Efficient Numerical Methods for Secrecy Capacity of Gaussian MIMO Wiretap Channel
aa r X i v : . [ c s . I T ] F e b Efficient Numerical Methods for Secrecy Capacityof Gaussian MIMO Wiretap Channel
Anshu Mukherjee ∗ , Björn Ottersten † , and Le Nam Tran ∗∗ School of Electrical and Electronic Engineering, University College Dublin, IrelandEmail: [email protected]; [email protected] † Interdisciplinary Centre for Security, Reliability and Trust, University of Luxembourg, LuxembourgEmail: [email protected]
Abstract —This paper presents two different low-complexitymethods for obtaining the secrecy capacity of multiple-inputmultiple-output (MIMO) wiretap channel subject to a sum powerconstraint (SPC). The challenges in deriving computationallyefficient solutions to the secrecy capacity problem are due to thefact that the secrecy rate is a difference of convex functions (DC)of the transmit covariance matrix, for which its convexity is onlyknown for the degraded case . In the first method, we capitalizeon the accelerated DC algorithm, which requires solving asequence of convex subproblems. In particular, we show that eachsubproblem indeed admits a water-filling solution. In the secondmethod, based on the equivalent convex-concave reformulationof the secrecy capacity problem, we develop a so-called partialbest response algorithm (PBRA). Each iteration of the PBRAis also done in closed form. Simulation results are provided todemonstrate the superior performance of the proposed methods.
Index Terms - MIMO, secrecy capacity, sum power constraint,convex-concave, closed form solution.
I. I
NTRODUCTION :Wireless communication is an integral part of our modernlife. Due to its broadcasting nature, information sent overwireless channels is vulnerable to security breach. Significantmeasures and techniques have been developed by both indus-try and academia to address this critical issue. Particularly,cryptography is a conventional method to ensure data securityin wireless networks. In recent years, physical layer securityhas received growing interest as a promising alternative toaddressing wireless security. While cryptographic methodsare based on computational complexity and implemented inhigh network layers, physical layer security is concerned withexploiting distinguishing properties of wireless channels toachieve secure communication.The wiretap channel (WTC), in which an eavesdropper aimsto decode to the message exchanged by a pair of legitimatetransceivers, represents a fundamental information-theoreticmodel for physical layer security. The secrecy capacity of theWTC was first studied by Wyner in [1]. Since Wyner’s seminalpaper, the WTC has been extended, covering various scenarios.In particular, the secrecy capacity of the Gaussian WTC wasstudied [2]. The use of multiple antennas at transceivers in con-temporary wireless communications systems naturally givesrise to the so-called multiple-input multiple-output (MIMO)Gaussian WTC. The secrecy capacity of Gaussian MIMOWTC has received significant interests since the late 2000s. In this regard, there have been many significant results in theliterature, which are discussed as follows.The analytical solution for the Gaussian multiple-inputsingle-out (MISO) WTC where both the eavesdropper and thelegitimate receiver have a single antenna was proposed in [3].When the channel state information is perfectly known, thesecrecy capacity of MIMO WTC was characterized in [4], [5],[6]. Particularly, explicit expressions for optimal signaling forGaussian MIMO WTC are possible under some special cases[7], [8], [9]. Power minimization and secrecy rate maximiza-tion for MIMO WTC was studied in [10] using a differenceof convex functions algorithm (DCA). More recently, a low-complex solution for Gaussian MIMO WTC was proposed in[11] using the equivalent convex-concave reformulation of thesecrecy capacity problem.In this paper we consider the problem of finding the secrecycapacity-achieving input covariance for Gaussian MIMO WTCsubject to a sum power constraint (SPC). The case of generallinear transmit covariance constraints such as per-antennapower constraint is studied in [12]. In particular, we developtwo low-complexity methods to solve the secrecy-capacityproblem for the Gaussian MIMO WTC. The first methodis an accelerated difference of convex functions algorithm(ADCA) [13] where each subproblem is found in closedform. In the second method, we propose an efficient iterativemethod to calculate the secrecy capacity, which is based onthe equivalent concave-convex reformulation of the secrecycapacity problem. We refer to this proposed method as thepartial best response algorithm (PBRA). The idea of PBRAis to find a saddle point of the concave-convex problem,for which efficient numerical methods are also derived. Weremark that the method presented in [11] is a double-loopiterative algorithm, while the proposed PBRA in this paperonly requires a single loop.
