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Secrecy Capacity Region of a Multi-Antenna Gaussian Broadcast Channel with Confidential Messages

Ruoheng Liu, H. Vincent Poor

arXiv:0709.4671v1cs.IT

TL;DR

The paper asks how two users can receive independent confidential messages over a multi-antenna Gaussian broadcast channel. It combines secret dirty-paper coding with a computable Sato-type outer bound, showing that the achievable boundary matches the bound and establishing the secrecy capacity region. Numerical examples show simultaneous positive rates for both users under information-theoretic secrecy.

  • Problem

    The paper studies reliable and confidential delivery of independent messages to two users over a multi-antenna Gaussian broadcast channel.

  • Method

    It develops a secret dirty-paper coding scheme and a computable Sato-type outer bound for the MGBC-CM.

  • Results

    The achievable secret dirty-paper coding boundary is consistent with the Sato-type outer bound, establishing the secrecy capacity region of the MGBC-CM.

  • Takeaways & Limitations

    Both users can achieve strictly positive rates simultaneously under the information-theoretic secrecy requirement.

Abstract

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In wireless data networks, communication is particularly susceptible to eavesdropping due to its broadcast nature. Security and privacy systems have become critical for wireless providers and enterprise networks. This paper considers the problem of secret communication over the Gaussian broadcast channel, where a multi-antenna transmitter sends independent confidential messages to two users with information-theoretic secrecy. That is, each user would like to obtain its own confidential message in a reliable and safe manner. This communication model is referred to as the multi-antenna Gaussian broadcast channel with confidential messages (MGBC-CM). Under this communication scenario, a secret dirty-paper coding scheme and the corresponding achievable secrecy rate region are first developed based on Gaussian codebooks. Next, a computable Sato-type outer bound on the secrecy capacity region is provided for the MGBC-CM. Furthermore, the Sato-type outer bound prove to be consistent with the boundary of the secret dirty-paper coding achievable rate region, and hence, the secrecy capacity region of the MGBC-CM is established. Finally, two numerical examples demonstrate that both users can achieve positive rates simultaneously under the information-theoretic secrecy requirement.

I. INTRODUCTION

The paper studies confidential communication from a multi-antenna transmitter to two users over a Gaussian broadcast channel, where each user must decode its own message reliably and confidentially. It develops a secret dirty-paper coding scheme, derives a computable outer bound, and establishes the secrecy capacity region.

  • Problem: The MGBC-CM models independent confidential messages sent by a multi-antenna transmitter to two users, each of whom must receive its own message reliably and confidentially.The model is motivated by wireless broadcast exposure and the need for secure communication.
  • Problem: In a single-antenna Gaussian broadcast channel, the inferior user's secrecy rate is zero because the channel is degraded.The problem then reduces to the scalar Gaussian wiretap channel for the superior user.
  • Main insight: Sufficiently separated multiple antennas allow the transmitter to communicate with both users at nonzero confidential rates under suitable conditions.This addresses whether simultaneous confidential communication to both users is possible.
  • Contributions: The paper develops a secret dirty-paper coding scheme with Gaussian codebooks and double-binning, yielding an achievable secrecy rate region.The scheme supports joint encoding while preserving confidentiality.
  • Contributions: A computable Sato-type outer bound is consistent with the boundary of the achievable region, establishing the secrecy capacity region of the MGBC-CM.The paper also presents numerical examples in which both users achieve positive rates simultaneously under information-theoretic secrecy.

B. Important Channel Parameters for the MGBC-CM

The MGBC-CM characterizes secrecy through generalized eigenvalue parameters and defines achievable secret communication using stochastic encoding and reliability and equivocation criteria. Its secrecy capacity region is established by matching secret dirty-paper coding achievability with a Sato-type converse, with positive rates for both users under a linear-independence condition.

