IEEE ICC 2012 - Communication and Information Systems Security Symposium
Multiuser Transmit Security Beamforming in Wireless Multiple Access Channels Jinsong Wu, Senior Member, IEEE, Jinhui Chen Bell Laboratories, Alcatel-Lucent Shanghai 201206, P.R. China Email:
[email protected],
[email protected]
Abstract— In this paper, we investigate beamforming approaches for multiple access secrecy channels. This paper investigates the equivalency between beamforming weight solutions under a fixed sum transmit power constraint and those under a fixed secrecy rate when multiple eavesdroppers are available. We propose an advantageous semidefinite-programming based beamforming solution for multiple eavesdroppers in multipleaccess wire-tap channels.
I. I NTRODUCTION To obtain perfect communication secrecy, it is necessary to have the conditional probability of the cryptogram given a message independent of the actual transmitted message [1]. The wire-tap communications for multiple-access channels have been recently discussed. For a common receiver, the multiple access channel with confidential messages has been investigated when two transmitters want their messages secret from each other [2] [3]. In [3]–[6], multiple-access wire-tap channels where transmitters communicate with an intended receiver have been investigated, when there is an external wire tapper from whom it is required to maintain the confidentiality of the messages. In [4]–[6], Tekin et al. considered the case of the wire tapper for a degraded version of a Gaussian multipleaccess (GMAC), and extended Wyner’s measure to multiuser channels through proposing two separate secrecy measures to reflect the level of trust the network may have in each node, and they showed that the secrecy sum capacity can be achieved using Gaussian inputs and stochastic encoders. In [7], two achievable rates were provided for two different scenarios, the Gaussian two-way wire-tap channel (GTWWT), and the binary additive two-way wire-tap channel. In [8], achievable secrecy rate regions are provided for both Gaussian multipleaccess wiretap channel (GGMAC-WT) and the Gaussian twoway wiretap channel (GTW-WT), and it has been shown that in multiple-access scenarios, users can help each other to collectively achieve positive secrecy rates. Recently, there are several secrecy precoding proposals in single user multiple-input multiple-output (MIMO) channels [9]–[11]. In single user multiple-input single-output (MISO) channels, Shafiee and Ulukus restricted channel input to Gaussian signalling, and found that the optimal communication strategy is beamforming. For the cases of imperfect channel state information, robust secrecy precoding approaches for single user MIMO channels have been discussed in [10], [11]. Although a number of fundamental issues on multiuser MIMO secure communications have been discussed [4]–[6], there are lack of investigations in relevant precoding solutions to achieve
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better sum secrecy rate on multiple-access wire-tap MIMO channels. This paper investigates the equivalency between beamforming weight solutions under a fixed sum transmit power constraint and those under a fixed secrecy rate when multiple eavesdroppers are available. One of main contributions of this paper is to provide an advantageous semidefinite-programming based security beamforming solution for multiple eavesdroppers in multiple-access wire-tap channels. The following notations are used: (·)T denotes matrix transpose, (·)∗ conjugate, (·)H matrix conjugate transpose, Hardmard product operator, [A]a,b the (a, b)-th entry (element) of matrix A, tr (·) matrix trace operation, Re (·) real part of the object (matrix or variable), Im (·) imaginary part of the object (matrix or variable), Eα (·) expectation over random variable or random variable set α, φ denotes empty set, and X 0 denotes that X is a positive semi-definite matrix. II. S YSTEM