Notation:
We use bold uppercase and lowercase lettersto denote matrices and vectors, respectively. C M × N denotesthe space of M × N complex matrices.To lighten the no-tation, I and define identity and zero matrices respec-tively, of which the size can be easily inferred from thecontext. H † and H T are Hermitian and ordinary transposeof H , respectively; H i,j is the ( i, j ) -entry of H ; | H | isthe determinant of H ; Furthermore, we denote the expectedvalue of a random variable by E [ . ] . For x ∈ R N [ x ] + = max( x , , max( x , , · · · max( x N , (cid:3) . The i th unitvector (i.e., its i th entry is equal to one and all other entriesare zero) is denoted by e i . The notation A (cid:23) ( ≻ ) B means A − B is positive semidefinite (definite). diag( x ) creates adiagonal matrix whose diagonal elements are taken from x . k H k denotes the Frobenius norm of H .II. S YSTEM M ODEL
A. MIMO Wiretap Channel Model
We consider a MIMO WTC, where Alice wants to transmitinformation to the legitimate receiver Bob in presence of Eve,the eavesdropper. The number of antennas at Bob, Alice andEve is denoted by N t , N r , and N e , respectively. H b ∈ C N r × N t is the channel matrix between Alice and Bob. The channelmatrix between Bob and Eve is denoted by H e ∈ C N e × N t .Thereceived signals at Bob and Eve are respectively expressed as y b = H b x + z b (1a) y e = H e x + z e (1b)where, x ∈ C N t × represents the transmitted signal ; z b ∈ C N r × ∼ CN ( , I ) and z e ∈ C N e × ∼ CN ( , I ) are theadditive white Gaussian noise at Bob and Eve, respectively.In this paper H b and H e are assumed to be quasi-static andperfectly known at Alice and Bob. For a given input covariancematrix X = E { xx † } (cid:23) , where E {·} is the statisticalexpectation, the maximum secrecy rate (in nat/s/Hz) betweenAlice and Bob is given by [6] C s ( X ) = ln | I + H b XH † b | | {z } f b ( X ) − ln | I + H e XH † e | | {z } f e ( X ) + . (2)The secrecy capacity under the sum power constraint (SPC)is written as maximize X (cid:23) C s ( X ) (3a) subject to tr( X ) ≤ P (3b)where P is the total transmit power. We remark that theproblem (3) is non-convex in general, and thus, it is verydifficult to find a globally optimal solution. However, if thechannel is degraded, i.e. H † b H b (cid:23) H † e H e , (3) becomes convexbut off-the-shelf solvers cannot be used to solve it. In thisregard, the equivalent minimax reformulation of (3) presentedin the next subsection is more numerically useful. B. Minimax Reformulation
It is interesting to note that the secrecy capacity of MIMOWTC in (2) is equivalent to the following minimax optimiza-tion problem C s = min Q ∈Q max X ∈X f ( Q , X ) , log | I + Q − ¯ HX ¯ H † || I + H e XH † e | (4)where ¯ H = [ H T b , H T e ] T and Q ∈ C N R × N E . The sets Q and X are defined as X = { X | X (cid:23) ; tr( X ) = P } (5)and Q = (cid:26) Q | Q (cid:23) ; Q = (cid:20) I ¯ Q ¯ Q † I (cid:21)(cid:27) . (6) Compared to (3), (4) is more tractable since the objective of (4)is concave with X for a given Q and convex with Q for a given X . In particular, we can compute the secrecy capacity and theoptimal signaling by finding the saddle point of f ( Q , X ) .III. P ROPOSED A LGORITHMS
In this section we present two low-complexity methods forfinding the secrecy capacity of the MIMO WTC. The firstmethod is a result of applying an ADCA to (3) and the secondone is based on a finding a saddle point of (4).
