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Optimal Cooperative Relaying Schemes for Improving Wireless Physical Layer Security

Jiangyuan Li, Athina P. Petropulu, Steven Weber

arXiv:1001.1389v1cs.IT

TL;DR

The paper addresses physical-layer security in cooperative wireless networks with one or more eavesdroppers, focusing on optimal design rather than prior suboptimal formulations. It optimizes relay weights and source power for DF and CJ under secrecy-rate and total-power objectives, obtaining closed forms or convergent search algorithms. Numerical results illustrate the proposed schemes and the advantages of cooperation over direct transmission.

  • Problem

    Existing cooperative-security designs used additional constraints that produced suboptimal solutions, motivating optimal weight and power allocation for multiple relays and eavesdroppers.

  • Method

    The paper optimizes predefined DF and CJ cooperative schemes for secrecy-rate maximization under a power constraint and power minimization under a secrecy-rate constraint.

  • Results

    The work provides explicit relay-weight and source-power constructions, using closed-form solutions or algorithms to search for solutions.

  • Takeaways & Limitations

    Numerical results illustrate the proposed solutions and show that cooperation can significantly improve system performance compared with direct transmission.

Abstract

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We consider a cooperative wireless network in the presence of one of more eavesdroppers, and exploit node cooperation for achieving physical (PHY) layer based security. Two different cooperation schemes are considered. In the first scheme, cooperating nodes retransmit a weighted version of the source signal in a decode-and-forward (DF) fashion. In the second scheme, while the source is transmitting, cooperating nodes transmit weighted noise to confound the eavesdropper (cooperative jamming (CJ)). We investigate two objectives, i.e., maximization of achievable secrecy rate subject to a total power constraint, and minimization of total power transmit power under a secrecy rate constraint. For the first design objective with a single eavesdropper we obtain expressions for optimal weights under the DF protocol in closed form, and give an algorithm that converges to the optimal solution for the CJ scheme; while for multiple eavesdroppers we give an algorithm for the solution using the DF protocol that is guaranteed to converge to the optimal solution for two eavesdroppers. For the second design objective, existing works introduced additional constraints in order to reduce the degree of difficulty, thus resulting in suboptimal solutions. In this work, either a closed form solution is obtained, or algorithms to search for the solution are proposed. Numerical results are presented to illustrate the proposed schemes and demonstrate the advantages of cooperation as compared to direct transmission.

I. INTRODUCTION

The paper studies cooperative relaying for wireless physical-layer security, optimizing relay weights and power allocation under secrecy-rate or total-power constraints. It targets optimal DF and CJ designs in networks with single or multiple eavesdroppers, extending prior suboptimal formulations.

  • Motivation: Wireless secrecy depends on both source–destination and source–eavesdropper channel conditions, motivating cooperation to improve achievable secrecy rates.Node cooperation via relays is presented as a low-cost approach to exploit or mitigate these channel effects.
  • Cooperative schemes: Cooperating relays either retransmit weighted decoded signals using DF or transmit weighted noise while the source transmits using CJ.Earlier work also considered AF, in which relays retransmit weighted versions of the signals they hear.
  • Prior limitations: Prior designs selected weights to maximize secrecy rate under total power constraints or minimize total power under a secrecy-rate constraint, but yielded suboptimal solutions.The difficulty of the associated optimization problems led to additional criteria and constraints, including signal or jamming nulling.
  • Contributions: This paper focuses on obtaining optimal solutions for DF and CJ, using closed forms when possible and convergent search algorithms otherwise.The AF solution is identified as more difficult and deferred to future work.
  • Contributions: For secrecy-rate maximization, the paper derives optimal DF weights for a single eavesdropper and treats multiple eavesdroppers with DF, while also studying CJ.The paper’s stated development covers single-eavesdropper DF and CJ and multiple-eavesdropper DF cases.
  • Evaluation: For total-power minimization under a secrecy-rate constraint, the paper studies both DF and CJ and presents numerical results comparing the proposed solutions with direct transmission.The introduction states that the numerical results illustrate the proposed solutions and that cooperation can improve performance relative to direct transmission.

B. Notation

The paper models cooperative wireless communication from a source to a destination with relays and multiple eavesdroppers, and formulates secrecy-rate and power-allocation objectives under DF and CJ protocols.

