On the Achievable Secrecy Diversity of Cooperative Networks with Untrusted Relays
11 On the Achievable Secrecy Diversity of CooperativeNetworks with Untrusted Relays
Mohaned Chraiti, Ali Ghrayeb, Chadi Assi and Mazen O. Hasna
Abstract —Cooperative relaying is often deployed to enhancethe communication reliability (i.e., diversity order) and con-sequently the end-to-end achievable rate. However, this raisesseveral security concerns when the relays are untrusted sincethey may have access to the relayed message. In this paper,we study the achievable secrecy diversity order of cooperativenetworks with untrusted relays. In particular, we consider anetwork with an N -antenna transmitter (Alice), K single-antennarelays, and a single-antenna destination (Bob). We consider thegeneral scenario where there is no relation between N and K ,and therefore K can be larger than N . Alice and Bob areassumed to be far away from each other, and all communicationis done through the relays, i.e., there is no direct link. Providingsecure communication while enhancing the diversity order hasbeen shown to be very challenging. In fact, it has been shownin the literature that the maximum achievable secrecy diversityorder for the adopted system model is one (while using artificialnoise jamming). In this paper, we adopt a nonlinear interferencealignment scheme that we have proposed recently to transmit thesignals from Alice to Bob. We analyze the proposed scheme interms of the achievable secrecy rate and secrecy diversity order.Assuming Gaussian inputs, we derive an explicit expression forthe achievable secrecy rate and show analytically that a secrecydiversity order of up to min p N, K q ´ can be achieved usingthe proposed technique. We provide several numerical examplesto validate the obtained analytical results and demonstrate thesuperiority of the proposed technique to its counterparts thatexist in the literature. Index Terms —Cooperative networks, interference alignment,interference dissolution, secrecy diversity, untrusted relays.
I. I
NTRODUCTION
A. Background
Owing to its great potential in improving the communicationreliability of wireless networks, cooperative relaying has beenadopted as a key technology in major wireless standards,including IEEE 802.11, among others [1]. One of the mainadvantages of cooperative relaying is that it increases thetransmission range [2], which is crucial for achieving ubiq-uitous coverage without compromising the quality of serviceprovided to customers [3]. Cooperative relaying will certainly
M. Chraiti is with the ECE Department, Concordia University, Montreal,Canada (email:m [email protected]).A. Ghrayeb is with the ECE Department, Texas A & M University at Qatar,Doha, Qatar and with Qatar Computing Research Institute (QCRI), HBKU,Qatar (e-mail: [email protected]).C. Assi is with the CIISE Department, Concordia University, Montreal,Canada (email:[email protected])M. O. Hasna is with Electrical engineering Department, Qatar University,Doha, Qatar (e-mail: [email protected])This work was supported by Qatar National Research Fund (a memberof Qatar Foundation) under NPRP Grant NPRP8-052-2-029, by an NSERCDiscovery Grant and Concordia University. The statements made herein aresolely the responsibility of the authors. continue to play a central role in shaping the backbone of nextgeneration wireless networks. This can be clearly seen fromthe proliferation of heterogeneous devices connected throughlarge-scale networks, including Ad-hoc network femto-cells,and the Internet of things (IoT), where information is deliveredto remote destinations through relaying. While cooperativerelaying has obvious benefits, it also poses serious securityrisks. Indeed, relays are usually decentralized and it is oftenthe case that relays can be as simple as hand-held devices,with limited computational capability. Consequently, relyingon cryptography-based techniques will be severely limitedand may not be an option [4]. Even when it is possible touse cryptography, it may not be possible to achieve securewireless communication. This is due to the fact that infor-mation in packet overheads is left unencrypted to facilitatecommunication, and such overheads could provide a wealthof information for eavesdroppers. This challenge has recentlyled to an overwhelming interest in physical layer security(PLS), which is considered as a first-line of defense againsteavesdropping [5]–[7].The concept of PLS was coined by Wyner in his seminalwork [8] in which he introduced the wiretap channel. Heproved that, for a point-to-point degraded wiretap channel,where an eavesdropper (Eve) receives a degraded versionof the signal sent from a transmitter (Alice) to a legitimatedestination (Bob), secure communication can be providedwithout sharing a secret key between the communicatingparties. This result is encouraging in the sense that it ispossible to establish secure communications independent ofhow powerful eavesdroppers can be in terms of computationalcapability. However, the situation is completely different forcooperative networks, especially when there is no direct linkbetween the transmitter and the legitimate destination. In sucha scenario, the signal received at the legitimate destination is adegraded version of the signal received by the relays. It couldbe the case that it is desired to not allow relays receiving thesignal to be able to extract any information contained by thesignal. To achieve this, the relays involved in relaying shouldbe considered as untrusted relays. Furthermore, one shouldassume that there could be eavesdroppers adjacent to the relayswho will receive a copy of the signal received by the untrustedrelays. To the end, it is easy to conclude that it is possible thatthe signal received at the destination is a degraded version ofthe one received by untrusted relays and eavesdroppers. Thisrenders PLS-related techniques/results obtained for point-to-point wireless networks irrelevant. This necessitates comingup with new techniques to deal with this problem. Among theperformance metrics that need to be revisited in light of the above mentioned challenge include the achievable secrecy rateand secrecy diversity order, which are the focus of this paper.