A. ADCA for Solving (3)
To solve (3), we propose a simple but efficient methodderived based on the obvious observation that C s ( X ) is aDC function, which naturally motivates the use of DCA. Inthis work we apply the ADCA presented in [13]. The ideais that from the current and previous iterates, denoted by X n and X n − respectively, we compute an extrapolated point Z n using the Nesterov’s acceleration technique: X n + ( t k − /t k +1 (cid:0) X n − X n − (cid:1) , where t is the acceleration parameter.We remark that the specific t -update in Line 4 is a condition toguarantee the convergence of the iterative process as describedin [14]. Since C s ( X ) is possibly non-convex for a generalMIMO WTC, Z n can be a bad extrapolation and a monitoris required. Specifically, if Z n is better than one of the last q iterates, then Z n is considered a good extrapolation andthus will be used instead of X n to generate the next iterate.Thus, the ADCA is generally non-monotone . The algorithmicdescription of ADCA for solving (3) is outlined in Algorithm1. Note that the subproblem in (7) is achieved by linearizing f e ( X ) around V n and by omitting the associated constantsthat do not affect the optimization. In Algorithm 1, q is anynon-negative integer and γ n is the minimum of the secrecyrate of the last q iterates. We remark that the case when q = 0 reduces to the conventional DCA, which is exactly the sameas the AO method in [15]. Algorithm 1
ADCA for solving (3) Initialization: W = X ∈ X , t = √ , q : integer. for n = 1 , , . . . do Update: X n = arg max X ∈X f b ( X ) − tr (cid:0) ∇ f e ( W n − ) X (cid:1)| {z } ¯ f ( X ; W n − ) (7)where ∇ f e ( X ) = H † e (cid:0) I + H e XH † e (cid:1) − H e t n +1 = √ t n Z n = X n + t n − t n +1 (cid:0) X n − X n − (cid:1) γ n = min (cid:0) C s ( X n ) , C s ( X n − ) , . . . , C ( X [ n − q ] + ) (cid:1) W n = ( Z n if C s ( Z n ) ≥ γ n and Z n is feasible X n otherwise end for Output: X n o implement Algorithm 1, we need to efficiently solve (7),which is explicitly written as maximize X (cid:23) ln (cid:12)(cid:12) I + H b XH † b (cid:12)(cid:12) − tr (cid:0) Φ n − X (cid:1) (8a) subject to tr( X ) = P (8b)where Φ n − = H † e (cid:0) I + H e W n − H † e (cid:1) − H e . We now showthat (8) admits a water-filling solution To begin with, let usform the partial
Lagrangian function associated with (7) as L ( X , µ ) = ln | I + H b XH † b | − tr (cid:0) ¯ Φ n − X (cid:1) + µP (9)where ¯ Φ n − = Φ n − + µ I N t and µ ≥ is the Lagrangianmultiplier. Let ¯ X , ¯ Φ / n − X ¯ Φ / n − and rewrite the Lagrangianfunction as a function of ¯ X as L ( ¯ X , µ ) = ln (cid:12)(cid:12) I + H b ¯ Φ − / n − ¯ X ¯ Φ − / n − H † b (cid:12)(cid:12) − tr( ¯ X ) . (10)To derive the dual function, the following lemma is in order[11]. Lemma 1.
