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Power and Subcarrier Allocation for Physical-Layer Security in OFDMA-based Broadband Wireless Networks

Xiaowei Wang, Meixia Tao, Jianhua Mo, Youyun Xu

arXiv:1106.6173v1cs.ITmath.IT

TL;DR

The paper studies how to allocate power and subcarriers in OFDMA networks that serve both secure users and normal users, while meeting average secrecy requirements. It uses dual decomposition to solve the allocation problem asymptotically optimally and derives channel-based conditions for secure-user allocation. The analysis also identifies a limitation when malicious eavesdroppers can falsify channel measurements.

  • Problem

    The paper asks how to allocate power and subcarriers in a downlink OFDMA network to maximize normal users’ average aggregate information rate while meeting each secure user’s average secrecy-rate requirement.

    Source: §I · pp. 2, 4

  • Method

    The paper solves the joint allocation problem in the dual domain using decomposition and also proposes a lower-complexity suboptimal allocation algorithm.

    Source: §I · pp. 4–5

  • Results

    An SU is a subcarrier candidate only when its channel is strongest and exceeds the second-largest channel by a threshold; NU power follows water-filling, while SU power depends on its gain and the strongest competing gain.

    Source: §VII · p. 18

  • Takeaways & Limitations

    The derived channel-based conditions support a suboptimal allocation scheme that decouples Lagrange-multiplier updates to achieve linear complexity in users and subcarriers.

    Source: §I · p. 5

    If a purely malicious eavesdropper does not care about its own transmission, false channel measurements can circumvent the network’s security.

    Source: §VII · p. 18

Abstract

from arXiv · show

Providing physical-layer security for mobile users in future broadband wireless networks is of both theoretical and practical importance. In this paper, we formulate an analytical framework for resource allocation in a downlink OFDMA-based broadband network with coexistence of secure users (SU) and normal users (NU). The SU's require secure data transmission at the physical layer while the NU's are served with conventional best-effort data traffic. The problem is formulated as joint power and subcarrier allocation with the objective of maximizing average aggregate information rate of all NU's while maintaining an average secrecy rate for each individual SU under a total transmit power constraint for the base station. We solve this problem in an asymptotically optimal manner using dual decomposition. Our analysis shows that an SU becomes a candidate competing for a subcarrier only if its channel gain on this subcarrier is the largest among all and exceeds the second largest by a certain threshold. Furthermore, while the power allocation for NU's follows the conventional water-filling principle, the power allocation for SU's depends on both its own channel gain and the largest channel gain among others. We also develop a suboptimal algorithm to reduce the computational cost. Numerical studies are conducted to evaluate the performance of the proposed algorithms in terms of the achievable pair of information rate for NU and secrecy rate for SU at different power consumptions.

I. INTRODUCTION

The paper addresses physical-layer security in OFDMA networks where secure users and best-effort users coexist, a setting not covered by earlier OFDMA allocation work or the cited two-user secrecy study. It formulates joint power and subcarrier allocation around average secrecy requirements for secure users and aggregate information rate for normal users.

I. INTRODUCTION

  • Earlier OFDMA resource-allocation studies did not account for security, while the cited OFDM secrecy-allocation study considered only two users and no other user types.The paper frames its setting as simultaneous service to multiple secure users (SUs) and normal users (NUs).Source: §I · p. 3
  • The system distinguishes SUs, who require nonzero secrecy rates, from NUs, whose best-effort traffic has no confidentiality requirement.Users are assumed honest, single-antenna, and passive when acting as potential eavesdroppers.Source: §I · p. 3
  • An SU can achieve nonzero secrecy on a subcarrier only when its channel gain is largest among users, and even then an NU may be preferred if the gain gap is too small.The secrecy rate is a difference of logarithmic terms, so a small main-channel/eavesdropper-channel gap can make NU service preferable.Source: §I · p. 4

II. SYSTEM MODEL AND PROBLEM FORMULATION

  • The allocation objective maximizes NUs’ average aggregate information rate while meeting each SU’s average secrecy-rate requirement under a base-station power constraint.The paper considers both long-term average and peak total transmit-power constraints; secrecy is averaged because fading can prevent nonzero instantaneous secrecy rates.Source: §I · p. 4, §II · p. 7
  • The model uses a downlink OFDMA system in which the base station knows users’ channel conditions and assigns each subcarrier to one user per frame.The channels are slowly time-varying and frequency-selective; the base station adapts allocations to instantaneous channel state information.Source: §II · pp. 5–6

III. OPTIMAL RESOURCE ALLOCATION UNDER AVERAGE POWER CONSTRAINT

Under the average-power constraint, the paper applies dual decomposition to the resource-allocation problem. The problem decomposes across subcarriers, and its solution is asymptotically optimal when the number of subcarriers is sufficiently large.

