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Enhancing the Physical Layer Security of Non-orthogonal Multiple Access in Large-Scale Networks

Yuanwei Liu, Zhijin Qin, Maged Elkashlan, Yue Gao, Lajos Hanzo

arXiv:1612.03901v2cs.IT

TL;DR

The paper addresses the security of NOMA in large-scale networks, where prior PLS research had paid comparatively little attention to NOMA. It uses stochastic geometry to analyze single- and multiple-antenna systems with protected zones, channel ordering, and artificial noise, deriving exact and asymptotic SOP results. The analysis shows that protected zones and artificial noise improve secrecy, while the worse user determines pairwise secrecy diversity and large-array asymptotics closely approximate exact SOP.

  • Problem

    Physical-layer security for NOMA in large-scale networks has received comparatively little research attention, despite NOMA’s distinct successive-interference-cancellation conditions.

  • Method

    The paper uses stochastic geometry to model randomly located NOMA users and eavesdroppers, analyzing protected zones and channel ordering for single-antenna transmission and artificial noise for multiple-antenna transmission.

  • Results

    The paper derives exact SOP expressions for both scenarios, finds that the worse user determines pairwise secrecy diversity, and obtains large-array asymptotic SOP results that closely approximate exact SOP.

  • Takeaways & Limitations

    Secrecy performance can be improved by extending the Eve-exclusion zone and generating artificial noise at the base station, with an optimal signal-to-noise power-sharing ratio in the multiple-antenna case.

Abstract

from arXiv · show

This paper investigates the physical layer security of non-orthogonal multiple access (NOMA) in large-scale networks with invoking stochastic geometry. Both single-antenna and multiple-antenna aided transmission scenarios are considered, where the base station (BS) communicates with randomly distributed NOMA users. In the single-antenna scenario, we adopt a protected zone around the BS to establish an eavesdropper-exclusion area with the aid of careful channel-ordering of the NOMA users. In the multiple-antenna scenario, artificial noise is generated at the BS for further improving the security of a beamforming-aided system. In order to characterize the secrecy performance, we derive new exact expressions of the security outage probability for both single-antenna and multiple-antenna aided scenarios. To obtain further insights, 1) for the single antenna scenario, we perform secrecy diversity order analysis of the selected user pair. The analytical results derived demonstrate that the secrecy diversity order is determined by the specific user having the worse channel condition among the selected user pair; and 2) for the multiple-antenna scenario, we derive the asymptotic secrecy outage probability, when the number of transmit antennas tends to infinity. Monte Carlo simulations are provided for verifying the analytical results derived and to show that: i)~The security performance of the NOMA networks can be improved by invoking the protected zone and by generating artificial noise at the BS; and ii)~The asymptotic secrecy outage probability is close to the exact secrecy outage probability.

I. INTRODUCTION

The paper addresses limited research on physical-layer security for NOMA in large-scale networks, analyzing randomly located users and eavesdroppers under single- and multiple-antenna transmission. It proposes protected zones, channel ordering, and artificial noise, and derives secrecy-outage and diversity results.

  • Motivation and Contribution: Physical-layer security for NOMA remains comparatively underexplored because successive interference cancellation creates receiver interference conditions distinct from OMA.The paper identifies this gap as its motivation.
  • Motivation and Contribution: The study covers both single-antenna and multiple-antenna base-station scenarios, using a protected zone in both and artificial noise in the multiple-antenna case.The protected zone is also called an eavesdropper-exclusion area.
  • Contributions: For the single-antenna case, exact SOP expressions and secrecy-diversity results are derived for a channel-ordered NOMA user pair.The selected pair consists of users indexed m and n, with m < n and more power assigned to the weaker-channel user.
  • Contributions: For the multiple-antenna case, exact and large-array SOP expressions are derived, showing that increasing antennas does not affect eavesdropper SINR in the large-array regime.The analysis also identifies an optimal desired-signal and artificial-noise power-sharing ratio that minimizes SOP.
  • System Model: The network model includes randomly located NOMA users served by a base station and eavesdroppers distributed over an infinite plane according to a homogeneous Poisson point process.Users are paired so that two users share each orthogonal resource block and can be separated using low-complexity SIC.
  • Assumptions and Scope: The analysis assumes perfect SIC, and the authors note that this assumption may overestimate attainable secrecy performance.Imperfect SIC and more sophisticated outage analysis are left for future work.

