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Multiple-Input Multiple-Output Gaussian Broadcast Channels with Confidential Messages
Ruoheng Liu, Tie Liu, H. Vincent Poor, Shlomo Shamai
TL;DR
The paper studies secret communication over a MIMO Gaussian broadcast channel and characterizes its secrecy capacity region. Its result shows that, under a matrix power constraint, both confidential messages can simultaneously achieve their respective maximal secrecy rates.
Problem
The paper studies secret communication over a MIMO Gaussian broadcast channel, including the maximum achievable secrecy rate for a confidential message.
Method
The paper characterizes the secrecy capacity region of the general MIMO Gaussian broadcast channel and uses this region to establish achievability.
Results
Under a matrix power constraint, both confidential messages W1 and W2 can be simultaneously transmitted at their respective maximal secrecy rates.
Takeaways & Limitations
The resulting secrecy capacity region is rectangular, making it suitable for proving simultaneous maximal-rate transmission and achievability.
Abstract
from arXiv · showhide
This paper considers the problem of secret communication over a two-receiver multiple-input multiple-output (MIMO) Gaussian broadcast channel. The transmitter has two independent messages, each of which is intended for one of the receivers but needs to be kept asymptotically perfectly secret from the other. It is shown that, surprisingly, under a matrix power constraint both messages can be simultaneously transmitted at their respective maximal secrecy rates. To prove this result, the MIMO Gaussian wiretap channel is revisited and a new characterization of its secrecy capacity is provided via a new coding scheme that uses artificial noise and random binning.
I. INTRODUCTION
The paper characterizes secrecy capacity for the general two-receiver MIMO Gaussian broadcast channel with confidential messages under a matrix power constraint. Its rectangular region implies that both messages can simultaneously attain their respective maximal secrecy rates, supported by wiretap-channel coding characterizations and S-DPC.
- Channel model: The matrix power constraint is a general positive-semidefinite constraint that includes average total and per-antenna power constraints as special cases.It requires the channel input covariance to satisfy a positive-semidefinite matrix inequality involving S.
- Problem and contribution: The general MIMO Gaussian broadcast channel secrecy capacity region remained open before this paper, which provides a precise characterization under a matrix power constraint.The channel has two confidential messages, each intended for one receiver and secret from the other.
- Main result: The secrecy capacity region under the matrix power constraint is rectangular, so both confidential messages can be simultaneously transmitted at their respective maximal secrecy rates.Each message is intended for one receiver and must remain secret from the other.
- Power-constraint scope: Under the average total power constraint, the secrecy capacity region is generally not rectangular and is described using positive semidefinite matrices B1 and B2 with Tr(B1 + B2) ≤ P.This contrasts with the rectangular matrix-power-constraint result.
- Proof strategy: The result is connected to the MIMO Gaussian wiretap channel, whose secrecy capacity can be achieved by Gaussian random binning or by appropriately chosen prefix coding.The paper uses two wiretap-channel characterizations together with secret dirty-paper coding based on double binning.
II. MIMO GAUSSIAN WIRETAP CHANNEL REVISITED
This section revisits the MIMO Gaussian wiretap channel under a matrix power constraint and develops an alternative secrecy-capacity characterization using prefix coding and artificial noise.
- The section studies the maximum achievable secrecy rate for a confidential message intended for one receiver and kept secret from the other.
- A prior characterization used Gaussian random binning without prefix coding as an optimal strategy for the MIMO Gaussian wiretap channel.
- The paper provides an alternative characterization whose achievability combines Gaussian random binning with prefix coding.
- The construction sets X = U + V with independent Gaussian vectors, where V serves as artificial noise to confuse the eavesdropper.
- Unlike the earlier formulation, the new characterization uses a full input covariance S and permits a nontrivial prefix channel.
- The converse is established using a channel-enhancement argument similar to prior work.
III. MIMO GAUSSIAN BROADCAST CHANNEL WITH CONFIDENTIAL MESSAGES
This section proves that secret dirty-paper coding achieves the rectangular secrecy-capacity region under the matrix power constraint, allowing both confidential messages to attain their maximal secrecy rates simultaneously.
- The converse bounds each confidential-message rate by reducing the broadcast problem to a MIMO Gaussian wiretap channel with the other receiver as eavesdropper.
- Every rate pair in the rectangular secrecy-rate region is achievable by proving achievability at its corner point.
- The achievability scheme uses independent Gaussian precoding signals with covariance matrices B and S − B under the matrix power constraint.
- Choosing B as an optimal solution simultaneously maximizes the two relevant secrecy-rate expressions, making the corner point achievable.
- S-DPC cancels the precoding signal for message W1 while using interference to increase protection for message W2's eavesdropper.
- Both confidential messages can be transmitted simultaneously at their respective maximal secrecy rates.
IV. NUMERICAL EXAMPLES
The numerical-examples section illustrates the secrecy-capacity region and discusses its matrix-optimization formulation under matrix and average total power constraints.
- The section provides numerical examples illustrating the secrecy capacity region of the MIMO Gaussian broadcast channel with confidential messages.
- Under both matrix and average total power constraints, the secrecy-capacity regions are expressed through matrix optimization programs.
- These optimization programs are generally nonconvex, making the region boundaries nontrivial to determine.