For a given µ ≥ , let ¯ Φ − / n − H † b H b ¯ Φ − / n − = VΣV † be the eigen value decomposition (EVD) of ¯ Φ − / n − H † b H b ¯ Φ − / n − , where V ∈ C N t × N t is unitary, Σ =diag( σ , σ , . . . , σ r , , . . . , , and r is the rank of ¯ Φ − / n − H b .Then the solution to the problem max ¯ X (cid:23) L ( ¯ X , µ ) is given by X = ¯ Φ − / n − V ¯ ΣV † ¯ Φ − / n − (11) where ¯Σ = diag (cid:0) [1 − σ ] + , . . . , [1 − σ r ] + , , . . . , (cid:1) . Next, to solve (8) we need to find the optimal value of µ which can be done by a bisection search. We skip thedetails here for the sake of brevity. The convergence proof ofAlgorithm 1 is provided in [12]. The idea is to show that thesequence { γ n } increasing and the feasible set X is compactand convex. Thus, there exists a convergent subsequence, theaccumulation point of which is then proved to be a stationarypoint. B. Partial Best Response Method for Solving (4)
The second proposed method is an iterative algorithmto find the saddle point of the concave-convex problem in(4). Suppose ( X n − , Q n − ) has been computed at the n -thiteration. Then X n is found as X n = arg max X ∈X f ( Q n , X )= arg max X ∈X log | Q n − + HXH † | − log | I + H e XH † e | . (12)In words, X k is the best response to Q n − as usual . Nowgiven X n , due to the concavity of the term log | Q + HXH † | ,the following inequality holds f ( Q , X n ) ≤ log | Q n − + HX n H † | + tr( Ψ n ( Q − Q n − )) − log( Q ) − log | I + H e X n H † e | , ∀ Q ∈ Q . , ¯ f ( Q , X n ) . (13)where Ψ n = ( Q n − + HX n H † ) − . Note that the aboveinequality is tight when Q = Q n − . Next, Q n is obtainedas Q n = arg min Q ∈Q ¯ f ( Q , X n ) = arg min Q ∈Q tr( Ψ n Q ) − log | Q | . (14) That is to say, Q n is found be the best response to X n using an upper bound of the objective. The proposed solution forfinding the secrecy capacity is summarized in Algorithm 2. Algorithm 2
PBRA for solving (4) Input: Q ∈ Q , ǫ > for n = 1 , . . . do Update X n according to (12) Update Q n +1 according to (14) end for Output: X n We remark that the iterative method presented in [11] alsoaims to find the saddle-point of (4). However, it is a double-loop algorithm where a lower bound of f ( Q n , X ) is used forthe X -update. In contrast, Algorithm 2 is a single-loop onewhere the X -update is exact. The efficient methods for the X -update and Q -update are detailed in the following subsections. X -update: To compute X n as in Line 3 of Algorithm2, we need to solve (12) which is a convex problem. Sincethe projection onto X can be done in closed form, we canapply an accelerated projected gradient method (APGM) [16]to solve it efficiently, which is described as follows. To avoidconfusion we use the superscript to denote the iteration countof the APGM. Suppose Y ( k ) , the extrapolated point at iteration k , is available. The next iterate X ( k ) is found as X ( k ) = p X (cid:16) Y ( k ) + 1 β ∇ f (cid:0) Q n − , Y ( k ) (cid:1)(cid:17) (15)where β is a step size and p X ( ¯ X ) denotes the projection of ¯ X onto X . The gradient of f ( Q n − , X ) is given by ∇ f (cid:0) Q n − , X (cid:1) = (cid:0) H † (cid:0) Q n − + HXH † (cid:1) − H (cid:1) − (cid:0) H † e (cid:0) I + H e XH † e (cid:1) − H e (cid:1) . (16)For a given point ¯ X , the projection p X ( ¯ X ) is mathematicallystated as maximize X (cid:23) (cid:8)(cid:13)(cid:13) X − ¯ X (cid:13)(cid:13) (cid:12)(cid:12) tr( X ) = P (cid:9) (17)which admits a closed-form solution as X = U diag (cid:0)(cid:2)(cid:2) ¯ σ (cid:3) + − τ (cid:3) + (cid:1) U † (18)where ¯ X = U diag( ¯ σ ) U † be the eigenvalue decomposition of ¯ X , ¯ σ = [¯ σ , ¯ σ , . . . , ¯ σ r ] where r is the rank of ¯ X , and τ isthe unique number such that P ri =1 max( (cid:2) ¯ σ i (cid:3) + − τ,