A. New Channel Statistics

This section develops channel statistics for legitimate users and the most detrimental eavesdropper, then uses them to formulate secrecy outage probabilities for the selected NOMA pair. The resulting theorems provide SOP expressions for both ordered users under the paper’s outage assumptions.

  • New Channel Statistics: New channel statistics for legitimate users and eavesdroppers are derived as inputs to the subsequent secrecy-outage analysis.The results include CDFs for ordered legitimate-user channel gains and a PDF for the most detrimental eavesdropper.
  • Legitimate-User Statistics: The legitimate-user analysis provides CDFs for the n-th and m-th ordered users in a disc containing randomly positioned NOMA users.The expressions include a complexity-versus-accuracy tradeoff parameter K for the n-th user result.
  • Eavesdropper Statistics: The eavesdropper analysis derives the PDF of the most detrimental Eve under a Poisson spatial distribution and an Eve-exclusion zone of radius r_p.This statistic is used for either selected user κ ∈ {m, n}.
  • Secrecy Outage Probability: Secrecy capacity is defined as C_n = [C_Bn − C_En]^+, with legitimate and eavesdropper capacities determined by their respective SINRs.The positive-part operator is [x]^+ = max{x, 0}.
  • Secrecy Outage Probability: A secrecy outage occurs when a user’s secrecy rate falls below its expected secrecy rate, and the SOP is evaluated for a typical independent user pair.The derivation conditions on successful connection establishment between the base station and legitimate users.
  • Secrecy Outage Probability: Theorems 1 and 2 give SOP expressions for the ordered n-th and m-th users, respectively, using the derived legitimate-user and eavesdropper statistics.Under the paper’s assumptions, outage events for the two users are treated as independent.

C. Secrecy Diversity Order Analysis

The paper derives high-SNR asymptotic secrecy outage behavior for ordered NOMA users and selected user pairs. The secrecy diversity order and asymptotic outage of a pair are governed by the user with the poorer channel condition.

  • The asymptotic SOP of the n-th user is derived under PPP-distributed ordered legitimate users.The derivation replaces the legitimate-user SINR CDF with its high-SNR asymptotic form.
  • The secrecy diversity order of the n-th user is n.
  • The asymptotic SOP of the m-th user is likewise derived for PPP-distributed ordered legitimate users.Its secrecy diversity order is m.
  • For m < n, the selected pair’s secrecy diversity order and asymptotic SOP are determined by the m-th user.Thus, the poorer-channel user controls the pair-level asymptotic behavior.
  • Pairing the best-channel user with the second-best user is identified as efficient for increasing secrecy diversity order.This follows because pair SOP is determined by the user with the poorer channel.

III. ENHANCING SECURITY WITH THE AID OF ARTIFICIAL NOISE

This section models a multiple-antenna BS that uses beamformed artificial noise to mask two-user NOMA transmissions and degrade eavesdropper reception. The design relies on null-space AN, power sharing, and spatial user regions, while acknowledging CSI and ordering limitations.

  • The BS uses NA > 2 antennas to generate artificial noise in the null space of the two NOMA users.The information-bearing signal is superimposed with AN to degrade eavesdroppers’ SNR.
  • The AN vector has NA − 1 independent Gaussian elements, and total transmit power satisfies PT = PS + PA.The information and AN powers are controlled through θ, with PS = θPT and PA = (1 − θ)PT.
  • The attainable secrecy performance is an upper bound because practical CSI estimation is treated as non-trivial.
  • The BS divides the user disc into D1 and D2 to create channel-quality differences and simplify channel ordering.The construction does not guarantee optimal MISO NOMA ordering.
  • The two NOMA users apply SIC and receive SINRs after accounting for inter-user interference, noise, and the beamformed transmission structure.
  • Supporting multiple NOMA pairs through signal alignment and more sophisticated precoding or detection is outside the paper’s scope.