- The paper generalizes earlier explicit or specialized results to the general MIMO Gaussian broadcast channel under the matrix power constraint.
- The examples use generalized eigenvalues of a matrix pencil as part of the analysis.
2 H⊺ 2H2S
This subsection states positivity properties for matrices involving the identity and the power-constraint matrix, implying strictly positive generalized eigenvalues.
- Consequently, the generalized eigenvalues φ_j satisfy φ_j > 0 for j = 1, . . . , t.
2 H⊺ 2H2S
The paper characterizes the secrecy capacity and capacity region under a matrix power constraint using generalized eigenvalues, yielding a computable expression and rate-pair conditions.
- 2 H⊺ 2H2S: Theorem 3 characterizes the secrecy capacity for a confidential message intended for receiver 1 and secret from receiver 2 under the matrix power constraint.The characterization uses the generalized eigenvalues greater than 1 of the associated pencil.
- 2 H⊺ 2H2S: The secrecy-capacity expression is computable from the generalized eigenvalues.The paper notes that the relevant generalized-eigenvalue computation can be reduced to standard eigenvalues of a related semidefinite matrix.
- 2 H⊺ 2H2S: The resulting secrecy capacity region consists of nonnegative rate pairs satisfying the stated bounds for the two confidential messages.Corollary 2 gives the region for W1 intended for receiver 1 and W2 intended for receiver 2 under the matrix constraint.
2 H⊺ 1H1S
The numerical comparisons examine secrecy regions under total-power and matrix-power constraints, contrasting S-DPC with zero forcing and standard DPC.
- 2 H⊺ 1H1S: Under the average total power constraint, no computable secrecy-capacity expression was found for the general MIMO case.The secrecy-capacity region can instead be obtained by exhaustive search over positive semidefinite covariance matrices with bounded trace.
- 2 H⊺ 1H1S: Zero forcing is strictly suboptimal to S-DPC in all four plotted channel scenarios.The comparison uses secrecy-capacity regions and zero-forcing achievable regions under the average total power constraint.
- 2 H⊺ 1H1S: If the unintended receiver's channel matrix has full row rank, zero forcing cannot achieve a positive secrecy rate for the corresponding message.S-DPC can still achieve positive secrecy rates for both messages unless the broadcast channel is degraded.
- 2 H⊺ 1H1S: Under the matrix power constraint, the zero-forcing region is contained in the secrecy-capacity region, which is contained in the nonsecrecy DPC capacity region.The plotted regions satisfy RZF_s(H1,H2,S) ⊂ Cs(H1,H2,S) ⊂ RDPC(H1,H2,S).
V. CONCLUDING REMARKS
The paper provides secrecy-capacity characterizations for two confidential messages and explains that matrix constraints permit simultaneous operation at both messages' maximal secrecy rates.
- V. CONCLUDING REMARKS: Each confidential message is intended for one receiver and must remain asymptotically perfectly secret from the other.This is the paper's two-receiver MIMO Gaussian broadcast-channel setting.
- V. CONCLUDING REMARKS: The paper provides precise secrecy-capacity-region characterizations under both matrix and average total power constraints.The conclusion distinguishes the two power-constraint settings explicitly.
- V. CONCLUDING REMARKS: Under the matrix power constraint, both confidential messages can be transmitted simultaneously at their respective maximal secrecy rates.This is the stated principal result for the two-message broadcast channel.
- V. CONCLUDING REMARKS: The proof revisits the MIMO Gaussian wiretap channel and introduces a coding scheme combining artificial vector Gaussian noise with random binning and prefix coding.The artificial-noise covariance is stated to coincide with the transmit-signal covariance.
- V. CONCLUDING REMARKS: The authors state that the resulting understanding of the MIMO Gaussian wiretap channel may help address other multiuser secret-communication problems.
A. Aligned MIMO Gaussian Wiretap Channel
The aligned-wiretap analysis derives a secrecy-capacity characterization under a matrix constraint and extends it to singular and general channel matrices through equivalent-channel constructions.
- A. Aligned MIMO Gaussian Wiretap Channel: For the aligned MIMO Gaussian wiretap channel, the secrecy capacity is characterized under the matrix power constraint.The paper denotes receiver 2 as legitimate and receiver 1 as the eavesdropper in this formulation.
- A. Aligned MIMO Gaussian Wiretap Channel: Achievability combines Gaussian random binning with prefix coding, while the converse uses a channel-enhancement argument.The converse is first proved for square invertible channel matrices and then broadened to the general case.
- A. Aligned MIMO Gaussian Wiretap Channel: The converse for the positive-definite case is completed by enhancing the legitimate receiver while retaining a degraded wiretap channel.The proof establishes the desired inequality and then handles the singular case through the equivalent-channel reduction.
- A. Aligned MIMO Gaussian Wiretap Channel: For singular covariance constraints, an equivalent lower-dimensional aligned channel preserves secrecy capacity and has a strictly positive definite constraint.Only the active θ antennas carry information under the transformed constraint.
- A. Aligned MIMO Gaussian Wiretap Channel: For noninvertible or rectangular channel matrices, singular-value decomposition constructs equivalent channels that preserve secrecy capacity under the same power constraint.The construction first produces square invertible matrices and then applies the characterization.