SECRECY RATE MAXIMIZATION OF A MISO CHANNEL WITH MULTIPLE MULTI-ANTENNA EAVESDROPPERS VIA SEMIDEFINITE PROGRAMMING Qiang Li and Wing-Kin Ma Department of Electronic Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong E-mail:
[email protected],
[email protected]
ABSTRACT The advances of multi-antenna techniques has recently led to renewed interest in physical-layer secrecy, a meaningful topic that enables us to prevent eavesdroppers from retrieving information intended for a legitimate user through physical layer designs. This paper address a secrecy-rate maximization problem for the scenario of a multi-input single-output channel listened by multiple multiantenna eavesdroppers; e.g., in downlink. This problem is nonconvex and has no analytical solution. Through a careful analysis and reformulation, we show that the secrecy-rate maximization problem has a convex equivalent in form of a semidefinite program (SDP). We also prove that the respective optimal transmit covariance generally can yield a rank-one structure, implying that transmit beamforming is secrecy-rate optimal in the considered scenario. Simulation results are also provided to illustrate that the optimal transmit design solved by our SDP approach can yield significantly improved secrecy rates than an existing closed-form design. Index Terms— secrecy capacity, convex optimization, semidefinite program (SDP). 1. INTRODUCTION Physical-layer secrecy is an information theoretic approach where we can guarantee that a legitimate receiver can achieve a reliable communication with a certain information rate (called secrecy rate) whilst preventing an illegitimate receiver from retrieving any information. Some important concepts of physical-layer secrecy were already developed in the 70’s, first by Wyner [1] and then later by Csiszar and Korner [2]. Recently, there has been a growing interest in this subject. There are two possible reasons for this. First, cryptographic encryption, a technique that has been popular in providing secure communications, would be faced with challenges in wireless networks. Those challenges basically arise from the open nature of the wireless medium, and physical-layer secrecy may offer an alternative to or perhaps complement cryptographic encryption. Second, the advances of the multi-input multiple-output (MIMO) technique today provide physical-layer secrecy with new and exciting opportunities of blocking eavesdropping attempts effectively by using the MIMO degree of freedom. In fact, many of the recent studies on physical-layer secrecy have been concentrated on the MIMO or multiple-input single-output (MISO) scenarios. In the information theory context, the secrecy capacity problem for the scenario of a MIMO channel listened by one MIMO eavesdropper has been considered and addressed in a This work is partially supported by a General Research Fund of Hong Kong Research Grant Council (2150599).
978-1-4244-4296-6/10/$25.00 ©2010 IEEE
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number of independent works; e.g., [3–5], and [6] for the MISO scenario. These advances have also triggered interest from the signal processing field very recently; see, e.g., [7, 8]. For physical-layer secrecy with multiple eavesdroppers, it appears that there are only several studies available at present. An information-theoretic work worth mentioning is that by Liang et al. [9], who extended one-eavesdropper results and formulated achievable secrecy rates for various multi-eavesdropper scenarios. However, those secrecy-rate results do not generally admit closedform solution. In particular, the achievable secrecy rate for MIMO channel listened by multiple multi-antenna eavesdroppers appears to be a hard problem from an optimization perspective. Despite this difficulty, opportunities for numerically tractable solutions of the secrecy rate problem exist when we examine specific scenarios. For MIMO channel with multiple single-antenna eavesdroppers, a concurrent work [10] has established an elegant quasi-convex optimization approach to numerically solve the problem. In this work we focus on the scenario of MISO channel with multiple multi-antenna eavesdroppers. We show that the respective secrecy-rate problem, which is nonconvex, can in fact be turned to a convex problem, namely, a semidefinite program (SDP) whose optimal solution can be efficiently and reliably computed by available algorithms or softwares [11]. The development to be presented also provides an important insight that transmit beamforming is a secrecy-rate optimal strategy for the scenario under consideration. The contributions of this work are specifically described in the next section, together the physical-layer secrecy problem statement. 2. PROBLEM STATEMENT AND SUMMARY OF THIS WORK Our scenario of interest is described as follows. A multi-antenna transmitter intends to send information to a legitimate receiver, which, in this work, is assumed to have one antenna. A number of multi-antenna eavesdroppers try to eavesdrop the transmitted signal. Following the convention in the secrecy capacity literature, we call the transmitter, legitimate receiver, and eavesdroppers as Alice, Bob, and Eves, respectively. The signal models for the Alice-to-Bob and Alice-to-Eves links are respectively given by yb (t) = hH x(t) + n(t), ye,k (t) =
GH k x(t)
+ vk (t),
(1a) k = 1, . . . , K.
