Secure Space–Time Communication - Electrical Engineering and

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003 1

Secure Space–Time CommunicationAlfred O. Hero, III, Fellow, IEEE

Abstract—Network security is important for information pro-tection in open, secure, or covert communications. One such re-quirement is to achieve high-rate communications between clients,e.g., terminals or sensors, in the network while hiding informa-tion about the transmitted symbols, signal activity, or other sen-sitive data from an unintended receiver, e.g., an eavesdropper. Forwireless links, the single-user capacity advantages of deploymentof multiple antennas at the transmitter is well known. One of theprincipal conclusions of this paper is that proper exploitation ofspace–time diversity at the transmitter can also enhance informa-tion security and information-hiding capabilities. In particular, weshow that significant gains are achievable when the transmitter andthe client receiver are both informed about their channel while thetransmitter and eavesdropper receiver are uniformed about theirchannel. More generally, we compare capacity limits for both in-formed and uninformed transmitter and informed receiver sce-narios subject to low probability of intercept (LPI) and low prob-ability of detection (LPD) constraints. For several general cases,we can characterize the LPI- and LPD-optimal transmitted sourcedistributions and compare them to the standard optimal sourcedistribution satisfying a power constraint. We assume the stan-dard quasi-static flat Rayleigh-fading channel model for the trans-mitter–receiver pairs. This paper is a step toward answering thefundamental question: what are the qualitative and quantitativedifferences between the information-carrying capabilities of openspace–time channels versus secure space–time channels?

Index Terms—Chernoff exponents, covert channels, flatRayleigh-fading models, information hiding, perfect secrecy,space–time channel capacity, space–time coding, wireless eaves-droppers.

I. INTRODUCTION

THIS paper proposes an information-theoretic frame-work for investigating information security in wireless

multiple-input multiple-output (MIMO) links. The resultspresented here have broad applications including: wirelesscommunications networks, wireless sensor networks, privatebroadcast networks, and steganography for multiphonic audio.For example, researchers in commercial wireless have pri-marily focused on quality of service (QoS) expressed in termsof deliverable information rates, channel capacity and outagecapacity, throughput, and delay. While these are relevant qualitymeasures there is increasing interest in network security bothfor assurance of data privacy, reliable user authentication, and

Manuscript received April 25, 2001. This work was supported in partby the Army Research Office under Grants ARO DAAH04-96-1-0337 andDAAD19-02-1-0262. The material in this paper was presented in part at theNational Radio Science Meeting of the International Union of Radio Sciences(URSI), Boulder CO, Jan. 2001.

The author is with the Department of Electrical Engineering and ComputerScience, The University of Michigan, Ann Arbor, MI 48109-2122 USA (e-mail:[email protected]).

Communicated by I. E. Telatar, Associate Editor for Shannon Theory.Digital Object Identifier 10.1109/TIT.2003.820010

protection of information from malicious eavesdroppers intenton discovering network vulnerabilities. The starting point ofthis paper is that a well-designed secure link should have lowprobability of intercept (LPI) and low probability of detection(LPD) with respect to an unauthorized eavesdropper. Animportant question which has motivated the work reported inthis paper is: what is the fundamental impact of implementinglink-level security measures on information rates and channelcapacity? This paper describes a theoretical framework foranswering such questions which is based on analyzing thefundamental impact on capacity imposed by different classesof link security constraints.

One cannot hope to ensure security without some cooperationof the transmitter and receiver (client) to put an eavesdropper at arelative disadvantage [30]. One of the most most common formsof cooperation is the use of a cipher [21] to encrypt each datastream transmitted which can only be deciphered at the client re-ceiver using a private shared key. One can refer to this method astemporal (single-channel) data encryption, an example of whichis the U.S. National Data Encryption Standard (DES) for sym-metric data encryption/decryption. Use of temporal encryptionis a very flexible measure for preventing unauthorized intercep-tion of private messages which can be applied to any messagesequence without considering the physical layer of the network.Another common form of cooperation is for the transmitter andreceiver to adopt information-hiding measures [26] to preventunauthorized detection of any signaling activity which could beused, for example, for geolocation of the transmitter. Informa-tion hiding is a form of covert encryption which encodes privatemessages in a background signal or noise process in such a waythat the presence of the messages is hidden from those withoutaccess to the private key. A well-known example is spread-spec-trum modulation for wireless channels which hides the spec-tral signature of the signal in the broad-band noise backgroundusing a pseudorandom convolution sequence as a private key.Another example is watermarking, where a owner-identifyingwatermark is hidden in an image or video signal [4]. The resultsof this paper can be applied to watermarking of space–time sig-nals. The thesis of this paper is that additional security againstdetection or interception can be achieved by space–time codingover multiple antennas (or acoustic transducers) at transmitterand receiver. In particular, when such information is availableto the transmitter, one can design the spatio-temporal modula-tion/demodulation to exploit known propagation and interfer-ence characteristics of the channel available to the client but notto the eavesdropper. For the memoryless channels consideredhere, this corresponds to spatial (multichannel) encryption andinformation hiding where the shared channel information playsthe role of a shared private key that can be used to unlock themessage.

0018-9448/03$17.00 © 2003 IEEE

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2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003

It is useful to place the contributions of this paper in thebackground of previous work. Shannon introduced the infor-mation-theoretic framework for studying secrecy in communi-cations [30]. As secrecy involves at least three terminals, thetransmitter, the client receiver, and the eavesdropper, the studyof achievable information rates for secure communications isa branch of multiterminal (more than two) information theory[6]. Wyner and coworkers [37], [25] developed the concept ofthe wire-tap channel for wired links and assessed the impact ofsecrecy on achievable information rate pairs. This work was ex-tended by Csiszár and Körner to more general broadcast chan-nels in [5]. The possibility of enhancing secrecy by incorpo-rating common knowledge of channel impulse response intothe data encryption was identified and exploited by Hassan andcoworkers [15] and was applied to single-antenna mobile radiolinks in [10] and [20]. This paper takes a different point of viewfrom previous work in that we evaluate the fundamental im-pact of transmission secrecy (LPI and LPD) on two-terminalchannel capacity in the setting of multiple-antenna spatio-tem-poral quasi-static Rayleigh-fading channels.

