Filter Design with Secrecy Constraints: The MIMO Gaussian Wiretap Channel
11 Filter Design with Secrecy Constraints:The MIMO Gaussian Wiretap Channel
Hugo Reboredo * , Student Member, IEEE,
Jo˜ao Xavier,
Member, IEEE, and Miguel R. D. Rodrigues,
Member, IEEE
Abstract
This paper considers the problem of filter design with secrecy constraints, where two legitimate parties(Alice and Bob) communicate in the presence of an eavesdropper (Eve), over a Gaussian multiple-input-multiple-output (MIMO) wiretap channel. This problem involves designing, subject to a power constraint,the transmit and the receive filters which minimize the mean-squared error (MSE) between the legitimateparties whilst assuring that the eavesdropper MSE remains above a certain threshold. We consider ageneral MIMO Gaussian wiretap scenario, where the legitimate receiver uses a linear Zero-Forcing (ZF)filter and the eavesdropper receiver uses either a ZF or an optimal linear Wiener filter. We provide acharacterization of the optimal filter designs by demonstrating the convexity of the optimization problems.We also provide generalizations of the filter designs from the scenario where the channel state is knownto all the parties to the scenario where there is uncertainty in the channel state. A set of numerical resultsillustrates the performance of the novel filter designs, including the robustness to channel modeling errors.In particular, we assess the efficacy of the designs in guaranteeing not only a certain MSE level at theeavesdropper, but also in limiting the error probability at the eavesdropper. We also assess the impact ofthe filter designs on the achievable secrecy rates. The penalty induced by the fact that the eavesdropperThis work was supported by Fundac¸ ˜ao para a Ciˆencia e Tecnologia through the research project CMU-PT/RNQ/0029/2009 and through the doctoral grant SFRH/BD/81543/2011. The work of J. Xavier is supportedby Fundac¸ ˜ao para a Ciˆencia e a Tecnologia under Grant [PEst- OE/EEI/LA0009/2011] and Grant CMU-PT/SIA/0026/2009.H. Reboredo is with the Instituto de Telecomunicac¸ ˜oes and the Dept. de Ciˆencia de Computadores da Faculdadede Ciˆencias da Universidade do Porto, Portugal ( email: [email protected] ).J. Xavier is with the Instituto de Sistemas e Rob´otica, Instituto Superior T´ecnico, Lisboa, Portugal ( email:[email protected] ).M. R. D. Rodrigues is with the Department of Electronic and Electrical Engineering, University College London,United Kingdom ( email: [email protected] ). a r X i v : . [ c s . I T ] M a y may use the optimal non-linear receive filter rather than the optimal linear one is also explored in thepaper. Index Terms
Filter Design, Physical-Layer Security, Secrecy, Wiretap, MIMO, ZF, Wiener, MSE, Mutual Infor-mation, Error Probability
I. I
NTRODUCTION
The issues of privacy and security in wireless communication networks have taken on an increasinglyimportant role as these networks continue to flourish worldwide. Traditionally, security is viewed as anindependent feature with little or no relation to the remaining data communication tasks and, therefore,state-of-the-art cryptographic algorithms are insensitive to the physical nature of the wireless medium.However, there has been more recently a renewed interest on physical-layer security which, motivatedby advances on information-theoretic security, calls for the use of physical-layer techniques exploiting theinherent randomness of the communications medium to guarantee both reliable communication betweentwo legitimate parties as well as secure communication in the presence of eavesdroppers.The basis of information-theoretic security, which builds upon Shannon’s notion of perfect secrecy [1],was laid by Wyner [2] and by Csisz´ar and K¨orner [3] who proved in seminal papers that there exist channelcodes guaranteeing both robustness to transmission errors and a certain degree of data confidentiality.In particular, Wyner considered the wiretap channel where two legitimate users communicate in thepresence of an eavesdropper. Wyner characterized the rate-equivocation region of the wiretap channeland its secrecy capacity. Ever since, the computation of the secrecy capacity of a range of communicationschannels has been an important research topic [4].For example, in [5] the authors considered a scenario where both the main and the eavesdropperchannels are additive white Gaussian noise (AWGN) channels. They showed that the secrecy capacity ofsuch so-called Gaussian wiretap channel is equal to the difference between the main and the eavesdropperchannel capacities and, therefore, confidential communications require the Gaussian main channel to havea better signal-to-noise ratio (SNR) than the Gaussian eavesdropper channel.Motivated by the emerging wireless applications, the evaluation of the secrecy capacity of wirelessfading channels with single or multiple antennas at the transmitters, receivers and/or eavesdroppers hasalso attracted considerable attention as well.Space-time signal processing techniques for secure communications over wireless links were introducedin [6]. The outage secrecy capacity of slow fading channels was characterized in [7], where it was shown that fading alone could guarantee information-theoretic security, even when the eavesdropper averageSNR is higher that the legitimate receiver average SNR. In turn, the ergodic secrecy capacity of fadingchannels was independently characterized in [8], [9] and [10]. In [11] Parada and Blahut consideredthe secrecy capacity of several degraded fading channels. The characterization of the secrecy capacity ofmultiple-input-multiple-output (MIMO) channels, which represent a model for multiple-antenna channels,can be found in [12], [13], [14] and [15]. The computation of optimal power allocation policies andinput covariances for the MIMO Gaussian wiretap channel are covered in [16] and [17], respectively.Another key aspect in the MIMO wiretap problem is the availability of channel state information (CSI).This problem is addressed in various works under different CSI assumptions. When the CSI about thevarious channels is assumed to be known to all the parties, several secrecy capacity achieving schemes,based on optimal beamforming designs that leverage the general singular value decomposition (GSVD)of the main and eavesdropper channel matrices, have been proposed (e.g. [15] and [18]). When the CSIabout the eavesdropper channel is assumed to be limited or not available, artificial noise schemes havebeen proposed instead [19], [20], where a fraction of the total power is used for reliable communicationbetween the legitimate transmitter and the legitimate receiver and the remaining fraction of the totalpower is used to jam the eavesdropper. For example, the authors in [21] and [22], set up a problemwhose objective is to determine the minimum transmit power necessary to guarantee a certain quality ofservice (QoS) between the legitimate transmitter and the legitimate receiver – the remaining power outof the total power budget is then used to jam the eavesdropper using artificial noise type of techniques.One key advantage of artificial noise transmission relates to the fact that the eavesdropper channelknowledge is not required. Nonetheless, the idea of transmitting artificial noise in the null space of themain channel in order to degrade the eavesdropper channel has also its limitations. On the one hand, thereis an inherent trade-off between data rate and the ability to impair the eavesdropper [19], so that one maynot take full advantage of the spatial multiplexing ability of MIMO systems. On the other hand, if thenull space of the main channel overlaps considerably with the null space of the eavesdropper channel,the artificial noise approach might lead to limited gains in security.This paper, at the heart of the novelty of the contribution, addresses the physical-layer securityproblem from the estimation-theoretic rather than the information-theoretic viewpoint. We consider theproblem of filter design with secrecy constraints in the classical MIMO wiretap scenario consisting oftwo legitimate parties that communicate in the presence of an eavesdropper, where the objective is toconceive transmit and receive filters that, subject to a power constraint, minimize the mean-squarederror (MSE) between the legitimate parties whilst assuring that the eavesdropper MSE remains above a certain threshold. Interestingly, this class of problems, which differs from previous approaches inphysical-layer security in the literature (see, e.g., [15], [18], [19], [21] and [22]), represents a naturalgeneralization of filter design without secrecy constraints for point-to-point communications systems(e.g., [23], [24], [25], [26], [27], [28]).One notable merit of this approach, in contrast to the information-theoretic work that relies on non-constructive random-coding arguments to demonstrate that there exist secrecy capacity achieving codes,is that it leads to realizable designs which can be easily implemented in practical systems. Instead,practical secrecy capacity achieving code designs are known only in some scenarios, which include: i)the main channel is noiseless and the eavesdropper channel is a binary erasure channel [29], [30]; ii)both channels are binary input symmetric discrete memoryless channels (DMC) and the eavesdropperchannel is degraded with respect to the main channel – where polar codes are used [31], [32]; and iii)the eavesdropper is constrained combinatorially [33].Nonetheless, it is relevant to pause to reflect on the operational relevance of this new metric, in viewof the fact that it is the norm, in the information-theoretic security literature, to use equivocation ratherthan MSE to measure security. In fact, the use of the MSE in lieu of equivocation does not guaranteeperfect information-theoretic security in the sense of [1], [2] and [3]. We view the design of the filtersbased on the MSE criteria as a means to provide additional confusion in a communications system.The rationale of the new design approach is then based on the fact that some applications require a MSEbelow a certain level to function properly, so that this approach would impair further the performance ofthe eavesdropper by imposing a threshold on its MSE level. Note also that the bit error rate (BER), whichis a very important figure of merit in a communications system, is typically monotonically increasingwith the MSE, so that a threshold on the MSE may also translate into a threshold in the BER.One particular scenario that suits this design approach relates to wireless broadcasting where a ser-vice provider provides different services, e.g. different video streams, to different users/subscribers (seeFigure 1). Here, the service provider (the legitimate transmitter) needs to guarantee that a user that hassubscribed to the service (the legitimate receiver) has access to a high quality version of the video streamwhereas a user that has not subscribed to the service (the so-called eavesdropper) has only access toa very poor quality version of the video stream. The use of a distortion metric, such as the MSE orthe BER, instead of equivocation, is then entirely appropriate for this class of applications, offering analternative to the cryptographic methods used by Content Access (CA) systems [34], [35], [36].It turns out thus that the filter design with secrecy constraints problem is to be understood broadlyas a filter design problem with distortion constraints. However, in order to connect this work with the large body of work of physical- and information-theoretic security whose overarching aim is to impairthe eavesdropper, we – in a somewhat abusive use of language – use the notion secrecy rather thandistortion.This paper is structured as follows: Section II defines the problem. Section III considers the designof the transmit filter when ZF filters are used at both the legitimate and the eavesdropper receivers. Inturn, Section IV considers the design of the transmit filter when the eavesdropper uses an optimal linearfilter while the legitimate receiver is restricted to the use of a ZF receive filter. Section V provides somegeneralizations of the problem of filter design with secrecy constraints, from the scenario where the stateof the channels is known exactly to all the parties (i.e., the legitimate transmitter, the legitimate receiverand the eavesdropper) to the scenario where there is uncertainty in the channel state. Section VI showsvarious numerical results to illustrate the impact of the filter designs on both the reliability and securitycriteria, evaluating, not only the MSE, but also the bit error rate and the achievable secrecy rates yieldedby the designs. The main contributions of the manuscript are summarized in Section VII.