B. Relevant Literature
The fundamental question concerning exploiting untrustedrelays to enhance the secrecy rate was explored in [4], [9]–[13] with favorable results. The authors in [4] considered acooperative system with an untrusted relay and they proposedartificial noise-based transmission schemes whereby the desti-nation sends artificial noise (AN) while the transmitter sendsthe information signal (i.e., destination-aided jamming). Thenthe (untrusted) relay amplifies and forwards the combinedsignal (the information and AN signals). The destination usesits perfect knowledge of the AN to subtract it form thereceived signal, and then decodes the intended signal. Thistechnique was extended in [9] to the case of multiple-inputmultiple-output (MIMO) systems with untrusted relays. Theauthors showed that it is beneficial in terms of the secrecyrate to treat relays as untrusted as opposed to treating themas eavesdroppers. However, the loss in the achievable secrecyrate, as compared to the case when the relays are trusted, couldbe significant.It is somewhat intuitive to assume that the secrecy rateis improved as the number of untrusted relays increases.However, the authors in [10], [11] showed the opposite; thatis, the secrecy rate decreases as the number of untrusted relaysincreases. Moreover, it was shown in [12] that the achievablespatial diversity associated with the confidential message islimited to one, regardless of the number of untrusted relaysused in relaying. This is bad news because it suggests thatthe loss incurred by treating the relays as untrusted degradesthe diversity significantly. As a remedy to this problem, theauthors in [13] proposed a scheme that involves using inter-relay jamming whereby relays jam each other in an effort tokeep information confidential. The system model consideredin [13] comprises an N -antenna transmitter, K single-antennarelays and a single-antenna destination, with the assumptionthat the number of transmit antennas is larger than the numberof relays. It was shown that the achievable secrecy diversityis improved to K ´ . However, while this is a positive result,it may not be useful because, in practical settings, the numberof relays is normally much larger than the number of transmitantennas. Furthermore, the schemes in [10], [11] use someforms of AN. C. Problem Statement
The techniques pertaining to the problem at hand that existin the literature, as mentioned above, achieve at most a secrecydiversity order of one for the practical scenario when the num-ber of relays is greater than the number of transmit antennas.This clearly limits exploiting the full potential of having manyuntrusted relays in cooperative networks. Moreover, the notionof using AN (whether relay-based or destination-based), whichis essentially used in [4], [9], [10], [12], [13], may result inthree main issues. First, in these works, AN is assumed tobe treated as noise by the eavesdroppers and this stems fromthe belief that two interfering signals are indistinguishable in a one-dimensional space unless a signal is treated as noisewhile decoding the other. However, recent works showed thatit is possible to jointly decode intended and interfering signalsif they are transmitted over different channels [14]–[17]. Forinstance, in [14], we proposed a scheme that is able to breakup a one dimensional space into two fractional dimensions. Assuch, a destination equipped with one antenna can perfectlyextract two interfering signals received via different channels.In [15]–[17], the authors proposed techniques dealing withinterference in a one-dimensional space, where they showedthat it is possible to jointly decode intended and interferingsignals received over different channels for almost all channelrealizations, which proves that interfering signals are naturallyaligned by the channel. They also showed that two signalsbelonging to a discrete constellation are inseparable only ifthey are transmitted simultaneously over the same channel,i.e., aligned by the same channel. Even when signals do notbelong to discrete constellations, they can be discretized tomake their effect less severe by applying real interferencealignment. In light of these results, it is reasonable to assumethat an eavesdropper, collocated with a relay, may have thepossibility of efficiently decoding an intended signal in thepresence of AN, rendering relying on AN-based techniquesinefficient.Second, AN-based approaches have the disadvantage ofconsuming power given that the relay amplifies and forwardsthe intended and AN signals at the same time. Third, whenthe number of untrusted relays is large, the destination, whenacting as a jammer, will have to generate AN with highpower to jam all the relays including the ones that are farfrom it. However, this results in the drawback of deterioratingthe quality of the intended signal at the destination. Toelaborate, considering that amplify-and-forward (AF) is used,the forwarded signal will comprise the intended signal and theAN signal. Given that the transmit power from the relays islimited, when the AN power is relatively large, the intendedsignal power will effectively be scaled down, which leads to alower signal-to-noise ratio (SNR) at the destination (after thejamming signal is subtracted). Consequently, the achievablesecrecy rate will deteriorate. This leads us to believe thatrelying solely on AN-based strategies may not be as efficientas one may think.