We investigate the client’s channel capacity and the capacity-achieving transmission strategy under LPI and LPD constraints.For this we consider MIMO links where both client and eaves-dropper access space–time Rayleigh-fading channels. For thediscrete time and space channels considered here, the trans-mitted signals are complex-valued matrices whose rowsspan time samples and whose columns span space samplesequal to the number of transmit antennas. We show that whenboth the transmitter and client receiver know the channel theycan exploit this knowledge to achieve improved LPI and LPDbeyond those achievable by single-channel systems. Such ex-ploitation is possible to a lesser extent when the transmitter doesnot know the client’s channel. Following the terminology usedin [2] we say that the client and/or transmitter are informed whenthey know their channel propagation coefficients, while we saythat the eavesdropper’s link is uniformed, i.e., neither trans-mitter nor eavesdropper know their propagation coefficients.

The LPI constraint can be imposed by constraining theeavesdropper’s channel capacity, cutoff rate, or decoding errorprobability. For example, when the eavesdropper’s channelcapacity is significantly lower than the client’s capacity theconverse to Shannon’s channel coding theorem implies that,by setting his rate between the capacities of the client and theeavesdropper, the transmitter can deprive the eavesdropperof arbitrarily low probability of decoding error while reliablycommunicating to the client. Here we show that when theeavesdropper is uninformed about his channel, the transmittercan enforce zero information rate to the eavesdropper whiledelivering positive information rate to the client. This LPIcondition is equivalent to the perfect secrecy condition incryptography [30]. We derive integral expressions for theperfect-secrecy capacity for the informed receiver and forcertain cases characterize the optimal signaling distributionswhich achieve it. The LPD constraint is imposed by con-straining the eavesdropper’s probability of correctly detectingthe presence of any signaling activity by the transmitter. Thisis closely related to the steganography problem [26]. We makeconservative assumptions on the information possessed by the

eavesdropper, e.g., the eavesdropper knows only the transmittedsignal distribution and the received signal-to-noise ratio (SNR).To obtain a tractable LPD constraint with which to derive acapacity expression we will rely on Chernoff error exponents,large eavesdropper standoff assumptions, and Edgeworthexpansions of the eavesdropper’s probability densities. TheChernoff exponent defines the asymptotic rate of decrease ofthe probability of detection error as the block length of the codegoes to infinity. The mathematical form of this exponent is usedto motivate a constraint on the transmitter which constrainsLPD as contrasted to total transmitted power.

Most of the results presented here apply to the case wherethe eavesdropper is at large standoff from the client’s link. Thisimplies that the eavesdropper has both low received SNR andapproximately Gaussian multiuser interference and noise sta-tistics. In addition, while many of these results can be general-ized, we assume that both the client and the eavesdropper accessthe transmitted energy through distinct mutually independentquasi-static Rayleigh-fading channels [1]. As mentioned above,exploiting channel information known to transmitter and clientreceiver can be viewed as a form of spatial encryption where theshared private key corresponds to the channel coefficients. As apractical matter, this private key is distributed to the transmitterand client receiver in the form of a training sequence, unknownto the eavesdropper. The initial portion of each coherent fade in-terval is reserved for transmission of the training sequence to theclient receiver which permits it to estimate the set of channel co-efficients. For the case of a link having informed transmitter andinformed receiver, these coefficients must be communicated tothe transmitter by some unspecified form of (secure) feedback.On the other hand, for the case in which only the receiver is in-formed no feedback is required. In both cases, it has been shownby us and others that the effect of channel estimation errors onchannel capacity may be significant [11], [3]. However, whilewe do investigate the effect of erroneous channel information atthe transmitter, we do not focus on broader channel estimationor feedback issues in this paper.

We present the following results.

1) Under the aforementioned space–time Rayleigh channelinformed/uninformed dichotomy it is possible for thetransmitter to communicate reliably to the client whiledepriving the eavesdropper of any transmitted informa-tion whatsoever. Thus, the transmitter attains perfectsecrecy as defined by Shannon [29]. This can be ac-complished by restricting the space–time modulation toa class of complex transmitted matrices whose spatialinner product is equal to a constant matrix.Two examples of such perfect-secrecy constellationsare square unitary space–time codes and quaternionspace–time codes [18], [17], [31]. The channel capacitywhen restricted to these signals is herein called the per-fect-secrecy capacity for which we give integral formsfor the case of an informed transmitter and receiver.

2) When the eavesdropper knows both the signal and hischannel exactly, constraining the eavesdropper’s Cher-noff exponent is equivalent to constraining the meanpower over the transmitter antennas, which we call a

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HERO: SECURE SPACE–TIME COMMUNICATION 3

mean-power constraint. Thus, we conclude that in thiscase no additional countermeasures beyond minimizingaverage transmitter power are required to enhance secu-rity of the client’s link.

3) When the channel is unknown but the signal is knownto the eavesdropper, constraining the Chernoff exponentis equivalent at low SNR to constraining the trace of thefourth moment of the signal matrix.

4) When both channel and signal are unknown (but thesignal distribution is known) to the eavesdropper, theChernoff exponent reduces to the sum of two terms:a function of the determinant of the spatio-temporalreceiver covariance matrix and a tensor product of thereceiver kurtosis and the signal covariance. The kurtosisis defined as the expectation of a four-fold product ofthe spatio-temporal signal amplitudes. The kurtosistensor product is nonnegative and equal to zero when thereceived signal is complex Gaussian. As the channel isGaussian, zero kurtosis is only possible when the trans-mitted signal is nonrandom. When the kurtosis tensorproduct increases from zero, as occurs, for example,when elements of the received signal matrices obey anincreasingly heavy tailed (super-Gaussian) distribution,the eavesdropper’s detection performance degrades. Thisresult is reminiscent of the well-known negative kurtosiscondition under which blind equalization is possible foran unknown single-input single-output (SISO) channelwith memory [28], [32].