A. Notation
We use the following notation: boldface upper-case letters denote matrices or column vectors ( X ) anditalics denote scalars ( x ); the context defines whether the quantities are deterministic or random. Thenotation M (cid:31) is used to denote a positive definite matrix and M (cid:23) denotes a positive semidefinitematrix. The symbol I represents the identity matrix. The operators (cid:107) · (cid:107) , tr {·} and ∇ represent the l -norm, the trace operator and the gradient operator, respectively. The operators ( · ) † and ( · ) + denote theHermitian transpose operator and the Pseudo-Inverse operator, respectively. The operator E ( · ) representsthe expectation. CN ( µ, Σ ) denotes a circularly symmetric complex Gaussian random vector with mean µ and covariance Σ . II. P ROBLEM S TATEMENT
We consider a communications scenario where a legitimate user, say Alice, communicates with anotherlegitimate user, say Bob, in the presence of an eavesdropper, Eve (see Figure 2).Bob and Eve observe the output of the MIMO channels given, respectively, by: Y M = H M H T X + N M (1) Y E = H E H T X + N E (2) where Y M ∈ C n M and Y E ∈ C n E are the vectors of receive symbols, X ∈ C m is the vector ofindependent, zero-mean and unit-variance transmit symbols, and N M ∈ C n M and N E ∈ C n E arecircularly symmetric complex Gaussian random vector with zero mean and identity covariance matrix .The n M × m matrix H M and the n E × m matrix H E contain the deterministic gains from each mainand eavesdropper channel input to each main and eavesdropper channel output, respectively. The m × m matrix H T represents Alice’s transmit filter.We assume that H M H T and H E H T are full column rank, which implies that n M ≥ m and n E ≥ m .This is necessary to guarantee the existence of some solutions. We further assume that, in a realisticscenario, the channel matrices H M and H E are not a multiple of each other. We also assume thatthe channel state is known by all the parties, i.e. Alice, Bob and Eve have perfect knowledge aboutthe channel matrices H M and H E . This is often a common assumption in the physical layer securityliterature (see e.g. [7] and [38]). The assumption that the legitimate receiver knows the state of the mainchannel and the eavesdropper receiver knows the state of the wiretap channel is realistic, because thereceivers can always estimate the channels in slow fading conditions. The assumption that the transmitterknows the state of the main channel and, more importantly, the wiretap channel or that the legitimatereceiver knows the state of the wiretap channel and the eavesdropper knows the state of the main channelcan be justified in wireless networks where the eavesdropper is another network active user (e.g. in thescenario of Figure 1). In particular, in time division duplex (TDD) environments Alice can estimate thestate of Bob’s and Eve’s channels and inform the receivers accordingly. However, we will also generalizethe framework to incorporate possible channel uncertainties in the sequel.Bob’s and Eve’s estimate of the vector of input symbols are, respectively, given by: ˆX M = H RM Y M (3) ˆX E = H RE Y E (4)where the m × n M matrix H RM and the m × n E matrix H RE represent Bob’s and Eve’s receive filters,respectively.In this setting, we take, as a performance metric, the MSE between the estimate of the input vector The models in (1) and in (2) follow from the more general models ˜Y M = ˜H M H T X + ˜N M and ˜Y E = ˜H E H T X + ˜N E , respectively,where ˜N M and ˜N E are circularly symmetric complex Gaussian random vectors with mean E (cid:16) ˜N M (cid:17) = 0 and E (cid:16) ˜N E (cid:17) = 0 , andcovariance matrices E (cid:16) ˜N M ˜N † M (cid:17) = Σ N M and E (cid:16) ˜N E ˜N † E (cid:17) = Σ N E , respectively, by using pre-whitening filters i.e., Y M = Σ − / N M ˜Y M = Σ − / N M ˜H M H T X + Σ − / N M ˜N M = H M H T X + N M and Y E = H E H T X + N E . These transformations are information lossless [37]. ˆX and the true input vector X given by: MSE = E (cid:104) (cid:107) X − ˆX (cid:107) (cid:105) (5)The objective is to design, for specific receive filter choices, the transmit filter that solves the opti-mization problem: min MSE M = E (cid:104) (cid:107) X − ˆX M (cid:107) (cid:105) (6)subject to the security constraint: MSE E = E (cid:104) (cid:107) X − ˆX E (cid:107) (cid:105) ≥ γ (7)where γ represents an MSE threshold, and to the total power constraint: tr (cid:110) H T H † T (cid:111) ≤ P avg (8)where P avg represents the available power.We restrict our attention to two specific design scenarios: i) the situation where both the legitimatereceiver and the eavesdropper receiver are constrained to obey ZF constraints; and ii) the situation wherethe legitimate receiver uses a ZF filter whereas the eavesdropper receiver uses the optimal linear Wienerfilter. For these receiver filter choices, the optimization problem in (6) – (8) is convex thus enabling thecharacterization of optimal designs; for other receiver filter choices, and to the best of our knowledge,the optimization problem in (6) – (8) is only convex for special scenarios, e.g. the degraded parallelGaussian wiretap channel, or the degraded MIMO wiretap channel (see [39] and [40]) .We recognize that our formulation assumes the so-called eavesdropper to perform a certain linearaction whereas the traditional information-theoretic formulation – in view of the fact that it is based onthe equivocation metric – does not assume the eavesdropper to perform any specific operation. However,in the scenario where the eavesdropper is another user of the network as in Figure 1, it seems appropriateto assume a certain action by this user. We also recognize the fact that a more sophisticated eavesdropperwould possibly leverage nonlinear techniques to estimate the information. This issue is also discussed inthe sequel. We prove the convexity of the filter design with secrecy constraints optimization problem by using the change of variables Z = (cid:16) H T H † T (cid:17) − .This change of variables leads to convex objective functions as well as convex feasible regions when both the legitimate receiver and theeavesdropper receiver use ZF filters (see (17), (18) and (19)) and when the legitimate receiver uses a ZF filter but the eavesdropper receiveruses a Wiener filter (see (43), (44) and (45)). However, such a change of variables does not lead immediately to a convex optimization problemwhen both the legitimate receiver and the eavesdropper receiver adopt the Wiener filter (the feasible region is still convex but the objectivefunction is concave rather than convex). Thus – with the exception of [39] and [40] – it is not entirely clear whether other change of variableslead to a convex optimization problem in such a case. It is also important to note that, and in contrast to the artificial noise approach in [19], [20], [21], [22]and [41], our filter design approach does not impose a limitation on the ability of transmitting informationalong all the dimensions that the MIMO channel has to offer and, therefore, we can expect to achievehigher data rates. However, by imposing a threshold on the eavesdropper MSE we may also naturallyconstraint the performance of the main channel.III. Z
ERO F ORCING F ILTERS AT THE R ECEIVERS
We now consider the scenario where both the legitimate receiver and the eavesdropper receiver useZF filters, thus obeying the ZF constraints given by: H RM H M H T = I (9) H RE H E H T = I (10)The rationale for including the ZF constraints in (9) and (10) is to eliminate crosstalk between the variousstreams (e.g. [42]). Note also that the performance of ZF linear receivers is equivalent to that of optimalWiener linear receivers in the regime of high SNR. Yet, one may still argue that a eavesdropper willalways adopt the optimal linear receive filter (or the optimal non-linear receive filter), rather than thesub-optimal ZF receive filter. These particular cases will be addressed in Sections IV and VII. A. Optimal Receive Filters