0) = P .The APGM for solving (12) is outlined in Algorithm 3. Notethat a proper step size can be found by a backtracking linesearch as done in Lines (4)-(7).Starting from the step size ofthe previous iteration, the idea of the backtracking line searchis to reduce it by a factor of θ until (7) is met. That is, we tryto find a lower quadratic approximation of the objective at thecurrent iterate. It is shown in [16] that 3 achieves the optimalconvergence rate of O (1 /k ) . Q -update: A closed-form solution is also possible forthe Q -update. Specifically,we can partition Ψ n into Ψ n = (cid:20) Ψ n, Ψ n, Ψ Hn, Ψ n, (cid:21) . (19)To lighten the notation, we will drop the subscript n onwards.Now, let Ψ Ψ † = U Ψ ¯ Σ Ψ U † Ψ be the eigenvalue decom- lgorithm 3 Accelerated projected gradient method for solv-ing (12) Input: Y (1) = X (0) = X n − , η > , θ > , ξ = 1 . for k = 1 , , . . . do β = η k − /θ repeat β ← θβ X ( k ) = p X (cid:0) Y ( k ) + β ∇ f (cid:0) Q n − , Y ( k ) (cid:1)(cid:1) until f ( Q n − , X ( k ) ) ≥ f ( Q n − , Y ( k ) ) + (cid:10) ∇ f ( Q n − , Y ( k ) ) , ( X ( k ) − Y ( k ) ) (cid:11) − β (cid:13)(cid:13) X ( k ) − Y ( k ) (cid:13)(cid:13) ξ k +1 = 0 . p ξ k ) ; η k = β Y ( k +1) = X ( k ) + ξ k − ξ k +1 (cid:0) X ( k ) − X ( k − (cid:1) end for position of Ψ Ψ † and ¯ Σ Ψ = diag( σ Ψ , σ Ψ , . . . , σ Ψ Nr ) .Then the optimal solution to (14) is given by ¯ Q n +1 = − U Ψ Ξ U † Ψ Ψ (20)where Ξ Ψ = 2 diag (cid:16)
11 + √ σ Ψ ,
11 + √ σ Ψ , . . . ,
11 + p σ Ψ Nr (cid:17) . (21)We refer the reader to [12] for the proof of (20).The main idea behind the convergence proof of Algorithm2 is show the monotonic decrease of the objective sequence f ( Q n , X n ) , which is due to the fact that the term log | Q + HXH † | − log | I + H e XH † e | is jointly concave with Q and X .We refer the interested reader to [12] for further details.IV. N UMERICAL R ESULTS
In this section we provide numerical results to evaluate theproposed algorithms. We adopt the Kronecker model in ournumerical investigation [17]. Specifically, the channel betweenAlice and Bob H b is modeled as H b = ˜ H b R / b , where ˜ H b is a matrix of i.i.d. complex Gaussian distribution with zeromean and unit variance and R / b the corresponding a transmitcorrelation matrix. Here we adopt the exponential correlationmodel whereby [ R b ] i,j = (cid:0) re jφ b (cid:1) | i − j | for a given r ∈ [0 , and φ b ∈ [0 , π ) . The channel between Alice and Eve ismodeled as H e = γ ˜ H e R / e for a given γ > and ˜ H e and R e are generated in the same way. The purpose of introducing γ is to study the secrecy capacity of the MIMO WTC withrespect to the relative average strength of H b and H e . For thesimulation purpose we use φ e = π/ , γ = 0 . and r = 0 . .The codes of all algorithms in comparison were written inMATLAB and executed in a 64-bit Windows PC with 16GBRAM and Intel Core i7, 3.20 GHz. Note that since the noisepower is normalized to unity and thus P is defined to be thesignal to noise ratio (SNR) in this section. In all simulationsresults, the parameter q for Algorithm 1 is taken as q = 5 .In Fig. 1 we show the convergence results of proposedalgorithms over two different SNRs dB and dB for a set ofrandomly generated channel where ( N t , N r , N e ) = (4 , , .For Algorithm 1 we plot the secrecy rate C s ( X n ) where X n . . . SNR = 10 dB C s ( X n ) / f ( Q n , X n ) [11, Alg. 2]Algorithm 1Algorithm 3 . SNR = 5 dBIteration index
Figure 1. Convergence results of iterative algorithms for different SNRsTable IC
OMPARISON OF RUN - TIME ( IN MILLISECONDS ) BETWEEN THE PROPOSEDMETHODS AND [11, A LG . 2]. ( N t , N r , N e )= (4 , ,
2) ( N t , N r , N e )= (4 , , Algorithm SNR 5dB 10dB 5dB 10dBAlgorithm 1 14.6 17.2