5) Under the scenario where channel and signal are un-known to the eavesdropper, at low SNR the constrainton the Chernoff exponent reduces to a constraint on thetrace of the square of the channel-averaged transmittedspatio-temporal signal covariance matrix, which we callthe mean-squared-power constraint. Unlike the standardmean-power constraint, this constraint penalizes largespatial power variation of the transmitted signals.

6) For informed transmitter and client receiver operatingunder a channel-dependent version of the mean-squaredpower constraint, the capacity of the client’s link isattained by a Gaussian signaling strategy, called theLPD-optimal strategy. In this signaling strategy, thetransmitted energy is distributed more evenly over themodes of the channel as compared to the water-pouringsolution, called the power-optimal strategy, which isoptimal under the standard mean-power constraint.

7) For uninformed transmitter but informed client receiveroperating under the mean-squared power constraint boththe capacity and the capacity-attaining signaling strategyare of identical form to the standard power-optimal ca-pacity obtained under a mean-power constraint. In thiscase, no additional countermeasures are required to en-hance security of the client’s link against eavesdropping.

8) The LPD-optimal and power-optimal signaling strategiesachieve different information transmission rates for equalsignal power or for equal LPD performance as measured

by the Chernoff exponent. For fixed Chernoff exponent,the power-optimal signal achieves lower information ratethan the LPD-optimal signal and conversely. We investi-gate the relative advantages of power-optimal and LPD-optimal signaling as a function of spatial diversity at thetransmitter and received SNR. In particular, while LPD-optimal signaling has no advantage over power-optimalsignaling for a single transmit antenna (no diversity), it isshown that almost a factor of two information rate advan-tage is achievable at low SNR with 16 transmit antennas.

We provide a brief outline of the paper. In Section II, theRayleigh-fading measurement model is introduced. In Sec-tion III, an integral expression is given for the perfect-secrecycapacity. In Section IV, we give Chernoff error exponents fordetection error probability for differing levels of channel andsignal information available to the eavesdropper. In Section V,we provide numerical comparisons illustrating the loss incapacity due to adoption of the LPD strategy.

II. BACKGROUND

An -element transmitter antenna array transmits asignal matrix over a time interval of time samples,

called the coherent fade sampling interval (Fig. 1). Let de-note the signal received by the client over channel andthe signal received by the eavesdropper over a channel(Fig. 2). For notational simplicity, throughout this paper su-perscripts and subscripts will be used interchangeably whenno confusion ensues. We will assume that the two receivershave and receive antennas, respectively. Similarly tomuch previously published research on space–time coding [33],[8], [16], [22], [34], [11], [13], [14], we will assume the multi-channel quasi-additive Rayleigh-fading models for the receivedsignals. Over independent frames of time samples each themodels are

(1)

where is the th transmitted signal, ,, are the normalized SNRs with the expected

SNRs at each receiver per transmit antenna, and aremutually independent and matrices of com-plex channel coefficients, and and are mutually un-correlated and matrices of complex circu-larly symmetric Gaussian noises. Note that we are assuming that

and have coherent fade intervals of identical dura-tion . When multiuser interference is present we can accountfor it in our capacity calculations by assuming the worst caseGaussian interference scenario. The theory developed here ap-plies to the case where the interferers have known covariances

and ; specifically, is known to the transmitter andthe client and is known to the transmitter and the eaves-dropper. In this case, the client and eavesdropper models can bereduced to the white noise models (1) by suitable prewhiteningat the receivers. Extension of the theory developed in the se-quel to unknown and is a difficult open problem whichwe do not consider here. We denote by common notation ,

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Fig. 1. Secure space–time link (M = 3, N = N = 2).

Fig. 2. A link in a wireless network between a transmitter (T) and a receiver(R) who have cooperated to learn their channel H . Eavesdropper Eattempts to detect a message (known signal), detect signaling activity (knownmodulation), or intercept data transported by the link H without knowingthe channel H . The eavesdropper and the client receiver must generallyperform these tasks in the presence of multiuser interference. In this paper, weassume Gaussian interferers with known spatial covariances.

the quantities , and , when no risk of confusion en-sues. The quasi-static Rayleigh flat-fading model corresponds totaking the elements of the matrices and

to be independent and identically distributed (i.i.d.)

complex zero mean (circularly symmetric) Gaussian randomvariables with unit variance.

Let and denote the se-quence of measurement and signal matrices, respectively, and

the channel matrix of either the client or the eavesdropperover the th frame. Under the assumption that the channel ma-trices are independent over each coherent fade interval, indexedby , the joint conditional probability density of the observationsfactors into a product of marginals

where, if the channel is known to the receiver

(2)while if the channel is unknown to the receiver

(3)

where is the identity matrix, anddenotes the magnitude determinant of square matrix .

A. Mean-Power Constrained Capacity

Following the standard random block coding construction ofchannel capacity, is interpreted as a blockcode consisting of statistically independent symbols drawn froma source distribution . Define the concatenation of the

rows of one of these symbols, denoted as matrix, into a -element row vector. The covariance ofis defined as the Hermitian symmetric matrix

. Let be a specified posi-tive constant. For an informed link where both transmitter and

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receiver know the channel over each frame of time sam-ples, the channel capacity can be derived under the per-framemean transmitted power constraint

(4)

Under this constraint, the capacity is [33], [34] as

(5)

where are the eigenvalues of . The capacity is at-tained by a zero mean circularly symmetric Gaussian source dis-tribution with covariance where

, are the (right) eigenvectors of , ,are given by water-filling

(6)

and is a parameter such that .In the sequel, we will call the T/R-informed power-ca-pacity and the informed capacity-achieving spatial signal co-variance will be called the T/R-informed power-optimalsignal covariance. Note that the capacity-achieving signal ma-trix has i.i.d. Gaussian rows each having (spatial) covariance

whose eigenvectors are the modes (columns of ) of .Note also that the water-filling strategy allocates transmitter en-ergy only to those channel modes which have the highest asso-ciated SNR. it can be shown that the optimal receiver applies abeamformer which is matched to the channel prior to max-imum a posteriori (MAP) decoding.

