Full Rank Solutions for the MIMO Gaussian Wiretap Channel with an Average Power ConstraintThe authors are with the Dept. of Electrical Engineering and Computer Science, University of California, Irvine, CA 92697-2625, USA. e-mail:{afakoori, swindle}@uci.eduThis work was supported by the U.S. Army Research Office under the Multi-University Research Initiative (MURI) grant W911NF-07-1-0318, and by the National Science Foundation under grant CCF-1117983.

Full Rank Solutions for the MIMO Gaussian Wiretap Channel with an Average Power ConstraintThe authors are with the Dept. of Electrical Engineering and Computer Science, University of California, Irvine, CA 92697-2625, USA. e-mail:{afakoori, swindle}@uci.eduThis work was supported by the U.S. Army Research Office under the Multi-University Research Initiative (MURI) grant W911NF-07-1-0318, and by the National Science Foundation under grant CCF-1117983.

Abstract

This paper considers a multiple-input multiple-output (MIMO) Gaussian wiretap channel model, where there exists a transmitter, a legitimate receiver and an eavesdropper, each equipped with multiple antennas. In this paper, we first revisit the rank property of the optimal input covariance matrix that achieves the secrecy capacity of the multiple antenna MIMO Gaussian wiretap channel under the average power constraint. Next, we obtain necessary and sufficient conditions on the MIMO wiretap channel parameters such that the optimal input covariance matrix is full-rank, and we fully characterize the resulting covariance matrix as well. Numerical results are presented to illustrate the proposed theoretical findings.

EDICS: WIN-PHYL, WIN-INFO, MSP-CAPC, WIN-CONT

1Introduction

The broadcast nature of a wireless medium makes it very susceptible to eavesdropping, where the transmitted message is decoded by unintended receiver(s). Recent information-theoretic research on secure communication has focused on enhancing security at the physical layer. The wiretap channel, first introduced and studied by Wyner [1], is the most basic physical layer model that captures the problem of communication security. Wyner showed that when an eavesdropper’s channel is a degraded version of the main channel, the source and destination can achieve a positive secrecy rate, while ensuring that the eavesdropper gets zero bits of information. The maximum secrecy rate from the source to the destination is defined as the secrecy capacity. The Gaussian wiretap channel, in which the outputs at the legitimate receiver and at the eavesdropper are corrupted by additive white Gaussian noise, was studied in [2].

Determining the secrecy capacity of a Gaussian wiretap channel is in general a difficult non-convex optimization problem, and has been addressed independently in [3]-[9]. Oggier and Hassibi [3] and Khisti and Wornell [4] followed an indirect approach using a Sato-like argument and matrix analysis tools. They considered the problem of finding the secrecy capacity of the Gaussian MIMO wiretap channel under the average total power constraint, and a closed-form expression for the secrecy capacity in the high signal-to-noise-ratio (SNR) regime was obtained in [4]. In [5], the rank property of the optimal input covariance matrix for the secrecy rate maximization problem is discussed but the authors were unable to characterize the solution for the general case. For some special cases of the MIMO wiretap channel, where the solution has rank one, the optimal input covariance matrix that achieves the secrecy capacity under the average total power constraint was obtained in [5]-[7].

In [8], Liu and Shamai propose a more information-theoretic approach using the enhancement concept, originally presented by Weingarten et al. [10], as a tool for the characterization of the MIMO Gaussian broadcast channel capacity. Liu and Shamai have shown that an enhanced degraded version of the channel attains the same secrecy capacity as does a Gaussian input distribution. From the mathematical solution in [8] it was evident that such an enhanced channel exists; however it was not clear how to construct such a channel until the work of [9], which provided a closed-form expression for the secrecy capacity under a covariance matrix power constraint. While this result is interesting since the expression for the secrecy capacity is valid for all SNR scenarios, there still exists no computable secrecy capacity expression for the MIMO Gaussian wiretap channel under an average total power constraint.