Let us consider the design of the receive filters. Bob uses the receive filter that, for any fixed transmitfilter H T , minimizes: MSE M = E (cid:104) (cid:107) X − ˆX M (cid:107) (cid:105) = E (cid:2) (cid:107) X − H RM Y M (cid:107) (cid:3) (11)subject to the ZF constraint in (9) and Eve uses the receive filter that, for any fixed transmit filter H T ,minimizes: MSE E = E (cid:104) (cid:107) X − ˆX E (cid:107) (cid:105) = E (cid:2) (cid:107) X − H RE Y E (cid:107) (cid:3) (12)subject to the ZF constraint in (10).In particular, the receive filters, which follow immediately from (9) and (10), are given by [37]: H ∗ R M = ( H M H T ) + = (cid:16) H † T H † M H M H T (cid:17) − H † T H † M (13) H ∗ R E = ( H E H T ) + = (cid:16) H † T H † E H E H T (cid:17) − H † T H † E (14) The MSEs in the main and eavesdropper channels, upon substituting (13) and (14) in (11) and (12),respectively, are then given by:
MSE M = E (cid:2) (cid:107) X − H ∗ R M Y M (cid:107) (cid:3) = tr (cid:26)(cid:16) H † T H † M H M H T (cid:17) − (cid:27) (15) MSE E = E (cid:2) (cid:107) X − H ∗ R E Y E (cid:107) (cid:3) = tr (cid:26)(cid:16) H † T H † E H E H T (cid:17) − (cid:27) (16) B. Optimal Transmit Filter
In view of (15) and (16), the form of the optimal transmit filter corresponds to the solution of theoptimization problem: min H T tr (cid:26)(cid:16) H † T H † M H M H T (cid:17) − (cid:27) (17)subject to the constraints: tr (cid:26)(cid:16) H † T H † E H E H T (cid:17) − (cid:27) ≥ γ (18) tr (cid:110) H T H † T (cid:111) ≤ P avg (19)and H T H † T (cid:31) (Note that H T H † T (cid:31) , because H M H T and H E H T are full column rank byassumption). Note that – due to the channel knowledge assumptions – the legitimate transmitter, thelegitimate receiver and the eavesdropper can all set up this optimization problem in order to determinethe transmit filter and hence the receive filters via (13) and (14).It is now possible to reduce this optimization problem to a standard convex optimization problem byadopting the change of variables Z = (cid:16) H T H † T (cid:17) − , thereby paving the way to the characterization ofthe optimal transmit filter.The following Theorem, which stems directly from the Karush-Kuhn-Tucker optimality conditions [43],defines the form of the optimal transmit filter. Theorem 1:
Assume that the legitimate transmitter, the legitimate receiver and the eavesdropper knowthe exact channel matrices H M and H E . Assume also that the legitimate receiver and the eavesdropperreceiver use ZF filters. Then, an optimal transmit filter that solves the optimization problem in (17) – (19) is, without loss of generality, given by: H ∗ T = (cid:115) P avg tr (cid:26) ( H † M H M ) − (cid:27) (cid:16) H † M H M (cid:17) − , tr (cid:26) ( H † M H M ) − (cid:27) P avg tr (cid:26)(cid:16) H † E H E (cid:17) − (cid:16) H † M H M (cid:17) (cid:27) > γ (cid:115) P avg tr (cid:26)(cid:104) [ H † M H M ] − − ν [ H † E H E ] − (cid:105) (cid:27) (cid:20)(cid:104) H † M H M (cid:105) − − ν (cid:104) H † E H E (cid:105) − (cid:21) , tr (cid:26) ( H † M H M ) − (cid:27) P avg tr (cid:26)(cid:16) H † E H E (cid:17) − (cid:16) H † M H M (cid:17) (cid:27) ≤ γ where the value of the Lagrange multiplier ν is such that: tr (cid:40)(cid:16) H † E H E (cid:17) − (cid:18)(cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − (cid:19) − / (cid:41) ×× tr (cid:40)(cid:18)(cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − (cid:19) / (cid:41) = γ · P avg (20)Note that the right multiplication of the transmit filter in Theorem 1 by any unitary matrix producesanother optimal filter. Proof:
By considering the change of variables Z = (cid:16) H T H † T (cid:17) − it is possible to rewrite theoptimization problem in (17) – (19) as follows: min Z tr (cid:26)(cid:16) H † M H M (cid:17) − Z (cid:27) (21)subject to the constraints tr (cid:26)(cid:16) H † E H E (cid:17) − Z (cid:27) ≥ γ , tr (cid:8) Z − (cid:9) ≤ P avg , and Z (cid:31) . Note that thisrepresents a standard convex optimization problem, so that the solution follows directly from the Karush-Kuhn-Tucker optimality conditions [43].The Lagrangian of the optimization problem is given by: L ( Z , ν, µ )= tr (cid:26)(cid:16) H † M H M (cid:17) − Z (cid:27) + ν (cid:18) γ − tr (cid:26)(cid:16) H † E H E (cid:17) − Z (cid:27)(cid:19) + µ (cid:0) tr (cid:8) Z − (cid:9) − P avg (cid:1) (22)where ν and µ are the Lagrange multipliers associated with the problem constraints. The Karush-Kuhn-Tucker optimality conditions are given by: ∇ Z L ( Z , ν, µ ) = (cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − − µ Z − = 0 (23) ν (cid:20) tr (cid:26)(cid:16) H † E H E (cid:17) − Z (cid:27) − γ (cid:21) = 0 , ν ≥ (24) µ (cid:2) P avg − tr (cid:8) Z − (cid:9)(cid:3) = 0 , µ ≥ (25)and Z (cid:31) , tr (cid:26)(cid:16) H † E H E (cid:17) − Z (cid:27) ≥ γ , tr (cid:8) Z − (cid:9) ≤ P avg .The Karush-Kuhn-Tucker optimality conditions reveal that the solution of this problem exhibits twodistinct regimes only: i) the regime where the secrecy constraint is not active ( ν = 0 ); and ii) the regimewhere the secrecy constraint is met with equality ( ν > ) .When ν = 0 , then (23) reduces to: (cid:16) H † M H M (cid:17) − − µ Z − = 0 (26)and the optimal solution is given by: Z ∗ = tr (cid:26)(cid:16) H † M H M (cid:17) − / (cid:27) P avg (cid:16) H † M H M (cid:17) / (27)This solution is valid if and only if: tr (cid:26)(cid:16) H † M H M (cid:17) − / (cid:27) P avg tr (cid:26)(cid:16) H † E H E (cid:17) − (cid:16) H † M H M (cid:17) / (cid:27) > γ (28)On the other hand, when ν > , then (23) reduces to: (cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − − µ Z − = 0 (29)and the optimal solution is given by: Z ∗ = tr (cid:40)(cid:20)(cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − (cid:21) (cid:41) P avg (cid:20)(cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − (cid:21) − (30)This solution is valid if and only if: tr (cid:26)(cid:16) H † M H M (cid:17) − / (cid:27) P avg tr (cid:26)(cid:16) H † E H E (cid:17) − (cid:16) H † M H M (cid:17) / (cid:27) ≤ γ (31) In each case the power constraint is met with equality i.e., µ > . Note that a scenario where the µ = 0 would require either the channelmatrices to be a multiple of each other ( ν > and µ = 0 ), or H † M H M = ( ν = 0 and µ = 0 ).2 Note that the optimal transmit filter obeys a simple operational interpretation. In the regime where thesecrecy constraint is inactive, i.e.: tr (cid:26)(cid:16) H † M H M (cid:17) − / (cid:27) P avg tr (cid:26)(cid:16) H † E H E (cid:17) − (cid:16) H † M H M (cid:17) / (cid:27) > γ (32)which typically occurs for low available powers, the filter performs two simple operations: i) conversion ofthe main channel (i.e. H † M H M ) into a set of parallel independent channels whose power gains correspondto the eigenvalues of the matrix H † M H M ; and ii) power allocation, by dividing the total power inverselyproportionally to the power gains of the set of parallel channels. This solution corresponds to the solutionin [37].In contrast, in the regime where the secrecy constraint is active, i.e.: tr (cid:26)(cid:16) H † M H M (cid:17) − / (cid:27) P avg tr (cid:26)(cid:16) H † E H E (cid:17) − (cid:16) H † M H M (cid:17) / (cid:27) ≤ γ (33)which typically occurs for high available powers, the filter can be seen to perform the operations: i)conversion of an equivalent channel (i.e. (cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − ) into a set of parallel independentchannels whose power gains correspond to the eigenvalues of the matrix (cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − and; ii) power allocation, by dividing the total power inversely proportionally to the power gains of theset of parallel channels. This result, which is based on the equivalent channels (rather than on the mainchannel), immediately generalizes the result in [37].Note also that, in the scenario where both receivers use ZF filters the power constraint is always active,i.e. the transmitter uses all the available power. We will observe in the sequel that this is not the case inother scenarios. C. Computational Procedure
