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Physical Layer Security in Millimeter Wave Cellular Networks

Chao Wang, Hui-Ming Wang

arXiv:1604.07506v1cs.IT

TL;DR

MmWave cellular networks offer substantial capacity potential, but their network-wide physical-layer secrecy had not been established. This paper uses stochastic geometry and mmWave blockage/path-loss models to analyze secure connectivity and perfect-link density under noise-limited and AN-assisted conditions. The results characterize colluding and non-colluding eavesdroppers and show that antenna pattern, eavesdropper intensity, and AN allocation shape secrecy performance.

  • Problem

    The network-wide secrecy performance of mmWave cellular communication remained unknown, although its channel characteristics differ from conventional microwave networks.

  • Method

    The paper uses stochastic geometry and mmWave channel models to analyze noise-limited and AN-assisted cellular secrecy with colluding and non-colluding eavesdroppers.

  • Results

    The analysis characterizes secure connectivity, perfect communication-link density, and received-SINR distributions across noise-limited and interference-limited settings.

  • Takeaways & Limitations

    Antenna pattern and eavesdropper intensity are important secrecy parameters, while optimal AN allocation depends on both.

Abstract

from arXiv · show

Recent researches show that millimeter wave (mmWave) communications can offer orders of magnitude increases in the cellular capacity. However, the secrecy performance of a mmWave cellular network has not been investigated so far. Leveraging the new path-loss and blockage models for mmWave channels, which are significantly different from the conventional microwave channel, this paper comprehensively studies the network-wide physical layer security performance of the downlink transmission in a mmWave cellular network under a stochastic geometry framework. We first study the secure connectivity probability and the average number of perfect communication links per unit area in a noise-limited mmWave network for both non-colluding and colluding eavesdroppers scenarios, respectively. Then, we evaluate the effect of the artificial noise (AN) on the secrecy performance, and derive the analysis result of average number of perfect communication links per unit area in an interference-limited mmWave network. Numerical results are demonstrated to show the network-wide secrecy performance, and provide interesting insights into how the secrecy performance is influenced by various network parameters: antenna array pattern, base station (BS) intensity, and AN power allocation, etc.

I. INTRODUCTION

This paper addresses the previously uninvestigated network-wide secrecy performance of mmWave cellular networks, whose channel characteristics differ from conventional microwave systems. It develops a stochastic-geometry analysis covering noise-limited and AN-assisted settings, eavesdropper cooperation, and key antenna and deployment parameters.

  • The paper applies stochastic geometry to characterize physical-layer security in mmWave cellular networks.
  • MmWave channels differ from microwave channels through blockage sensitivity and distinct LOS and NLOS fading characteristics.
  • The network-wide secrecy performance of mmWave cellular communication remains unknown despite prior work on rate and reliability.
  • For noise-limited networks, the analysis covers secure connectivity and perfect communication links with colluding and non-colluding eavesdroppers.
  • For AN-assisted networks, the paper analyzes interference-limited secrecy using received-SINR distributions and derives secrecy probabilities and perfect-link densities.
  • Simulation results identify antenna pattern and eavesdropper intensity as important secrecy parameters, while optimal AN allocation depends on antenna pattern and eavesdropper intensity.

C. Paper Organization and Notations

The paper models downlink secure communication with stochastic spatial distributions of base stations and eavesdroppers, directional beamforming, fading, blockage, and quasi-static channels. It then organizes the analysis around these system assumptions and secrecy metrics.

  • The paper proceeds from system and channel modeling to noise-limited and AN-assisted secrecy analysis, numerical results, and conclusions.
  • The downlink contains spatially distributed base stations transmitting confidential information to authorized users amid multiple malicious eavesdroppers.
  • Base stations and eavesdroppers are modeled as independent homogeneous Poisson point processes with intensities λB and λE.
  • Authorized users and eavesdroppers are assumed to use single omnidirectional antennas, while eavesdroppers can cancel interference from other information signals.
  • Base-station antenna arrays use a sectored directional-beamforming model with main-lobe gain Ms, sidelobe gain ms, and beam width θb.
  • Perfect channel-state information lets base stations steer antenna boresights toward intended receivers and maximize directivity gains.

C. Small-scale fading

The channel model combines independent Nakagami fading with LOS/NLOS-specific path loss and a blockage model that separates nearby LOS and NLOS base stations from an exclusively NLOS exterior region.

  • Small-scale fading: Each link uses independent Nakagami fading, with separate positive-integer parameters NL and NN for LOS and NLOS links.
  • Blockage Model: Within the ball b(o, D), base stations are divided into independent LOS and NLOS Poisson processes; outside it, only the NLOS process remains with intensity λB.
  • Path loss: LOS and NLOS links use distinct path-loss laws with exponents αL and αN and intercepts CL and CN.
  • Path loss: At 28 GHz, the example parameters are βL = 61.4, αL = 2, βN = 72, and αN = 2.92.
  • Path loss: The model satisfies CL > CN and αL < αN, distinguishing LOS from NLOS propagation.
  • User association: The authorized user associates with the base station offering the lowest path loss, and the analysis derives LOS/NLOS association probabilities and serving-distance distributions.