  • Important Channel Parameters: λ1 and λ2 are defined as largest generalized eigenvalues of reciprocal matrix pencils involving the two users’ channel vectors.λ1 uses (I + PhhH, I + PggH), while λ2 reverses the pencil order.
  • Definitions: A secret code consists of two confidential message sets, a stochastic encoder, and deterministic decoding functions at the two receivers.The encoder maps message pairs probabilistically to channel input sequences.
  • Definitions: Reliability is measured by maximum error probability, while secrecy is measured through equivocation rates for each confidential message at the other user.Each receiver decodes its own message while serving as the secrecy observer for the other message.
  • Main Result: The secrecy capacity region is established by secret dirty-paper coding achievability and a Sato-type outer-bound converse.The outer bound is consistent with the boundary of the achievable region.
  • Main Result: Both users achieve positive secrecy rates if and only if λ1 > 1 and λ2 > 1; linear independence of h and g ensures this condition.The corresponding single-user corner rates are log2 λ1 for user 1 and log2 λ2 for user 2.

IV. SECRET DPC CODING SCHEME AND ACHIEVABILITY PROOF

The paper develops an achievable secrecy-rate region for the MGBC-CM by combining prior BC-CM coding results with double-binning and a secret dirty-paper coding construction.

  • Secret DPC construction: The secret DPC construction employs Gaussian codebooks and identifies a capacity-achieving input covariance matrix for the MGBC-CM.The construction begins by separating the channel input into two random vectors and selecting corresponding auxiliary variables.
  • Prior BC-CM inner bound: Prior BC-CM results use joint encoding with Slepian-Wolf binning and random binning to preserve message confidentiality.The achievable secrecy rates include mutual-information penalties associated with the unintended receiver and with jointly encoded auxiliaries.
  • Prior BC-CM inner bound: The double-binning inner bound is defined over auxiliary variables V1 and V2 with factorization p(v1,v2)p(x|v1,v2)p(y1,y2|x).The resulting bounds constrain each confidential rate by a Marton-type sum-rate expression minus information leaked to the other user.
  • Achievability result: Confidentiality makes joint encoding cost an additional mutual-information penalty compared with Marton’s general broadcast-channel region.The paper characterizes this as paying “double” for joint encoding at the transmitter.

B. Secret DPC Scheme for the MGBC-CM

The secret DPC scheme converts the double-binning inner bound into a Gaussian construction whose covariance-based achievable region can be evaluated directly.

  • Secret DPC scheme: The general double-binning achievable region does not clearly specify how to choose auxiliary variables V1 and V2 constructively.The paper notes that implementing the scheme therefore requires guessing the density p(v1,v2,x).
  • Secret DPC scheme: The paper combines DPC with double-binning to develop the secret DPC achievable secrecy-rate region for the MGBC-CM.The construction uses Gaussian codebooks and separates the channel input into U1 and U2.
  • Secret DPC scheme: The S-DPC region is parameterized by covariance matrices KU1 and KU2 associated with the two channel-input components.The achievable region is obtained by selecting these matrices and evaluating the resulting rate inequalities.
  • Secret DPC scheme: The paper uses the same auxiliary random-variable choices as classical DPC but changes the codebook and coding structure to use double-binning.Thus, the secrecy construction preserves the DPC variable design while adding a secrecy-oriented binning structure.
  • Achievability proof: The selected covariance construction satisfies the channel input power constraint.The achievability proof derives the corresponding rate expressions after substituting the covariance choices into the S-DPC bounds.
  • Achievability proof: The resulting rate pairs are achievable using the S-DPC scheme, with the proof also covering the reversed user roles.The paper further reports a rank-2 capacity-achieving input covariance matrix in the relevant result.

V. SATO-TYPE OUTER BOUND AND CONVERSE PROOF

The paper constructs a computable Sato-type outer bound for the MGBC-CM and proves that it coincides with the secret DPC achievable boundary, establishing the secrecy capacity region.