III. OPTIMAL RESOURCE ALLOCATION UNDER AVERAGE POWER CONSTRAINT

  • For sufficiently many subcarriers, the dual approach yields an asymptotically optimal solution under the average-power constraint.The stated basis is the time-sharing condition: the objective is concave and the secrecy-rate constraint is convex.Source: §III · pp. 7–8
  • Dual decomposition separates the Lagrangian maximization into independent subproblems, one for each subcarrier.The dual variables correspond to the secrecy-rate and average-power constraints.Source: §III · p. 8
  • For fixed dual variables, each subcarrier’s subproblem is a single-carrier, multiple-user power-allocation problem.The solution maximizes the expected per-subcarrier objective over feasible power allocations.Source: §III · p. 8

A. Optimality Condition of Power Allocation

The power-allocation conditions distinguish normal and secure users. Normal users follow water-filling, while an SU’s allocation depends on its channel relative to the strongest other user and the dual-variable threshold.

A. Optimality Condition of Power Allocation

  • NU power allocation follows conventional water-filling, with its water level determined by the NU weight and average-power constraint.The power condition is obtained by applying KKT conditions to the concave per-subcarrier objective.Source: A. Optimality Condition of Power… · p. 9
  • The SU power-allocation condition has the same form as the result for conventional fading wire-tap channels cited as.Source: A. Optimality Condition of Power… · p. 9
  • An SU receives nonzero power only when its channel gain exceeds the strongest other user’s by the threshold λ/µk.The SU’s allocation depends on both its own channel gain and the largest channel gain among the other users.Source: A. Optimality Condition of Power… · p. 9

B. Optimality Condition of Subcarrier Assignment

For fixed dual variables, each subcarrier is assigned to the user with the highest assignment value. This criterion makes an SU eligible only when its channel is strongest and clears a threshold over the second-largest channel gain.

B. Optimality Condition of Subcarrier Assignment

C. Dual Update

The dual objective is handled with subgradients because discontinuous subcarrier assignment makes it nondifferentiable; the optimal dual variables are then substituted into the allocation conditions. When statistical averages are too complex to compute, time averages can be used if fading is ergodic.

C. Dual Update

  • The dual function is nondifferentiable because subcarrier assignment is discontinuous, so the method uses a subgradient for dual-variable updates.The supplied passages state that the dual problem is convex and that its gradient does not exist.Source: C. Dual Update · p. 10
  • When statistical-average computation is too complex, time averages can replace it if the channel fading process is ergodic.Source: C. Dual Update · p. 11
  • After the optimal dual variables are found, the allocation policy is obtained by substituting them into the optimality conditions.Source: C. Dual Update · p. 11

D. Discussion of Feasibility

Feasibility requires each secure user's average secrecy-rate target to remain below an upper bound determined under Rayleigh fading. In the stated example, a target above 3.5 nat/OFDMA symbol is infeasible.

D. Discussion of Feasibility

  • The feasibility test compares each secure user's secrecy-rate requirement with an upper bound on its achievable average secrecy rate.A requirement equal to or greater than that bound cannot be satisfied.Source: D. Discussion of Feasibility · pp. 11–12
  • The upper bound is derived using Rayleigh fading and the distribution of the largest and second-largest CNRs on a subcarrier.The integrals are evaluated numerically because they are difficult to express in closed form.Source: D. Discussion of Feasibility · p. 12
  • C_k above 3.5 nat/OFDMA symbol makes the problem infeasible in the stated example.Source: D. Discussion of Feasibility · p. 12

IV. OPTIMAL RESOURCE ALLOCATION UNDER PEAK POWER CONSTRAINT

Under a peak transmit-power constraint, dual decomposition again separates the optimization into per-subcarrier sub-functions. The resulting power allocations have a similar structure to the average-power case, but the power-controlling dual variable depends on the system channel.