  • Relays use decode-and-forward or cooperative jamming protocols, with weights w and source power Ps selected jointly.
  • The designs either maximize achievable secrecy rate under total power P0 or minimize total transmit power under a secrecy-rate constraint.
  • The system contains a source S, destination D, N relays, and J passive eavesdroppers.
  • The model assumes single-antenna, half-duplex nodes, flat fading, Gaussian noise, and global CSI including eavesdropper channels.
  • The paper assumes public source encoding, decoding methods, and cooperative protocol information, and focuses on memoryless relay channels using common source and relay codewords.
  • For DF and CJ, achievable secrecy rates are obtained from destination and eavesdropper rates as functions of relay weights, source power, noise, and channel gains.

III. SECRECY RATE MAXIMIZATION UNDER POWER CONSTRAINT

This section addresses secrecy-rate maximization under a total power constraint.

  • Problems 1 and 2 maximize achievable secrecy rate under a total power constraint for DF and CJ protocols.
  • The optimization concerns relay weights and source power within the stated power-constrained design objective.
  • The section precedes separate treatments of single and multiple eavesdropper cases.

A. One Eavesdropper (J = 1)

For one eavesdropper, the paper derives closed-form DF solutions and an alternating optimization algorithm for CJ, while identifying conditions for positive secrecy rate.

  • 1) DF-based protocol: Theorem 1 gives the closed-form solution for the single-eavesdropper DF optimization problem.The solution is characterized through the theorem and its associated function J(Ps).
  • 1) DF-based protocol: Depending on channel-related values, the DF optimum either uses minimum source power with relay weights combining normalized channel vectors or uses all power with no relays.
  • 2) CJ-based protocol: Positive secrecy rate requires α2 > α4 and a total-power threshold involving P0, α2, α4, α3, and G(z0).
  • 2) CJ-based protocol: If the source-destination channel is weaker than the source-eavesdropper channel, relays and power above a threshold are required for positive secrecy rate.
  • 2) CJ-based protocol: For CJ, alternating optimization fixes z and Ps in turn, and the proposed procedure converges to the optimal solution.For fixed variables, candidate optima include endpoints and zero-derivative points; Newton's method can search for the relevant root with quadratic convergence.
  • 2) CJ-based protocol: The CJ subproblem for fixed Ps has a unique root z′, making Newton's method effective for locating the optimizer.

B. Multiple Eavesdroppers

For multiple eavesdroppers, the paper treats DF optimization with alternating updates, semidefinite relaxation, and special guarantees for two eavesdroppers; CJ remains more difficult.

  • The multiple-eavesdropper analysis restricts attention to the DF protocol.
  • Adding the nulling constraint w†G = 0 yields a suboptimal solution and requires more relays than eavesdroppers for sufficient degrees of freedom.
  • The proposed DF algorithm alternates optimization of source power Ps and relay-related variable x.
  • The CJ protocol with multiple eavesdroppers is more difficult, and the alternating procedure is not guaranteed to converge optimally for J > 2.
  • For J = 2, the semidefinite relaxation is always tight and the optimal solution can be constructed in polynomial time.
  • For J > 2, non-rank-one relaxed solutions are handled with Gaussian randomization to obtain an approximate solution.

IV. TRANSMIT POWER MINIMIZATION UNDER SECRECY RATE CONSTRAINT

This section addresses Problems 3 and 4 within transmit power minimization under a secrecy rate constraint.

  • Problems 3 and 4 are addressed in this section.
  • The section concerns transmit power minimization under a secrecy rate constraint.
  • The section presents the treatment of two numbered optimization problems.

A. DF-based protocol

The DF-based analysis derives closed-form optimal relay weights and develops an alternating optimization procedure for the transmit-power problem.

  • DF-based protocol: The closed-form solution makes the optimal weight vector a linear combination of h and g.
  • DF-based protocol: The relevant matrix has only two nonzero eigenvalues, one positive and one negative, with simple expressions for them.
  • DF-based protocol: For fixed z, γ is obtained explicitly as a function of z, reducing the original problem to single-variable optimization over z ∈ [0, 1].
  • DF-based protocol: For transmit-power minimization, the method alternates between optimizing γ with fixed z and optimizing z with fixed γ.
  • DF-based protocol: The alternating procedure converges to the optimal γ◦ and z◦.
  • DF-based protocol: Newton’s method is effective for finding the unique root z′ and has quadratic convergence.

V. NUMERICAL SIMULATIONS

The simulations compare direct transmission with DF and CJ under varying node positions and eavesdropper counts. Cooperation maintains secrecy-rate and transmit-power advantages in the reported configurations.