  • Outer bound: The paper develops a Sato-type outer bound applicable to discrete memoryless and Gaussian broadcast channels with confidential messages.A computable Gaussian version is then derived specifically for the MGBC-CM.
  • Converse proof: The resulting equality establishes the secrecy capacity region of the MGBC-CM.The argument uses orthogonal generalized-eigenvector directions and bounds on the two outer-bound terms.
  • Outer bound: The outer bound depends only on the marginal channel distributions, so channels with identical marginals have the same secrecy capacity region.The Gaussian family is obtained by allowing arbitrarily correlated noise variables with correlation parameter ρ satisfying 0 ≤ |ρ| ≤ 1.
  • Outer bound: The bound evaluates secrecy at each user individually while allowing the users to decode their messages cooperatively.For user 1, a genie supplies the eavesdropped signal to the intended receiver, enabling a wiretap-channel argument.
  • Gaussian specialization: The Gaussian outer bound is computable as a function of the noise correlation ρ and input covariance matrix KX.The paper defines the corresponding rate region by taking the union of rate pairs satisfying the outer-bound inequalities.
  • Converse proof: The Sato-type outer bound coincides with the secrecy capacity region after choosing an appropriate correlation parameter ρo.The proof establishes the needed relationship between the input covariance matrix KX and the scalar parameter α.
  • Converse proof: For every feasible input covariance matrix KX with tr(KX) ≤ P, the paper identifies an α ∈ [0,1] linking the outer-bound and achievable parameterizations.Generalized eigenvalues and their normalized eigenvectors are used to complete the correspondence.

VI. NUMERICAL EXAMPLES

Two numerical examples illustrate the MGBC-CM secrecy capacity region and compare it with time-sharing and simplified DPC schemes.

  • Examples: Under the real-alphabet condition, all calculated rate values are divided by 2.The paper assumes real input and output alphabets and real attenuation vectors for this normalization.
  • Examples: The examples calculate and depict secrecy capacity regions for two MGBC-CM channel instances with total power P = 10.The first instance uses h = [1.5, 0]T and g = [1.801, 0.872]T; the second uses h = [1.414, 1.414]T and g = [0.4, 1.959]T.
  • Simultaneous secrecy rates: Both users can achieve positive rates simultaneously under the information-theoretic secrecy requirement.This observation holds in the example where every component of user 1’s attenuation vector is strictly less than the corresponding component for user 2.
  • Comparisons: Both figures compare the Sato-type outer bound with secrecy-rate regions achieved by time-sharing and simplified DPC schemes.The comparisons are shown for the channel instances in (81) and (82).
  • Comparisons: Time-sharing is strictly suboptimal for providing the secrecy capacity region.In each time fraction, the MGBC-CM reduces to a Gaussian MISO wiretap channel, with τ1 + τ2 = 1 and τ1P1 + τ2P2 = P.

VII. CONCLUSION

The paper establishes the secrecy capacity region for the multi-antenna Gaussian broadcast channel with confidential messages. It shows that multiple transmit antennas can support strictly positive confidential rates for both users under suitable channel conditions.

  • VII. CONCLUSION: The model considers a two-user Gaussian broadcast channel with confidential messages and a transmitter equipped with t antennas.Each user has a single antenna.
  • VII. CONCLUSION: A secret dirty-paper coding scheme and a computable Sato-type outer bound are developed for the MGBC-CM.These provide the achievable-region and converse components of the capacity analysis.
  • VII. CONCLUSION: The boundary of the secret dirty-paper coding rate region is consistent with the Sato-type outer bound, establishing the secrecy capacity region.The conclusion identifies this agreement as the basis for obtaining the capacity region.
  • VII. CONCLUSION: Unlike the single-antenna Gaussian BC-CM case, both users can achieve strictly positive confidential rates through a multiple-antenna Gaussian broadcast channel.The result applies when the attenuation vectors imposed on the users are linearly independent.
  • VII. CONCLUSION: Multiple transmit antennas make information-theoretic secrecy more practical and attractive at the physical layer of wireless networks.The conclusion connects the channel result to wireless-network design.

APPENDIX I

The appendix reviews generalized eigenvalue properties and Rayleigh’s quotient, then applies them to secrecy-rate expressions. It derives positivity and monotonicity properties used in the paper’s analysis.