D. Contributions and Outline
Motivated by the above discussion, we propose in thispaper a novel PLS scheme for cooperative networks withmultiple untrusted relays. It is assumed that there is no directlink between the transmitter and the intended destinationand all communication is done through the untrusted relays.This mimics a situation when the destination is far from thetransmitter. The proposed scheme does not use any form ofAN, which makes it completely different from its counterpartschemes that have been proposed in the literature. In devel-oping the proposed scheme, we make use of an interferencealignment scheme that we proposed in [14], which involvesprecoding signals in a nonlinear fashion with the channel gainssuch that only the destination will able to efficiently separateinterfering signals.
The system model considered in this paper comprises an N -antenna transmitter, K single-antenna untrusted relays anda single-antenna destination. We consider that N, K ě .However, there is no additional constraints concerning therelation between N and K , implying that K may be arbitrarilylarge and hence greater than N . As often considered in theliterature, we assume the worst case scenario where each relayis collocated with an eavesdropper that is able to get thesame signal received by the relay it is collocated with. Weanalyze the performance of the proposed technique in termsof the achievable secrecy rate for which we drive a closed-form expression that explicitly shows the impact of the numberof transmit antennas and number of untrusted relays on theachievable secrecy rate. We use the derived secrecy rate toderive the achievable secrecy diversity order. In particular,we invoke the notion of the secrecy outage probability toderive the achievable secrecy diversity order. Although theobtained expression is exact, it does not reveal explicitly thesecrecy diversity order. To get around this, we derive anupper bound on the outage probability, which shows that theachievable secrecy diversity order is up to p min p N, K q ´ q while keeping the messages confidential with respect to alluntrusted nodes. In contrast, the technique proposed in [12]offers secrecy diversity order one, and the one proposed in[13] is not applicable to the system model considered in thispaper since K can be larger than N .Note that the proposed technique does not use AN, whichhelps to overcome the three issues related to the use of ANlisted above. As a result, the proposed technique offers theadvantage of significant power savings as compared to itscounterpart schemes, and this directly translates by a highertransmission rate for the same total transmit power.The rest of the paper is organized as follows. In SectionII, we present the system model and we describe in detailsthe proposed technique. In Section III, we analyze the perfor-mance of the proposed technique in terms of the achievablesecrecy rate. In Section IV, we derive the achievable secrecydiversity order. Numerical results are provided in Section V.We conclude the paper in Section VI.Throughout the rest of the paper, we use | ¨ | , p¨q ˚ , p¨q T and x¨ , ¨y to denote the 2-norm, the transpose conjugate, thetranspose operators and the inner product between two vectors,respectively. We use E r¨s to denote the expectation operator.In this paper, I p¨q and P r r¨s denote the mutual informationand the probability of an event respectively.II. P ROPOSED S CHEME
A. System Model
The system model considered in this paper is depicted inFig. 1, which comprises an N -antenna transmitter (Alice), asingle-antenna destination (Bob) and K single-antenna relaysdenoted by t R , R , . . . , R K u . Knowledge of the whereaboutsof Eves is unknown to the relays nor to Alice or Bob.Therefore, it is assumed that each relay is collocated withan eavesdropper (Eve). Each Eve receives the same signalreceived by the relay it is collocated with. This assumptionis adopted to account for the worst case scenario in terms of the achievable secrecy rate, and it is commonly used inthe literature. As it will be elaborated on later in Section III,among the available K relays, ď L ď min p N, K q realysare selected at random to relay the confidential message toBob. The remaining K ´ L relays are treated as Eves. We areaware that there are other relay selection criteria available inthe literature, but