G. Secrecy Performance Metric

The paper evaluates secrecy using outage-based metrics suited to quasi-static fading: secure connectivity and the density of links that simultaneously achieve connection and secrecy. It also considers noise-limited mmWave conditions motivated by directional, short-range links.

  • Quasi-static fading prevents perfect secrecy from being guaranteed on every transmission, motivating outage-based secrecy metrics.
  • Secure connectivity probability is the probability that the secrecy rate is nonnegative for a randomly chosen base station and intended user.
  • A perfect communication link simultaneously has reliable connection and perfect secrecy, with its network density measured as the average number per unit area.
  • Connection succeeds when code rate Rb does not exceed legitimate-link capacity; otherwise, connection outage occurs.
  • Perfect secrecy occurs when eavesdropper capacity is below redundancy Re = Rb − Rs; otherwise, secrecy outage occurs.
  • With fixed Rb and Rs, average achievable secrecy throughput per unit area is ω = NpRs.
  • Highly directional transmission and short cell radius make links noise-dominated in many blocked or medium/sparse deployments, so the analysis first omits inter-cell interference.

1) Secure Connectivity Probability:

This section develops secure-connectivity analysis for non-colluding eavesdroppers using a path-loss process with fading, then validates the theoretical result against simulations. It also gives analytical expressions for connection and secrecy probabilities.

  • PLPF modeling: The path-loss process with fading (PLPF) is introduced to represent eavesdroppers’ wiretapping capability through ordered path-loss values.The process incorporates small-scale fading and the spatial distribution of eavesdroppers.
  • PLPF modeling: The PLPF is modeled as a one-dimensional nonhomogeneous PPP, enabling secure-connectivity analysis through its intensity measure.The construction separates line-of-sight and non-line-of-sight components and uses antenna gains in the intensity expression.
  • Non-colluding eavesdroppers: Theorem 1 gives the secure connectivity probability for non-colluding eavesdroppers.The derivation conditions on whether the serving base station is line-of-sight or non-line-of-sight and uses the PPP void probability.
  • Validation: Theoretical curves coincide well with simulation curves for secure connectivity probability versus λE, validating Theorem 1.The simulations use 100000 trials.
  • Additional performance metrics: For non-colluding eavesdroppers, Theorem 2 provides analytical results for both connection probability pcon and secrecy probability psec,n.These results support the analysis of perfect communication links per unit area.

B. Colluding Eavesdroppers

This section analyzes colluding eavesdroppers, deriving an exact secure-connectivity expression and a compact approximation for secrecy probability. Theoretical bounds and the approximation are validated numerically.

  • Model and exact analysis: Colluding eavesdroppers combine intercepted information using maximal-ratio combining, representing the worst-case wiretapping scenario.The resulting secure-connectivity probability is denoted τc.
  • Model and exact analysis: Theorem 3 gives an exact analytical expression for τc in the presence of colluding eavesdroppers.The expression is general but computationally unwieldy.
  • Bounds and approximation: A tight upper bound for τc is obtained using a lower bound on the CDF of a normalized gamma random variable.The bound is designed to provide a more compact expression than the exact result.
  • Validation: Theoretical curves coincide well with simulations for τc versus λE, showing that Theorem 4’s upper bound is tight.The comparison uses the colluding-eavesdropper setting and multiple base-station intensities.
  • Secrecy-probability approximation: Theorem 5 approximates secrecy probability psec,c for multiple colluding eavesdroppers because direct inverse-Laplace-transform evaluation can be computationally intensive.The approximation uses N terms and is accurate when N = 5.
  • Secrecy-probability approximation: For N = 5, the approximation accurately matches the secrecy-probability behavior shown in Fig. 3.Figure 3 plots secrecy probability versus Te under the stated system parameters.

IV. SECRECY PERFORMANCE OF THE INTERFERENCE-LIMITED MMWAVE NETWORK WITH AN

This section analyzes AN-assisted mmWave secrecy while accounting for inter-cell interference. It derives a tight upper bound for connection probability and shows that increasing BS intensity need not improve connectivity because interference also increases.

  • Model and metric: Artificial noise increases network interference, so AN-assisted mmWave secrecy is analyzed in the interference-limited setting.The analysis focuses on the average number of perfect communication links per unit area for non-colluding eavesdroppers.
  • Model and metric: The sectorized AN model assigns φPt to confidential information and (1 −φ)Pt to AN transmitted outside the intended sector.Confidential signals use main-lobe gain Ms and spread θb, while AN uses gain Ma over the complementary angular region; the sectors do not overlap.
  • Analytical model: Interfering BSs are mapped into independent PPPs for confidential-signal and AN transmissions, with intensities λBPrx(Ms) and λBPrx(Ma), respectively.This decomposition supports the interference analysis for a typical authorized user.
  • Connection probability: Theorem 6 gives a tight upper bound on the typical communication link’s connection probability, validated by close agreement between approximation and simulation.The reported approximation is specifically evaluated in Fig. 4 against Tc.
  • Connection probability: Connection probability is not monotonic in λB over the full Tc range because denser BS deployment shortens serving distance while increasing network interference.The paper concludes that AN-assisted mmWave communication is interference-limited.