When only the receiver has information about the channel,the transmitter cannot exploit the highest SNR modes and theaverage power constrained channel capacity takes the form [33]

We call this capacity the R-informed power capacity. The ca-pacity achieving source is a matrix with i.i.d. zero meancircularly symmetric Gaussian elements having identical vari-ances equal to .

For an uninformed link where neither transmitter and receiverknow the channel, the channel capacity under an average trans-mitted power constraint was first investigated in [22].

While approximations have been investigated [22], [38], noclosed-form expression exists for either the capacity or thecapacity-achieving source. However, it was shown in [22] thatthe capacity-achieving source has the abstract form

where is an isotropically distributed matrix and isan independent nonnegative diagonal matrix.

III. LOW PROBABILITY OF INTERCEPT:THE PERFECT-SECRECY CAPACITY

Here we focus on the LPI strategy of designing transmittersignaling to zero out the channel information rate available tothe eavesdropper while maintaining high information rate com-munication to the client. We motivate this section by consideringcutoff rates.

A. Motivation: Channel Cutoff Rate

The channel cutoff rate is a lower bound on the Shannonchannel capacity . Cutoff rate analysis has frequently beenadopted to establish practical coding limits [35], [9] as the cutoffrate specifies the highest information rate beyond which sequen-tial decoding becomes impractical [27], [36] and as it is fre-quently simpler to calculate than channel capacity. The cutoffrate for an uninformed link with quasi-static Rayleigh channelwas derived in [14], [13].

B. Single-Link Cutoff Rates

For a space–time channel , the cutoff rate has the generalexpression [14]

where the maximization is over suitably constrained source dis-tributions and is a signal dissimilarity measurebetween pairs of transmitted signals and . The cutoff rateincreases as dissimilarity between pairs of signals increases, i.e.,as the average of increases. Thus, is di-rectly related to the information transport and decoding limita-tions imposed by a particular channel.

The following expressions are easily derived for receiversand received SNR .

1) Trasmitter/receiver (T/R) informed cutoff rate: knownto both T/R

2) Receiver (R) informed cutoff rate: known to R only

3) Uninformed cutoff rate: unknown to either T/R [14]

Note that in the T/R-informed case, the channel cutoff ratedepends on the dissimilarity of the signal pair after they arereceived, i.e., the difference squared between and ,while in the R-informed case, the cutoff rate depends on the dif-ference squared between the pair of transmitted signals. On theother hand, in the uninformed case the cutoff rate depends onthe difference between the determinant of the arithmetic mean(numerator) and the geometric mean (denominator) of the con-ditional received covariances and

. Thus, only temporal information

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can be used to distinguish between different signals. A user oreavesdropper on an uninformed channel cannot use any spatialinformation to help decode the symbols since this spatial infor-mation is completely unknown to him.

When the source has constant spatial inner product , theuninformed receiver’s absolute blindness to all spatial informa-tion can also be deduced directly from the form of the receiver’slikelihood function , given in (3), as this func-tion is constant. This also implies that the channel capacity willbe equal to zero and the minimum probability of decoding errorwill be equal to one if a source with constant is transmittedover a uninformed Rayleigh-fading link.

C. Perfect-Secrecy Signaling

We conclude that if the eavesdropper has a uninformedchannel, his information rate can be reduced to zero if the trans-mitter adopts a signaling strategy which uses a constellation

having constant spatial inner product

(7)

where is a prespecified nonrandom matrix. Whenis diagonal, many known signal constellations satisfy thisperfect-secrecy property.

• Doubly unitary codes ( )

Some instances of such codes are as follows.– Square unitary codes ( ) [31]:

– Space–time quaternary phase-shift keying (QPSK):Quaternion codes [19]: ( ):

• Constant spatial modulus (CM) codes :

D. Perfect-Secrecy Capacity

Of obvious interest is the channel capacity of a T/R-informedlink with signaling limited to the class of signals satisfying theconstant spatial inner product condition (7), which we defineas the T/R-informed perfect-secrecy capacity. In Appendix A,we give an integral expression for this capacity. The capacity-achieving source density satisfies an equalization condi-tion that says in essence that the optimal source should makethe instantaneous per-symbol mutual information independent

of the particular transmitted symbol . In the special case thatand the eigenvalues of are identical, we show

that the optimal source distribution is the uniform codeword dis-tribution supported on where

is the perfect secrecy constraint set. Whenand this set is the hypersphere

which is also known as the Stiefel manifold. This source dis-tribution is similar to the isotropically random unitary sourceintroduced in [22] and suggests that if a tessellation of the hy-persphere is possible the equispaced constellation constructedon each lattice point in the tessellation might be close to op-timal.

An integral expression for the R-informed perfect-secrecy ca-pacity is not available. However, we make the following conjec-ture: for and the case of an uninformed transmitter butinformed receiver a uniform is close to optimal for anyfor large T and large SNR. The intuition is that in this limitingregime the transmitter knows that the receiver can accuratelyestimate the channel and diagonalize it, thereby converting thechannel to for which the uniform source distribution

is optimal. Recent techniques such as that of Hassibi andMarzetta [12] for deriving compact integral expressions for themutual information for isotropically random unitary transmittedsource matrices may be useful here. This is an interesting openproblem.

IV. LOW PROBABILITY OF DETECTION: IMPACT ON CAPACITY

In this scenario, the eavesdropper attempts to detect the pres-ence of a transmitted signal against noise alone based on ob-servations of his channel output. Formally, definetwo hypotheses and ,

. For any strictly positive prior probabilitiesand of these hypotheses, the minimumattainable probability of decision error of the eavesdropperhas the following large sample limiting behavior [7], shown atthe bottom of the page. The nonnegative constant is calledthe Chernoff error exponent and is the error rate which deter-mines how quickly the decision error decays exponentially tozero. This error rate is the minimum (unnormalized) -diver-gence between the eavesdropper’s densities (the alternativedensity) and (the null density) which is a measure of theease of discrimination between the two statistical distributions.This constant must be negative for a patient eavesdropper to beable to correctly detect signal presence with arbitrarily low prob-ability of error as number of time frames increases. The objec-tive of LPD-secure modulation is to design signaling strategieswhich constrain to a large value (small negative value nearzero if possible) and achieve highest possible information ratesto the client. To this aim, we will compute the informed channelcapacity of the client under such an LPD constraint for low SNRand for several eavesdropper scenarios.