determining the optimal selection criterionwith respect to some performance metric is out of the scopeof this paper. Fig. 1: System model.All sub-channels are assumed to be quasi-static and thechannel gains follow the Rayleigh distribution with varianceone. The channel gains remain constant during a coherencetime and change independently from one coherence timeto another. For a given coherence time, the channel gainsvector between the transmitter and R k is denoted by h k “t h k, , h k, , . . . , h k,N u . The channel gain between R k relayand Bob is denoted by g k . We assume that all untrusted relayshave all the channel state information (CSI) of all sub-channelswhich is the worst case scenario. Moreover, we assume thatAlice has the CSI of the channel between Alice and theselected relays, and between the selected relays and Bob. Theselected relays remain the same during a coherence time. B. Interference Dissolution for Cooperative Networks
We proposed in [14] an interference management technique,referred to as interference dissolution (ID), with the objectiveof managing interference in a one-dimensional space overtime-invariant channels. We showed that ID can achieve arate of two symbols per channel use. More importantly, itwas shown that ID achieves a non-zero degree-of-freedom(dof) for all transmitted symbols, implying that all symbolsare perfectly separable at the destination. Building on thosepromising results, we borrow in this paper the core idea of IDand adapt it to the underlying system model, i.e., cooperativenetworks with multiple untrusted relays.Without loss of generality, let us assume that Bob in-tends to transmit to Alice mL pairs of symbols, namely, pt x , , x , u , t x , , x , u , . . . , t x L, m ´ , x L, m uq . Note thatthe restriction on grouping the transmitted symbols into pairsstems from the fact that ID was developed to achieve a rateof two symbols per channel use [14]. The transmit power persymbol is assumed to be P , i.e., E r| x l,i | s “ P @t l, i u Pr , L s ˆ r , m s .Note that the mL symbols will be transmitted through theselected L relays. These symbols are divided into L sets,where each set contains m symbols. As it will be shownbelow, p mL ` q channel uses are required to transmit the mL symbols. During the first channel use, the transmitterbeamforms each sum of m pairs of symbols in the directionof a selected relay while nulling it out in the direction ofthe remaining relays. This is performed through zero forcingbeamforming (ZFBF) [18]. That is, the sum of the l th symbolpairs ř mi “ x l,i ( l P r , L s ) is beamformed in the direction ofR l . This is done by multiplying it by the l th column vector b l of the pseudo-inverse matrix p b T , b T , . . . , b TL q “ H ˚ p H H ˚ q ´ , where H “ p h l,i q t l,i uPr ,L sˆr ,N s is the channel matrix. The l th row of the channel matrix is denoted by h l . We note herethat the multiplication with a beamforming vector may resultin a power amplification [18]. To deal with this, the transmitternormalizes each beamforming vector with respect to its norm,and therefore the signal received by textR l is expressed as z l, “ h l,l m ÿ i “ x l,i ` n l, , (1)where h l,l “ } b l } h l b Tl and n l, is AWGN with zero meanand variance σ . We use h “ t h l, , h l, , . . . , h l,L u to denote the vector resulting from beamforming t ř mi “ x ,i , ř mi “ x ,i , . . . , ř mi “ x L,i u to R l ( l P r , K s ). In(1), the elements h l,j “ } b j } h l b Tj ( j ‰ l ) do not appearbecause they are equal to zero due to ZFBF.In the second channel use, the relays simultaneously amplifyand forward their respective received signals. The averagepower available at each relay is assumed to be mP . Eachrelay has to normalize the received signal before transmis-sion. The l th relay normalizes the received signals by factor α l “ b | h l,l | ` σ mP . Consequently, the signal received byBob can be written as y “ L ÿ l “ ˜ h l,l g l α l m ÿ i “ x l,i ` g l α l n l, ¸ ` n , (2)where n is AWGN with zero mean and variance σ . Thereceived signal can be written also in the following form. y “ L ÿ l “ g l m ÿ i “ x l,i ` n , (3)where g l “ h l,l g l α l and n “ ř Ll “ g l α l n l, ` n is AWGN withzero mean and variance σ ´ ` ř Ll “ | g l | α l ¯ .In the third channel use, the transmitter precodes t x , , x , u , according to the ID technique [14], in