PaperlayerView exact revision on arXiv

Source-linked AI summary

Secure Communication over Fading Channels

Yingbin Liang, H. Vincent Poor, Shlomo Shamai

arXiv:cs/0701024v1cs.ITcs.CR

TL;DR

The paper investigates secrecy for fading broadcast channels with confidential messages, where information for one receiver must remain secret from the other. It first studies parallel BCC models, then establishes secrecy-capacity regions and characterizes optimal power allocations for Gaussian and fading settings.

  • Problem

    The paper studies fading BCCs in which confidential information for one receiver must be perfectly secret from the other receiver.

  • Method

    The paper first analyzes a general parallel BCC and examines optimization cases and independent inputs across subchannels.

  • Results

    The paper establishes secrecy-capacity regions for parallel and parallel Gaussian BCCs and characterizes power allocations achieving the Gaussian region boundary.

  • Takeaways & Limitations

    Independent inputs and characterized power allocations provide capacity-achieving designs for the studied parallel Gaussian BCC setting.

  • Takeaways & Limitations

    Outage probability cannot be prevented for channel states where the source-to-receiver-1 channel is worse than the corresponding competing channel.

Abstract

from arXiv · show

The fading broadcast channel with confidential messages (BCC) is investigated, where a source node has common information for two receivers (receivers 1 and 2), and has confidential information intended only for receiver 1. The confidential information needs to be kept as secret as possible from receiver 2. The broadcast channel from the source node to receivers 1 and 2 is corrupted by multiplicative fading gain coefficients in addition to additive Gaussian noise terms. The channel state information (CSI) is assumed to be known at both the transmitter and the receivers. The parallel BCC with independent subchannels is first studied, which serves as an information-theoretic model for the fading BCC. The secrecy capacity region of the parallel BCC is established. This result is then specialized to give the secrecy capacity region of the parallel BCC with degraded subchannels. The secrecy capacity region is then established for the parallel Gaussian BCC, and the optimal source power allocations that achieve the boundary of the secrecy capacity region are derived. In particular, the secrecy capacity region is established for the basic Gaussian BCC. The secrecy capacity results are then applied to study the fading BCC. Both the ergodic and outage performances are studied.

1 Introduction

The paper studies a fading broadcast channel with confidential messages under fading, Gaussian noise, and transmitter–receiver CSI, using parallel BCCs as a general model. It establishes secrecy-capacity regions and optimal power allocations, then applies these results to ergodic and outage performance.

  • Parallel BCC: The parallel BCC with L independent subchannels provides a general information-theoretic model that includes the fading BCC as a special case.The paper establishes its secrecy-capacity region and proves that independent input distributions across subchannels are optimal.
  • Gaussian BCC: The secrecy-capacity region is specialized to parallel BCCs with degraded subchannels and then established for the parallel Gaussian BCC.For the Gaussian case, the region is obtained as a union over source power allocations, with optimal allocations characterized for its boundary.
  • Gaussian BCC: The parallel Gaussian BCC result also establishes the secrecy-capacity region of the basic Gaussian BCC.This complements the discrete memoryless BCC secrecy-capacity result cited in the paper.
  • Fading BCC: The results are applied to fading BCC ergodic performance, where secrecy rates are averaged over channel states and CSI enables state-dependent power allocation.The fading model can be viewed as a parallel Gaussian BCC with each fading state corresponding to one subchannel.
  • Fading BCC: The outage analysis uses block fading and derives power allocations minimizing outage when common or confidential target rates are not achieved.The analysis considers a long-term source power constraint and CSI at both transmitter and receivers.

2 Parallel BCCs

The paper models confidential broadcasting across independent parallel subchannels and characterizes the resulting secrecy capacity regions. It specializes the result to degraded subchannels and shows how coding across subchannels and subchannel ordering affect common and confidential transmission.