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A. -Informed Eavesdropper

Assume that the eavesdropper knows his channel sequenceand also knows the matrix valued amplitudes

of the signals sequence transmittedover the frames. The eavesdropper’s null and alternative den-sities become

An exact analysis of minimum probability of error is pos-sible in this Gaussian case from which it can be shown thatis monotone decreasing in the detectability index that is lin-early proportional to the magnitude Chernoff error exponent

. The -divergence is simply computed

which is minimized over by the choice .Thus,

(8)

where

Since the eavesdropper would not normally be cooperating withthe transmitter to provide feedback of his channel coefficients,a reasonable LPD signaling strategy would be to try to constrainthe channel-averaged Chernoff error exponent

For example, the transmitter could constrain the magnitude ofthe channel-averaged Chernoff exponent for each signal frame

where denotes “ linearly proportional to .” We identifythe above as an instantaneous power constraint.

An alternative strategy would be for the transmitter to gen-erate i.i.d. signal matrices from a source having sourcedistribution and satisfying the mean power constraint

(9)

where . By the strong law of large numbersthis is equivalent to constraining the exponent (8) under theassumption that and are i.i.d. sequences of matrices.

Recall that under the mean power constraint

the informed channel capacity is attained by zero mean complexGaussian with covariance . Hence, (9)is an equivalent constraint on and we conclude that when the

eavesdropper knows both the channel and the signal, the stan-dard mean transmit power constraint also ensures a modicum ofLPD performance.

B. -Informed Eavesdropper

Assume that the eavesdropper knows the signal amplitudesbut not the channel . In this case, the

eavesdropper’s densities become

As both densities are multivariate Gaussian the -divergence isagain simply computed

A simple asymptotic development gives

Thus, after substituting the minimizing value , theChernoff exponent has the low-SNR representation

(10)

We conclude that for the -informed eavesdropper and lowSNR, an appropriate “instantaneous LPD” constraint for thetransmitter is

a constraint which we call the instantaneous fourth moment con-straint.

If the transmitted signal matrices are i.i.d. realiza-tions of a source with source distribution , then withprobability one, from the strong law of large numbers appliedto (10), constraining is equivalent to constraining the meanfourth moment of the source

(11)

where is a specified constant.

C. Uninformed Eavesdropper

In this case, the eavesdropper does not know the amplitudesof the transmitted signals nor the channel .

We will assume that is a realization of an i.i.d. sourcefor which the source distribution is known to the eaves-dropper. This is a conservative assumption—in the absence ofsuch knowledge, the eavesdropper can only have worse detec-tion error rates than predicted below.

The eavesdropper’s densities are

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The -divergence is not closed form for the uninformedeavesdropper since it involves the difficult marginalization over

required for computation of . However, we can apply themethod of Edgeworth expansion to develop abouta Gaussian density to obtain the low-SNR approximation (seeAppendix B)

(12)

where is the covariancematrix associated with the source matrix ,

is the fourth-orderkurtosis tensor of the matrix and Einstein sum-mation notation is used for the tensor product in the second termof (12). For a more precise definition of see Appendix B.

As the additive eavesdropper noise is Gaussian, the kur-tosis of the eavesdropper’s received signal satisfies

If an element of is negative, then the received signal ma-trix has a sub-Gaussian (light-tailed) component while if an el-ement of is positive, this matrix has a super-Gaussian(heavy-tailed) component. The kurtosis tensor product in (12)can be explicitly expressed as a function of the moments of thetransmitted signal matrix using the fact that the entries ofare i.i.d. complex Gaussian (see equation at the bottom of thepage). As for fixed

(13)

is a nonnegative-definite function in the pairs of indexesand , the kurtosis tensor product is nonnegative and in-creases in the centralized fourth momentof the source. This reflects the fact that under the assumptionof a random Gaussian channel, the received signal is alwayssuper-Gaussian, i.e., its kurtosis is greater than zero, unless thesignal has zero variance.

Note that the -divergence (12), and hence the error rate ,is an increasing function of the received kurtosis tensorfor all . We conclude that the best countermeasureto thwart eavesdropper signal detection is for the transmitter totransmit signals leading to as high positive kurtosis of as pos-sible. In particular, for fixed nonzero transmitted power, an ef-fective LPD signaling scheme would transmit signals havinglarge centralized fourth-moment tensor (13). This strategy may

be closely related to diminishing the ability of the eavesdropperto perform blind equalization which, for the case of a scalarchannel with memory, is known to be possible only when thesource’s fourth moment is sufficiently small to make the kur-tosis negative valued [32]. The choice of a signal distribution

which minimizes the -divergence (12) or maximizesthe tensor product therein is an interesting open problem.

Considerable simplification occurs when the SNR is very lowand terms of order can be neglected. In this case, the Cher-noff error exponent, i.e., the minimum of the -divergence (12),becomes

(14)

V. LPD-CONSTRAINED CAPACITY

Here we use the asymptotic error rate (14) to motivate an LPDconstraint under which we derive the channel capacity forthe case where both transmitter and receiver know their channel(T/R-informed) and for the case that only the receiverknows the channel (R-informed).

The asymptotic LPD constraint (14) is equivalent to the meansquared power constraint

(15)

where is a prespecified maximum tolerable mean-squaredpower and are the eigenvalues of the matrix

.