(1b)
where x(t) ∈ CNt
transmit vector by Alice;
Nt
number of transmit antennas of Alice;
ICASSP 2010
h ∈ CNt
3. MINIMIZING POWER WITH FIXED SECRECY RATE
MISO Alice-to-Bob channel;
Gk ∈ CNt ×Ne,k
MIMO channel from Alice to kth Eve;
n(t) ∈ C, vm ∈ C
Ne,k
K, Ne,k
zero-mean additive white Gaussian noise; number of Eves, and number of receive antennas of kth Eve, respectively.
Without loss of generality, we assume unit variance of all the noise terms n(t) and vk (t); i.e., E{|n(t)|2 } = 1 and E{vk (t)vkH (t)} = I. Moreover, we denote the transmit covariance by W = E{x(t)xH (t)}. This work considers a secrecy rate maximization problem for (1), formulated by Liang et al. [9] and stated as follows: R (P ) = max W
min
k=1,...,K
log(1 + hH Wh) − log det(I + GH k WGk )
s.t. Tr(W) ≤ P,
W0
(2) where we use W 0 to denote that W is Hermitian positive semidefinite. Here, R (P ) denotes the optimal secrecy rate corresponding to (2) (in bps/Hz), given an average transmit power limit P . The goal of Problem (2) is to find an optimal transmit covariance, denoted here by W , such that the mutual information differences of Bob and Eves are maximized in a worst case sense. Suppose that W is solved. Then, according to recent advances of physical-layer secrecy in information theory [2, 9], one can guarantee that there exist codes such that Bob can enjoy a perfectly secure message from Alice at a rate R (P ) while all Eves can retrieve almost nothing about the message. This work focuses on solving Problem (2). Our main contributions are twofolds: 1. We show that Problem (2) generally has a rank-one optimal transmit covariance solution W . This result is irrespective of the numbers of receive antennas employed by Eves. This implies that transmit beamforming is an optimal strategy under the MISO Alice-to-Bob scenario. 2. We show that Problem (2) can be transformed to an SDP, which is convex and can be numerically solved by available, efficient SDP algorithms [11]. As a side-product, we also solve a variation of Problem (2) where the problem is to minimize the average transmit power subject to secrecy rate requirements. We should mention two related works. In [6], it was shown that Problem (2) has a rank-one, closed-form transmit covariance solution, but for the special case of one Eve only. The case of multiple single-antenna Eves was investigated in [10] (note though that the main strength of that work is actually on MIMO Alice-to-Bob channels with single-antenna Eves). In that case it was also shown that the optimal transmit covariance is generally of rank one. Our work here may be regarded as an extension of those two cases, though we have to resort to a different technique to cope with issues arising from multiple multi-antenna Eves in order to reach the conclusion of rank-one optimal transmit covariance. Furthermore, [10] solves Problem (2) (with single-antenna Eves) via a bisection methodology where a sequence of SDPs need to be solved. We employ a linear fractional SDP reformulation [12] to turn the problem to a single SDP.
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The secrecy-rate maximization problem in (2) is a nonconvex optimization problem, due mainly to the presence of multi-antenna Eves which result in the nonconcave terms − log det(I + Gk WGk ). Let us consider another secrecy-rate problem that will shed light into how the main problem (2) can be solved in a convex fashion: P (R) = min Tr(W) W0
s.t.
min
k=1,...,K
log
1 + hH Wh det(I + GH k WGk )
≥ R,
(3)
that is, fixing a secrecy rate specification R, we minimize the power subject to the constraint that the secrecy rate is at least R. Problem (3) is nonconvex, due to the same reason as in (2). We consider a relaxation of (3) (which will be shown to be tight) by using the following lemma: Lemma 1 Let A 0, A = 0. It holds true that det(I + A) ≥ 1 + Tr(A).
