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Physical Layer Security for Two-Way Untrusted Relaying with Friendly Jammers

Rongqing Zhang, Lingyang song Zhu Han, Bingli Jiao

arXiv:1202.3824v1cs.GTcs.NI

TL;DR

The paper addresses secure two-way communication when an essential relay is also an eavesdropper. It analyzes friendly-jammer assistance through a Stackelberg power-control game and reports non-zero secrecy without jammers, with improvement from suitable jamming power.

  • Problem

    Physical layer security for two-way relay networks with an untrusted but necessary relay has not been well investigated.

  • Method

    The paper formulates a Stackelberg buyer/seller game between the sources and friendly jammers for secrecy-oriented power control.

  • Results

    A non-zero secrecy rate is available without jammers, and friendly jammers can improve the secrecy rate with proper jamming power.

  • Takeaways & Limitations

    The proposed game provides a distributed solution for coordinating source payments and friendly-jammer services to obtain a Stackelberg equilibrium.

Abstract

from arXiv · show

In this paper, we consider a two-way relay network where two sources can communicate only through an untrusted intermediate relay, and investigate the physical layer security issue of this two-way relay scenario. Specifically, we treat the intermediate relay as an eavesdropper from which the information transmitted by the sources needs to be kept secret, despite the fact that its cooperation in relaying this information is essential. We indicate that a non-zero secrecy rate is indeed achievable in this two-way relay network even without external friendly jammers. As for the system with friendly jammers, after further analysis, we can obtain that the secrecy rate of the sources can be effectively improved by utilizing proper jamming power from the friendly jammers. Then, we formulate a Stackelberg game model between the sources and the friendly jammers as a power control scheme to achieve the optimized secrecy rate of the sources, in which the sources are treated as the sole buyer and the friendly jammers are the sellers. In addition, the optimal solutions of the jamming power and the asking prices are given and a distributed updating algorithm to obtain the Stakelberg equilibrium is provided for the proposed game. Finally, the simulations results verify the properties and the efficiency of the proposed Stackelberg game based scheme.

I. INTRODUCTION

The paper studies physical layer security in two-way relay networks where an untrusted relay is necessary for communication. It establishes secure transmission without external jammers and develops a Stackelberg power-control scheme using friendly jammers.

  • Physical layer security for the relay in two-way relay networks has not been well investigated.
  • The two sources communicate only through an untrusted relay that acts as both an essential forwarding node and a malicious eavesdropper.
  • A non-zero secrecy rate is achievable even without friendly jammers, with optimal power allocation derived for the relay and sources.
  • Proper jamming power from friendly jammers can improve the secrecy rate of the two-way relay system.
  • The proposed Stackelberg game models the sources as the buyer and friendly jammers as sellers in a distributed power-control scheme.The paper gives optimal jamming powers and asking prices, and provides a distributed updating algorithm for the Stackelberg equilibrium.
  • Simulations verify the properties and efficiency of the proposed Stackelberg game-based scheme.

II. SYSTEM MODEL

The system is a half-duplex two-way relay network with two sources, one untrusted amplify-and-forward relay, and friendly jammers. The sources have no direct link and are assumed to know the jamming signals.

  • The network contains two source nodes, one untrusted relay, and N friendly jammer nodes, with no direct link between the sources.
  • The system operates half-duplex with single omni-directional antennas, and channel fading coefficients remain constant over each frame.
  • The sources and friendly jammers transmit during phase 1, while the relay amplifies and broadcasts a combined signal during phase 2.
  • The relay is necessary for data transmission and is modeled as both a cooperative forwarding node and a malicious eavesdropper.
  • Sources are assumed to have perfect knowledge of the friendly jammers’ signals, with one-time signaling information causing trivial bandwidth cost.
  • The secrecy rate is defined from the source-to-source capacities and is achievable for two-hop communication with an untrusted relay.

III. SECRECY RATE OF TWO-WAY RELAY CHANNEL WITHOUT JAMMERS

The paper first analyzes the two-way relay channel without friendly jammers and shows that positive secrecy is possible. It also derives power allocation that maximizes the secrecy rate.

