An MMSE Approach to the Secrecy Capacity of the MIMO Gaussian Wiretap Channel
© Ronit Bustin et al. 2009
Received: 26 November 2008
Accepted: 21 June 2009
Published: 27 July 2009
This paper provides a closed-form expression for the secrecy capacity of the multiple-input multiple output (MIMO) Gaussian wiretap channel, under a power-covariance constraint. Furthermore, the paper specifies the input covariance matrix required in order to attain the capacity. The proof uses the fundamental relationship between information theory and estimation theory in the Gaussian channel, relating the derivative of the mutual information to the minimum mean-square error (MMSE). The proof provides the missing intuition regarding the existence and construction of an enhanced degraded channel that does not increase the secrecy capacity. The concept of enhancement has been used in a previous proof of the problem. Furthermore, the proof presents methods that can be used in proving other MIMO problems, using this fundamental relationship.
The secrecy capacity of a wiretap channel, defined by Wyner , as "perfect secrecy" capacity is the maximal rate such that the information can be decoded arbitrarily reliably by the legitimate recipient, while insuring that it cannot be deduced at any positive rate by the eavesdropper.
The problem of characterizing the secrecy capacity of the MIMO Gaussian wiretap channel remained open until the work of Khisti and Wornell  and Oggier and Hassibi . In their respective work, Khisti and Wornell  and Oggier and Hassibi  followed an indirect approach using a Sato-like argument and matrix analysis tools. In  Liu and Shamai propose a more information-theoretic approach using the enhancement concept, originally presented by Weingarten et al. , as a tool for the characterization of the MIMO Gaussian broadcast channel capacity. Liu and Shamai have shown that an enhanced degraded version attains the same secrecy capacity as does the Gaussian input distribution. From the mathematical solution in  it is evident that such an enhanced channel exists; however it is not intuitive why, or how to construct such a channel.
and regardless of the input distribution, the mutual information and the minimum mean-square error (MMSE) are related (assuming real-valued inputs/outputs) by
where stands for the conditional mean of given . This fundamental relationship and its generalizations [8, 9], referred to as the I-MMSE relations, have already been shown to be useful in several aspects of information theory: providing insightful proofs for entropy power inequalities , revealing the mercury/waterfilling optimal power allocation over a set of parallel Gaussian channels , tackling the weighted sum-MSE maximization in MIMO broadcast channels , illuminating extrinsic information of good codes , and enabling a simple proof of the monotonicity of the non-Gaussianness of independent random variables . Furthermore, in  it has been shown that using this relationship one can provide insightful and simple proofs for multiuser single antenna problems such as the broadcast channel and the secrecy capacity problem. Similar techniques were later used in  to provide the capacity region for the Gaussian multireceiver wiretap channel.
Motivated by these successes, this paper provides an alternative proof for the secrecy capacity of the MIMO Gaussian wiretap channel using the fundamental relationship presented in [8, 9], which results in a closed-form expression for the secrecy capacity, that is, an expression that does not include optimization over the input covariance matrix, a difficult problem on its own due to the nonconvexity of the expression . Thus, another important contribution of this paper is the explicit characterization of the optimal input covariance matrix that attains the secrecy capacity. The proof presented here provides the intuition regarding the existence and construction of the enhanced degraded channel which is central in the approach of . Furthermore, the methods presented here could be used to tackle other MIMO problems, using the fundamental relationships shown in [8, 9].
2. Definitions and Preliminaries
where is the secrecy capacity under a total power constraint (2), and is the secrecy capacity under a per antenna power constraint. As shown in [2, 7], characterizing the secrecy capacity of the general MIMO Gaussian wiretap channel (1) can be reduced to characterizing the secrecy capacity of the canonical version (7). For full details the reader is referred to , and [17, Theorem 3].
Our first observation regarding the relationship given in (12) is detailed in the following lemma.
The proof of the lemma is given in Appendix A.
3. The Degraded MIMO Gaussian Wiretap Channel
This is due to the independence of the line integral (A.3) on the path in any open connected set in which the gradient is continuous .
The error covariance matrix of any optimal estimator is upper bounded (in the positive semidefinite partial ordering between real symmetric matrices) by the error covariance matrix of the joint Gaussian estimator, , defined in (11), for the same input covariance. Formally, , and thus one can express as follows: , where is some positive semidefinite matrix.
4. The General MIMO Gaussian Wiretap Channel
where , and . Since the matrix is positive definite, the problem of calculating the generalized eigenvalues and the matrix is reduced to a standard eigenvalue problem . Choosing the eigenvectors of the standard eigenvalue problem to be orthonormal, and the requirement on the order of the eigenvalues, leads to an invertible matrix , which is -orthonormal. Using these definitions we turn to the main theorem of this paper.
Following [7, Lemma 2], we may assume that is (strictly) positive definite. We divide the proof into two parts: the converse part, that is, constructing an upper bound, and the achievability part-showing that the upper bound is attainable.
Equality will be attained when the second integral equals zero. Using the upper bound in (29) we present two possible proofs that result with the upper bound given in (30). The more information-theoretic proof is given in the sequel, while the second, the more estimation-theoretic proof, is relegated to Appendix B.
The upper bound given in (29) can be viewed as the secrecy capacity of an MIMO Gaussian model, similar to the model given in (7), but with noise covariance matrices and and outputs and respectively. Furthermore, this is a degraded model, and it is well known that the general solution given by Csiszár and Körner , reduces to the solution given by Wyner  by setting . Thus, (29) becomes
where the third inequality is according to (15), and the last two transitions are due to Theorem 1, (16). This completes the converse part of the proof.
We now show that the upper bound given in (30) is attainable when is Gaussian with covariance matrix , as defined in (23). The proof is constructed from the next three lemmas. We first prove that is a legitimate covariance matrix, that is, it complies with the input covariance constraint (8).
Proof of Lemma 3.
which is the result attained in (33). This concludes the proof of Lemma 3.
Proof of Lemma 4.
Thus, concluding the proof of Lemma 4.
where the first equality is due to Lemma 3, and the second equality is due to Lemma 4. Thus, the upper bound given in (30) is attainable using the Gaussian distribution over , and , defined in (23). This concludes the proof of Theorem 2.
5. Discussion and Remarks
The alternative proof we have presented here uses the enhancement concept, also used in the proof of Liu and Shamai , in a more concrete manner. We have constructed a specific enhanced degraded model. The constructed model is the "tightest" enhancement possible in the sense that under the specified transformation, the matrix is the "smallest" possible positive definite matrix, that is, both and .
The specific enhancement results in a closed-form expression for the secrecy capacity, using . Furthermore, Theorem 2 shows that instead of we can maximize the secrecy capacity by taking an input covariance matrix that "disregards" subchannels for which the eavesdropper has an advantage over the legitimate recipient (or is equivalent to the legitimate recipient). Mathematically, this allows us to switch back from to , and thus to show that , explicitly defined, is the optimal input covariance matrix. Intuitively, is the optimal input covariance for the legitimate receiver, since under the transformation, , it is for the sub-channels for which the legitimate receiver has an advantage and zero otherwise.
This work has been supported by the Binational Science Foundation (BSF), the FP7 Network of Excellence in Wireless Communications NEWCOM++, and the U.S. National Science Foundation under Grants CNS-06-25637 and CCF-07-28208.
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