In this paper, we first investigate the rank property of the optimal input covariance matrix that achieves the secrecy capacity of the general Gaussian multiple-input multiple-output (MIMO) wiretap channel under the average total power constraint, where the number of antennas is arbitrary for both the transmitter and the two receivers. Next, we obtain the optimal input covariance matrix for the case that this optimal covariance matrix is full-rank. Necessary and sufficient conditions to have a full-rank optimal input covariance matrix are characterized as well.

The rest of this paper is organized as follows. In the next section, we describe the assumed mathematical model and revisit the current solution for the wiretap channel under the matrix power constraint. The rank property of the optimal input covariance matrix under the average power constraint is investigated in Section III, and in Section IV we characterize the conditions under which the input covariance matrix that achieves the secrecy capacity of a wiretap channel under the average power constraint is full-rank. In Section V, we discuss some interesting facts regarding the optimal solution, and in Section VI we present numerical results to illustrate the proposed solutions. Finally, Section VII concludes the paper.

Notation:

Vector-valued random variables are written with non-boldface uppercase letters (e.g., ), while the corresponding non-boldface lowercase letter () denotes a specific realization of the random variable. Scalar variables are written with non-boldface (lowercase or uppercase) letters. The Hermtian (i.e., conjugate) transpose is denoted by , the matrix trace by Tr(.), and I indicates an identity matrix. Inequality means that is Hermitian positive semi-definite. The Euclidean norm of the vector is written as . Mutual information between the random variables and is denoted by , is the expectation operator, and represents the complex circularly symmetric Gaussian distribution with zero mean and variance .

2System Model and Prior Works

We begin with a multiple-antenna wiretap channel with transmit antennas and and receive antennas at the legitimate recipient and the eavesdropper, respectively:

where is a zero-mean transmitted signal vector, and are additive white Gaussian noise vectors at the receiver and eavesdropper, respectively, with i.i.d. entries distributed as . The matrices and represent the channels associated with the receiver and the eavesdropper, respectively. Similar to other papers considering the perfect secrecy rate of the wiretap channel, we assume that the transmitter has perfect channel state information (CSI) for both the legitimate receiver and the eavesdropper. For the Gaussian channel, where Gaussian inputs are an optimal choice, the secrecy capacity is given by [3]

where , and is the input covariance matrix.

In [9], the above secret communication problem was analyzed under the matrix power-covariance constraint, defined as

where is a positive semi-definite matrix. An explicit expression for the secrecy capacity under (Equation 3) was obtained via applying the generalized eigenvalue decomposition to the following two positive definite matrices

In particular, there exists an invertible generalized eigenvector matrix such that [14]

where is a positive definite diagonal matrix and represent the generalized eigenvalues. Without loss of generality, we assume the eigenvalues are ordered as

so that a total of are greater than 1. Hence, we can write as

where and . We can partition similarly:

where is the submatrix representing the generalized eigenvectors corresponding to and is the submatrix representing the generalized eigenvectors corresponding to . Using the above notation, the secrecy capacity of the MIMO wiretap channel under the matrix power constraint (Equation 3) can be expressed as [9], [13]:

In this paper, we consider the secrecy capacity problem in (Equation 2) under the average power constraint:

For this constraint, no computable secrecy capacity expression has been derived to date for the general MIMO case. In principle, one would have to find the secrecy capacity through an exhaustive search over the set [10], [13]:

where for any given semidefinite , should be computed as given by ( ?).

In the next section, we investigate the rank of the optimal input covariance matrix that attains . Next, in Section IV, we obtain the optimal under the average power constraint for the case that is full-rank.

3Rank Property of the Optimal Solution under an Average Power Constraint

First, we note that the problem under is equivalent to that under [3]1. Also note that in (Equation 10), this implies that we have instead of .

We are interested in finding the optimal which maximizes the problem (Equation 10). Let us assume that we have found the optimal . Consequently, from ( ?), the optimal input covariance matrix that attains is given by

where and have respectively the same definitions as those of and , given by (Equation 5)-(Equation 8), but here for the pencil . Note that can be rewritten as

where is the projection matrix onto the space of . Moreover, let be the projection onto the space orthogonal to . We have

where (Equation 13) comes from the fact that , and ( ?) is because . Similarly we have

where ( ?) results from ( ?).