  • A positive secrecy rate exists in the two-way relay channel even without friendly jammers.
  • The sources and relay have an optimal power allocation that maximizes the secrecy rate without jammers.
  • The jammer-free case provides the baseline for comparing secrecy-rate gains from friendly jammers.

A. Existence of Non-zero Secrecy Rate

The paper explains when non-zero secrecy is possible and formulates power optimization for the source and relay. Under suitable channel and power conditions, secure transmission can occur.

  • A. Existence of Non-zero Secrecy Rate: When the relay’s eavesdropper channels are degraded, the equivalent main channel can be better, enabling non-zero secure rates in both directions.
  • A. Existence of Non-zero Secrecy Rate: For transmission from S1 to S2, S2’s own signal acts as a jamming signal at the relay but can be removed at S2.
  • A. Existence of Non-zero Secrecy Rate: The existence of positive secrecy depends on power vectors (p_r, p_1, p_2) satisfying the derived channel and power conditions.
  • B. Maximizing the Secrecy Rate: The optimization seeks a power vector (p_r, p_1, p_2) that maximizes the secrecy rate subject to individual secrecy-rate and power constraints.
  • B. Maximizing the Secrecy Rate: The objective can be transformed into maximizing ˜F(p_r, p_1, p_2), which has the same monotonic behavior as the secrecy rate under the stated conditions.
  • B. Maximizing the Secrecy Rate: The optimal source powers are obtained for the channel cases gS1,R > gS2,R, gS1,R < gS2,R, and gS1,R = gS2,R using KKT conditions.

1. Otherwise, we have p1

The proposed optimal power solutions are evaluated across different cases, and simulation results agree well with the analytical solutions.

  • Simulations in different cases agree well with the derived optimal power solutions.The verification is shown in Fig. 2 and Fig. 3.

IV. PHYSICAL LAYER SECURITY WITH FRIENDLY JAMMERS

Friendly jamming can improve the sources’ secrecy rate, motivating a Stackelberg game that controls jammer power while accounting for payments and synchronization assumptions.

  • Proper friendly-jammer power can effectively improve the sources’ secrecy rate.
  • The sources and friendly jammers are modeled as a buyer and sellers in a Stackelberg power-control game.The sources act as leader, while friendly jammers act as followers.
  • The game investigates optimal jamming powers and asking prices and provides a distributed updating algorithm.A centralized scheme is also proposed for performance comparison.
  • When jammer power increases, the secrecy-rate behavior can first improve in some regions, but relevant rates approach zero after further increases.
  • The analysis assumes perfect synchronization among the sources and friendly jammers, although synchronization is not the paper’s key issue.The paper notes that synchronization can be addressed using methods similar to prior distributed-network schemes.

B. Source Side Game

The source-side game treats the two sources as buyers seeking secrecy-rate gains from friendly-jammer power while minimizing payment, and derives power-selection strategies under general and special conditions.

  • The two sources jointly act as buyers that choose jammer power to maximize their utility and secrecy rate.The utility accounts for the economic gain from confidential data transmission and the cost paid for jamming power.
  • The optimal jamming-power solution depends on the jammer’s price, other jammers’ powers, and system parameters.
  • Candidate roots of the high-order equation must be real, satisfy power constraints, and yield higher source utility than other real roots.If no real roots exist, the boundary strategy is selected by comparing utility values.
  • Under high interference and high signal-to-noise-ratio assumptions, the source utility is approximated to obtain a simpler optimal-power expression.The approximation uses log(1 + x) ≈ x when x is sufficiently small.
  • If D1 < 0, source utility decreases with jammer power and the jammer does not participate; in the special case, optimal power is convex in price.
  • Simulations indicate that sources prefer one sufficiently effective jammer offering a proper price, supporting the severe-interference special case.The special case also yields a monotonic relationship between optimal power consumption and price under the stated analysis.

C. Friendly Jammer Side Game

The friendly-jammer side models each jammer as a seller that selects its price to maximize utility from payments after accounting for transmission costs and the power purchased by the sources.

  • The power bought from each jammer depends on the full vector of prices because source demand responds jointly to all asking prices.
  • Each friendly jammer chooses an asking price to maximize utility from payments while covering its transmitting cost and obtaining profit.
  • The jammer-side optimization derives prices by differentiating utility and solving the resulting condition, with positive optimal power required for participation.

