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On the Design of Secure Non-Orthogonal Multiple Access Systems

Biao He, An Liu, Nan Yang, Vincent K. N. Lau

arXiv:1612.06961v2cs.IT

TL;DR

The paper addresses secure NOMA design against an external eavesdropper when instantaneous eavesdropper CSI is unknown. It uses secrecy outage probability and optimizes decoding order, rates, and power allocation for power minimization and fair confidential-rate maximization. The resulting designs retain conventional NOMA’s decoding order, shift more power toward weaker users under stricter secrecy, and outperform OMA in the studied problems.

  • Problem

    Secure NOMA design against external eavesdroppers with appropriate secrecy and QoS metrics had not been investigated for unknown instantaneous eavesdropper CSI.

  • Method

    The paper models power-domain NOMA with wiretap coding and secrecy outage probability, then solves transmit-power minimization and minimum confidential-information-rate maximization under secrecy, QoS, and power constraints.

  • Results

    The secrecy outage constraint leaves the optimal decoding order unchanged, stricter secrecy increases power allocated to the relatively weak user, and NOMA outperforms OMA in the studied problems.

  • Takeaways & Limitations

    Secure NOMA can preserve conventional channel-strength-based decoding order while providing better secrecy-constrained performance than OMA in the studied settings.

Abstract

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This paper proposes a new design of non-orthogonal multiple access (NOMA) under secrecy considerations. We focus on a NOMA system where a transmitter sends confidential messages to multiple users in the presence of an external eavesdropper. The optimal designs of decoding order, transmission rates, and power allocated to each user are investigated. Considering the practical passive eavesdropping scenario where the instantaneous channel state of the eavesdropper is unknown, we adopt the secrecy outage probability as the secrecy metric. We first consider the problem of minimizing the transmit power subject to the secrecy outage and quality of service constraints, and derive the closed-form solution to this problem. We then explore the problem of maximizing the minimum confidential information rate among users subject to the secrecy outage and transmit power constraints, and provide an iterative algorithm to solve this problem. We find that the secrecy outage constraint in the studied problems does not change the optimal decoding order for NOMA, and one should increase the power allocated to the user whose channel is relatively bad when the secrecy constraint becomes more stringent. Finally, we show the advantage of NOMA over orthogonal multiple access in the studied problems both analytically and numerically.

I. INTRODUCTION

The paper designs secure power-domain NOMA against an external eavesdropper when instantaneous eavesdropper CSI is unavailable, using secrecy outage probability and confidential information rates. It studies decoding order, power allocation, and rate design, and compares NOMA with OMA.

  • Motivation: Secure NOMA design against external eavesdroppers had not been adequately investigated with appropriate secrecy and QoS metrics.The paper uses secrecy outage probability because perfect secrecy is generally unavailable without instantaneous eavesdropper CSI.
  • System and design: The paper jointly treats decoding order, transmission rates, and user power allocation as design variables under secrecy outage constraints.The setting is a single-antenna transmitter serving multiple users while distinguishing an external eavesdropper.
  • System and design: The optimal decoding order remains the descending order of channel gains normalized by noise, as in conventional NOMA.The paper analytically proves that the secrecy outage constraint does not change this order.
  • Contributions: The transmit-power minimization problem under secrecy outage and confidential-rate QoS constraints has a closed-form solution.The confidential information rate is used because it represents useful informative data rather than secrecy redundancy and total codeword rate.
  • Contributions: For minimum confidential-rate maximization, the paper proposes an iterative algorithm and obtains a closed-form solution for the two-user special case.As secrecy constraints become more stringent, the power ratio allocated to the relatively weak user should increase.
  • Contributions: NOMA analytically always outperforms OMA in the studied secrecy-constrained problems, with numerical gains increasing almost linearly as the number of users grows.The comparison is made for both transmit-power minimization and minimum confidential-information-rate maximization.

B. Secure Encoding

The secure NOMA formulation uses wiretap coding with adaptive user-side rates and secrecy outage probability, then imposes decoding, power, QoS, and secrecy constraints in the design problem.

  • Secure encoding: Wiretap coding assigns each user a codeword transmission rate R_t,k and confidential information rate R_s,k, with their difference providing secrecy redundancy.The positive rate difference R_t,k − R_s,k is the secrecy cost against the eavesdropper.
  • Secure encoding: The transmitter can adapt R_t,k and R_s,k to users’ instantaneous CSI, while the eavesdropper’s instantaneous CSI remains unknown.This setting prevents perfect secrecy, motivating secrecy outage probability as the secrecy metric.
  • Secure encoding: The secrecy outage probability measures whether the eavesdropper’s channel capacity exceeds the secrecy redundancy for a message.The eavesdropper capacity for message s_π(k) is denoted C_e,π(k).
  • Optimization formulation: The transmit-power minimization design uses confidential information rates as QoS targets because they represent the actual rate of useful data received by users.Codeword rate also includes redundancy, so a high codeword rate with low confidential rate may provide slow useful information.
  • Optimization formulation: The formulation constrains positive user powers, successful decoding by intended and SIC users, confidential-rate QoS, and maximum secrecy outage probability.The decoding-order vector π specifies the common SIC order, while Q and ǫ denote the minimum confidential rate and tolerable outage probability.