IV. OPTIMAL RESOURCE ALLOCATION UNDER PEAK POWER CONSTRAINT

  • The peak-power dual function decomposes into N independent sub-functions, one for each subcarrier.Source: §IV · p. 13
  • For each channel realization, the subcarrier sub-function is maximized after reversing the expectation and maximization operations; differentiating the objective gives the power-allocation conditions.Source: §IV · p. 13
  • Peak- and average-power constraints yield power allocations with similar structure, but λ is channel-dependent under the peak constraint and constant under the average constraint.Source: §IV · p. 13

V. SUBOPTIMAL POWER AND SUBCARRIER ALLOCATION ALGORITHM

The suboptimal scheme reduces the cost of jointly optimizing dual variables by allocating resources to secure users first and distributing any residual resources to normal users. It uses separate binary searches for secure-user parameters and the normal-user water level.

V. SUBOPTIMAL POWER AND SUBCARRIER ALLOCATION ALGORITHM

  • Joint optimization of the Lagrange multipliers can be computationally unfavorable when the number of secure users is large, motivating a lower-complexity scheme.The ellipsoid method is stated to require O((K1 + 1)^2 log 1/ε) iterations.Source: §V · p. 14
  • The scheme first allocates resources to secure users as if normal users were eavesdroppers, then distributes residual subcarriers and power to normal users.Source: §V · p. 14
  • Separate binary searches find secure-user parameters ν_k and the normal-user water level L0, whose allocation follows water-filling with levels proportional to normal-user weights.Source: §V · pp. 14–15

P SU, LUB

The suboptimal procedure separates secure-user threshold searches from normal-user water-level allocation, using the remaining power for normal users after secure-user powers are set.

P SU, LUB

  • For each secure user, a CNR threshold defines its candidate subcarriers; disjoint candidate sets permit independent binary searches for thresholds meeting secrecy targets.The search tests whether the secure user's CNR exceeds the second-largest CNR by νk.Source: P SU, LUB · p. 15
  • After the secure-user thresholds and power allocations are found, the normal-user water level is searched until the remaining total power is used.Source: P SU, LUB · p. 15
  • Each iteration of both optimal and suboptimal schemes has computational load linear in KN|α|, where |α| is the number of training channel realizations.Source: P SU, LUB · p. 15

VI. NUMERICAL RESULTS

The numerical studies compare adaptive power and subcarrier allocation with fixed assignments across secrecy targets, transmit SNRs, and power constraints. Adaptive schemes outperform fixed assignments, and the suboptimal method stays close to the optimal one.

VI. NUMERICAL RESULTS

  • The simulation uses 64 subcarriers and eight users, split evenly between SUs and NUs, with identical SU secrecy targets and independent unit-mean Rayleigh fading.FSA-1 assigns eight subcarriers per user; FSA-2 assigns 12 per SU and four per NU.Source: §VI · p. 16
  • At 30 dB, adaptive schemes remain feasible to RSU=3.5 nat/OFDM symbol, versus 0.44 for FSA-1 and 0.66 for FSA-2.Source: §VI · pp. 16–18
  • At fixed total transmit SNR=30dB, for the same RSU, the suboptimal algorithm incurs less than 20% loss in RNU compared with the optimal algorithm; RNU decreases as RSU increases.Source: §VI · p. 16
  • As RSU rises, SU power consumption is higher under the optimal scheme, while both adaptive schemes assign up to 32 subcarriers to SUs at RSU=3.5.The optimal scheme assigns fewer subcarriers to SUs than the suboptimal scheme; 32 is half of the 64-subcarrier total.Source: §VI · p. 17
  • For RSU=0.4 nat/OFDM symbol, FSA-2 outperforms FSA-1 below 14 dB SNR, but underperforms it above 14 dB.Source: §VI · p. 17
  • Under average and peak power constraints, the RNU curves differ slightly at low SNR and nearly coincide at high SNR for both algorithms.Source: §VI · p. 18

VII. CONCLUSIONS AND DISCUSSIONS

The paper concludes that dual decomposition yields asymptotically optimal joint resource allocation for coexisting secure and normal users, with a lower-complexity alternative. Its discussion identifies accurate channel knowledge and centralized CSI as practical boundaries.

VII. CONCLUSIONS AND DISCUSSIONS

  • The asymptotically solved allocation maximizes NU information rates subject to SU secrecy targets; SU candidacy depends on beating the second-largest CNR by a threshold.SU power depends on its own channel gain and the largest gain among other users.Source: §VII · p. 18
  • The analysis assumes accurate BS channel information; malicious eavesdroppers could circumvent security by sending false channel measurements.Source: §VII · p. 18
  • The centralized algorithm requires full CSI, while distributed algorithms using partial or local channel knowledge are left for future work.Source: §VII · p. 19
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