  • Channel Model: The parallel BCC has one source, two receivers, common and confidential messages, and secrecy of receiver 1's confidential message from receiver 2.The model uses finite input and output alphabets with a product transition distribution across independent subchannels.
  • Channel Model: Perfect secrecy requires the equivocation rate to equal the confidential-message rate, so receiver 2 obtains no information about that message.The secrecy capacity region contains rate pairs achievable under this perfect-secrecy condition.
  • Secrecy Capacity Region: Theorem 1 characterizes the secrecy capacity region of the parallel BCC through common-rate and confidential-rate bounds summed across subchannels.The result applies when each subchannel is a general broadcast channel and need not be degraded.
  • Secrecy Capacity Region: Independent input distributions for each subchannel are optimal, a fact requiring a converse beyond the single-letter BCC result.The theorem also implies the parallel wire-tap specialization when only confidential information is transmitted.
  • Secrecy Capacity Region: Coding across all parallel subchannels can achieve a larger secrecy capacity region than summing the individual subchannel regions.The paper illustrates the broader parallel-channel capacity with a two-subchannel deterministic example whose total capacity is 9.
  • Degraded Subchannels: For degraded subchannels, the common message uses all subchannels, while the confidential message uses only subchannels where receiver 2 is degraded relative to receiver 1.Although every subchannel is degraded, the overall channel may not be degraded when subchannels have opposite degradation orders.

3 Parallel Gaussian BCCs

The parallel Gaussian BCC is analyzed through an equivalent physically degraded representation, yielding its secrecy capacity region and boundary-achieving power allocations. The single-subchannel Gaussian BCC follows as a special case.

  • Channel model: The parallel Gaussian BCC is modeled with independent Gaussian subchannels and additive Gaussian noise processes.
  • Equivalent degraded representation: An equivalent physically degraded channel has the same marginal distributions and therefore the same secrecy capacity region.
  • Secrecy capacity region: The secrecy capacity region of the parallel Gaussian BCC is established in Theorem 2.
  • Power allocation: Power allocation vectors divide each subchannel’s power between common and confidential messages, or allocate power only to common messages.
  • Gaussian BCC: When L = 1, the parallel Gaussian BCC reduces to the Gaussian BCC, whose secrecy capacity region follows as Corollary 4.
  • Boundary characterization: Every boundary point is characterized by a max-min optimization whose solution provides the power allocation achieving that point.
  • Optimal allocation: The optimal power allocation has three cases, with λ selected to satisfy the average power constraint.

4 Fading BCCs: Ergodic Secrecy Capacity Region

The fading BCC is treated as a parallel Gaussian BCC whose subchannels are fading-state realizations, with instantaneous CSI available at the transmitter and receivers. This yields an ergodic secrecy capacity region for broad stationary fading processes and exposes the role of joint channel-gain statistics.

  • System model: The fading BCC includes multiplicative fading gains and additive Gaussian noise, with instantaneous CSI known to the transmitter and receivers.
  • Ergodic performance: With no delay constraints, the secrecy capacity region is averaged over channel states and is called the ergodic secrecy capacity region.
  • Capacity reduction: Each fading realization forms a Gaussian BCC, so the fading BCC can be viewed as a parallel Gaussian BCC with fading states as subchannels.
  • General fading processes: Corollary 5 establishes the fading BCC secrecy capacity region for general stationary ergodic fading processes, including temporal correlation and non-Gaussian fading.
  • Correlation effects: The capacity region depends on the joint correlation between the two channel-gain processes, even when their marginal distributions are identical.
  • Secrecy opportunity: Positive secrecy rate can be achieved whenever the set of states where receiver 1 is stronger than receiver 2 has nonzero probability.
  • Optimal power allocation: CSI enables instantaneous power adaptation, and Corollary 6 gives three cases of optimal allocation for the secrecy-capacity boundary.

5 Fading BCCs: Outage Performance

The outage analysis addresses block-fading systems where target rates must be met within each block, allowing outages when a target rate pair is not achieved. The paper derives power allocations that minimize outage under long-term power constraints and considers variants with confidential or common rates.

  • Outage formulation: Outage is declared when a target common-confidential rate pair is not achieved in a fading block.
  • Power constraint: The source adapts power to each instantaneous channel state while satisfying a long-term average power constraint across blocks.
  • Optimization principle: The outage-minimizing allocation prioritizes channel states requiring less power and then serves states requiring more power until the total power is exhausted.
  • Optimal allocation: For a given target rate pair, the optimal power allocation is a threshold solution that serves only selected fading states.
  • Confidential-only case: The outage formulation also yields the optimal allocation for transmitting only confidential information by removing the common-rate requirement.
  • Constant common rate: When both common and confidential rates are required in every block, the common rate imposes a power requirement that must hold for all channel states.

6 Numerical Results

Numerical results illustrate how the three allocation cases shape the secrecy-capacity boundary and how channel-aware power allocation affects secrecy and outage performance. The results show that secrecy benefits from favorable channel states but remains constrained by unfavorable receiver ordering.