B. Problem Simplification

The problem is simplified by optimizing transmission and confidential rates for a fixed decoding order, then deriving the secrecy outage constraint. The resulting formulation remains non-convex because power variables are coupled by that constraint.

  • For fixed decoding order and power allocation, each transmission rate is chosen at its largest value satisfying the relevant constraints.
  • Each confidential information rate is set to the minimum value satisfying the QoS constraint, yielding Rs,π(k) = Q for every user.
  • The secrecy outage probability is derived from these optimized rates and the eavesdropper’s average channel gain.
  • The optimal decoding order is descending channel gains normalized by noise, identical to conventional NOMA without secrecy constraints.
  • The reduced power-allocation problem is non-convex because the secrecy constraint couples all user powers, preventing transformation into a sequence of linear programs.

C. Optimal Solution

This section derives the optimal power allocation for minimizing transmit power under QoS and secrecy constraints, and characterizes feasibility and user selection. It also identifies conditions under which transmission must be suspended.

  • C. Optimal Solution: The optimal power solution is obtained sequentially, beginning with the last user and then iteratively determining the remaining users’ powers.
  • C. Optimal Solution: The problem is feasible if and only if the derived power allocation satisfies the associated feasibility conditions.
  • C. Optimal Solution: Users whose channel conditions do not satisfy the secrecy-related threshold cannot be served under the stated constraints.
  • C. Optimal Solution: When no user satisfies the threshold, the transmitter suspends transmission for that time slot.
  • C. Optimal Solution: The formulation optimizes confidential information rates and power under secrecy, QoS, positivity, and maximum instantaneous transmit-power constraints.
  • C. Optimal Solution: The resulting reduced problem maximizes the minimum confidential information rate subject to the stated secrecy and power constraints.
  • C. Optimal Solution: The coupled design parameters remain non-convex in the secrecy constraint.

B. Optimal Solution

For the fairness-oriented rate-maximization problem, feasibility is reduced to the earlier power-minimization problem and solved algorithmically. In the two-user case, closed-form solutions reveal how secrecy conditions shift power toward the weaker user.

  • B. Optimal Solution: The rate-maximization problem is feasible if and only if its corresponding power-minimization problem is feasible under the transmit-power constraint.
  • B. Optimal Solution: The rate-maximization problem may be infeasible when a positive common confidential rate cannot satisfy the secrecy constraint.
  • B. Optimal Solution: For arbitrary user counts, bisection search obtains the optimal minimum confidential rate and transmit power using a closed-form feasibility solution.
  • B. Optimal Solution: The algorithm outputs the optimal minimum confidential rate and power-allocation vector after iterating over the feasibility condition.
  • B. Optimal Solution: For two users, closed-form expressions are derived for the optimal minimum confidential rate, transmit power, and power-allocation ratio.
  • B. Optimal Solution: The optimal power ratio assigned to the weaker user increases when the eavesdropper’s channel improves or the secrecy constraint becomes more stringent.

V. ANALYTICAL COMPARISON BETWEEN NOMA AND OMA

The paper analytically compares NOMA with an OMA benchmark using the maximum achievable minimum confidential information rate under secrecy outage and transmit-power constraints.

  • V. ANALYTICAL COMPARISON BETWEEN NOMA AND OMA: The comparison evaluates NOMA and OMA by the maximum minimum confidential information rate achievable under secrecy outage and transmit-power constraints.

A. Benchmark Scheme

The benchmark is a TDMA orthogonal-access scheme with designable time allocation, whose maximum minimum confidential information rate is compared against NOMA. NOMA achieves a strict advantage when user channel conditions differ, but the schemes tie when they are equal.

  • Benchmark construction: TDMA is used as the benchmark OMA scheme, with time resources divided among users; OFDMA is mathematically equivalent in this analysis.Unlike the conventional equal-sharing benchmark, the analysis allows designable time allocation ratios.
  • Benchmark construction: The TDMA benchmark maximizes the minimum confidential information rate by optimizing the time allocation variables t1 and t2.The comparison focuses on the two-user case and solves for the allocation that maximizes RTDMA_s,min.
  • Comparison result: When γ1 ≠ γ2, NOMA always achieves a higher maximum minimum confidential information rate than TDMA.This is stated as Proposition 4 for unequal user channel conditions.
  • Comparison result: When γ1 = γ2, NOMA and TDMA achieve the same maximum minimum confidential information rate.Thus, the strict NOMA advantage in the proposition is specific to unequal channel conditions.