MODEL AND SECRECY RATE FORMULATION
Consider a communication channel with M source nodes {Sm }, 1 destination node D, L eavesdropper nodes {El }, and each source node Sm with K transmitting antennas for beamforming. The M source nodes {Sm } simultaneously try to transmit confidential messages to destination D while keeping the eavesdropper {El } ignorant of the information. We denote the channel coefficient between the k-th transmit antenna of source (m) Sm and the destination D node as hk , while we denote the channel coefficient between the k-th transmitting antenna of (m,l) source node Sm and the eavesdropper node El as gk . nodes {Sm } transmit {sm } to D, where The source E sm 2 = 1, and the corresponding received signal y at the D node is given by M
T (m) h + w, (1) αm sm y= m=1
and the source nodes {Sm } transmit {sm } to El , and the corresponding received signal zl at the El node is given by M
T (m,l) g + vl , αm sm (2) zl = m=1
T (m) (m) = h1 , ..., hK , g(m,l) = where h(m) T T (m,l) (m,l) (m) (m) (m) , ..., gK , αm = α1 , ..., αK , αk is g1
the beamforming coefficient for the k-th antenna in the source node Sm , w and vl are the complex, circularly symmetric 2 Gaussian noises with zero mean and variance of σ (D) and 2 (E) at the D node and the {El } node, respectively. σl For the case of multiple eavesdroppers, an achievable secrecy rate is given by
, (3) Rs = max 0, min Rd − Re(l) l
where the maximum is again taken over possible input covariance matrices, Rd is the achievable rate of the source(l) destination link and Re is the achievable rate of the link between the source and the l-th eavesdropper [12]. The achievability of 3 for the case of a single source and a single eavesdropper was shown by the use of Gaussian inputs, and the relevant secrecy rate can be interpreted as a worst-case result [12]. It is worth to mention that the above works considered memoryless MIMO channels, in other words, channels in which the outputs at time depend only on the current inputs. Under the context of our model with M source nodes {Sm }, (3) is no longer a secrecy rate but a bound of secrecy sum rate. The relevant achievable sum rate of the source-destination link is given by Rd = I (s1 , ..., sM ; y) ⎞ ⎛ 1+ ⎧ ⎫ ⎜ M ⎨ 2 ⎬⎟ T = log ⎜ ⎟ (m) ⎝ 1 E h αm sm ⎠ 2 ⎩ ⎭ σ (D) m=1
M T ∗ = log 1 + (αm ) A(D) m (αm ) m=1 (D) Am
(m)
m=1 (E,l)
=
subject to: ⎧ ⎛ ⎞⎫ 1+ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ ⎪ ⎟⎪ M ⎪ ⎪ ⎪ ⎜ ⎪ ⎟ T ∗ ⎪ ⎪ ⎜ ⎪ (αm ) A(D) (αm ) ⎟⎪ ⎪ m ⎬ ⎨ ⎜ ⎟⎪ ⎜ m=1 ⎟ log ⎜ min ⎟ Rs , ⎜ 1+ ⎟⎪ l=1,...,L ⎪ ⎪ ⎜ ⎪ ⎟⎪ ⎪ ⎪ ⎜ ⎪ ⎟⎪ ⎪ M ⎪ ⎪ ⎪ ⎝ ⎠ T ∗ ⎪ ⎪ (E,l) ⎪ ⎪ (αm ) Am (αm ) ⎪ ⎭ ⎩
1 2 g(m,l) (E) σl
(6a)
(6b)
m=1
where l = 1, ..., L, M
T
∗
(αm ) (αm ) P (Sum) .
(6c)
m=1
Problem 2: Given P (Sum) , find {αm } , m = 1, ..., M , maximize Rs , subject to: ⎧ ⎛ ⎞⎫ 1+ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ ⎪ ⎪ ⎟⎪ M ⎪ ⎪ ⎜ ⎪ ⎪ ⎟ T ∗ (D) ⎪ ⎪ ⎜ ⎪ ⎪ ⎟ (α ) A (α ) m m ⎪ m ⎬ ⎨ ⎜ ⎟⎪ ⎜ m=1 ⎟ log ⎜ min ⎟ Rs , ⎜ 1+ ⎟⎪ l=1,...,L ⎪ ⎪ ⎜ ⎪ ⎪ ⎟⎪ ⎪ ⎜ ⎪ ⎪ ⎟⎪ M ⎪ ⎪ ⎪ ⎝ ⎠ T ∗ ⎪ ⎪ ⎪ ⎪ ⎪ (αm ) A(E,l) (α ) m ⎭ ⎩ m M
(m) H . The relevant achievable h
Re(l) = I (s1 , ..., sM ; zl ) = ⎞ ⎛ 1+ ⎫ ⎧ ⎟ ⎜ M 2 ⎬⎟ ⎨ ⎜ T 1 log ⎜ ⎟ (m,l) g αm sm ⎠ ⎝ 2 E ⎭ ⎩ (E) m=1 σl
M T ∗ (E,l) = log 1 + (αm ) Am (αm ) where Am
minimize P (Sum) ,
(7a)
(7b)
m=1
2h (σ(D) ) sum rate of the link between the source and the l-th eavesdropper is given by
where
=
1
(4)
Problem 1: Given Rs , find {αm } , m = 1, ..., M ,
(5)
(m,l) H . g
T
∗
(αm ) (αm ) P (Sum) .