A. T/R-Informed LPD-Capacity

Motivated by constraint (15), consider the simpler constrainton -conditioned mean-squared transmitted power

(16)

where are the eigenvalues of the conditionalcovariance matrix . Under constraint (16) we can de-rive (Proposition 3 in Appendix C) the following expression forT/R-informed capacity:

(17)

where are the eigenvalues of and is anunitary matrix whose columns are the (right) eigenvec-

tors of . The capacity (17) is attained by a zero-mean

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Gaussian source with covariancewhere ,

(18)

and is a parameter such that . Inthe sequel, we will call the T/R-informed LPD-capacityand the capacity-achieving signal covariance will be called theT/R-informed LPD-optimal signal covariance.

Observe the following.

1) Like the power-optimal source (6) which achieves T/R-in-formed power-capacity , the LPD-optimal sourcehas i.i.d. Gaussian rows and each row has (spatial) covari-ance whose eigenstructure is matched to the eigen-structure (modes) of the channel.

2) In contrast to the water-filling strategy, distributing trans-mitted energy only to the highest SNR channel modes isnot optimal for attaining the LPD capacity.

3) The T-informed mean-squared-power and mean-powerconstraints can be related to each other by the Schwarzinequality

Thus, the mean-squared-power constrains the meanpower of the transmitted signal. However, the two con-straints produce qualitatively different optimal sourcecovariances.

4) The eavesdropper’s Chernoff exponent (14) only dependson his antennas through his received SNR . Hence,if the eavesdropper’s Chernoff exponent is to be con-trolled via the mean-square-power constraint (15), thetransmitter’s LPD-optimal signaling strategy depends on

only through the mean-square-power constraint level. In particular, if the transmitter knows that has

increased he will only need to reduce his transmit powerto ensure the same eavesdropper Chernoff exponent.

5) The LPD constraint (15) is weaker than the -dependentconstraint (16). Specifically, for the capacity-achievingsource Jensen’s inequality asserts that ,so that the constraint (16) guarantees the LPD constraint(15). The form of the capacity under constraint (15) is anopen problem.

B. R-Informed LPD Capacity

In Proposition 4 of Appendix C we establish that when thetransmitter does not know the channel but the receiver doesknow the channel, the mean-squared-power constraint (15) andthe mean-power constraint (4) produce the same optimal sig-naling strategy and result in identical forms for the channel ca-pacity. Thus, we conclude that when the eavesdropper has lowreceived SNR, his Chernoff exponent is controlled by averagetransmitter power and no special countermeasures are requiredto enhance security of the client’s link.

Fig. 3. Spectra of optimal source covariance matrices under T/R-informedLPD (mean-squared-power) and mean-power constraints: SNR= 20 dB; M =

N = 32.

VI. NUMERICAL COMPARISONS

Here we compare the T/R-informed power capacities andLPD capacities derived in the previous section. Simulations of aRayleigh-fading channel were performed and the T/R-informedcapacities under both and constraints were computedempirically. The number of transmit antennas was chosenequal to the number of the client’s receive antennas. InFig. 3, we show the eigenvalues (diagonal entries) of theoptimal signal covariance matrices which achieve each one ofthe capacities. These eigenvalues are indexed by the modes ofthe channel and are denoted as such in the figure. Both signalcovariances are power normalized, i.e., they have the sametrace. The LPD-optimal eigenvalue distribution is flatter andits peaks are much less prominent than the standard power-op-timal eigenvalue distribution. This reflects the intuitive factthat an eavesdropper can less easily detect the presence ofa flat signal eigenvalue profile and hence the LPD-optimalsignaling strategy better hides the signal information than thepower-optimal signaling strategy.

In Figs. 4 and 5 are plotted the T/R-informed standard powercapacity and LPD capacity as a function of SNR forvarious numbers of antenna elements .

Next we investigated tradeoffs between the T/R-informedLPD-optimal signaling strategy and the standard power-optimalwater-filling signaling strategy. Define

the average information rate attained by a zero-mean Gaussiansource with covariance matrix satis-fying the constraint denoted by . The notation means that if

then satisfies and ifthen it satisfies . Thus, the standard powercapacity is

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Fig. 4. T/R informed power capacity C (N = M).

Fig. 5. T/R informed LPD capacity C (N = M).

where the power-optimal signal covariance is specified by(6) and the LPD capacity is where LPD-optimal signal covariance is specified by (18).

The loss in power capacity due to using the LPD-optimalsignal covariance structure is defined as

(19)

while the loss in LPD capacity due to using the power-optimalsignal covariance structure is

(20)

In (19), both the LPD-optimal and the power-optimal covari-ances are forced to satisfy the same mean-power constraintwhile in (20) they both satisfy the same mean-squared-powerconstraint. Figs. 6 and 7 plot the capacity losses as a functionof mean power and mean-squared power, respectively. Notice

Fig. 6. Loss in power capacity due to mean-squared-power (LPD) constraint(N = M).

Fig. 7. Loss in LPD capacity due to mean-power constraint (N =M).

that the loss increases as more antennas are deployedby transmitter and client. This is because a higher proportionof the signal covariance eigenspectrum is flattened out by theLPD-optimal signaling strategy as compared to the power-op-timal strategy. Also note, that as the client’s SNR increasesthe relative capacity loss becomes negligible while asdecreases to 20 dB, the losses flatten out. This is because atvery low , the power-optimal water-filling strategy requiresthe transmitter to use only a single transmit antenna while theLPD-optimal signal applies energy to all antennas no matterhow low gets. Finally, for a single transmitter antennaelement there is no loss in capacity since in this casethe average-power and the mean-squared-power constraints areequivalent up to a scale factor.

