single-Antenna Case · p. 22, Figure 2 · p. 23
  • CJ outage approaches one when the source-to-relay gain is extremely small or large, so CJ performs best at intermediate gains.At a very large gain, the untrusted relay is nearly colocated with Alice and secure transmission is impossible; at a very small gain, a reliable relay link is difficult to establish.Source: A. Single-Antenna Case · pp. 22–23, Figure 3 · p. 24
  • When the direct-link gain is small, CJ is best; as it grows large, DT and AF outage decay to zero at a rate proportional to 1/γ̄A,B.In the weak-relay-hop regime, DT asymptotically outperforms AF.Source: A. Single-Antenna Case · p. 23, Figure 4 · p. 24
  • When the direct and relay-to-destination gains increase together, AF outperforms the other schemes because its outage decays faster, while CJ SOP converges to a constant.The improving second hop benefits AF but does not benefit DT.Source: A. Single-Antenna Case · pp. 23–24, Figure 5 · p. 25

B. Multi-Antenna Case · VI. CONCLUSIONS

The paper derives exact and asymptotic secrecy-outage analyses for direct transmission, amplify-and-forward relaying, and cooperative jamming. It finds that using all relay antennas drives SOP toward unity as their number grows, while antenna selection can preserve secrecy, particularly for cooperative jamming when second-hop CSI is concealed.

B. Multi-Antenna Case

  • The multi-antenna analytical SOP results for direct transmission and amplify-and-forward agree well with simulations, as do the antenna-selection analyses for direct transmission, amplify-and-forward, and cooperative jamming.The antenna-selection results are given for the respective analyses in Eqs. (51), (54), and (64).Source: B. Multi-Antenna Case · p. 25
  • When all relay antennas are used, cooperative-jamming SOP rises rapidly from K = 1 to 2 because relay beamforming suppresses Bob’s intentional interference.This makes cooperative jamming ineffective as the relay gains multiple receive antennas.Source: B. Multi-Antenna Case · p. 25, Figure 6 · p. 26
  • As ρ increases with antenna selection, direct transmission and amplify-and-forward SOPs approach constants, while cooperative-jamming SOP decays to zero, faster when second-hop CSI is available.With that CSI, the relay selects its best transmit antenna, making the diversity gain more apparent at larger ρ.Source: B. Multi-Antenna Case · p. 26, Figure 7 · p. 27
  • Antenna selection yields lower SOP than using all antennas for amplify-and-forward and cooperative jamming, and markedly improves cooperative-jamming secrecy relative to relay beamforming.Selection removes relay array gain, which benefits the relay more than Bob.Source: B. Multi-Antenna Case · p. 26, Figure 8 · p. 27
  • With antenna selection, cooperative jamming using second-hop CSI first improves and then worsens as K grows, whereas withholding that CSI makes its SOP decrease monotonically.The second-hop diversity gain initially outweighs the relay’s first-hop diversity, but its marginal benefit diminishes for larger K; concealing CSI is recommended to maintain confidentiality at large K.Source: B. Multi-Antenna Case · pp. 26–27

VI. CONCLUSIONS

  • The study derives exact SOPs for direct transmission, conventional non-regenerative relaying, and destination cooperative jamming, then uses asymptotic analysis to identify policies for different operating regimes.The SOP analysis determines performance transitions among the three policies across scenarios.Source: §VI · p. 28
  • With many relay antennas, all three schemes’ SOPs converge to unity; with antenna selection, secrecy improves and cooperative jamming can gain significant diversity with moderate antenna growth.The conclusion adds that concealing the destination’s CSI from the relay prevents secrecy from being compromised as the antenna count grows.Source: §VI · p. 28

APPENDIX A PROBABILITY OF POSITIVE SECRECY RATE FOR AF … APPENDIX D

The appendices derive probability and outage expressions for AF and cooperative-jamming schemes, then use those results in a continuity argument supporting Lemma 1.

APPENDIX A PROBABILITY OF POSITIVE SECRECY RATE FOR AF

  • Appendix A gives the c.d.f. and p.d.f. of U and the c.d.f. of V for the AF positive-secrecy-rate derivation.The p.d.f. of U is obtained by differentiating its c.d.f.Source: Appendix A · p. 28
  • Using independence of U and V, Appendix A calculates P_AF and applies a change of variables and an identity from [19, eq. 3.353.5].The stated transformation is x = v/(1−v).Source: Appendix A · p. 29

APPENDIX B

  • Appendix B applies the results in (69)–(70) to obtain (72), using a change of variables and a specified parameter β2.The passage gives β2 in terms of the channel averages and R.Source: Appendix B · p. 30

APPENDIX C

  • Appendix C states the outage-probability expression for a given secrecy rate R.Source: Appendix C · p. 30

APPENDIX D

  • Appendix D gives the cooperative-jamming SOP using the exponential distribution of |hR,B|².The supplied passage identifies the distribution and introduces the SOP expression.Source: Appendix D · p. 31
  • Combining (78) and (79), the conclusion in Lemma 1 can be inferred, completing its proof.Source: Appendix D · p. 31
Loading 1302.1216v1…