Secure Beamforming for MIMO Two-Way Communications with an Untrusted Relay

A. Motivation

• The study asks when an untrusted two-way relay should be used rather than treated as a passive eavesdropper, how to jointly optimize beamformers, and whether network coding improves security.In two-way relaying, the relay decodes a network-coded message rather than each individual message, which can itself provide some security.Source: A. Motivation · p. 2

B. Related Work; 1) Beamforming in MIMO Two-way Relay Systems:; 2) Trusted Relay:

• For multi-antenna relaying, proposed alternating relay/source precoding to maximize achievable weighted sum rate, whereas proposed iterative joint source-relay precoding based on mean-square error (MSE).Traditional one-way secret communication was studied in –, while only a few recent works considered two-way secret communication –; studied MIMO two-way beamforming to maximize secrecy sum rate, proven achievable in.Source: 1) Beamforming in MIMO Two-way Relay… · p. 3, 2) Trusted Relay · p. 3

3) Untrusted Relay:

• Although prior work considered untrusted-relay secrecy in several settings, joint source and relay beamforming for MIMO two-way untrusted relaying had not been studied.The cited work included one-way MIMO amplify-and-forward beamforming with alternating optimization.Source: 3) Untrusted Relay · p. 3

C. Contribution

• The paper derives an optimal relay-beamformer structure for fixed source beamformers and proposes convergent alternating-optimization algorithms, which cannot guarantee global optimality because the problems are nonconvex.The relay structure reduces the number of unknowns in the relay beamformer and simplifies joint design.Source: C. Contribution · p. 4
• At high source and relay powers, two-phase relaying achieves the maximum secrecy rate when the received signals align at the relay; at vanishing source powers, three-phase relaying performs best and two-phase performs worse than direct transmission.The high-power result also identifies signal alignment as beneficial against eavesdropping and as lowering the source-antenna requirement for security.Source: C. Contribution · p. 4

D. Organization and Notations; II. SYSTEM MODEL AND PROBLEM FORMULATION

• The paper organizes its system model, beamformer designs, asymptotic analysis, and simulations across Sections II–V; it assumes AF relaying and includes direct transmission as a comparison.The relay is an essential helper and potential eavesdropper, and the model excludes decode-and-forward because the untrusted relay is not expected to decode.Source: D. Organization and Notations · pp. 4–5, §II · pp. 5–6

A. Two-Phase Two-Way Relay Scheme

• In the two-phase scheme, both sources transmit to the relay simultaneously in phase one; in phase two, the relay amplifies the received signal with a precoding matrix and broadcasts it to both sources.Each source subtracts the back-propagated self-interference to obtain its equivalent received signal.Source: A. Two-Phase Two-Way Relay Scheme · p. 6
• The two-phase design maximizes the bidirectional secrecy sum rate by jointly optimizing relay and source beamformers under source and relay power constraints, accounting for the relay's eavesdropping rate.The relay's achievable eavesdropping rate is modeled as the maximum sum rate of a two-user MIMO multiple-access channel.Source: A. Two-Phase Two-Way Relay Scheme · pp. 6–7

B. Three-Phase Two-Way Relay Scheme

• In phase one, source A transmits, and the relay and source B receive signals.Source: B. Three-Phase Two-Way Relay Scheme · p. 7
• In phase two, source B transmits, and the relay and source A receive signals.Source: B. Three-Phase Two-Way Relay Scheme · p. 8
• In phase three, the relay amplifies the received signals with F_A and F_B, respectively, and broadcasts the resulting signal.Source: B. Three-Phase Two-Way Relay Scheme · p. 8
• After self-interference subtraction, the source signals are combined to obtain the information rate from B to A.Source: B. Three-Phase Two-Way Relay Scheme · p. 8

A + FBFH

• The formulation sequence gives the A-to-B information rate, relay leakage sum rate, and resulting secrecy sum rate.It then formulates secrecy sum-rate maximization for the three-phase scheme.Source: A + FBFH · pp. 8–9

+ FAFH

• The relay power constraint is part of the formulation, and source A is assumed to coordinate the joint beamformer design and share the designed beamformers with B and the relay.The two sources and relay then use their assigned beamformers to process transmitted signals.Source: + FAFH · p. 9

III. SECURE BEAMFOMRING DESIGNS

• The secure beamforming-design section presents algorithms after formulating the two-phase and three-phase secrecy sum-rate maximization problems.The two-phase algorithm initializes A, q_A, and q_B, then updates them iteratively until the rate stops increasing.Source: §III · p. 9, A. Secure Beamforming in Two-Phase… · p. 10

A. Secure Beamforming in Two-Phase Two-Way Relay Scheme

• The optimal two-phase relay matrix F has a reduced-parameter structure, cutting its unknowns from N_R^2 to 2 min{N_R, N_A+N_B}.This reduction lowers the computational complexity of joint source and relay beamforming design.Source: A. Secure Beamforming in Two-Phase… · p. 10
• The source and relay beamformers are jointly optimized by alternating over q_A, q_B, and the unknown matrix A because the reduced problem remains nonconvex.With q_A and q_B fixed, A is searched by a gradient method; source vectors are optimized with F and the other source vector fixed.Source: A. Secure Beamforming in Two-Phase… · p. 10
• The iterative two-phase algorithm converges because each iteration does not decrease the secrecy sum rate and that rate is finite.The iterations stop when the secrecy sum rate no longer increases.Source: A. Secure Beamforming in Two-Phase… · pp. 10–11