: The proof is obtained using ( ?), and by noting that for the optimal we must have . This means that we must have , or equivalently , which completes the proof.

Using Lemma ? in (Equation 12) we have

The following lemma reveals another property of the optimal input covariance matrix under the average power constraint.

We note that any vector which lies in the null space of can be a generalized eigenvector of the pencil , with a generalized eigenvalue equal to 1. Such vectors span the null space of , i.e., . On the other hand, from Lemma ?, . Thus for the optimal , all generalized eigenvectors of the pencil correspond to generalized eigenvalues either bigger than or equal to 1.

Let denote number of generalized eigenvalues of the pencil that are strictly bigger than 1, where again represents the optimal input covariance matrix that attains the secrecy capacity under the average power constraint given by (Equation 9). From Lemma ?, we have .

Subtracting ( ?) from ( ?), a straightforward computation yields

From (Equation 16), we note that , from which it follows that .

The following lemma will be used for the computations in the next section.

The claim is easily proved by considering the rank of both sides of (Equation 16).

4Characterization of the Optimal Full-Rank Solution

In this section, we characterize the secrecy capacity under the average power constraint for a particular class of MIMO Gaussian wiretap channel where the optimal solution is full rank. While necessary conditions for a full-rank were characterized in the previous section, here we derive sufficient conditions as well.

We begin by rewriting problem (Equation 2) here:

The Lagrangian associated with this problem is given by

where and are the Lagrange multipliers. The optimal must satisfy the following KKT conditions:

Using the matrix inversion lemma [14], (Equation 19) can be written as

Left multiplication by and right multiplication by of both sides of (Equation 20) yields

where in obtaining (Equation 21) we have used the KKT condition ( ?), and Eq. ( ?) comes from the fact that (Equation 21) is Hermitian.

We are considering problem (Equation 17) for the case that is strictly positive definite, i.e. , since this is the necessary condition for having a full rank optimal . As we characterize the full rank , the sufficient conditions are revealed as well.

Thus for the case of interest, the KKT conditions ( ?), ( ?) and (Equation 21) are necessary and sufficient conditions for the optimality of . In other words, any that satisfies those conditions is an optimal solution for the problem (Equation 17).

By the KKT condition ( ?), a full rank follows that . Thus, (Equation 21) and ( ?) simplify to

Eq. ( ?) comes directly from ( ?). Please refer to Appendix A for details on obtaining ( ?).

Using ( ?) and ( ?) in (Equation 22), after some simplification we have

where in (Equation 23) we defined

From (Equation 23), we get

Let and be two diagonal matrices, which will be defined soon. Left multiplication by and right multiplication by of both sides of (Equation 25) gives

We find and by solving the following set of equations

which results in

Substituting (Equation 28) into (Equation 26), we have

where we have used (Equation 28) in obtaining (Equation 29).

Using an approach similar to what we used to obtain (Equation 29) from (Equation 22), one can show that2

Define as

where and are respectively given by ( ?) and (Equation 24). Moreover, let

Using (Equation 31) and (Equation 32) in (Equation 29) and (Equation 30), we have

We note from (Equation 33) that

This result implies that, for the optimal , , and commute and have the same eigenvectors [14].

Denote the eigenvalue decomposition of as

Based on the argument made after (Equation 34), we have

where one can easily confirm that . By replacing (Equation 35) and (Equation 36) in (Equation 33), and noting that , and that , and are all diagonal, we have

Recall that the unknown parameters in (Equation 37) are the diagonal matrix and the scalar . Let and denote the th diagonal element of and , respectively. From (Equation 37), we can solve for and obtain

where the Lagrange multiplier is chosen to satisfy the power constraint , as will be explained below.

The proof is obtained by obtaining from (Equation 36), and substituting it back into (Equation 31) and (Equation 24) via straightforward computations.

