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The Secrecy Capacity of the MIMO Wiretap Channel
Frédérique Oggier, Babak Hassibi
TL;DR
The paper asks how to compute perfect secrecy capacity for arbitrary-antenna MIMO wiretap channels, where confidential communication must remain unknown to an eavesdropper. It develops matching achievability and converse arguments for the non-degraded broadcast channel and obtains the capacity as the legitimate user’s capacity minus the eavesdropper’s capacity. The result also identifies regimes where the legitimate channel dominates or no positive secrecy capacity is achievable.
Problem
The paper addresses perfect secrecy capacity for multiple-antenna wiretap channels, extending beyond prior degraded-channel and special-case results.
Method
The paper combines an achievability analysis with a different converse technique for non-degraded MIMO broadcast channels, optimizing over transmit covariance and noise correlation.
Results
The perfect secrecy capacity equals the legitimate user’s capacity minus the eavesdropper’s capacity for arbitrary transmit and receive antenna numbers.
Takeaways & Limitations
When all legitimate links are better, the capacity is the difference of the two capacities; when all eavesdropper links are better, no positive secrecy capacity can be achieved.
Takeaways & Limitations
The converse’s correlation construction requires A to satisfy I − AA∗ ≻ 0, while an alternative bound uses I − AA∗ ⪰ 0.
Abstract
from arXiv · showhide
We consider the MIMO wiretap channel, that is a MIMO broadcast channel where the transmitter sends some confidential information to one user which is a legitimate receiver, while the other user is an eavesdropper. Perfect secrecy is achieved when the the transmitter and the legitimate receiver can communicate at some positive rate, while insuring that the eavesdropper gets zero bits of information. In this paper, we compute the perfect secrecy capacity of the multiple antenna MIMO broadcast channel, where the number of antennas is arbitrary for both the transmitter and the two receivers.
1 Introduction
The paper studies perfect secrecy in arbitrary-antenna MIMO wiretap channels, where a transmitter communicates confidentially with a legitimate receiver despite an eavesdropper. It develops a converse for the non-degraded broadcast channel and computes the secrecy capacity.
- Motivation: Wireless communication uses a shared medium favorable to eavesdropping, motivating an information-theoretic security model based on Wyner’s wire-tap channel.
- Information-theoretic confidentiality: Perfect secrecy requires reliable confidential communication while the eavesdropper obtains zero information about the message.
- Previous work: Prior results established capacity differences for degraded channels and several special antenna configurations, but the general MIMO case remained unresolved.
- Contribution: The paper computes perfect secrecy capacity for any numbers of transmit and receive antennas and any SNR regime.
- Contribution: The non-degraded MIMO broadcast channel is handled with a converse proof different from Wyner’s degraded-channel argument.
- Main result: Theorem 1 states the secrecy capacity, while the paper’s achievability and converse sections establish the matching characterization.
2 On the Achievability
The achievability analysis characterizes secrecy rates through the difference between legitimate-user and eavesdropper mutual informations. In the non-degraded case, optimization can be restricted to low-rank transmit covariance matrices.
- Achievability: The achievability section states the perfect secrecy rate and characterizes its maximizing covariance matrices.
- Achievability: For a chosen transmit covariance KX, the difference between legitimate-user and eavesdropper mutual informations can be transmitted secretly.
- Achievability: The optimal covariance matrix ˜KX is low rank when the relevant Hermitian matrix is indefinite or semidefinite.
- Achievability: The low-rank conclusion follows because the optimization has no interior stationary solution and therefore reaches the boundary of the positive semidefinite cone.
3 Proof of the Converse
The converse upper-bounds the secrecy rate through I(X;Y|Z), using a correlated-noise formulation whose objective is concave in KX and convex in A. Optimizing these variables yields a low-rank input covariance and an upper bound matching achievability.
- 3.1 Bound on I(X; Y |Z) and result for the degraded case: The converse reduces the problem to finding an upper bound on I(X;Y|Z), including through an equivalent correlated-noise formulation.The auxiliary noise correlation A can tighten the bound without changing secrecy capacity.
- 3.1 Bound on I(X; Y |Z) and result for the degraded case: Gaussian inputs optimize the conditional mutual information used in the converse upper bound.The optimization is expressed through conditional and joint differential entropies before applying the Gaussian optimality result.
- 3.1 Bound on I(X; Y |Z) and result for the degraded case: In degraded cases, the secrecy capacity equals the difference of the two capacities when the legitimate channel dominates, and is zero when the eavesdropper channel dominates.The latter case occurs when H∗_M H_M ≼ H∗_E H_E; the non-degraded case is the remaining indefinite case.
- 3.2 Minimization over A and maximization over KX: The function ˜I(X;Y|Z) is concave in KX and convex in A, enabling a max-min upper-bound argument.The feasible set for A is convex under I−AA∗≻0.
- 3.2 Minimization over A and maximization over KX: The minimization over A is characterized using invariant subspaces and an algebraic Riccati equation, while the optimal KX is shown to be low rank.The Riccati formulation supplies the structure of the optimizing A and KX.
- 3.3 Conclusion of the converse: The converse is completed when the optimized upper bound matches the achievable secrecy rate.This establishes the required equality between the converse and achievability arguments.
4 Conclusion
The paper computes the perfect secrecy capacity of the MIMO wiretap channel for arbitrary numbers of transmit and receive antennas. It proves that this capacity equals the legitimate receiver’s capacity minus the eavesdropper’s capacity.
- The perfect secrecy capacity is computed for a MIMO broadcast wiretap channel with arbitrary transmit and receive antenna counts.
- The resulting capacity equals the legitimate user’s capacity minus the eavesdropper’s capacity.
Appendix
The appendix develops optimization and matrix-analysis tools for the MIMO secrecy-capacity derivation. It uses Gaussian extremality, unitary invariance, conditional Gaussian distributions, and positive-eigenvalue properties of positive-definite matrices.
- The appendix formulates an entropy optimization under a trace constraint and identifies Gaussian distributions as optimal in the relevant setting.The cited expressions involve maximizing h(X + A, X + B) − h(X + B) subject to Tr(KX) = P.
- Unitary transformations are used because multiplication by a unitary matrix does not change entropy.
- Conditional Gaussian distributions support the entropy calculations, with an auxiliary Gaussian vector U specified through its covariance matrix.
- Products of positive-definite Hermitian matrices have positive eigenvalues, a property established through a similarity transformation to a positive-definite matrix.