The computation of the optimal transmit filter embodied in Theorem 1 requires finding the solution ofthe non-linear equation in (20), in order to determine the value of the Lagrange multiplier ν . We shallnow put forth a simpler procedure to design the optimal transmit filter and hence the receive filters via(13) and (14), based on the dual of the optimization problem.Consider again the Lagrangian of the optimization problem in (22). Consider also the dual function ofthe optimization problem in (21): L ( ν, µ )= inf Z (cid:23) L ( Z , ν, µ ) (34) where ν ≥ and µ ≥ . It is straightforward to show that the dual function reduces to: L ( ν, µ ) = √ µ tr (cid:40)(cid:18)(cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − (cid:19) (cid:41) − µP avg + νγ, (cid:18)(cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − (cid:19) ≥ −∞ , otherwiseThe dual problem of the optimization problem in (21) is now given by: max µ,ν √ µ tr (cid:40)(cid:18)(cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − (cid:19) (cid:41) − µP avg + νγ (35)subject to ν ≥ , µ ≥ and (cid:18)(cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − (cid:19) (cid:23) . We can now employ a two stepprocedure to express the solution of this optimization problem: i) optimization over µ for a fixed ν ; ii)optimization over ν for the optimal µ . It is straightforward to show that the optimal value of µ , for afixed ν , is given by: µ = 1 P avg (cid:32) tr (cid:40)(cid:18)(cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − (cid:19) (cid:41)(cid:33) (36)Consequently, the dual optimization problem reduces to: max ν P avg (cid:32) tr (cid:40)(cid:18)(cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − (cid:19) (cid:41)(cid:33) + νγ (37)subject to ν ≥ and (cid:18)(cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − (cid:19) (cid:23) or, equivalently: max ν P avg (cid:32) tr (cid:40)(cid:18)(cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − (cid:19) (cid:41)(cid:33) + νγ (38)subject to: ≤ ν ≤ λ min (cid:18)(cid:16) H † E H E (cid:17) (cid:16) H † M H M (cid:17) − (cid:16) H † E H E (cid:17) (cid:19) (39)This is due to the fact that the positive semidefinite constraint (cid:18)(cid:16) H † M H M (cid:17) − − ν (cid:16) H † E H E (cid:17) − (cid:19) (cid:23) is equivalent to the constraint ν ≤ λ min (cid:18)(cid:16) H † E H E (cid:17) (cid:16) H † M H M (cid:17) − (cid:16) H † E H E (cid:17) (cid:19) , where λ min ( M ) denotes the minimum eigenvalue of the positive definite matrix M . The solution to the optimizationproblem (38) – (39) can be computed in a straightforward manner using, for example, the bisectionmethod [44], which represents a much simpler procedure than any method that solves the non-linearequation in (20).The optimal values of µ in (36) and ν , which corresponds to the solution of (38) subject to (39) thendefine the optimal transmit filter. In turn, the optimal transmit filter defines the ZF receive filters through(13) and (14). IV. O
PTIMAL L INEAR R ECEIVE F ILTER AT THE E AVESDROPPER
We now consider the scenario where the legitimate receiver uses a ZF filter, whilst the eavesdropperreceiver uses the optimal linear Wiener filter. This corresponds to a generalization of the previous scenariowhere both the receivers are restricted to obey ZF constraints.
A. Optimal Linear Receive Filter Design
Let us consider the design of the eavesdropper optimal linear receive filter. Eve now uses the receivefilter that, for any fixed transmit filter H T , minimizes: MSE E = E (cid:2) (cid:107) X − H RE Y E (cid:107) (cid:3) (40)This corresponds to the Wiener filter given by (see e.g. [45]): H ∗ RE = H † T H † E (cid:16) I + H E H T H † T H † E (cid:17) − (41)In turn, the MSE in the eavesdropper channel, upon substituting (41) in (40), is given by: MSE E = tr (cid:26)(cid:16) I + H † E H E H T H † T (cid:17) − (cid:27) (42)Note that the expressions for the legitimate receive filter and for the MSE in the the main channel arealready given in (13) and (15). B. Optimal Transmit Filters
We now consider the design of the optimal linear transmit filter. This, in view of (15) and (42),corresponds to the solution of the optimization problem given by: min H T tr (cid:26)(cid:16) H † T H † M H M H T (cid:17) − (cid:27) (43)subject to the secrecy constraint: tr (cid:26)(cid:16) I + H † E H E H T H † T (cid:17) − (cid:27) ≥ γ (44)and to the power constraint: tr (cid:110) H T H † T (cid:111) ≤ P avg (45)with H T H † T (cid:31) . Note that – due to the channel knowledge assumptions – the legitimate transmitter,the legitimate receiver and the eavesdropper can also all set up this optimization problem to compute thetransmit filter and receive filters via (13) and (41). It is also possible to reduce this optimization problem to a standard convex optimization problem, byadopting the change of variables Z = (cid:16) H T H † T (cid:17) − together with the Woodbury matrix identity [46].Thus, the optimization problem reduces to: min Z tr (cid:26)(cid:16) H † M H M (cid:17) − Z (cid:27) (46)subject to the constraints: tr { I } − tr (cid:26)(cid:16) H † E H E (cid:17) (cid:16) Z + (cid:16) H † E H E (cid:17)(cid:17) − (cid:27) ≥ γ (47) tr (cid:8) Z − (cid:9) ≤ P avg (48)and Z (cid:31) . The solution follows from the Karush-Kuhn-Tucker optimality conditions given by: (cid:16) H † M H M (cid:17) − − ν (cid:20)(cid:16) Z + (cid:16) H † E H E (cid:17)(cid:17) − (cid:16) H † E H E (cid:17) (cid:16) Z + (cid:16) H † E H E (cid:17)(cid:17) − (cid:21) − µ Z − = 0 (49) ν (cid:26) tr { I } − tr (cid:26)(cid:16) H † E H E (cid:17) (cid:16) Z + (cid:16) H † E H E (cid:17)(cid:17) − (cid:27) − γ (cid:27) = 0 , ν ≥ (50) µ (cid:2) P avg − tr (cid:8) Z − (cid:9)(cid:3) = 0 , µ ≥ (51)and Z (cid:31) , tr { I } − tr (cid:26)(cid:16) H † E H E (cid:17) (cid:16) Z + (cid:16) H † E H E (cid:17)(cid:17) − (cid:27) ≥ γ , tr (cid:8) Z − (cid:9) ≤ P avg , where ν ans µ are theLagrange multipliers associated with the secrecy and power constraints, respectively.It is clear from the Karush-Kuhn-Tucker conditions above that there are three operational regimes: i)the scenario where the transmitter can use all the available power without violating the secrecy constraint,so that the secrecy constraint is not active ( ν = 0 ) and the power constraint is active ( µ > ); ii) thescenario where both the secrecy and power constraints are active ( ν > and µ > ); and iii) the scenariowhere the transmitter cannot use all the available power without violating the secrecy constraint, so thatthe secrecy constraint is active ( ν > ) and the power constraint is inactive ( µ = 0 ). Note that thissituation differs from the previous scenario (with ZF filters at both receivers) where it was possible touse all the power available without violating the secrecy constraint. The difference derives from the useof a more powerful receive filter by the eavesdropper.It is difficult to extract a characterization of the optimal filter design from the Karush-Kuhn-Tuckeroptimality conditions above in the general scenario, even though the problem is convex. Consequently,we concentrate on scenarios i) and iii) only.
1) Power constraint active / secrecy constraint inactive:
This situation arises typically in a regime oflow available power, due to the fact that the power, injected into the channel, is not enough to meet orviolate the secrecy constraint.The following Theorem, which stems directly from the Karush-Kuhn-Tucker optimality conditionsabove, defines the form of the optimal transmit filter, in such a regime.
Theorem 2:
Assume that the legitimate transmitter, the legitimate receiver and the eavesdropper knowthe exact channel matrices H M and H E . Assume also that the legitimate receiver uses a ZF filter whereasthe eavesdropper receiver uses the optimal linear Wiener filter. Then, an optimal transmit filter in thescenario where the power constraint is active whilst the secrecy constrain is inactive is, without loss ofgenerality, given by: H ∗ T = α (cid:16) H † M H M (cid:17) − (52)where α = (cid:115) P avg tr (cid:26) ( H † M H M ) − (cid:27) .Note that the right multiplication of the transmit filter in (52) by any unitary matrix produces anotheroptimal filter. Proof:
This Theorem follows from the Karush-Kuhn-Tucker conditions by using the fact that ν = 0 ,so that we can rewrite (49) as follows: (cid:16) H † M H M (cid:17) − − µ Z − = 0 (53)Note that, as expected, this solution corresponds to the solution embodied in Theorem 1, when thesecrecy constraint is inactive.