  • Noise-level trade-off: Reducing σ2 improves confidential rate R1 but decreases common rate R0 because receiver 2’s channel becomes worse.
  • Confidential-message allocation: When only confidential information is transmitted, power concentrates on states with small |h2|2 and large |h1|2.
  • Secrecy-favorable states: The source transmits only when receiver 1’s channel is better than receiver 2’s channel.
  • CSI and power allocation: Uniform power allocation is not close to secrecy capacity at the examined SNRs, whereas exact CSI is important for achieving higher secrecy rate.
  • Outage behavior: Outage probability decreases as the confidential target rate decreases but has a nonzero lower threshold for fixed target rate.
  • Outage comparisons: A small positive common rate can cause a large increase in outage probability, while optimized power allocation significantly reduces outage relative to equal allocation.

7 Conclusions

The paper establishes secrecy-capacity results for parallel, Gaussian, and fading broadcast channels with confidential messages, including optimal power allocations and outage performance.

  • Parallel BCC: The secrecy capacity region of the parallel BCC is established using a converse proof showing independent inputs across subchannels are optimal.This result supports the region characterization for parallel subchannels.
  • Parallel Gaussian BCC: The secrecy capacity region of the parallel Gaussian BCC is established, with optimal power allocations characterized for its boundary.The allocations achieve the boundary of the secrecy capacity region.
  • Gaussian BCC: The secrecy capacity region of the Gaussian BCC is established, complementing the discrete memoryless BCC result of Csiszár and Körner.The Gaussian result is presented alongside the earlier discrete memoryless characterization.
  • Fading BCC: The results are applied to obtain the ergodic secrecy-capacity region of the fading BCC and the power allocations achieving its boundary.The paper states that these results generalize recently obtained secrecy-capacity results.
  • Fading BCC: The fading BCC outage performance is studied through power allocation minimizing the probability that target rates are not achieved.The outage criterion concerns target-rate achievement.

A Proof of Theorem 1

The proof of Theorem 1 derives an outer bound for the parallel BCC and shows that product-form distributions with independent subchannel components suffice.

  • Independence conclusion: Because each bound term depends only on a single-subchannel distribution, product-form distributions lose no optimality.The converse concludes by restricting attention to distributions with independent components and per-subchannel channel laws.
  • Auxiliary-variable construction: The proof introduces auxiliary variables for each subchannel and imposes the Markov chain Q_l → U_l → X_l → (Y_l, Z_l).The variables are constructed using a time-sharing random variable and per-subchannel auxiliaries.
  • Rate bounds: The common and confidential rates are bounded using Fano’s inequality, chain-rule arguments, and the perfect-secrecy condition.The resulting bounds are collected into the outer-bound region.
  • Outer bound: The outer bound consists of rate pairs satisfying the stated inequalities over admissible joint distributions.The union is taken over distributions involving the parallel channel variables.

B Proof of Theorem 2

The proof of Theorem 2 applies Gaussian-input constructions and entropy-power and Jensen inequalities to establish the relevant Gaussian parallel-BCC bounds.

  • Achievability: Achievability uses Gaussian input distributions with separate common and confidential-message power components on selected subchannels.For subchannels in A, Q_l and X′_l are assigned Gaussian distributions with powers p_l0 and p_l1.
  • Converse: The converse applies the theorem’s rate bounds and follows analogous steps to derive the Gaussian upper bounds.The proof invokes the entropy power inequality for one term.
  • Converse: Jensen’s inequality and convexity of log(2x + c) justify intermediate inequalities in the Gaussian converse.The cited steps also use a prior bound to complete the derivation.
  • Power decomposition: The converse proof concludes after defining power components for subchannels in A and its complement.The component definitions separate common-message and total powers according to subchannel membership.

C Proof of Theorem 3

The proof of Theorem 3 solves the power-allocation optimization by considering three cases and selecting roots or boundary values under a power constraint.

  • Case analysis: The optimization is divided into three cases, each using an objective function whose maximizing power allocation is determined separately.The proof applies a stated lemma before treating the cases.
  • Root selection: In the cases, optimal power components are selected as nonnegative roots, largest roots, or boundary values of specified equations.The proof distinguishes positive roots, zero or negative roots, and cases requiring maximization over the remaining component.
  • Power constraint: The Lagrange parameter λ is chosen to satisfy the power constraint in each optimization case.The Lagrangian is explicitly introduced in the case analyses.
  • Case 3: For the third case, a parameter α between 0 and 1 is chosen to satisfy the common-rate condition after the preceding cases are considered.The proof states that such an α must exist in the relevant situation.
Loading cs/0701024v1…