Finally, we investigated the sensitivity of the T/R-informedpower capacity and LPD capacity due to errors in the trans-mitter’s channel estimate. The receiver is assumed to have

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Fig. 8. T/R informed power capacity loss due to transmitter-channel mismatch(N = M).

decode symbols with zero channel estimation error. Thisasymmetric channel error scenario is an idealization of thesituation where channel estimation errors occur during trainingwhich are then fed back to the transmitter. While we offer noproof, we believe that the effect on capacity of using erroneouschannel information at the transmitter is greater than usingequivalent error estimates at the receiver and therefore theseresults should approximate actual information rate reductionsdue to training. This would have to be verified by doing moreextensive simulations to determine the mutual informationloss due to training errors at both transmitter and receiver.In our simulation, the total number of samples in a coherentfade was 1024 and 128 of these samples were used forestimating the channel at the receiver. 500 realizations of dif-ferent Gaussian channels were generated for various numbersof antennas and SNRs. Over each frame, the channel wasestimated at the receiver via the exact least squares estimatorbased on 128 -element snapshots generated by transmittedzero mean i.i.d. Gaussian training symbols. These channelestimates were then substituted into the power-optimal andLPD-optimal covariances and and substituted intothe capacity (5) and (17), respectively. Figs. 8 and 9 showsthe resultant degradation in these two capacities. Observethat for the example simulated here, for moderate-to-largeSNR, the relative loss due to transmitter-channel mismatch issignificantly less than the loss due to not accounting for theeavesdropper LPD constraint (compare Fig. 9 to Fig. 7).

VII. CONCLUSION

This paper has presented a study of capacity under link se-curity constraints corresponding to low probability of intercept(LPI) and low probability of detect (LPD). We have establishedthat optimal signaling for LPD- and LPI- constrained securechannels is qualitatively different from open channels. We havealso shown that constraining moment quantities, such as traceof the fourth-moment matrix, are relevant for eluding detection

Fig. 9. T/R informed LPD capacity loss due to transmitter-channel mismatch(N = M).

by eavesdroppers who have only limited knowledge about thechannel and transmitter modulation. A smart eavesdropper withinformation on data or training sequences can be handled simi-larly by constraining the fourth moment of the transmitted signalmatrices. The analysis in this paper holds only for doubly in-formed links for which the receiver and transmitter know theirchannel exactly. Extensions of these results to the case of net-work-wide QoS metrics such as min- and sum-capacity are ofinterest. This paper has treated the case of Gaussian noise andknown receiver noise covariance matrices. Generalizations tothe case of non-Gaussian multiuser interference (MUI) wouldbe worthwhile for answering questions such as: to what extentare LPD and MUI resistance compatible goals in wireless net-works. Finally, another interesting avenue for exploring the in-formation-hiding capabilities of space–time channels would bethe information-theoretic framework of Moulin [24].

APPENDIX A

Propositions 1: Assume that is an matrix of i.i.d.zero mean and unit variance circularly symmetric Gaussian el-ements. If the density defined below exists, the T/R-in-formed perfect-secrecy capacity is where

(21)

where

• is the probability density of a standard zeromean and identity covariance complex normal ma-trix.

• is a probability density over the perfect secrecy con-straint set , a nonrandom matrix,which makes the right-hand side of (22) functionally in-dependent of over

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If then, with the singulara-valuedecomposition (SVD) of

(22)

where the matrix is defined by the following columnpartition of the matrix : . Furthermore, if

for then is theuniform density over .

Proof: For fixed , the T/R-informed perfect secrecy ca-pacity is defined as

where the maximization is over distributions supported on .As

we focus on the entropy function

where . As is concave as a function of, a standard calculus of variations argument can be used

to derive a sufficient condition for the maximum of .Define the Lagrangian

where is an undetermined positive multiplier which enforcesthe constraint that . The stationary point condi-tion on the maximizing follows from consid-ering

For the above to be zero for all such thatremains a valid distribution over we require

which, upon substituting , givesthe expression (21) of the proposition.

Specializing to the case that , and using the givenSVD of and definitions of and , observe that

Plugging this back into the expression for and usingsimple algebra this establishes (22).

Finally, assuming , from (22), we must show that

is equal to a constant over independent of . Note that

and observe that by change of variable in the integrals above,for any unitary matrix : . Hence, byVinograd’s theorem, only depends on through . Since

over , must, in fact, be constant over and theoptimality of the uniform distribution is established.

APPENDIX B

As in Section II, let be a matrixof complex amplitudes measured at antennas over timesamples at the output of an i.i.d. zero-mean complex Gaussian

channel with additive complex Gaussian noiseand input signal (source) with complex amplitudes .While cooperation between client and transmitter may lead todependency between and , we assume there is no such co-operation between eavesdropper and transmitter. Thus, as thefollowing results are applied to the eavesdropper, we assumehere that is independent of and .

Let be a complex-valued matrix whereand are real matrices. Define the real-valued -ele-

ment vector as the concatenation of the columns of thematrix . Following the notation of [23], for zero mean

, define the tensors , , and as the variance,skewness, and kurtosis of

Note that as is Gaussian

Further, define the element of the inverse of the covari-ance matrix and the positive-definite square-rootfactor of , i.e., using the Einstein summation convention

, the Kronecker delta function, and .

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Propositions 2: Assume that is independent of andand that and are mutually independent zero-mean

complex Gaussian matrices. The (normalized) -divergencebetween and has the asymptoticexpression

(23)

(24)

where , is the kurtosis of the whitenedand variance normalized measurement , and is thevariance of the prewhitened signal .

Proof: Using the expression in [23] for the Edge-worth expansion of a zero-mean multivariate density abouta Gaussian multivariate density with zero mean and co-variance we have the representation shownat the bottom of the page, where is the

-variate Gaussian density with zero mean and covariance, and , , etc., are Hermite tensors given in [23].

As and are independent zero-meanGaussian random matrices, and is independent of and :

and the representation reduces to

Using , after some algebra,the substitution of the Edgeworth expansion into the -diver-gence expression gives

(25)

where is the matrix inverse ofand

Apply the small argument formula to performthe integration (25)

Next we use the Hermite tensor expression [23]

where

Substituting this back into (25) and noting that, as tensorshave the same eigenvectors they commute

This establishes (24).

Noting that andso that we have the following.

Corollary 1:

(26)

where is the kurtosis tensor of thematrix .

APPENDIX C

For a realization of the channel, define

the mutual information for a T/R-informed link over anquasi-static Rayleigh channel which is constant over the co-herent fade interval of time samples. Note thatis a function of and is not the standard conditional mutualinformation. The LPD capacity of this link is defined as

where the expectation is over and the maximization is overconditional source distributions which satisfy themean-squared-power constraint

(27)

and is the source’sconditional covariance matrix and is a -ele-

ment row vector constructed by concatenating the rows of .