(4)
Moreover, equality in (4) holds if and only if A is of rank one. The proof of Lemma 1 is given in the Appendix. By using Lemma 1 to relax the secrecy rate constraint in (3), we formulate a relaxation of (3) as follows P (R) ≥ min Tr(W) W0
s.t.
min
k=1,...,K
log
1 + hH Wh 1 + Tr(GH k WGk )
≥ R,
(5)
where equality in (5) holds when the optimal solution of (5) is of rank one. Our idea is to show that excluding trivial cases, (5) always yields a rank-1 solution. Problem (5) is convex. By some appropriate manipulations of the variables, one can show that (5) is equivalent to min Tr(W)
W0
s.t. 1 + Tr(hhH W) ≥ 2R (1 + Tr(Gk GH k W)), k = 1, . . . , K
(6)
which is an SDP. We inspected the Karush-Kuhn-Tucker (KKT) conditions of (6) and obtain the following important proposition: ˆ be an optimal solution of the SDP (6). SupProposition 1 Let W ˆ must be of rank ˆ = 01 . Then W pose that (6) is feasible, and that W ˆ one. As a result, W is also an optimal solution to the fixed-secrecyrate power minimization problem in (3). We give the proof of Proposition 1 in the Appendix. In conclusion, this section shows that the fixed secrecy-rate problem (3) generally has a rank-one optimal solution, and that such a rank-one solution can be found by solving the SDP (6). 1 The case of W ˆ = 0 is trivial. In order to have W ˆ = 0 as a feasible solution of (6), one must choose R = 0. Subsequently the optimal solution of (5) must also be 0; i.e., transmitting nothing.
4. SECRECY-RATE MAXIMIZATION VIA SDP Now we turn our attention back to the secrecy-rate maximization in (2), showing how the previously studied fixed secrecy-rate problem can lead us to an effective SDP approach to solving (2). For convenience, we rewrite (2) as γ (P ) =
min
max
W0, k=1,...,K Tr(W)≤P
det(I + GH k WGk ) 1 + hH Wh
(7)
where 0 < γ (P ) ≤ 1 is related to the optimal secrecy rate through the relation R (P ) = log(1/γ (P )). Following the same spirit as in the last section, we apply Lemma 1 to (7) to obtain a lower bound: γ (P ) ≥
min
max
W0, k=1,...,K Tr(W)≤P
1 + Tr(GH k WGk ) γˆ (P ) 1 + hH Wh
min
s.t.
γˆ (P ) ≥
max
k=1,...,K
1
for some Z 0, ζ > 0, and re-express (8) as γˆ (P ) = min Z,ζ
s.t.
ζ + Tr(Gk GH k Z) k=1,...,K ζ + Tr(hhH Z) Z 0, ζ > 0, Tr(Z) ≤ ζP max
min t
(10)
(11a)
Z,ζ,t
Tr(W) + Tr(GH k WGk ) 1 + hH Wh
W = Z/ζ
Problem (10) can be further reformulated as an SDP: (8)
where the bound is tight (or γ (P ) = γˆ (P )) if Problem (8) can be proven to yield a rank-one solution. To show that a rank-one solution for (8) does exist, assume that γ ˆ (P ) has been computed (this aspect will be tackled later) and consider the following problem: W0
Having shown the rank-one optimality of the secrecy-rate maximization, we proceed to solving the secrecy-rate maximization problem, by solving its equivalent form in (8). Problem (8) is a quasiconvex optimization problem, but it can be further simplified to a convex problem. The idea is to apply a transformation
(9)
The aim of (9) is to find a minimum-power W whose secrecy rate is no less than log(1/ˆ γ (P )). One can verify that (8) is equivalent to the fixed-secrecy-rate power minimization problem in (5) or (6), with the rate specification R = log(1/ˆ γ (P )). The insight is that by Proposition 1, Problem (9) generally has a rank-one optimal solution. With this in mind, we prove that: Proposition 2 Suppose that γ ˆ (P ) < 1.2 Then, Problem (8) has a rank-one optimal solution. It also follows that γ (P ) = γˆ (P ), and that a rank-one optimal solution of (8) is also optimal to the secrecy-rate maximization problem in (7) or in (2). ¯ be an optimal solution of (8). We note that W ¯ is also Proof: Let W ˆ feasible to (9). By letting W be an optimal solution of (9), one may deduce that ¯ ≥ Tr(W). ˆ P ≥ Tr(W) ˆ is also feasible to (8). This implies that W ˆ achieves an Hence, W objective value in (8) no less than the optimal value γˆ (P ). Moreover, ˆ must satisfy the constraint in (9). as an optimal solution of (9), W ˆ in (8) is no greater One can then see that the objective value of W ˆ imply than γ ˆ (P ), by the constraint in (9). The two conditions on W ˆ ˆ that W must achieve γˆ (P ) in (8); i.e., W is optimal to (8). ˆ Assume γˆ (P ) < 1. The remaining issue is with the rank of W. Then, W = 0 is not feasible to (9). This enables us to apply Proposition 1 to (9), where we conclude that one can find a rank-one ˆ for (9). As such a rank-one solution is also optimal to (8), W it fulfills the equality γ (P ) = γˆ (P ) (recall Lemma 1) thereby serving as an optimal solution to the original problem (7) as well. 2 The case of γ ˆ (P ) = 1 is trivial. This case corresponds to the zero secrecy rate, and W = 0 is optimal to (8) as well as the main problem (7).