(7c)
m=1
Both Problems 1 and 2 are with sum power constraints (6c) and (7c). We may have the following relation: Proposition 1: The solutions of the optimization Problems 1 and 2 are equivalent up to constant coefficients: 1) Find the weights that maximize secrecy rate Rs for a fixed sum transmit power constraint. 2) Find the weights that minimize the transmit power for a fixed secrecy rate Rs . The proof of Proposition 1 is provided in Appendix. According to Proposition 1, if the solution of Problem 1 is found, we can also easily find the solution of Problem 2. IV. O PTIMAL B EAMFORMING
WEIGHT DESIGN FOR MULTIPLE EAVESDROPPERS
The main problems to solve in this paper are to design the beamforming weights of multiple source nodes to maximize the bound of secrecy sum rate for a fixed transmit power, or minimize transmit power for a fixed secrecy capacity. III. E QUIVALENCY
BETWEEN DESIGNS OF MINIMIZING SUM POWER AND MAXIMIZING SECRECY RATE UNDER SUM POWER CONSTRAINT
In this section, we investigate the relations of two problems of sum power minimization and secrecy rate maximization for a sum power constraint.
904
T
∗
Using (αm ) M (αm ) = tr (Xm M), where Xm = (αm )∗ (αm )T , we rewrite Problem 2 as Problem 3: Given P (Sum) , find {αm } , m = 1, ..., M , maximize Rs ,
(8a)
subject to: 1+ 1+
M m=1 M m=1
(D) tr Xm Am
t, (E,l) tr Xm Am
(8b)
where l = 1, ..., L, M
V. P ERFORMANCE
tr (Xm ) == P (Sum) ,
(8c)
m=1
rank (Xm ) = 1, m = 1, ..., M, H
(8d)
Xm == (Xm ) , m = 1, ..., M,
(8e)
Xm 0, m = 1, ..., M,
(8f)
where t = 2Rs . Problem 3 is non-convex. We may perform semidefinite relaxation (SDR) through removing the rank constraints (8d). The problem further becomes Problem 4: Given P (Sum) , find {αm } , m = 1, ..., M , maximize t, subject to: M
(E,l) tr Xm A(D) − tA m m
(9a)
t − 1,
(9b)
m=1
This section shows some relevant numerical results for secrecy beamforming approaches discussed in Section IV. The sum power of all transmitters are calculated using P (Sum) = M Pm . In the case of uniform power allocation, the power of m=1
(k)
the(Sum) k-th transmit antenna of the m-th transmit user is Pm = P MK , where k = 1, ..., K and m = 1, ..., M . Assume that all channel paths experience frequency non-selective Rayleigh fading. Based on semidefinite programming approaches in Section IV, Figure 1 shows the advantageous performance of sum secrecy rate using beamforming under given sum power constraints when two eavesdroppers are available. VI. C ONCLUSION In this paper, we discuss the secrecy beamforming system model for multiple access channels. When multiple eavesdroppers are available, we prove the equivalency between beamforming weight solutions under a fixed sum transmit power constraint and those under a fixed secrecy rate. This paper provides a semidefinite-programming based beamforming solution with advantageous performance for multiple eavesdroppers in multiple-access wire-tap channels.