Algorithm 2 is the solution returned at the n th iteration. For Algorithm2 we plot the objective f ( Q n , X n ) in (4). We also plot theconvergence of the outer loop of [11, Algorithm 2] for compar-ison. It is clearly seen that the proposed algorithms convergevery fast and all algorithms converge to the same objective.For Algorithm 2, we can also see that f ( Q n , X n ) is indeedan upper bound of C s ( X n ) and it keeps decreasing untilconvergence as expected. We remark that while Algorithm 1 isdeveloped based on a local optimization method, our extensivenumerical results show that Algorithm 1 always achieves thesame solution as Algorithm 2 which is optimal.In Fig. 1 it also appears that all algorithms in comparisonachieve similar convergence rate performance in terms of therequired number of iterations. However, the complexity periteration of each algorithm is different. To achieve a moremeaningful comparison, we present their average actual runtime in Table I. For this purpose, the stopping criterion ofall algorithms is when the corresponding objective is notimproved during the last iterations. The average run timein Table I is obtained from 1000 random channel realizations.We can see that the proposed algorithms, i.e. Algorithms 1 and2, outperform [11, Algorithm 2]; and 2 is slightly better thanAlgorithm 1 when N t > N e and vice versa when N t < N e .We now study how the secrecy capacity scales with thenumber of transmit antennas at Bob and Alice. Fig. 2 plotsthe average secrecy capacity for various numbers of antennasat Eve. The number of transmit antennas at Alice is N r = 4 .As can be seen in Fig. 2, the secrecy capacity increases withthe number of receive antennas at Alice, which is expected. N r A v e r a g e s ec r ec y ca p ac it y N e = 4 N e = 8 N e = 12 N e = 16 N e = 20 Figure 2. Secrecy capacity as a function of N r for different values of N e .The number of transmit antennas is N t = 4 .
10 20 30 40 5033 . N e = 16 N e = 12 SNR(dB) A v e r a g eca p ac it y ( bp s / H z ) Secrecy CapacityAsymptotic Capacity
Figure 3. Impact of N e and SNR on the secrecy capacity and asymptoticcapacity at N t = 6 , N r = 4 Simultaneously, we also observe that the secrecy capacity isreduced when the number of antennas at Eve increases. Inparticular, Eve can significantly decrease the secrecy capacitywhen N e is much larger than N t . This is because the null spaceof H b will increasingly intersect with the space spanned by H e .Finally, Figure 3 plots the average secrecy capacity as afunction of SNR for different numbers of antennas at Eve.The purpose is to understand the gap between the true secrecycapacity and the asymptotic capacity obtained in [5]. Asexpected, the true secrecy capacity converges to the asymptoticcapacity when the SNR is sufficiently high. Again, we canobserve the secrecy capacity decreases when the number ofreceive antennas at Eve increases.V. C ONCLUSION
In this paper, we have proposed two efficient numericalmethods for computing the secrecy capacity and the optimalsignaling of MIMO WTC. In the first method, the secrecycapacity problem is viewed as a DC program and we haveapplied an accelerated version of the celebrated DCA, referredto as the ADCA. In the second method, we have drawn on the convex-concave reformulation of the secrecy capacityproblem and developed the PBRA in which each iteration isdone in closed form. Numerical results have been providedto demonstrate that the proposed solutions can reduce therun time of a known solution by 5 times for the consideredscenarios. Moreover, through extensive numerical experiments,we have observed that the ADCA, albeit inherently a localoptimization method, always achieve the optimal solution. Ourconjecture is that the proposed ADCA is indeed a globaloptimization method, the proof of which is left for future work.A
CKNOWLEDGMENT
This publication has emanated from research supported bya Grant from Science Foundation Ireland under Grant number17/CDA/4786. R
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