  • Generalized eigenvalue preliminaries: A generalized eigenvalue problem determines nontrivial solutions of Ae = λBe, with generalized eigenvectors associated with the satisfying eigenvalues.The appendix assumes A is Hermitian and B is Hermitian and positive definite.
  • Generalized eigenvalue preliminaries: For Hermitian A and positive-definite Hermitian B, generalized eigenvalues are real and eigenvectors are B-orthogonal.These are stated as properties of the generalized eigenvalue pencil.
  • Rayleigh quotient: The Rayleigh quotient is maximized and minimized by generalized eigenvectors corresponding to the largest and smallest generalized eigenvalues, respectively.The appendix denotes these vectors by emax and emin and the eigenvalues by λmax and λmin.
  • Application to secrecy analysis: When h and g are linearly independent, the relevant generalized eigenvalues satisfy λ1 > 1 and λ2 > 1.The weaker inequalities λ1 ≥ 1 and λ2 ≥ 1 also hold without that linear-independence condition.
  • Application to secrecy analysis: The appendix proves that γ1(α) is nondecreasing on α ∈ [0, 1].The proof uses endpoint properties and continuity of the associated expression.
  • Secret DPC: The secret DPC scheme differs from classical DPC by using a double-binning code structure.The appendix introduces this structure before detailing encoding and covariance-based derivations.

(Double-Binning Scheme):

The double-binning scheme organizes randomly generated codewords into message bins and sub-bins, then uses joint stochastic encoding and a mapping to channel inputs. Its derivation checks power and rate expressions using covariance matrices.

  • Codebook construction: Independent codebooks are generated according to p(vk), and each message bin is partitioned into sub-bins containing additional codewords.The construction uses message indices wk, sub-bin indices jk, and codeword indices lk.
  • Encoding: To transmit a message pair, the encoder randomly selects a sub-bin and codeword for user 1, then selects a sub-bin and jointly typical codeword for user 2.The joint stochastic selection implements the double-binning structure.
  • Encoding: The encoder succeeds with probability close to 1 for large n when the selected codewords satisfy the required joint typicality condition.Each sub-bin contains 2^nI(V1;V2) codewords under the stated construction.
  • Encoding: The channel input sequence is generated from the selected codewords through the mapping p(x|v1, v2).This completes the stochastic encoding operation for the message pair.
  • Rate and power analysis: The derivation verifies the power constraint using independent U1 and U2 and their covariance matrices KU1 and KU2.The subsequent rate calculations use covariance-dependent expressions involving h and g.
  • Rate and power analysis: The appendix applies successive dirty-paper encoding results and Lemma 2 to obtain the desired secrecy-rate expression.The mutual-information identity and bounds (101)–(104) are combined in this step.

SECTION V DERIVATIONS

The derivations establish a Sato-type outer bound, show Gaussian inputs optimize the resulting expressions, and connect the bound to the closed-form secrecy-rate region. Auxiliary lemmas analyze eigenvalue-based rate terms.

  • Sato-type outer bound: The Sato-type argument replaces the original outputs with channels having the same relevant marginal error and equivocation behavior under a common codebook.The replacement yields the Sato-type outer bound on R1, with the R2 bound following by symmetry.
  • Sato-type outer bound: The secrecy requirement and Markov-chain relations are used to bound confidential-message rates through mutual-information expressions involving both users’ outputs.The derivation applies conditioning and entropy inequalities before invoking the Sato replacement.
  • Gaussian optimality: The Gaussian optimality result restricts attention to zero-mean Gaussian X with covariance matrix KX when evaluating the outer bounds.The resulting bounds are then written in terms of ρ and KX.
  • Capacity-region conclusion: The derived rate region is written as RMG, and the paper obtains a closed-form secrecy-capacity result under information-theoretic secrecy.The derivation includes the corresponding rate and equivocation analyses.
  • Eigenvalue-based analysis: Continuity and endpoint inequalities guarantee an α ∈ [0, 1] satisfying L(KX, α) = 0 for a given KX.This lemma uses L(KX, 0) ≥ 0 and L(KX, 1) ≤ 0.
  • Eigenvalue-based analysis: Rayleigh-quotient arguments characterize γ1(α), γ2(α), and their corresponding normalized eigenvectors through generalized eigenvalue pencils.The proof uses the largest generalized eigenvalue and associated vector for the relevant pencil.
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