To fully characterize , one must obtain the Lagrange multiplier such that . As (Equation 38) shows, is monotonically decreasing with :

Thus for any transmit power , there exists a Lagrange multiplier for which . The appropriate value of can be easily found using, for example, the bisection method.

Note that Theorem ? also reveals the necessary and sufficient conditions for having a full rank optimal . While for any transmit power one can find a Lagrange multiplier for which and , to have a full rank , ( ?) must be satisfied. Also recall from ( ?) that . The flowchart in Fig. ? summarizes the steps required to calculate the optimal full-rank for the case of .

In the following we show that for the case of , i.e. when at least one of the eigenvalues of is zero, there exists an equivalent wiretap channel with the same secrecy capacity of the original wiretap channel. For this case, the secrecy capacity is characterized too.

Please refer to Appendix B for the proof and the characterization of the equivalent channels and .

Although at this time a proof is unavailable, we conjecture that the secrecy capacity of any general MIMO wiretap channel is given by an equivalent wiretap channel with and such that , and . If one cannot find such an equivalent wiretap channel, the secrecy capacity of the main channel is zero, i.e., . Besides the case of , the case for which the optimal input covariance matrix is rank 1 is another case supporting this conjecture. For the rank 1 case, it is shown in [6] that the optimal input covariance matrix is given by , where is the normalized principal eigenvector corresponding to the largest eigenvalue of the pencil . For this case, if , the secrecy capacity is , and the equivalent wiretap channel is defined such that and .

5Remarks Regarding

This section discusses some interesting points regarding the optimal solution in ( ?). For the following observations, one can assume when required that both conditions for a full-rank given in Theorem ( ?) are satisfied. Let , , be the generalized eigenvalues of the pencil . Then from the definition [15], , where is the th generalized singular value of .

From (Equation 48) we have:

where (Equation 39) comes from ( ?). Thus, from the definition [14], the generalized eigenvalue matrix of is , which completes the proof.

Note that the definition of singular values here is slightly different from what is given in [4]. Here may be , while this is not the case in [4]. More precisely, from ( ?) for the case of , diagonal elements of are equal to one, as mentioned in Remark ?. The corresponding to tends to .

For , the Lagrange parameter satisfies , as mentioned after Theorem ?. Moreover, for the case of , the matrix given by (Equation 32) will have finite-valued eigenvalues. Thus as , the elements of the diagonal matrix , given by (Equation 38), go to (). Consequently, the first term in ( ?) disappears as .

It is also interesting to consider the optimal solution in ( ?) for the case that the eavesdropper’s channel is very weak, e.g. . For this specific case, the wiretap channel simplifies to a point-to-point MIMO Gaussian link, where the optimal input covariance matrix under the average power constraint is known to be , and is found via the standard water-filling solution, where unitary and diagonal are obtained from the eigenvalue decomposition .

Using (Equation 48) and via simple calculations, we note that for any , Eq. ( ?) can be rewritten as

From ( ?), when then . Next from (Equation 48), . Using these facts in (Equation 32), and after some straightforward calculations, we have , and in (Equation 35). Using these in (Equation 40), we have

where in obtaining (Equation 41) we used the fact that when .

6Numerical Results

Figure 1: Secrecy Capacity versus P_t for n_t=n_e=2 and n_r=3. Solid curve represents secrecy capacity and dotted curve indicates the achievable secrecy rate using a rank-one input covariance matrix.
Figure 1: Secrecy Capacity versus for and . Solid curve represents secrecy capacity and dotted curve indicates the achievable secrecy rate using a rank-one input covariance matrix.

In the first example, we consider a MIMO wiretap channel with , and channel matrices given by

which satisfy . Figure 1 shows the secrecy capacity as a function of transmit power . For comparison, the figure also depicts the achievable secrecy rate using the input covariance matrix , which results to , as shown in [4]-[6]. Note that in this example, the optimal is not full-rank for .

In Figure 2 we consider another example of the case of , here with , and channel matrices given by