B. Secure Beamforming in Three-Phase Two-Way Relay Scheme

• The optimal three-phase relay matrices have a three-part structure: matching received signals, processing information-bearing signals, and beamforming toward intended receivers.This structure reduces the design of two relay matrices to two vectors, with 2 min{NR, NA+NB} unknowns.Source: B. Secure Beamforming in Three-Phase… · p. 11
• For three-phase relaying, the relay beamforming vectors and source beamformers are alternatively optimized until secrecy sum rate stops increasing.Source: B. Secure Beamforming in Three-Phase… · p. 11

IV. ASYMPTOTIC ANALYSIS

• The nonconvex joint beamforming problems do not guarantee global optimality under the proposed iterative algorithms, motivating analysis of zero- and infinite-power limits.Source: B. Secure Beamforming in Three-Phase… · pp. 11–12

A. The Case of High Relay and Source Powers

• As relay and source powers tend to infinity, the two-phase scheme’s secrecy sum rate is unbounded when NA+NB>NR, but bounded when NA+NB≤NR.When the source column spaces intersect, source beamformers can align the received signals at the relay; the paper attributes the reduced antenna requirement to signal alignment.Source: A. The Case of High Relay and Source… · pp. 15, 17
• When the relay has more antennas than both sources, the three-phase and direct-transmission secrecy sum rates each reach a floor.Source: A. The Case of High Relay and Source… · p. 16
• With PR→∞, NA≤NR, NB≤NR, and NA+NB>NR, two-phase secrecy sum rate keeps increasing with source powers while three-phase and direct-transmission rates approach constants.Source: A. The Case of High Relay and Source… · p. 17
• When each source has more antennas than the relay, direct transmission performs better than three-phase relaying at high SNR.Source: A. The Case of High Relay and Source… · p. 17

B. The Case of High Relay Power and Low Source Powers

• Proposition 4 characterizes the optimal two-phase source beamformers and maximum secrecy sum rate as relay power tends to infinity and both source powers tend to zero.The supplied passage omits the beamformer and rate expressions, so their specific forms cannot be stated here.Source: B. The Case of High Relay Power and… · p. 18
• Under the same power limits, Proposition 5 characterizes the three-phase scheme’s maximum secrecy sum rate.The supplied passage omits the expression that specifies its asymptotic behavior.Source: B. The Case of High Relay Power and… · p. 18
• Unlike the two-phase scheme, the direct-transmission and three-phase secrecy sum rates are closely related to the term TH.The excerpt does not provide the definition of TH or the corresponding rate expressions.Source: B. The Case of High Relay Power and… · p. 18
• In this regime, the three-phase scheme is better than direct transmission, and the two-phase scheme’s maximum secrecy sum rate approaches zero faster than the other schemes.The paper concludes that the three-phase scheme is best for secure transmission when both source powers are extremely low.Source: B. The Case of High Relay Power and… · p. 19

C. The Case of Low Relay Power

• As PR → 0, the two-phase scheme’s maximum secrecy sum rate goes to zero because its relay-link information rate vanishes while leakage to the untrusted relay does not.The leakage rate is not related to relay power and therefore does not approach zero.Source: C. The Case of Low Relay Power · p. 19
• Direct transmission is best when relay power is low.The passage says a similar conclusion holds for relay systems without a secrecy constraint.Source: C. The Case of Low Relay Power · p. 19
• A four-phase one-way relay scheme can be best when PR → ∞, PA → 0, and PB → 0; otherwise, it is suboptimal or its comparison depends on the channel realization.Source: C. The Case of Low Relay Power · pp. 19–20

B

• Simulations for the cases discussed in Sections IV and V assume channel reciprocity (HA = GT) and use channel-coefficient matrices whose entries are generated from CN(0, 1) to show asymptotical performance.If a needed channel matrix is smaller than the dimension of the generated matrices, the simulation selects the corresponding matrix’s upper-left portion.Source: §V · pp. 20–21
• Signal-alignment-based initialization makes Algorithm 1 converge faster than random initialization, and its asymptotically optimal beamformers initialize the subsequent simulations.Because Algorithms 1 and 2 are not guaranteed to find an optimal solution and may converge far from it, the authors also suggest trying multiple random initializations and selecting the best-performing one.Source: §V · p. 21
• Algorithms 1 and 2 are not guaranteed to find the optimum, and their convergence points may be far from it.The proposed coping method is to run multiple random initializations and choose the one with the best performance.Source: §V · p. 21