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Let have eigendecomposition where.

Propositions 3: Let be an channel matrix withzero mean and unit variance i.i.d. complex circularly symmetricGaussian entries. For the case that both transmitter and receiverknow the channel capacity under the mean-squared-powerconstraint (27) is

(28)

which is attained by a zero-mean Gaussian source distributionwith covariance where

,

and is a parameter such that .Proof: The argument is similar to that used in [33] in

proving optimality of the water-filling solution for the caseof informed transmitter and receiver under a mean-powerconstraint. The matrix observation model over a single frame

has the equivalent vectorized form

(29)

where is a -element row vector fromconcatenating rows of and, similarly, for ,

is a similarly defined -element row vector,and is a block-diagonal matrix withidentical diagonal blocks . Invoking the maximum entropyproperty of the Gaussian distribution for having fixed covari-ance , we have the following inequality:

(30)with equality when is a zero-mean complex Gaussiandistribution with a covariance matrix .It remains to maximize the right-hand side of this inequalityover nonnegative definite symmetric matrices subject to

.With the eigendecomposition , the eigende-

composition of is simply

Let

have diagonal elements . Then, by Hadamard’s in-equality

(31)

with equality when . Note that

The maximizer of the right-hand side of the preceding equationsubject to the inequality constraintachieves the constraint with equality as (31) is increasing in .The Lagrangian for this constrained optimization problem is

where is an undetermined multiplier. This concave func-tion has a unique unconstrained maximum which occurs when

or equivalently

There is one positive root

where . Thus, , the optimum sourcecovariance is , and plugging this into(31), the capacity is (28) as claimed.

Propositions 4: Let be an channel matrix withzero mean and unit variance i.i.d. complex circularly symmetricGaussian entries. For the case that only the receiver knows thechannel capacity under the mean-squared-power constraint (27)is identical to the standard mean-power constrained capacity forthis case

(32)

which is attained by a source matrix whose elementsare zero mean i.i.d. circularly symmetric Gaussian random vari-ables with variances .

Proof: The proof parallels the proof of the mean-powerconstrained capacity in [33]. The capacity for the case that onlythe receiver knows the channel is defined as

where satisfies the power constraint

Using the vectorized signal representation (29) and (30) ob-tained in the proof of Proposition 3 we have

(33)

where equality is achieved when is zero-mean circularlysymmetric complex Gaussian with covariance ma-trix . As in [33], for any matrix the function

is concave and for any unitary matrix ,. Thus, specializing to the eigenvector

matrix in the eigendecomposition , wehave so that, as ,without loss of generality we can assume that the capacityachieving covariance is a nonnegative diagonal matrix

. Next, specializing to , a permutation matrix

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. Thus, by Jensen’s inequality, summingover all permutation matrices

where which is a scaledidentity matrix. It follows from the following inequality that theconstraints

and

are equivalent so that is the optimal sourcecovariance

where the last line follows from Jensen’s inequality. This estab-lishes the proposition

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Alfred O. Hero, III (S’79–M’80–SM’96–F’98) was born in Boston, MA, in1955. He received the B.S. degree in electrical engineering (summa cum laude)from Boston University in 1980 and the Ph.D degree from Princeton University,Princeton, NJ, in 1984, both in electrical engineering.

Since 1984, he has been a Professor with the University of Michigan, AnnArbor, where he has appointments in the Department of Electrical Engineeringand Computer Science and, by courtesy, in the Department of Biomedical Engi-neering and in the Department of Statistics. He has held visiting positions at I3SUniversity of Nice, Sophia-Antipolis, France (2001), Ecole Normale Supérieurede Lyon , Lyon, France (1999), Ecole Nationale Supérieure des Télécommu-nications, Paris, France (1999), Scientific Research Labs of the Ford MotorCompany, Dearborn, MI (1993), Ecole Nationale Superieure des TechniquesAvancees (ENSTA), Ecole Superieure d’Electricite, Paris (1990), and MIT Lin-coln Laboratory, Lexington, MA (1987-1989). His research has been supportedby NIH, NSF, AFOSR, NSA, ARO, ONR, DARPA, and by private industryin the areas of estimation and detection, statistical communications, bioinfor-matics, signal processing, and image processing.

Prof. Hero has served as Associate Editor and Guest Associate Editor forthe IEEE TRANSACTIONS ON INFORMATION THEORY (1995–1998, 1999) and theIEEE TRANSACTIONS ON SIGNAL PROCESSING (2002, 2003). He was Chairmanof the Statistical Signal and Array Processing (SSAP) Technical Committee(1997–1998) and Treasurer of the Conference Board of the IEEE Signal Pro-cessing Society (1997–1999). He was Chairman for Publicity of the 1986 IEEEInternational Symposium on Information Theory (Ann Arbor, MI) and GeneralChairman of the 1995 IEEE International Conference on Acoustics, Speech,and Signal Processing (Detroit, MI). He was Co-Chair of the 1999 IEEE Infor-mation Theory Workshop on Detection, Estimation, Classification and Filtering(Santa Fe, NM) and the 1999 IEEE Workshop on Higher Order Statistics (Cae-saria, Israel). He Chaired the 2002 NSF Workshop on Challenges in PatternRecognition. He co-chaired the 2002 Workshop on Genomic Signal Processingand Statistics (GENSIPS). He was Vice President (Finance) of the IEEE SignalProcessing Society (1999–2002). He was Chair of Commission C (Signals andSystems) of the U.S. National Commission of the International Union of RadioScience (URSI) (1999–2002). He is a member of Tau Beta Pi, the AmericanStatistical Association (ASA), the Society for Industrial and Applied Mathe-matics (SIAM), and the U.S. National Commission (Commission C) of the In-ternational Union of Radio Science (URSI). He has received the 1998 IEEESignal Processing Society Meritorious Service Award, the 1998 IEEE SignalProcessing Society Best Paper Award, and the IEEE Third Millenium Medal.

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