B. Secrecy Probability

This section characterizes secrecy probability for AN-assisted mmWave communication and evaluates noise-limited secrecy under non-colluding and colluding eavesdroppers. The results show that eavesdropper density and collaboration reduce secrecy, whereas more directional beamforming improves it.

  • AN-assisted secrecy probability: Only eavesdroppers inside the serving BS’s intended sector are modeled as wiretapping the confidential information, forming a mapped PPP with density λEPrx(Ms).The analysis assumes eavesdroppers can eliminate interference from information signals sent by other interfering BSs, leaving AN as the degrading interference.
  • AN-assisted secrecy probability: Theorem 7 provides a tight lower bound for the secrecy probability of AN-assisted mmWave communication.The bound agrees well with simulation results in Fig. 5.
  • Noise-limited secrecy: In noise-limited mmWave networks, colluding eavesdroppers have greater wiretapping capability than non-colluding eavesdroppers, reducing secrecy connectivity probability.The deterioration increases as λE grows.
  • Noise-limited secrecy: Increasing beamforming directionality improves secrecy performance by reducing information leakage and improving the authorized user’s reception.The paper reports that narrow-beam antennas increase secure connectivity probability.
  • Noise-limited secrecy: For highly directional arrays, colluding-case performance deteriorates more rapidly with increasing λE than non-colluding performance.The comparison is reported for the average number of perfect communication links per unit area.
  • Noise-limited secrecy: For non-colluding eavesdroppers with θb = 9o, Ms = 15dB, Ma = 3dB, and λE = 4 × 10−4, Np ≈1.1 × 10−4 and more than half of communication links is perfect, on average.Other array patterns produce much smaller Np because their main lobes are wider and intended-sector array gains are lower.

B. Secrecy performance evaluation of interference-limited mmWave cellular networks with AN

The paper evaluates AN-assisted secrecy in interference-limited mmWave networks and shows that antenna directivity and eavesdropper intensity shape the optimal AN allocation and secrecy performance.

  • The optimal fraction of power allocated to AN decreases as eavesdropper intensity decreases and antenna-array directivity improves.The reduced allocation is attributed to weaker eavesdropper reception and lower information leakage.
  • The optimal φ for maximizing Np increases with decreasing λE and decreasing θb.
  • The mmWave network achieves better secrecy performance than the microwave network under the stated comparison parameters.The paper attributes this to blockage effects and highly directional beamforming, which reduce eavesdropper reception quality and information leakage.
  • The analysis characterizes received SINR distributions and the average number of perfect communication links per unit area for non-colluding eavesdroppers.
  • Array pattern and eavesdropper intensity are identified as important system parameters for improving mmWave secrecy performance.

APPENDIX A PROOF OF LEMMA 1

This appendix derives distributions for distances and path-loss-related quantities using the mmWave blockage model and Poisson point-process properties.

  • The derivation obtains distance distributions for LOS and NLOS base-station processes using conditional void probabilities.
  • The LOS and NLOS base-station processes are treated as two independent Poisson point processes.
  • The eavesdropper path-loss process is modeled as a transformed point process with an intensity measure on R+.
  • The intensity measure accounts for blockage states, directivity gains, polar coordinates, and LOS-dependent fading distributions.

APPENDIX E PROOF OF THEOREM 3

This appendix derives the secure connectivity analysis by conditioning on LOS or NLOS serving links and incorporating interference through Laplace-transform calculations.

  • The secure connectivity probability is derived from the received SINR distribution at the intended receiver.
  • The derivation separates cases in which the serving base station is LOS or NLOS and averages over those conditions.
  • Interference contributions are evaluated using probability generating functionals, integration by parts, and Laplace-transform properties.
  • The resulting conditional SINR probabilities use a tight lower bound for the gamma distribution and gamma-variable Laplace transforms.

APPENDIX G PROOF OF THEOREM 7

This appendix derives secrecy probabilities for eavesdroppers by modeling LOS and NLOS path-loss processes and accounting for correlated artificial-noise reception.

  • Because artificial-noise signals received at multiple eavesdroppers are not independent, the secrecy probability is bounded using conditioning and Jensen’s inequality.
  • The derivation distinguishes LOS and NLOS eavesdroppers when evaluating their received SINR probabilities.
  • The path loss process with fading is introduced as a point process on R+ and ordered by its path-loss values.
  • The eavesdropper SINR analysis uses intensity measures and Laplace transforms for interference and artificial noise.
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