A. New Channel Statistics

The section derives channel statistics needed for secrecy-outage analysis when artificial noise is present. Separate results characterize legitimate users in D1 and D2 and eavesdroppers outside an exclusion zone.

  • The section derives new legitimate-user and eavesdropper channel statistics for subsequent SOP analysis with AN.
  • Lemma 4 gives the CDF for an AN-affected channel when user n is randomly located in disc D1.
  • Lemma 5 gives the corresponding CDF when user m is randomly located in ring D2 for θ ≠ 1.
  • Lemma 6 characterizes the eavesdropper-channel PDF under a PPP and an Eve-exclusion zone of radius rp.The expressions use incomplete Gamma functions.

B. Secrecy Outage Probability

The paper derives exact secrecy outage probabilities for both users in the multiple-antenna NOMA scenario with artificial noise. The results are obtained from the preceding legitimate-user and eavesdropper channel statistics.

  • The multiple-antenna AN scenario is analyzed through secrecy outage probability expressions for the selected user pair.
  • Theorem 3 gives the SOP of user n under PPP-distributed legitimate users and eavesdroppers with AN generated at the BS.The expression is given in (38).
  • Theorem 4 gives the SOP of user m under the corresponding PPP and AN assumptions.The expression is given in (39), with the stated case distinction involving NA.
  • The SOP expressions for users n and m are obtained by substituting the preceding channel-statistics results into the secrecy-outage formulations.

C. Large Antenna Array Analysis

The large-antenna analysis develops asymptotic distributions and secrecy-outage expressions as the number of BS antennas tends to infinity, avoiding the exponential complexity of exact formulas. The resulting asymptotic SOP characterizes the selected NOMA user pair under artificial-noise transmission.

  • Large-array motivation: As NA increases, the exact SOP expressions require exponentially more summations, motivating lower-complexity asymptotic approximations.The analysis targets tractable large-array expressions because the exact formulas become excessively complex.
  • Asymptotic user distributions: For NA →∞, the analysis derives asymptotic CDFs for users n and m under their respective spatial regions.User n is modeled in disc D1, while user m is modeled in ring D2.
  • Eavesdropper distribution: With a PPP eavesdropper distribution, an Eve-exclusion radius rp, and BS-generated AN, the analysis derives the asymptotic Eve PDF.The derivation uses the large-antenna limit together with generating-function and polar-coordinate methods.
  • Eavesdropper distribution: The asymptotic Eve PDF is independent of the number of antennas NA in the large-antenna analysis.This result describes the eavesdropper-side distribution after taking NA →∞.
  • Secrecy-outage analysis: For NA →∞, the paper derives asymptotic SOP expressions for users n and m and combines them into the selected user-pair SOP.The user-specific results are obtained from the corresponding asymptotic CDFs and Eve distribution.

IV. NUMERICAL RESULTS

The numerical-results section characterizes large-scale-network performance using Monte Carlo simulations with a complexity-versus-accuracy parameter K = 20.

  • Simulation setup: Monte Carlo simulations use complexity-vs-accuracy tradeoff parameter K = 20.The section also summarizes its simulation parameters in Table I.
  • Simulation setup: BPCU is defined as bit per channel use in the simulation parameters.

A. Secrecy outage probability with channel ordering

The paper evaluates secrecy outage in NOMA networks under channel ordering and artificial-noise transmission. Results show that protected zones, lower eavesdropper density, antenna diversity, and artificial noise can improve secrecy, while the paired user's poorer channel determines single-antenna secrecy diversity.