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s.t. ζ + Tr(Gk GH k Z) ≤ t, k = 1, . . . , K H
ζ + Tr(hh Z) = 1 Tr(Z) ≤ ζP Z 0, ζ ≥ 0
(11b) (11c) (11d) (11e)
In the formulation (11), (11a)-(11b) is a standard epigraph reformulation, and (11b) is additionally introduced to fix the denominator of the objective function in (10) which is without loss of generality. The more crucial part lies in replacing ζ > 0 by ζ ≥ 0 in (11e), a mild relaxation. We show that this does not cause a problem: Proposition 3 The SDP (11) solves Problem (10), or equivalently, the secrecy-rate maximization problem in (8) through the relation W = Z/ζ. Proof: The remaining, nontrivial part is when ζ = 0 in (11). Suppose that this is true. Then, by (11d) and Z 0, we must have Z = 0. The constraint (11c) is then violated as a consequence. Thus, a feasible point of (11) must not have ζ = 0. We should mention again that (11), as an SDP, can be very conveniently solved by available convex optimization software [11]. 5. SIMULATION RESULTS AND CONCLUSIONS We provide two simulation examples to test the performance of the proposed algorithm and compare it with projected-channel SVD (PSVD) [10, 13], a closed-form, but suboptimal, algorithm. In the following simulations, the elements of h and Gk are i.i.d. complex Gaussian distributed with mean 0 and variance 1. All results were averaged over 1000 independent channel realizations. Fig. 1 evaluates the relationship between the secrecy rate and the transmit power. The transmit antenna size at Alice is Nt = 10, and there are 4 Eves each using 2 receive antennas (i.e., K = 4, Ne , k = 2 for all k). In the legend, ‘SDP’ stands for the result of secrecy-rate maximization in (2), obtained by solving its SDP equivalent in (11). From the figure we can see that SDP outperforms P-SVD over the entire range of powers tested, and that the performance gap can be as large as 1 bps/Hz for transmit powers between -1dB and 6dB. Fig. 2 shows the impact of the number of Eves on the secrecy rate. We assume Nt = 20, P = 3dB, and Ne,k = 2 for all k. We see that the secrecy rate of P-SVD drops rapidly with the number of Eves K. In fact, P-SVD yields zero secrecy rate for K = 10. This is owing to the fact that P-SVD has used all the degree of freedom
6 P−SVD SDP
4
Secrecy rate (bps/Hz)
Secrecy rate (bps/Hz)
5
3 2 1 0 −11
−8
−5 −2 1 4 Transmit power (dB)
7
5 4 3 2 1 0 1
10
SDP P−SVD
5
9 13 17 Number of Eves (K)
21
Fig. 1: Secrecy rate versus transmit power. Nt = 10, Ne,k = 2 for all k, K = 4.
Fig. 2: Secrecy rate versus the number of Eves. Nt = 20, Ne,k = 2 for all k, P = 3dB.
(DOF) to kill Eves, leaving no DOF for Bob. We also see that for the optimal secrecy rate provided by SDP, its decrease with respect to K is much slower than that of P-SVD. In fact, the optimal secrecy rates give reasonable values even when Nt < K k=1 Ne,k , i.e., when DOF of Alice is smaller than the combined DOF of all Eves. To conclude, this paper has addressed the transmit covariance design problem of maximizing the MISO secrecy rate with multiple multi-antenna eavesdroppers, using an effective SDP approach. We have also shown a physically useful property that transmit beamforming is a secrecy-rate optimal strategy for the considered scenario.
i.e., rank(Y) is either Nt or Nt − 1. For rank(Y) = Nt , (13) can only be satisfied by W = 0. For rank(Y) = Nt − 1, (13) is achieved only when W lies in the nullspace of Y, the dimension of which is one. This means that W has to be of rank one.