where l = 1, ..., L, M
tr Xm A(D) P (Sum) , m
P ROOF (9c)
m=1 H
Xm == (Xm ) , m = 1, ..., M,
(9d)
Xm 0, m = 1, ..., M,
(9e)
Proof: Note that (αm )
T
and
where t = 2Rs . Combined with the bisection algorithm as discussed in [13], Problem 4 can be solved iteratively, since it is quasiconvex in each loop within each bisection procedure, where t acts as a constant. Problem 4 can now be efficiently solved by standard interior point algorithms based on semi-definite programming [14]. Note that the above solution is obtained through removing the rank-1 constraint (8d), which might lead to an increased problem dimension. If there are rankone solutions for all {Xm }, the beamforming weights can be obtained using the eigenvector corresponding to the largest eigenvalue of the rank-one matrix, otherwise we may find solutions through randomization methods as discussed in [15]. To discuss the possibilities of rank-one solutions, we have the following result. Proposition 2: For L < M , if the semidefinite Problem 3 is feasible, there is always at least one maximum secrecy rate solution where all {Xm } are rank-one. Problems 3 and 4 can be transformed into the similar forms of (18.17) and (18.24) of [16]. Following the proof strategies in [16], the proof can be completed. If all {Xm } are rank-one, this actually says that Problem 4 is not a real relaxation but an equivalent formulation of of Problem 2, and the solution is an optimal result.
905
(αm )T
A PPENDIX I OF P ROPOSITION 1
∗ + aI A(D) K (αm ) 0 m
(10)
∗ A(D) m + aIK (αm ) 0,
(11)
where a 0 and m = 1, ..., M . This above facts are useful in this proof and the proofs of the rest of the paper. This proposition can be proved through contradiction. We assume that Assumption 1: For Problem 2, there exist a group of optimal weights {αm } , m = 1, ..., M to reach the sum power constraint P (Sum) and achieve optimal secrecy rate Rs . Thus, the M T ∗ sum power constraint agrees (αm ) (αm ) = P (Sum) , m=1
and there exists l1 such that ⎧ ⎛ ⎞⎫ M (D) T ∗ ⎪ ⎪ ⎪ ⎪ 1 + (αm ) Am (αm ) ⎨ ⎜ ⎟⎬ m=1 ⎜ ⎟ log2 ⎝ min ⎠ M l=1,...,L ⎪ (E,l) T ∗ ⎪ ⎪ ⎭ ⎩ 1+ (αm ) Am (αm ) ⎪ ⎛ ⎜ = log2 ⎜ ⎝
1+
m=1
M m=1
1+
M
m=1
⎞
(D)
(αm )T Am (αm )∗ T
(E,l1 )
(αm ) Am
(αm )
∗
⎟ ⎟ = Rs . ⎠
Assumption 2: For Problem 1, given the secrecy rate Rs , there exist a group of optimal weights {αm } , m = 1, ..., M (Sum) . to and reach the minimal sum power constraint P
Thus, the sum power constraint agrees
M m=1
T
∗
(αm ) (αm ) =
(Sum) , and there exists l such that P 2 ⎧ ⎛ ⎞⎫ M (D) T ∗ ⎪ ⎪ ⎪ ⎪ 1+ (αm ) Am (αm ) ⎨ ⎜ ⎟⎬ m=1 ⎟ log2 ⎜ min ⎝ ⎠ M l=1,...,L ⎪ (E,l) T ∗ ⎪ ⎪ ⎭ ⎩ 1+ (αm ) Am (αm ) ⎪
⎛ ⎜ = log ⎜ ⎝
m=1
M
1+
m=1 M
1+
m=1
(D)
T
⎞
∗
(αm ) Am (αm ) (E,l2 )
T
(αm ) Am
(αm )
∗
⎟ ⎟ = Rs . ⎠
(Sum) holds, Assumption 3: The relation of P (Sum) = P and αm = αm , m = 1, ..., M . (Sum) into Now we partition the condition of P (Sum) = P two cases: (Sum) : This contradicts Assumption 1, since 1) P (Sum) < P the secrecy rate Rs could be achieved under sum power (Sum) is not the minimal sum constraint P (Sum) . Thus, P power constraint. (Sum) : There exists a constant a such that 2) P (Sum) > P (Sum) = P (Sum) , where a > 1. Consider the function aP M (D) T ∗ 1 , where b = (αm ) Am (αm ) and f (x) = 1+xb 1 1+xb2
b2 =
m=1
M
T
m=1
(αm )
(E,l ) Am 2
(αm )∗ .