  • Channel ordering: Smaller user zones reduce SOP by lowering path loss, while the n-th user has a steeper SOP slope than the m-th user.The analytical and simulation results closely agree.
  • Channel ordering: The selected pair's secrecy diversity order is determined by the user with the poorer channel condition.For the ordered pair, the m-th user has secrecy diversity order m.
  • Protected zones: Increasing the Eve-exclusion radius reduces the selected pair's SOP, while lower eavesdropper density improves physical-layer security.A lower λe reduces the multiuser diversity gain available to the most detrimental eavesdropper.
  • Artificial noise: Increasing the number of antennas improves secrecy through multi-antenna diversity, while asymptotic analysis approaches the exact SOP for large antenna arrays.The asymptotic results closely agree with Monte Carlo simulations.
  • Artificial noise: Artificial noise enhances PLS because legitimate users experience only each other's artificial noise, whereas eavesdroppers experience artificial noise from both users.The selected pair's SOP also decreases as the Eve-exclusion radius increases.
  • Artificial noise: The SOP can first decrease and then increase with transmit SNR because stronger desired signaling competes with increased inter-user interference.The power-sharing factor affects the optimal SOP, and optimizing transmit SNR and power sharing can further improve performance.

APPENDIX A: PROOF OF LEMMA 1

This appendix derives the ordered-channel-gain CDF by relating ordered and unordered channel distributions, then incorporating the homogeneous PPP user-location model and numerical quadrature.

  • APPENDIX A: PROOF OF LEMMA 1: The derivation starts from the CDF of the ordered channel gain for the n-th user and relates it to the unordered-channel CDF.Order statistics and a binary series expansion are used in this relationship.
  • APPENDIX A: PROOF OF LEMMA 1: A homogeneous PPP model and polar coordinates are used to formulate the channel-gain distribution for randomly located users.The resulting expression is not readily available in an easily implemented insightful form.
  • APPENDIX A: PROOF OF LEMMA 1: Gaussian-Chebyshev quadrature approximates the spatial integral before substitution into the ordered-channel CDF.The multinomial theorem is then applied to obtain the CDF expression used in the lemma.
  • APPENDIX A: PROOF OF LEMMA 1: The appendix separately derives the CDF associated with artificial noise using intermediate integrals and Gamma-distribution properties.The final substitution yields the CDF of FγAN.

APPENDIX C: PROOF OF LEMMA 5

This appendix derives the artificial-noise-related CDF by combining channel-norm distributions, interference terms, spatial integration, and closed-form integral identities.

  • APPENDIX C: PROOF OF LEMMA 5: The derivation begins from the CDF of the artificial-noise quantity and uses the Gamma distribution of the channel norm.The norm ∥hm∥^2 follows a Gamma distribution with parameters (NA, 1).
  • APPENDIX C: PROOF OF LEMMA 5: Binary series expansions, polar-coordinate transformations, and integral identities are applied to evaluate Q1 and related expressions.The distance dm is determined by the location of ωm.
  • APPENDIX C: PROOF OF LEMMA 5: The channel is decomposed into Xm and Ym, with Xm exponential and Ym Gamma distributed, to obtain the PDF of the artificial-noise interference term.The resulting PDF is used in the integral Q1.
  • APPENDIX C: PROOF OF LEMMA 5: After substituting the evaluated integral into the preceding expression, the appendix obtains the CDF of FγAN.The result is identified as equation (37).

APPENDIX D: PROOF OF LEMMA 6

This appendix derives the CDF of the artificial-noise term by modeling projected channel components with Gamma distributions, evaluating their sum, and differentiating the result.

  • APPENDIX D: PROOF OF LEMMA 6: The derivation first reformulates the relevant artificial-noise quantity using projections onto beamforming vectors.The analysis introduces component norms Ye,m and Ye,n for the projected channels.
  • APPENDIX D: PROOF OF LEMMA 6: The projected-channel norms Ye,m and Ye,n each follow a Gamma (NA −1, 1) distribution.This distributional property provides the basis for evaluating the interference integral Q2.
  • APPENDIX D: PROOF OF LEMMA 6: The sum of the two projected-channel contributions is modeled with a generalized integer Gamma distribution.Its PDF is then inserted into the preceding integral expression.
  • APPENDIX D: PROOF OF LEMMA 6: Substitution and integral evaluation produce the CDF of FγAN, after which differentiation yields equation (37).The derivation relies on the cited integral identity before taking the derivative.
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