  • The simulations use LOS channels, path-loss exponent c = 3.5, and 500 independent Monte-Carlo trials.
  • When the destination moves beyond the eavesdropper, direct transmission cannot sustain positive secrecy rate.
  • DF and CJ maintain positive secrecy rate even when the destination is farther from the source than the eavesdropper.
  • DF yields the higher secrecy rate, while optimal and suboptimal CJ produce the same result in the reported comparison.
  • As eavesdropper count increases, the suboptimal DF solution becomes inferior to the optimal solution.
  • For source-eavesdropper distances below 65 m, direct transmission cannot meet the secrecy-rate constraint at any transmit-power level.
  • For distances above 85 m, direct transmission and CJ require equivalent minimum transmit power, whereas DF requires significantly less.

VI. CONCLUSION

The paper constructs optimal relay weights and source powers for secrecy-rate maximization and total-power minimization under DF and CJ protocols. Numerical results show that cooperation can significantly improve performance over direct transmission.

  • The paper gives explicit constructions for optimal relay weights and source transmission power under two secrecy-related objectives.
  • The designs cover DF and CJ protocols with single or multiple eavesdroppers.
  • Numerical results compare optimal secrecy rates with suboptimal solutions from -.
  • The results illustrate that cooperation can significantly improve system performance compared with direct transmission.

APPENDIX A

The appendix derives structural properties and closed-form components for optimization problems involving relay weights, source power, and secrecy-rate objectives.

  • The relevant eigenvector is a linear combination of r and s, with coefficients determined by normalization and the associated eigenvalue equations.The eigenvalue equation yields the representation a = π1r + π2s; the resulting characteristic equation has one positive and one negative root.
  • The solution z◦ of problem (14) is a linear combination of d1 and d2, and its phase can be chosen so that c2 ≥ 0.This reduction supports solving the constraints in terms of c1 and c2, including the optimal phase and magnitude of c1.
  • For fixed source power Ps ≥ P min, the optimal weight vector can be obtained in closed form as z(Ps).The appendix then optimizes over Ps using the associated problem and the quasi-linearity of the objective.
  • The optimization can be reduced to a single variable z, whose optimum is characterized by a zero derivative because the derivative changes sign at the interval endpoints.Specifically, M′(0) > 0 and M′(1) < 0, so an interior optimum satisfies M′(z) = 0.
  • Because the relevant objective is quasi-linear in Ps, the optimum occurs at one of the two endpoint powers, including P min.The appendix evaluates endpoint candidates and then obtains the corresponding optimal w according to Lemma 2.

APPENDIX D

The appendix establishes feasibility and optimization reductions for secrecy-rate problems by analyzing a concave auxiliary function and reducing selected cases to one-dimensional searches.

  • The auxiliary function K(z) is positive at zero, strictly concave, and has a unique interior maximizer z0 determined by K′(z0) = 0.These properties follow from K(0) = α3(1 −η2) > 0, K′′(z) < 0, and the endpoint derivative behavior.
  • When α2 < α4, the feasibility condition holds for some Ps and z because K(0) > 0.When α2 > α4, feasibility instead requires P0K(z0) > α2 − α4.
  • Fixing z = 0 reduces the problem to a single-variable optimization over Ps ∈ [0, P0].The maximum is attained at an endpoint or at a point where the derivative is zero.
  • Adding the constraint w†G = 0 allows w to be obtained from w†w = P0 − Ps and w†G = 0, after which the problem is further reduced.The resulting reduced objective uses the maximizer z = E†h/∥E†h∥ and a quasi-linear function f2(Ps).
  • The maximum of the reduced f2(Ps) objective occurs at one of the two endpoint powers, including P min.This follows from the quasi-linearity of f2(Ps).

APPENDIX H

The appendix proves equivalent formulations and characterizes optimal relay weights through eigen-directions, scalar optimization, and case-based solutions under power and secrecy constraints.

  • Introducing a slack variable y transforms the reciprocal minimum objective into an equivalent normalized problem, with u◦ recovered by normalizing the solution v◦.The transformation uses u†u = 1 and ∥v∥2 = 1/y to establish equivalence between the formulations.
  • The optimal relay weight vector w◦ lies in the span of the eigenvectors u1 and u2 of the relevant matrix.It has the form d1u1 + d2u2, and the resulting scalar problem depends on the eigenvalues and the constraint threshold.
  • One case yields zero coefficients when ζ ≥ P min and λ2 ≥ −1, while another case determines the coefficients from the active constraint.The appendix separates the solution according to the eigenvalue condition and the relation between ζ and P min.
  • Fixing z = 0 reduces problem (47) to a single-variable optimization over γ, with candidates given by zero and stationary points.The maximum is obtained by evaluating these candidate points.
  • The appendix uses concavity of F and an inequality at an interior point to derive the desired result for the reduced optimization.The argument invokes F′′(z′) < 0 together with γα3F(z′) + α4 > γα1z′ + α2.
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