order toallow the destination to properly separate them from the other symbols. To this end, the transmitter calculates a dissolutionfactor β by solving [14] g x , ` β g x , “ L ÿ l “ g l m ÿ i “ x l,i , (4)which gives β “ ` g ř mi “ x ,i ` ř Ll “ g l ř mi “ x l,i g x , . (5)As detailed in [14], the transmitter sends a nonlinear combi-nation of s , s and β and hence, the received signal has thefollowing form. y “ g x , ´ β g x , ` n , (6)where n is AWGN. The noiseless part of y is beamed in thiscase in the direction of all selected relays. To guarantee thatthe used power is mP , the transmitted signal is normalizedby ̺ “ b mP E r| g x , ´ β g x , | s “ bř L | g l | . Thereceived signal by R l can be then written as z l, “ h l,l ̺ p g x , ´ g β x , q ` n l, . (7)In the fourth channel use, the selected relays precode thenforward their respective received signals. Indeed, R l precodesits received signals by multiplying it by p g l q ˚ α l | g l | . α l is usedto have average transmit power mP and p g l q ˚ | g l | is used toguarantee that the signals forwarded from the selected relaysadd constructively at Bob. The received signal by Bob can bewritten as given by (8), where the variance of n is equal to σ p ` ř L | g l | α l q . The signals received during the second andfourth channel uses can be written in vector form as y “ « y ̺ ř l | g l | y ff “ „ g x , g x , ` β „ g x , ´ g x , ` « n ̺ ř l | g l | n ff . (8)One can easily conclude that the signals other than x , and x , are confined to (i.e., aligned by) the sub-space formed bythe signal vector p g x , , ´ g x , q T which is orthogonal to p g x , , g x , q T . Since β is aligned by p g x , , ´ g x , q T ,it was shown in [14] that the destination can use the re-ceived signals p y , y q to efficiently decode the symbol pair t x , , x , u .In addition to the two first channel use required to de-liver the signal y to Bob, two channel uses are requiredto transmit the pair p x , x q , as argued above. Since y is used in the decoding process of all symbols, the sameapplies to the other symbol pairs, that is, each pair requirestwo channel uses to be delivered to Bob. To elaborate, thesymbol pair p x , x q is precoded and transmitted to therelays in the fifth channel use. To achieve this, Alice usesagain the noiseless part of y , but this time to dissolve g ř mi “ ,i ‰ x ,i ` ř Kk “ g k ř mi “ x k,i in g x , , and it alsocalculates the dissolution factor β . Alice beamforms g x , ´ β g x , in the direction of t R , R , . . . , R L u . In turn, theserelays amplify and forward the received signal during the sixth channel use. Bob uses the signals p y , y q to decodethe second signal pair t x , , x , u . Alice, the selected relaysand Bob proceed similarly for the remaining signal pairs. Ingeneral, during the p m p l ´ q ` i ´ q th channel use for( t i, l u P r , m s ˆ r , L s ), the p m p l ´ q ` i q th symbol pairis precoded then bemformed in the direction of all selectedrelays. During the p m p l ´ q ` i ` q th channel use, eachselected relay amplifies and forwards its respective receivedsignal. The destination then uses p y , y m p l ´ q` i ` q to decodethe p m p l ´ q ` i q th symbol pair.Based on the above discussion, we conclude that p mL ` q channel uses are required to transmit mL symbols, resultingin rate mL p mL ` q Ñ m Ñ8 symbol per channel use. We stresshere that ID achieves rate two symbols per channel use fora point-to-point system [14], whereas, the achievable rate forthe underlying system is halved since it is a two-hop link andthe relays are half-duplex.As per the ID scheme, for each symbol pair, the remainingsignals are nonlinearly precoded in order to be aligned by theintended symbol vector. Since the channel gain vector betweenthe transmitter and any relay (selected or not) is differentfrom the transmitter-relays-destination’s channel vector, theremaining signals will not be aligned at these relays. Moreover,during the first channel use, the transmitter communicates thesums of m symbols t ř mi “ x ,i , ř mi “ x ,i , . . . , ř mi “ x L,i u which implies that each symbol is aligned, i.e., received viathe same channel, with m ´ other symbols on the samechannel. Therefore, using interference management techniquessuch as real interference alignement [15]–[17], a relay can onlydecode the sum of m symbols but it cannot separate them.The implication here is that any of the selected relays cannotdecode symbols individually and hence can not extract anyuseful information.III. A CHIEVABLE S ECRECY R ATE