2) Power constraint inactive / secrecy constraint active:
This is a situation that typically arises in aregime of high available power; in fact, the use of all the available power would immediately violate thesecrecy constraint.The following Theorem, which also stems directly from the Karush-Kuhn-Tucker optimality conditions,defines the form of the optimal transmit filter, in such a regime. In particular, we use the fact that thereexists a non-singular m × m matrix C that diagonalizes both H † M H M and H † E H E simultaneously [46],i.e. C † H † E H E C = Λ E and C † H † M H M C = Λ M , where Λ M and Λ E are m × m positive definitediagonal matrices, with diagonal elements λ M i , i = 1 , , . . . , m and λ E i , i = 1 , , . . . , m , respectively. Theorem 3:
Assume that the legitimate transmitter, the legitimate receiver and the eavesdropper knowthe exact channel matrices H M and H E . Assume also that the legitimate receiver uses a ZF filter whereasthe eavesdropper receiver uses the optimal linear Wiener filter. Then, an optimal transmit filter in thescenario where the power constraint is inactive whilst the secrecy constrain is active is, without loss ofgenerality, given by: H ∗ T = C (cid:16) α Λ M Λ E − Λ E (cid:17) − (54)where α = tr (cid:26) Λ E Λ − M (cid:27) tr { I }− γ .Note that the right multiplication of the transmit filter in (54) by any unitary matrix produces anotheroptimal filter. Proof:
This Theorem also follows from the Karush-Kuhn-Tucker conditions by using the fact that µ = 0 , so that we can rewrite (49) as follows: (cid:16) H † M H M (cid:17) − − ν (cid:20)(cid:16) Z + (cid:16) H † E H E (cid:17)(cid:17) − (cid:16) H † E H E (cid:17) (cid:16) Z + (cid:16) H † E H E (cid:17)(cid:17) − (cid:21) = 0 (55)or equivalently: Λ − M − ν (cid:20)(cid:16) C † ZC + Λ E (cid:17) − Λ E (cid:16) C † ZC + Λ E (cid:17) − (cid:21) = 0 (56)
3) Interpretation:
It is interesting to contrast the operational principle of the optimal transmit filterdesign when the secrecy constraint is inactive (in Theorem 2) to that when the secrecy constraint is active(in Theorem 3).In the regime where the power constraint is active and the secrecy constraint is inactive, the optimaltransmit filter decomposes the MIMO main channel into a set of parallel channels using an orthonormaltransformation that does not affect the transmit power. The optimal transmit filter then weighs theindividual subchannels, such that the power constraint is met with equality. The optimal weights dependonly on the eigenvalues of the matrix H † M H M .In the regime where the power constraint is inactive and the secrecy constraint is active, the optimaltransmit filter decomposes simultaneously the MIMO main channel and the MIMO eavesdropper channelinto a set of parallel channels using an in general non-orthonormal transformation. Note that, even thoughsuch a transformation may affect the transmit power, this is not a concern in this regime. The optimaltransmit filter then weighs the individual subchannels further, such that the secrecy constraint is met with equality. Interestingly, the optimal weights now depend on the generalized eigenvalues of the matrices H † M H M and H † E H E .It is also interesting to contrast the transmit filter design when the eavesdropper employs a ZF filter(in Theorem 1) to that when the eavesdropper employs a Wiener filter. In the ZF case, when the secrecyconstraint is active, the transmit filter uses an orthonormal transformation to decompose an equivalentchannel in view of the fact that the power constraint is always active. In the Wiener case, when thesecrecy constraint is active, the transmit filter uses a non-singular matrix to decompose simultaneouslyboth channels. C. A Note on the Validity of the Operational Regimes
It is now relevant to establish conditions, which are a function of the system parameters, that identifythe exact regions of validity of the operational regimes unveiled in the previous subsection.
1) Power constraint active / secrecy constraint inactive:
To identify the validity of this regime weminimize the objective function in (43), subject to the power constraint in (45) only. Note that thisconstitutes a relaxation of the original optimization problem so the solution of this new optimizationproblem can never lead to a worse MSE than the solution of the original problem. In turn, this solutionis also a solution of the original optimization problem provided that it does not violate the secrecyconstraint.It is easy to show that this regime is valid if, for a fixed set of system parameters, P avg , γ , H M and H E , the following condition holds: tr { I } − tr (cid:40) H † E H E (cid:20)(cid:16) H ∗ T H ∗† T (cid:17) − + H † E H E (cid:21) − (cid:41) ≥ γ (57)where H ∗ T corresponds to the design embodied in Theorem 2 given by: H ∗ T = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) P avg tr (cid:26)(cid:16) H † M H M (cid:17) − (cid:27) (cid:16) H † M H M (cid:17) − (58)Note that (57) and (58) can be used to determine a threshold secrecy constraint, γ max reg , below whichwe operate under this regime, or equivalently, a threshold power constraint, P avg maxR , below which weoperate under this same regime. The threshold secrecy constraint is given by: γ max reg = tr { I } − tr H † E H E tr (cid:26)(cid:16) H † M H M (cid:17) − (cid:27) P avg (cid:16) H † M H M (cid:17) + H † E H E − (59)
2) Power constraint inactive / secrecy constraint active:
To identify the validity of this regime we nowminimize the objective function in (43), subject to the secrecy constraint in (44) only. This also constitutesa relaxation of the original optimization problem so the solution of this new optimization problem cannever lead to a worse MSE than the solution of the original problem. Moreover, this solution is also asolution of the original optimization problem provided that it does not violate the power constraint.It is also straightforward to show that this regime is valid if, for a fixed set of system parameters, P avg , γ , H M and H E , the following condition holds: tr (cid:110) H ∗ T H ∗† T (cid:111) ≤ P avg (60)where H ∗ T corresponds to the design embodied in Theorem 3, given by: H ∗ T = C tr (cid:110) Λ E Λ − M (cid:111) tr { I } − γ Λ M Λ E − Λ E − (61)Similarly to the previous case, (60) and (61) can be used to determine a threshold secrecy constraint, γ min reg , above which we operate under this regime, or equivalently, a threshold power constraint, P avg minR , above which we operate in the same regime. The threshold power constraint is given by: P avg minR = tr C tr (cid:110) Λ E Λ − M (cid:111) tr { I } − γ Λ M Λ E − Λ E − C † (62)V. G ENERALIZATIONS
It is also of interest to generalize the filter design problem to scenarios that involve some degree ofchannel uncertainty. We consider two cases:1) The legitimate receiver knows the exact state of the main channel and the statistics of the eaves-dropper channel, the eavesdropper receiver knows the exact state of the eavesdropper channel andthe statistics of the main channel, and the transmitter knows only the statistics of the main andeavesdropper channels;2) The legitimate receiver knows the exact state of the main channel and the statistics of the eaves-dropper channel, the eavesdropper receiver knows the exact state of the eavesdropper channel andthe statistics of the main channel, and the transmitter knows the exact state of both channels.These scenarios arise naturally in the ”secure” video broadcasting model depicted in Figure 1, whereboth receivers – even though they may have subscribed to different services – are active users of thenetwork: in case 1), it is assumed that the receivers convey information about the statistics of their own channels to the transmitter via a feedback path (this information is then relayed to the other receivers); incase 2), it is assumed that the receivers convey information about the exact state of their own channels tothe transmitter also via a feedback path (this information is not relayed to the other receivers though) .In addition, these scenarios can also be used to capture some of the uncertainty about the state of theeavesdropper channel leading to filter designs with considerable operational significance.We also comment on more efficient mechanisms to use the available resources, due to the fact thatsome of the solutions unveiled earlier have demonstrated that the transmitter does not always use all theavailable power in order to meet the security constraints.The ensuing formulations are based on the assumption that the so-called eavesdropper adopts a linearreceiver. Once again, the implications of the use, by the eavesdropper, of a non-linear rather than linearestimator are also discussed in the Section VI. A. Scenario 1