[4] E.Tekin, S.Serbetli, and A.Yener, “On secure signaling for the Gaussian multiple access wire-tap channel,” in Proc. Asilomar Conf. Signal Syst. Comput., Nov. 2005. [5] E.Tekin and A.Yener, “The Gaussian multiple-access wire-tap channel with collective secrecy constraints,” in Proc. IEEE Int. Symp. Inform. Theory, July 2006. [6] ——, “The Gaussian multiple access wire-tap channel,” IEEE Trans.Inform.Theory, vol. 54, no. 12, Dec. 2008. [7] ——, “Achievable rates for two-way wire-tap channels,” in Proc. IEEE Int. Symp. Inform. Theory, June 2007. [8] ——, “The general Gaussian multiple-access and two-way wiretap channels: Achievable rates and cooperative jamming,” IEEE Trans.Inform.Theory, vol. 54, no. 6, June 2008. [9] S. Shafiee and S. Ulukus, “Achievable rates in gaussian MISO channels with secrecy constraints,” in Proc. IEEE International Symposium on Information Theory, June 2007, pp. 2466–2470. [10] A.Mukherjee and A.L.Swindlehurst, “Robust beamforming for security in MIMO wiretap channels with imperfect CSI,” IEEE Trans.Sig.Proc., vol. 59, no. 1, pp. 351–361, Jan. 2011. [11] Q. Li and W.-K. Ma, “Optimal and robust transmit designs for MISO channel secrecy by semidefinite programming,” IEEE Trans.Sig.Proc., vol. 59, no. 8, pp. 3799–3812, Aug. 2011. [12] Y.Liang, G.Kramer, H.V.Poor, and S. (Shitz), “Compound wire-tap channels,” in Proc. 45th Annual Allerton Conf. on Communication, Control, and Computing, Sept. 2007. [13] R. L.Burden and J. Faires, Numerical Analysis, 7th ed. Brooks-Cole, 2000. [14] C.Helmberg, F.Rendl, R.Vanderbei, and H.Wolkowicz, “An interior point method for semidefinite programming,” SIAM J.Optimization, vol. 6, no. 2, pp. 342–361, May 1996. [15] N.D.Sidiropoulos, T.N.Davidson, and Z.-Q. Luo, “Transmit beamforming for physical-layer multicasting,” IEEE Trans.Sig.Proc., vol. 54, no. 6, pp. 2239–2251, June 2006. [16] M.Bengtsson and B.Ottersten, “Optimal and suboptimal transmit beamforming,” Handbook of Antennas in Wireless Communications, CRC Press, Aug. 2001, .
According to (3), we know Rs ≥ 0. Calculate the derivative of (12),
7
(12)
1 where the last inequality incurs due to the fact 1+b 1+b2 = Rs 2 > 1. This determines that the function f (a) is monotonically increasing, and thus f (a) > f (1) and ⎛ log2 (f (a)) > log2 (f (1)) ⎞ = Rs . This leads
log2 ⎝
1+
1+
M
m=1 M
m=1
( αm )T A(D) αm )∗ m ( (E,l2 )
( αm )T Am
α! m =
( αm )∗
6 Average sum secrecy rate
∂f (x) b1 (1 + xb2 ) − b2 (1 + xb1 ) = 2 ∂x (1 + xb2 ) b1 − b2 = 2 > 0, (1 + xb2 )
8
5 4
Optimal beamforming under sum power constraint Uniform power allocation
3 2 1
⎠ > Rs , where
0 14
√
aαm . (13) √ Actually, that is to say α! aαm is another solution of m = Problem 2, in other words, {αm } , m = 1, ..., M are not a group of optimal solution of Problem 2, which creates a contradiction. The proof is done.
R EFERENCES [1] C.E.Shannon, “Communication theory of secrecy systems,” Bell Sys.Tech.Journal, vol. 29, pp. 656–715, Apr. 1949. [2] Y. Liang and H.V.Poor, “Generalized multiple access channels with confidential messages,” in Proc. IEEE International Symposium on Information Theory, July 2006. [3] R.Liu, I.Maric, R.D.Yates, and P.Spasojevic, “The discrete memoryless multiple access channel with confidential messages,” July 2006, pp. 957– 961.
906
16
18
20 22 24 Sum transmit power (dB)
26
28
Fig. 1. Performance of average sum secrecy rate under given sum transmit power for two eavesdroppers, K = 4, M = 3