A. High Relay Power and High Source Powers

• In Case 1, the two-phase scheme’s maximum secrecy sum rate grows without bound as source powers increase, while the other schemes reach floors; signal alignment is important for security.The direct-transmission and three-phase upper bounds are about 1.82bps/Hz and 1.48bps/Hz, respectively.Source: A. High Relay Power and High Source… · p. 21
• In Case 2, all schemes’ maximum secrecy sum rates approach infinity as powers increase, but direct transmission is best and outperforms the two-phase scheme.The direct-transmission and three-phase schemes have degrees of freedom one and 2, respectively.Source: A. High Relay Power and High Source… · pp. 21–22
• In Case 3, all schemes have upper-bounded secrecy sum rates, and the two-phase scheme is best, including when signal alignment cannot be achieved.Approaching its upper bound requires all three nodes’ powers to increase, with relay power much larger than source powers; at PR = 50dB, its curve approaches the bound more closely than at PR = 40dB.Source: A. High Relay Power and High Source… · p. 22
• Increasing the number of relay antennas reduces performance, contrary to relay systems without secrecy constraints, where more relay antennas improve performance.Source: A. High Relay Power and High Source… · p. 22

B. High Relay Power and Low Source Powers

• With relay power at 40dB and low source powers, the two-phase scheme is much worse than the other schemes, while three-phase outperforms direct transmission.The reported asymptotic results are quite accurate at low source powers.Source: B. High Relay Power and Low Source… · p. 22

C. Low Relay Power

• At relay power of -20dB, the two-phase scheme’s maximum secrecy sum rate is near zero, and direct transmission outperforms the three-phase scheme.The passage attributes this to a weak two-phase relay path and strong direct links in the other schemes at high source powers.Source: C. Low Relay Power · p. 22

D. General Relay SNR and Fading Channels

• With source powers fixed at 15dB and NA = NB = 2, NR = 3, the two-phase average rate rises from zero to 2.2 bps/Hz as relay power increases and outperforms the other schemes at high relay power.The passage attributes the high-power advantage to achievable signal alignment when NA + NB > NR.Source: D. General Relay SNR and Fading Channels · p. 23
• With three antennas at each source and powers of 25dB at the relay and 15dB at the sources, the two-phase average rate peaks at NR = 4; direct and three-phase rates decrease as NR increases.The passage explains that small NR limits relay assistance, while large NR strengthens the relay’s ability to decode received signals.Source: D. General Relay SNR and Fading Channels · p. 23

VI. CONCLUSION

• For both two-phase and three-phase relaying, the paper proposes iterative algorithms to jointly design secure source and relay beamformers, and reports that simulations validate the asymptotic analysis.The analysis covers low- and high-power regimes for sources and relay.Source: §VI · p. 23
• At vanishing relay power, direct transmission is preferred; at vanishing source powers, three-phase relaying performs best; and at high powers, two-phase relaying performs best with signal alignment.The paper also states that signal alignment lowers the source-antenna requirement for security.Source: §VI · pp. 23–24

APPENDIX A SECURE BEAMFORMING OF TWO-WAY DIRECT TRANSMISSION SCHEME

• In direct transmission, A transmits while B and R listen in the first time slot, then B transmits while A and R listen in the second.Source: Appendix A · p. 24
• The direct-transmission design maximizes secrecy sum rate R_DT subject to the source power constraints.Source: Appendix A · p. 24
• The maximum secrecy sum rate includes a factor of 1/2 because the direct scheme uses two orthogonal phases.Source: Appendix A · p. 24

APPENDIX B SEARCH A USING GRADIENT METHOD

• For fixed q_A and q_B, the search for A uses a logarithmic barrier with parameter μ > 0 and gradient descent.Source: Appendix B · p. 25

APPENDIX C SEARCH OPTIMAL qB GIVEN F AND qA IN TWO-PHASE TWO-WAY RELAY SCHEME

• To find optimal q_B, the method relaxes the homogenized problem into fractional SDP, applies Charnes–Cooper transformation, then uses rank-one matrix decomposition.Source: Appendix C · p. 25

APPENDIX D

• For N_R > N_A + N_B, the information rates in both directions are independent of c_i and D_i and attain upper bounds when B_i = 0.Source: Appendix D · p. 26

A + FBFH

• The relay-power argument sets B_i = 0, c_i = 0, and D_i = 0, and a related proof also establishes optimal B_i = 0.Source: A + FBFH · p. 27
• The beamforming derivation identifies V as coming from (23) and U_i as H_iq_i.Source: A + FBFH · p. 27

APPENDIX E

• The Appendix E fragment states that an expression involving BHAqA∥2 yields Proposition 4.Source: Appendix E · p. 28

APPENDIX F

• Substituting the optimal relay beamforming structure (24) into (16) yields its third term.Source: Appendix F · p. 28
• The third term in (16) has a lower bound obtained by setting a_1 = a_2 = ¯a.Source: A + FBFH · p. 29

A + FBFH

• The Proposition 2 proof substitutes upper and lower bounds into (18) and follows the procedure of [24, Lemma 7].Source: A + FBFH · p. 29
• The Proposition 2 proof assumes channel-matrix entries are generated from a continuous distribution.Source: A + FBFH · p. 29

• When relay power PR → 0, the three-phase scheme has only direct links between the source nodes.Source: Appendix G · p. 29