A possible formulation of the filter design problem when the receivers know the exact state of their ownchannels and the distribution of the other channels, whereas the transmitter knows only the distributionof the channels, is given by: min H T MSE M = E H M , H E { MSE M ( H M , H E ) } (63)subject to the security constraint: MSE E = E H M , H E { MSE E ( H M , H E ) } ≥ γ (64)and the total power constraint: tr (cid:110) H T H † T (cid:111) ≤ P avg (65)where MSE M is the expected value, with respect to H M and H E , of the MSE in the main channelfor fixed channel matrices H M and H E , i.e. MSE M ( H M , H E ) , and MSE E is the expected value, withrespect to H M and H E , of the MSE in the eavesdropper channel for fixed channel matrices H M and H E , i.e. MSE E ( H M , H E ) .By assuming that the legitimate receiver uses a ZF filter and the eavesdropper uses either a ZF filteror a Wiener filter, then the optimization problem reduces to: Note that the transmitter may also be able to capture an estimate of the statistics of the channels or the state of the channels in time divisionduplex (TDD) environments.1 min H T E H M (cid:26) tr (cid:26)(cid:16) H † T H † M H M H T (cid:17) − (cid:27)(cid:27) (66)subject to: tr (cid:110) H T H † T (cid:111) ≤ P avg (67)and: E H E (cid:26) tr (cid:26)(cid:16) H † T H † E H E H T (cid:17) − (cid:27)(cid:27) ≥ γ (68)or: E H E (cid:26) tr (cid:26)(cid:16) I + H † E H E H T H † T (cid:17) − (cid:27)(cid:27) ≥ γ (69)depending on whether it is assumed that the eavesdropper adopts a ZF or a Wiener filter, respectively.The significance of this formulation relates to the fact that the legitimate transmitter, the legitimatereceiver and the eavesdropper receiver all have the necessary information to set up this optimizationproblem in order to conceive the transmit filter and therefore the receive filters via (13) and (14) or (41),respectively. In addition, as long as the legitimate transmitter and the legitimate receiver agree to usethis formulation to perform the legitimate transmit and receive filter designs, there is no incentive for theeavesdropper to adopt any other formulation beyond this one to design its own filter.In particular, assume that the legitimate transmitter and the legitimate receiver adopt the formulationbased on the use of a Wiener filter by the eavesdropper. If the eavesdropper adopted another linear filter,the average value of the MSE of the eavesdropper channel would still be above γ in view of the optimalityof the Wiener filter.In contrast, assume that the legitimate transmitter and the legitimate receiver adopt the formulationbased on the use of a ZF filter by the eavesdropper. In the regime of high available power, and once againif the eavesdropper used another linear filter, then the average value of the MSE of the eavesdropperchannel would still be above γ in view of the fact that the performance of a ZF filter approaches thatof a Wiener filter in such a regime. In the regime of low available power, if the eavesdropper used aWiener filter instead, then the average value of the eavesdropper MSE could be evidently below γ . Thisconcern can be bypassed by operating at high enough available powers. B. Scenario 2
A formulation of the filter design problem when the receivers know the exact state of their own channelsand the distribution of the other channels, where as the transmitter knows the exact state of the channels,is given by: min H T MSE M ( H M , H E ) (70)subject to the security constraint: MSE E = E H M , H E { MSE E ( H M , H E ) } ≥ γ (71)and the total power constraint: tr (cid:110) H T H † T (cid:111) ≤ P avg (72)By assuming once again that the legitimate receiver uses a ZF filter and the eavesdropper uses eithera ZF filter or a Wiener filter, then the optimization problem reduces to: min H T tr (cid:26)(cid:16) H † T H † M H M H T (cid:17) − (cid:27) (73)subject to: tr (cid:110) H T H † T (cid:111) ≤ P avg (74)and: E H E (cid:26) tr (cid:26)(cid:16) H † T H † E H E H T (cid:17) − (cid:27)(cid:27) ≥ γ (75)or: E H E (cid:26) tr (cid:26)(cid:16) I + H † E H E H T H † T (cid:17) − (cid:27)(cid:27) ≥ γ (76)depending on whether it is assumed that the eavesdropper adopts a ZF or a Wiener filter, respectively.Note now that the legitimate transmitter and the legitimate receiver can also set up this optimizationproblem in order to determine the transmit filter and therefore the legitimate receive filter via (13). Incontrast, the eavesdropper – in view of the absence of knowledge of the legitimate receiver channel– cannot set up this optimization problem, so it is bound to use a mismatched filter. In view of theprevious rationale, as long as the eavesdropper uses a linear filter and independently of whether thelegitimate parties use the ZF or Wiener based formulation, we can thus argue that in the regime of highavailable power the average value of the eavesdropper MSE is always above γ whereas in the regime oflow available power the average value of the eavesdropper MSE can in principle be below γ , e.g. in theextremely unlikely event that the linear filter chosen (perhaps randomly) by the eavesdropper correspondsto the Wiener filter, but the legitimate parties assume that the eavesdropper uses a ZF rather than a Wienerfilter in the design formulation.Note also that this formulation does not explore the transmitter knowledge about the exact state ofthe eavesdropper channel per se. It is not clear whether or not such knowledge can be exploited in anoperational meaningful way. C. Towards the solution of the new formulations
These problems appear to be difficult to solve in general in view of the expectation operations in (63)– (64) in scenario 1 and in (71) in scenario 2. However, it is possible to conceive a solution for theformulations that are based on the use of a ZF filter by the eavesdropper.By adopting the change of variables Z = (cid:16) H T H † T (cid:17) − the optimization problem in (63), (64) and (65)reduces to: min Z tr (cid:26) E H M (cid:26)(cid:16) H † M H M (cid:17) − (cid:27) Z (cid:27) (77)subject to: tr (cid:26) E H E (cid:26)(cid:16) H † E H E (cid:17) − (cid:27) Z (cid:27) ≥ γ (78)and: tr (cid:8) Z − (cid:9) ≤ P avg (79)and H T H † T (cid:31) , whereas the optimization problem in (70), (71) and (72) reduces to: min Z tr (cid:26)(cid:16) H † M H M (cid:17) − Z (cid:27) (80)subject to: tr (cid:26) E H E (cid:26)(cid:16) H † E H E (cid:17) − (cid:27) Z (cid:27) ≥ γ (81)and: tr (cid:8) Z − (cid:9) ≤ P avg (82)The availability, when H M is such that its n M rows are independent CN (0 , Σ M ) circularly sym-metric complex Gaussian random vectors and when H E is such that its n E rows are also independent CN (0 , Σ E ) circularly symmetric complex Gaussian random vectors, of closed form expressions for E (cid:26)(cid:16) H † M H M (cid:17) − (cid:27) and E (cid:26)(cid:16) H † E H E (cid:17) − (cid:27) , which are given by [47]: E H M (cid:26)(cid:16) H † M H M (cid:17) − (cid:27) = 1 n M − m − Σ − M , for n M − m − > (83)and E H E (cid:26)(cid:16) H † E H E (cid:17) − (cid:27) = 1 n E − m − Σ − E , for n E − m − > (84)enable us to solve the optimization problem using the previous techniques [47].The availability of closed for expressions for E H M (cid:26)(cid:16) H † M H M (cid:17) − (cid:27) and E H E (cid:26)(cid:16) H † E H E (cid:17) − (cid:27) when H M and H E follow more general distributions would allow us to solve the optimization problem in otherscenarios too. D. A discussion about effective use of resources
Another relevant aspect relates to the fact that some of the filter designs are such that the transmitterdoes not use the entire available power budget in order to meet the secrecy constraint (see Section IV).One could thus argue that there is not an effective use of the available resources.There are various possible generalizations to address this issue:
1) Enter artificial noise:
Artificial noise is an effective approach to provide some degree of distortionat the eavesdropper ( [19], [20], [21] and [22]), so it is interesting to reflect whether it might be possibleto integrate elements of the filter design approach with elements of the artificial noise paradigm wherebythe fraction of the unused power is also explored to further jam the eavesdropper.In general, it is not possible to integrate directly the artificial noise approach with our filter designapproach because the transmitter does not signal over the null space of the main channel.However, it is possible to conceive more elaborate scenarios that involve the use of an additional friendlyjammer that shares the available power budget with the transmitter. This jammer is also constrained toconvey artificial noise over the null space of the MIMO channel that links the jammer to the legitimatereceiver.The action of the jammer – which adds additional noise to the eavesdropper channel – translatesinto a new eavesdropper channel between the transmitter and the eavesdropper receiver incorporatingthe effect of the artificial noise, that replaces the original eavesdropper channel. Therefore, one canpose immediately an optimization problem akin to the previous filter design with secrecy constraintsoptimization problems that – in addition to involve the design of the transmit filter – also involves thedetermination of the fraction of power to be used by the legitimate transmitter and the fraction of powerto be used by the friendly jammer subject to the available power budget. The determination of the solutionof this optimization problem entails the extra level of complexity associated with how to share the powerbudget though.