VI. NUMERICAL RESULTS

The numerical results evaluate secure NOMA under fixed and randomly generated channels, comparing it with TDMA/OMA across power, QoS, secrecy, and user-count settings. NOMA consistently outperforms the benchmark and offers stronger gains as the number of users increases.

  • Power and QoS tradeoff: As the QoS requirement increases, total transmit power increases, while infeasible QoS requirements cannot be satisfied even with infinite transmit power.The feasible QoS range for NOMA is characterized analytically.
  • Power and QoS tradeoff: The minimum confidential information rate increases with the transmit-power constraint but approaches an upper bound equal to the maximum feasible QoS requirement.This behavior is reported for the designed NOMA scheme in Figure 3.
  • Power allocation: The optimal power share for the weak user decreases as the secrecy-outage constraint loosens and increases as the eavesdropper’s channel condition improves.User 1 is the relatively weak user; the transmit power is fixed at P = 20 dBm.
  • Average performance: NOMA achieves a better QoS–secrecy tradeoff than TDMA, and its performance-gain ratio over TDMA increases almost linearly with the number of users.This advantage persists when users have either better or identical channel statistics relative to the eavesdropper.

APPENDIX A PROOF OF PROPOSITION 1

The proof establishes the optimal NOMA decoding order by showing that swapping adjacent users with stronger normalized channel gain before weaker gain cannot worsen performance and may improve secrecy outage.

  • Exchange argument: Swapping adjacent users ordered with γπ(x) > γπ(x+1) leaves other effective channel terms unchanged and increases or preserves the later term.The secrecy outage probability decreases as the relevant effective channel term increases.
  • Exchange argument: Because the swap does not worsen performance, repeatedly exchanging out-of-order adjacent users yields the optimal decoding order.The resulting order satisfies the required monotonic channel-gain condition.
  • Power optimization: The minimum-power solution is characterized through monotonicity: the minimum Pk occurs when one constraint is active and decreases as the remaining-user power increases.These derivative properties support the iterative construction of the optimal power vector.
  • Feasibility: The feasibility proof shows that the minimum-power problem is infeasible when its QoS or secrecy conditions cannot be met under the governing constraints.The argument separately rules out satisfaction of the relevant constraints when conditions (17) or (18) fail.

APPENDIX D PROOF OF PROPOSITION 2

The proof characterizes the max–min confidential rate by first determining feasibility, then identifying the largest feasible common rate and the corresponding power allocation.

  • Feasibility: The max–min problem is infeasible if any constraint remains unsatisfied even as the common confidential rate approaches zero.Conversely, it is feasible when all constraints can be satisfied for a sufficiently small common rate.
  • Optimal rate: The optimal common confidential rate is the maximum QoS value for which the minimum-power problem remains feasible under the total power constraint.This links the max–min rate problem to the feasibility conditions of the minimum-power formulation.
  • Power allocation: For a given QoS value, the associated optimal powers are obtained from the closed-form solutions of the minimum-power problem.The power expressions are substituted into the total-power constraint.
  • Two-user specialization: In the two-user case, the most stringent constraint determines the optimum, so the optimal common rate is found by solving 2^Q = b3.The proof establishes b3 as the minimum among the relevant constraint bounds.

APPENDIX F PROOF OF COROLLARY 2

The proof analyzes how the optimal weak-user power ratio changes with the secrecy parameter by reparameterizing the expression and studying its derivative.

  • Derivative analysis: The derivative of the optimal weak-user power ratio is difficult to analyze directly because its expression is complicated.The proof introduces normalized variables to simplify the derivative analysis.
  • Derivative analysis: The proof rewrites the ratio using ρ1 = γ1P, ρ2 = γ2P, and ϕ = φP before evaluating its derivative.These substitutions normalize the channel and secrecy parameters by the transmit power.

APPENDIX G PROOF OF PROPOSITION 4

The appendix characterizes the confidential-information rate-pair boundaries for TDMA and NOMA. It shows that NOMA's boundary is concave when users have unequal channel parameters and affine when they are equal, determining when NOMA strictly outperforms or matches TDMA.

  • Boundary construction: The available resources define the upper rate-pair boundaries: t1+t2 = 1 for TDMA and P1 + P2 = P for NOMA.The boundary is reached when all available resources are used.
  • NOMA–TDMA comparison: With a concave NOMA boundary, every TDMA rate pair is strictly smaller than some achievable NOMA rate pair.The TDMA rate pairs occupy a convex region with an affine boundary.
  • NOMA–TDMA comparison: With an affine NOMA boundary, TDMA rate pairs coincide with achievable NOMA rate pairs.The appendix uses this geometric distinction to establish the comparison between the two schemes.
  • Boundary curvature: For γ1 < γ2, the NOMA upper boundary is concave.The result follows from the negative second derivative with respect to the rate variable.
  • Boundary curvature: When γ1 = γ2, the NOMA upper boundary is affine.This is the equal-channel-parameter case treated separately in the proof.
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