6. APPENDIX Proof of Lemma 1: Let λ1 ≥ λ2 ≥ . . . ≥ λr > 0 denote the non-zero eigenvalues of A, with r ≥ 1. We have that det(I + A) = ri=1 (1 + λi ) = 1 + ri=1 λi + i=k λi λk + . . . ≥ 1 + ri=1 λi = 1 + Tr(A) and it can be seen that the equality above holds if and only if r = 1. Proof of Proposition 1: For Problem (6), let Y 0 be the Lagrangian dual variable for the constraint W 0, and μ1 , . . . , μK ≥ 0 be those for the fixed secrecy-rate constraints. The KKT conditions for a primal-dual point (W, Y, μ) to be optimal can be shown to be K H H Y = I + 2R K (12) k=1 μk Gk Gk − ( k=1 μk )hh YW = 0 (13) 1 + Tr(hhH W) ≥ 2R (1 + Tr(Gk GH k W)), ∀k W 0, Y 0, μk ≥ 0, k = 1, . . . , K
(14) (15)
The key to showing the rank-one structure of W lies in (12). Let H B = I + 2R K k=1 μk Gk Gk We see thus has full rank. By letting that B is positive definite, and 1/2 ρ= K as a square root of B, we k=1 μk ≥ 0 and by denoting B show that rank(Y) ≡ rank(B−1/2 YB−1/2 ) = rank(I − ρ(B−1/2 h)(B−1/2 h)H ) ≥ Nt − 1,
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7. REFERENCES [1] A. D. Wyner, “The wiretap channel,” The Bell System Technical Journal, vol. 54, pp. 1355–1387, Oct. 1975. [2] I. Csiszar and J. Korner, “Broadcast channels with confidential messages,” IEEE Trans. Info. Theory, vol. 24, no. 3, pp. 339–348, 1978. [3] A. Khisti and G. Wornell, “The MIMOME channel,” in Proc. 45th Annu. Allerton Conf. Commun., Control, Comput., 2007, pp. 625–632. [4] F. Oggier and B. Hassibi, “The secrecy capacity of the MIMO wiretap channel,” in IEEE Int’l Symp. on Info. Theory, July 2008, pp. 524–528. [5] R. Bustin, R. Liu, H. V. Poor, and S. Shamai (Shitz), “An MMSE approach to the secrecy capacity of the MIMO Gaussian wiretap channel,” in IEEE Int’l Symp. on Info. Theory, June-July 2009, pp. 2602– 2606. [6] A. Khisti and G. W. Wornell, “Secure transmission with multiple antennas: The MISOME wiretap channel,” submitted to IEEE Trans. Inf. Theory, Aug. 2007. Available online in arxiv.org. [7] L. Dong, Z. Han, A. P. Petropulu, and H. V. Poor, “Amplify-andforward based cooperation for secure wireless communications,” in ICASSP 2009, April 2009, pp. 2613–2616. [8] A. L. Swindlehurst, “Fixed SINR solution for the MIMO wiretap channel,” in Proc. ICASSP 2009, April 2009, pp. 2437–2440. [9] Y. Liang, G. Kramer, H. V. Poor, and S. Shamai (Shitz), “Compound wire-tap channels,” in Proc. 45th Annual Allerton Conf. Commun., Control, and Computing, Sept. 2007, pp. 136–143. [10] L. Zhang, R. Zhang, Y.-C. Liang, Y. Xin, and S. Cui, “On the relationship between the multi-antenna secrecy communications and cognitive radio communications,” submitted to IEEE Trans. Commun., Jan. 2009, Available online in arxiv.org. [11] M. Grant, S. Boyd, and Y. Ye, “CVX: Matlab software for disciplined convex programming,” online: http://www.stanford.edu/∼ boyd/cvx. [12] C.-W. Hsin, T.-H. Chang, W.-K. Ma, and C.-Y. Chi, “A linear fractional semidefinite relaxed ML approach to blind detection of 16-QAM orthogonal space-time block codes,” in Proc. IEEE ICC2008, May 2008. [13] R. Zhang and Y.-C. Liang, “Exploiting multi-antennas for opportunistic spectrum sharing in cognitive radio networks,” IEEE J. Select. Topics in Signal Processing, vol. 2, no. 1, pp. 88–102, Feb. 2008.