D. Stackelberg Equilibrium of the Proposed Game

The proposed game’s Stackelberg equilibrium coincides with optimal jamming powers and asking prices. A unique equilibrium is proved in a special high-interference case, while general cases rely on simulation evidence.

  • The equilibrium pair consists of optimal jamming powers for the sources and asking prices for the friendly jammers.
  • In the special high-interference case, an efficient jammer’s optimal purchased power decreases convexly with its asking price.
  • Sources prefer buying jamming power only from the efficient jammer in that special case.
  • A unique Stackelberg equilibrium exists in the special case and equals the optimal jamming-power and asking-price solutions.
  • For general cases, theoretical proof is intractable because the optimal solutions have extremely complex closed-form expressions.
  • Simulations show that the proposed game converges to a unique equilibrium optimizing source and jammer utilities.

E. Distributed Updating Algorithm

The paper proposes a distributed price-update algorithm for the Stackelberg game. Simulations indicate convergence and show that its secrecy-rate solution approaches the centralized solution under sufficiently large utility gain.

  • The distributed algorithm updates friendly jammers’ prices using information obtained from the sources.
  • Each jammer’s price converges from any feasible initial price vector to a fixed point identified as the game’s Stackelberg equilibrium.
  • Higher jammer prices reduce the amount of power purchased by the sources.
  • Monotonicity and scalability of the update analysis are established only for the high-interference case.
  • For general cases, analysis is intractable, but simulations show that the distributed scheme converges and outperforms the no-jammer case.
  • The distributed and centralized secrecy-rate solutions become asymptotically identical when the unit-rate gain a is sufficiently large.
  • The centralized solution requires all channel information in each time slot, whereas the distributed algorithm is more efficient in practical applications.

V. SIMULATION RESULTS

Simulations examine optimal power allocation, jammer pricing, multiple-jammer selection, and distributed-versus-centralized performance. They show that suitable jammer placement and pricing improve secrecy, while additional jammers help only when no single jammer is sufficiently effective.

  • (0.22pmax, pmax) is the optimal source power vector in the no-jammer case.
  • The secrecy rate first increases and then decreases with jamming power, yielding a location-dependent optimum.
  • A friendly jammer close to the malicious relay is more effective at improving secrecy rate.
  • Purchased jamming power decreases as the asking price rises, and sources eventually stop buying after a price threshold.
  • With sufficiently effective jammers, sources select one jammer, and the optimal secrecy rate does not increase with jammer count.
  • When both jammer prices exceed the sources’ payment ability, secrecy rate falls to the no-jammer lower bound.
  • Without a sufficiently effective jammer, increasing the number of jammers improves the optimal secrecy rate up to its maximal value.
  • Distributed and centralized optimal secrecy rates become asymptotically the same as gain factor a increases.

VI. CONCLUSIONS

The paper studies physical-layer security in two-way relay communications with and without friendly jammers, then formulates a Stackelberg game for distributed power control. It finds non-zero secrecy without jammers, improved secrecy with suitable jamming, and similar distributed and centralized game performance when the gain factor is sufficiently large.

  • VI. CONCLUSIONS: The study addresses two-way relay communications with friendly jammers and develops a distributed Stackelberg game between sources and jammers.The game targets the optimal secrecy rate through distributed interaction between the two parties.
  • VI. CONCLUSIONS: An optimal power allocation vector is found for the two-way relay system without jammers before analyzing friendly-jammer assistance.This establishes the jammer-free case as the baseline for the subsequent secrecy-rate analysis.
  • VI. CONCLUSIONS: A non-zero secrecy rate is available without friendly jammers, and friendly jammers can improve the secrecy rate with appropriate jamming power.The conclusions identify both jammer-free secrecy and a positive gain from suitable friendly-jammer power.
  • VI. CONCLUSIONS: There is an optimal jamming-power allocation and a tradeoff in the price set by each jammer.If a jammer sets too high a price, the sources buy from other jammers instead.
  • VI. CONCLUSIONS: The distributed algorithm and centralized scheme have similar performances, especially when the gain factor is sufficiently large.Simulation results support the efficiency of the proposed distributed game solution.
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