2) Enter the time and frequency dimension:
Another approach that points towards a more efficientuse of the resource relates to scenarios where one leverages the variability of the channel in the timedomain (as in MIMO wireless channels) or in the frequency domain (as in MIMO-OFDM channels) inconjunction with available power constraints that operate along the multiple dimensions, i.e. long-term– rather than short-term – power constraints (e.g. [48], [49] and [50]). As an example, by assuming thatall the parties know the state of the various time and/or frequency channels, it is possible to put forth the optimization problems: min H T ( i ) , i =1 , ··· ,n n n (cid:88) i =1 MSE M ( H M ( i ) , H E ( i )) (85)subject to: n n (cid:88) i =1 MSE E ( H M ( i ) , H E ( i )) ≥ γ (86)and: tr (cid:110) H T ( i ) H T ( i ) † (cid:111) ≤ P avg , i = 1 , · · · , n (87)assuming a short-term power constraint, or: n n (cid:88) i =1 tr (cid:110) H T ( i ) H T ( i ) † (cid:111) ≤ P avg (88)assuming a long-term power constraint, where H T ( i ) is the transmit filter at time/frequency i and H M ( i ) and H E ( i ) contain the gains from each main and eavesdropper channel input to each mainand eavesdropper channel output, respectively, at time/frequency i .The use of the long-term power constraint – instead of the short-term one – now offers the means todistribute the available power more efficiently over the time or frequency dimensions in order to obtaina better performance. Note that the short-term power constraint filter design problem can leverage theprevious techniques (see Sections III and IV); on the other hand, the long-term power constraint problemmay require more sophisticated techniques.VI. N UMERICAL R ESULTS
We now present a set of numerical results in order to provide further insight into the problem offilter design with secrecy constraints. In particular, we present the performance of the filter designs inthe presence of perfect and imperfect channel knowledge, as well as in the presence of eavesdroppersthat adopt non-linear rather than linear estimation. We also present the impact of the filter designs onother relevant metrics, that include the error probability and achievable secrecy rates. We consider forsimplicity a × MIMO Gaussian wiretap channel where the main channel and the eavesdropper channelmatrices are, respectively, given by: H M = −
11 2 , H E = −
11 1 This constitutes a degraded scenario because H † M H M (cid:31) H † E H E , therefore, in general the MSE in theeavesdropper channel will be higher than the one in the main channel. A. Performance of the Filter Designs in the Presence of Perfect Channel Knowledge
We first consider the scenario where the channels are known perfectly by all the nodes – as assumedin Theorems 1, 2 and 3 – in order to test the performance of our designs. Figure 3 depicts the MSEsin the main and in the eavesdropper channels and the input power to the channels vs . the secrecyconstraint for P avg = 1 when ZF filters are used at both the receivers. The solution clearly depicts thetwo operational regimes unveiled in Theorem 1: i) the regime where the power constraint is active but thesecurity constraint is inactive (for smaller values of γ ); and ii) the regime where the power and securityconstraints are active and met with equality (for larger values of γ ). Figure 3 also depicts the MSEs inthe main and in the eavesdropper channels and the input power to the channels vs. the secrecy constraintfor P avg = 1 when the optimal linear Wiener filters are used at both receivers, in order to provide furtherinsight. Surprisingly, in the relevant regime of large γ , the use of ZF filters rather than Wiener filtersleads to a better MSE in the main channel without the violation of the security constraint. This is dueto the fact that – via the use of ZF filters in lieu of the Wiener ones – the transmitter can use all of theavailable power in such a scenario, in order to drive the MSE to a lower value.Figure 4 now shows the values of the MSEs in the main and in the eavesdropper channels and theinjected power into the channels vs . the secrecy constraint for P avg = 1 , when the eavesdropper uses theoptimal linear filter instead. The solution exhibits the three operational regimes characterized in SectionIV-B. Below γ max reg , the optimal transmit filter, which is given by Theorem 2, minimizes the MSE inthe main channel subject to the power constraint only. We can indeed verify that the available poweris not sufficient to meet or violate the secrecy constraint. In-between γ max reg and γ min reg ,the transmitfilter minimizes the MSE in the main channel while meeting the power and the secrecy constraint withequality. Above γ min reg , the optimal transmit filter, which is given by Theorem 3, minimizes the MSEin the main channel subject to the secrecy constraint only. Note that it is not possible to use all theavailable power, otherwise the secrecy constraint would be violated. This power restriction results in amuch higher MSE in the main channel than in the eavesdropper channel for large values of γ becauseas the injected power tends to zero the MSE that results from the ZF receiver grows very rapidly.Finally, in view of the fact that we have motivated the filter design problem with secrecy constraintsproblems in scenarios where a provider seeks to guarantee that users that have subscribed to a service To the best of our knowledge, the problem of filter design with secrecy constraints when Wiener filters are used at both receivers is not aconvex in general. Therefore, an approximate solution has been determined through numerical methods. The solution in this regime, which has not been derived, was obtained through numerical methods.7 have a reasonable quality of service, whereas users that did not do not experience such quality of service,it is relevant to understand whether or not there are circumstances where the MSE in the main channelcan in fact be higher than the MSE in the eavesdropper channel.In the presence of channel degradedness the main channel MSE can be higher than the eavesdropperchannel MSE for low available power P avg for a fixed target γ when the legitimate receiver uses a ZFfilter and the eavesdropper receiver uses the Wiener filter. However, with the increase in the availablepower the performance of the ZF filter approaches that of the Wiener filter, so that – in view of channeldegradedness - the main channel MSE eventually becomes lower than the eavesdropper channel MSE.In contrast, in the absence of channel degradedness the main channel MSE can be higher than theeavesdropper channel MSE when both the legitimate receiver and the eavesdropper receiver use ZF filtersor when the legitimate receiver uses a ZF filter and the eavesdropper receiver uses the Wiener one. Thisaspect is highlighted for a scenario where H M = (cid:2) −
11 2 (cid:3) and H E = (cid:2) . −
11 3 (cid:3) in Figure 5 – note thatMSE of the eavesdropper obeys the secrecy constraint though.However, with the emergence of MIMO-OFDM systems in a variety of wireless standards, it is possibleconceive approaches that bypass the absence of degradedness. For example, one can in principle selectsets of sub-carriers whose MIMO channels obey the degradedness property in order to assure that theMSE in the main channel is significantly lower than the MSE in the eavesdropper channel.
B. Performance of the Filter Designs in the Presence of Imperfect Channel Knowledge
We now consider the scenario where the channels are only known imperfectly by the nodes in orderto test the robustness of the designs embodied in Theorems 1, 2 and 3. In particular, we assume that thenodes have only access to an estimate of the main channel ˜ H M = H M + Φ M , where H M represents thetrue main channel matrix and Φ M models the main channel estimation error (with i.i.d. elements thatfollow a Gaussian distribution with mean zero and variance σ M ), as well as access to an estimate of theeavesdropper channel ˜ H E = H E + Φ E , where H E represents the true eavesdropper channel matrix and Φ E models the eavesdropper channel estimation error (also with i.i.d. elements that follow a Gaussiandistribution with mean zero and variance σ E ). We also assume, for simplicity, that all the nodes haveaccess to exactly the same estimates of the main and eavesdropper channel. The transmit and receivefilters are designed based on the estimate of the channels rather than the true channels, via Theorems 1,2 and 3.Figures 6 and 7 depict the MSEs in the main and eavesdropper channels (averaged over 2000 realiza-tions of the matrices that model the channel estimation errors) vs. the secrecy constraint for P avg =1 , for the scenario where the legitimate and eavesdropper receivers use ZF filters and the scenario where thelegitimate receiver uses a ZF filter but the eavesdropper uses a Wiener filter, respectively.We observe that channel modelling errors have an impact on the MSE of the main channel and – ofparticular relevance – on the MSE of the eavesdropper channel. The higher the deviation of the channelestimate from the true channel, which is modelled by the variances σ M and σ E , the higher the deviationof the new MSEs from the original ones.However, we also observe that the filter designs exhibit a certain degree of robustness. In the scenariowhere the eavesdropper uses the Wiener filter, the corresponding MSE appears to be reasonably robustto the channel modelling errors. In contrast, in the scenario where the eavesdropper uses a ZF filter, thecorresponding MSE is more sensitive to the channel modelling errors.In general, for low to moderate channel estimation errors, the filter designs still guarantee that thesecrecy constraint is not violated for a reasonable large set of γ . C. Linear vs. Nonlinear Estimation
It is also relevant to consider the situation where the eavesdropper is not restricted to choose a linearfilter. One could in principle argue that the eavesdropper (even if another user of a network as in Figure 1)could use the optimal nonlinear receive filter, instead of the optimal linear one, to process the informationin order to derive a lower MSE. This involves using a conditional mean estimator (CME), that deliversthe estimate given by: ˆX E = E { X | Y E } = (cid:82) x P X ( X = x ) P Y E | X ( Y E | X = x ) d x (cid:82) P X ( X = x ) P Y E | X ( Y E | X = x ) d x (89)where P X ( X ) is the probability density function of the input and P Y E | X ( Y E | X ) is the conditionalprobability density function of the eavesdropper receive vector Y E given the input vector X .We thus assess the performance penalty incurred by the use of a conditional mean estimator by theeavesdropper, but the transmitter designs its filter based on the assumption that the eavesdropper uses theoptimal linear filter. We study scenarios where the elements of the input vector X are either BPSK or16-PAM. Figure 8 shows the values of the MSEs in the main and in the eavesdropper channels and theinjected power into the channels vs . the secrecy constraint for P avg = 1 . We can observe that designingthe transmit filters based on the assumption that the eavesdropper is using an optimal linear receivefilter results, as expected, in a lower eavesdropper MSE, when the input is not Gaussian (note that forGaussian signals the conditional mean estimator is, in fact, linear). However, and interestingly, in regimes of greatest operational interest of large γ , the penalty that we pay by assuming that the eavesdropper usesan optimal linear filter rather than the optimal non-linear one vanishes, so that the eavesdropper does nothave any real advantage in using the considerably more complex conditional mean estimator. This is dueto the fact that the power injected in the channel approaches zero as the values of γ increases, in orderto meet the secrecy constraint. D. Impact of the Filter Designs on Other Metrics
It is also of interest to assess the impact of the filter designs on other metrics of operational relevance,including the Bit Error Rate (BER) in the main and eavesdropper channels as well as achievable secrecyrates.Figure 9 and 10 depict the Bit Error Rates (BER) of the main and the eavesdropper channels for thescenarios where i) ZF filters are used at both receivers and ii) a ZF receiver is used at the legitimatereceiver and a Wiener filter is used at the eavesdropper receiver, respectively. These BER results areobtained through Monte Carlo simulations, assuming that the transmitter uses BPSK modulation and thatthe receiver uses a simple slicer to detect the information at the filters output. We can observe that byimposing a constraint on the MSE of the eavesdropper we also restrict the BER of the eavesdropper to beabove a certain threshold. The resulting BER in the main channel, though, is also slightly degraded dueto the secrecy restriction. We can also observe that the BERs that we can achieve when both receiversuse ZF filters are lower than those when the legitimate receiver uses a ZF filter and the eavesdropperuses a Wiener filter (cf. Figures 9 and 10). We argue that this seemingly counterintuitive behavior is dueto the fact that in the scenario where the eavesdropper uses a Wiener filter instead of the ZF one, thetransmitter cannot use all the available power.Finally, Figure 11 compares the achievable secrecy rates yielded by our filter designs to the secrecycapacity of the MIMO Gaussian wiretap channel, which is given in [15]. It is clear that the filter designsresult in a loss of secrecy rate, which is more pronounced at high than at low available power levels,both for scenarios where the eavesdropper uses a ZF filter as well as scenarios where the eavesdropperuses a Wiener filter.However, we note that our designs can be immediately realized in practice in order to impair theeavesdropper. In contrast, practical secrecy capacity achieving codes, which are known only for somespecial channels, have to be developed in order to achieve the secrecy capacity of the MIMO Gaussianwiretap channel. VII. C
ONCLUSION
We have considered the problem of filter design with secrecy constraints in the classical wiretapscenario, where the objective is to conceive, subject to a power constraint, transmit and receive filtersthat minimize the MSE between the legitimate parties whilst guaranteeing that the eavesdropper MSEremains above a certain threshold.In particular, we have provided characterizations of the form of the receive and transmit filters forMIMO Gaussian channels, considering the situation where both receivers use Zero-Forcing filters or theeavesdropper uses a Wiener filter. We have also provided efficient computational procedures to designthe optimal transmit and receive filters.In particular, we have shown that the transmit filter designs are resilient to channel modeling errors aswell as to the use of more powerful nonlinear receive filters, rather than the optimal linear Wiener filter,by the eavesdropper. We have also shown that the designs limit not only the eavesdropper MSE but alsothe error probability.We have also provided a framework to generalize this filter design problem from the scenario whereall parties are assumed to know the exact state of the channel to scenarios where there is some channeluncertainty. This generalization is applicable not only to wireless systems subject to various channel stateinformation regimes as well as to systems where there is uncertainty about the state of the eavesdropperchannel. The generalization of the designs to cases where both receivers use optimal linear Wiener filtersappear to be open in general. A
CKNOWLEDGMENT
The authors thank the anonymous reviewers for their valuable comments and suggestions that signifi-cantly contributed to improving the quality of the paper. M. R. D. Rodrigues also thanks Kai-Kit Wongfor very detailed comments about an earlier draft of the work.R
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ServiceProvider(Alice) Legitimate Subscriber(Bob)UnsubscribedUser(Eve) ... ......
Figure 1. A possible application scenario of the problem of filter design with secrecy constraints: ”Secure” video broadcasting. x N M Y M X M ^Alice H T H M H RM BobN E Y E X E ^H E H RE Eve
Figure 2. MIMO Gaussian wiretap channel model. γ M SE / P o w e r MSE M (Wiener Filters)MSE E (Wiener Filters)P in (Wiener Filters)MSE M (ZF Filters)MSE E (ZF Filters)P in (ZF Filters) Figure 3. Main and eavesdropper channel MSEs vs . secrecy constraint and input power vs . secrecy constraint, for the optimaltransmit filter design and either ZF filters at both receivers or Wiener filters at both the receivers ( P avg = 1 ). γ M SE / P o w e r MSE M MSE E P used γ max reg1 γ min reg3 Figure 4. Main and eavesdropper channel MSEs vs . secrecy constraint and input power vs . secrecy constraint, for the optimaltransmit filter design with a ZF filter at the legitimate receiver and a Wiener filter at the eavesdropper receiver ( P avg = 1 ). γ M SE MSE M (ZF filters at both receivers)MSE E (ZF filters at both receivers)MSE M (Wiener Filter at the eavesdropper)MSE E (Wiener Filter at the eavesdropper) Figure 5. Main and eavesdropper channel MSEs vs . secrecy constraint, for the optimal transmit filter design with ZF filters atboth receivers and Wiener filters at the eavesdropper receiver, in a non-degraded scenario ( P avg = 1 ). γ M SE / A v e r age M SE MSE M (Perfect channel estimation)MSE E (Perfect channel estimation)MSE M ( σ M2 = 0.1, σ E2 = 0.1)MSE E ( σ M2 = 0.1, σ E2 = 0.1)MSE M ( σ M2 = 0.5, σ E2 = 0.5)MSE E ( σ M2 = 0.5, σ E2 = 0.5)MSE M ( σ M2 = 1, σ E2 = 1)MSE E ( σ M2 = 1, σ E2 = 1) Figure 6. Main and eavesdropper channel average MSEs vs . secrecy constraint, in the presence of channel error estimation,for the optimal transmit filter design with ZF filter at both receivers ( P avg = 1 ). γ M SE / A v e r age M SE MSE M (Perfect channel estimation)MSE E (Perfect channel estimation)MSE M ( σ M2 = 0.1, σ E2 = 0.1)MSE E ( σ M2 = 0.1, σ E2 = 0.1)MSE M ( σ M2 = 0.5, σ E2 = 0.5)MSE E ( σ M2 = 0.5, σ E2 = 0.5)MSE M ( σ M2 = 1, σ E2 = 1)MSE E ( σ M2 = 1, σ E2 = 1) Figure 7. Main and eavesdropper channel average MSEs vs . secrecy constraint, in the presence of channel error estimation,for the optimal transmit filter design with a ZF filter at the legitimate receiver and a Wiener filter at the eavesdropper receiver( P avg = 1 ). γ M SE / P o w e r MSE M MSE E (Wiener)P used MSE E (CME − BPSK)MSE E (CME − 16PAM) γ min reg3 γ max reg1 Figure 8. Main and eavesdropper channel MSEs vs . secrecy constraint and input power vs . secrecy constraint, for the transmitfilter design based on the use of a ZF filter at the legitimate receiver and a Wiener filter at the eavesdropper ( P avg = 1 ). MSE E ( Wiener ) corresponds to the eavesdropper MSE associated with the linear Wiener filter. MSE E ( CME − BPSK ) cor-responds to the eavesdropper MSE associated with the CME for BPSK inputs. MSE E ( CME − ) corresponds to theeavesdropper MSE associated with the CME for 16PAM inputs. −4 −3 −2 −1 P avg B i t E rr o r R a t e ( BE R ) BER in the main channelBER in the eavesdropper channelBER in the main channel − no secrecy constraintBER in the eavesdropper channel − no secrecy constraint
Figure 9. Bit error rate vs . available power for the scenario where both receivers use ZF filters ( γ = 0 . ). −4 −3 −2 −1 P avg B i t E rr o r R a t e ( BE R ) BER in the main channelBER in the eavesdropper channelBER in the main channel − no secrecy constraintBER in the eavesdropper channel − no secrecy constraint
Figure 10. Bit error rate vs . available power for the scenario where the legitimate receiver uses a ZF filter and the eavesdropperreceiver uses the optimal linear filter ( γ = 0 . ). avg S e c r e cy C apa c i t y / A c h i e v ab l e S e c r e cy R a t e Secrecy Capacity of MIMO Gaussian wiretap channelAchievable Secrecy Rate (ZF filters at both receivers)Achievable Secrecy Rate (Wiener filter at the eavesdropper)
Figure 11. Secrecy capacity of the MIMO Gaussian wiretap channel vs . available power and achievable secrecy rate vs . availablepower, for the optimal transmit filter design with ZF filters at both receivers and Wiener filters at the eavesdropper receiver( γ = 0 .5