In this paper, we make use of the following expression forthe achievable secrecy rate [10], [11]. R s “ τ max ˆ , I p x m ; y m q ´ max k Pr ,K s I p x m ; z mk q ˙ , (9)where τ is the number of channel uses and x mk is the channelinput. The channel output is denoted by the pair t z mk , y m u ,which represent the received signals by R k and Bob, respec-tively. We note that this secrecy rate can be achieved in thesense of strong secrecy by using the channel resolvability-based method to code the message [19]. To analyze (9), we need to analyze the achievable rate on allsubchannels, including the Alice-Bob, Alice-relay channels.The latter includes the selected and non-selected relays. Asshown above, the symbols are precoded and decoded in pairs.Moreover, the precoding and decoding processes are similarfor all symbol pairs. Therefore, without loss of generality,we consider the symbol pair t x , , x , u in our analysis. We Precoding and coding in this paper are used to denote two different signalprocessing stages. In the case of PLS, coding is used to map a binary sequenceto a sequence of symbols in order to achieve weak or strong secrecy. Whereasprecoding is used to maximize Alice-Bob’s mutual information and to degradeAlice-Eve’s channel. then generalize the result to the remaining mL ´ symbolpairs. Based on this result, we provide a lower bound on theachievable secrecy rate at high SNR. These results are usedto provide a lower bound on the achievable secrecy diversityorder. A. The Achievable Rate for the Alice-Bob Channel
The transmitted symbols and the noise components areassumed to be Gaussian. Therefore, the signals p y , y q tend tobe Gaussian and the achievable rate associated to the symbolpair t x , , x , u at the destination is lower bounded as givenby the following lemma. Lemma 1.
The achievable rate associated to the symbol pair t x , , x , u on the Alice-Bob channel is lower bounded as R Bob p x , , x , q ě log ¨˝ ` mP ř Ll “ | g l | σ ´ ` ř Ll “ | g l | α l ¯ ˛‚ ´ . (10) Proof.
See Appendix A.It is clear from (10) that the lower bound is independentof the symbol pair index, and it only depends on the selectedrelays. As such, the same bound applies to all symbol pairs.Recall that the number of channel uses required to transmitthe mL symbols is p mL ` q . Consequently, the averageachievable rate at Bob for large values of mL is given as R Bob “ p mL ` q ˜ L ÿ l “ m ÿ i “ R p x l, i ´ , x l, i q ¸ ě p mL ` q L ÿ l “ m ÿ i “ »– log ¨˝ ` mP ř Ll “ | g l | σ ´ ` ř Ll “ | g l | α l ¯ ˛‚ ´ fifl »
12 log ¨˝ ` mP ř Ll “ | g l | σ ´ ` ř Ll “ | g l | α l ¯ ˛‚ ´ . (11)Note that the transmission of the mL symbols is performedwithin the same (finite) coherence time, which might appearas a contradiction with the assumption of having large valuesof mL . The assumption of having large values of mL isconsidered merely to determine the multiplexing gain (pre-log factor). The original expression is mL p mL ` q . When mL is very large, this expression approaches . This is true evenif mL is not very large. For example, when mL “ , theexpression becomes ´ » . It should be emphasizedhere that this assumption has nothing to do with the achievablesecrecy diversity order, which is the focus of this paper.In the sequel, we are interested in determining the achiev-able secrecy diversity order, which is valid at high SNR. Inthis case, we have α l “ | h l,l | ` σ mP » | h l,l | and | g l | “ | g l | | h l,l | α l » | g l | . Consequently, the expression in (11) simplifies at high SNRas R Bob ě
12 log ¨˚˚˝ ` mP ř Ll “ | g l | σ ˆ ` ř Ll “ | g l | | h l,l | ˙ ˛‹‹‚ ´ . (12) B. The Achievable rate for the Alice-Selected Relay channels
In this subsection, we first derive the achievable rateassociated to a selected relay considering the symbol pair t x , , x , u . We then generalize it to the remaining symbolpairs. Recall that the received signals by R l during the firstand third channel uses can be written as „ z l, z l, “ »– h l,l ř mi “ x l,ih l,l ř Ll “ | g l | p g x , ´ g β x , q fifl ` „ n l, n l, , (13)where n l, and n l, are both AWGN with zero mean andvariance σ .Since β in (11) is a nonlinear combination of signals andthe vector channel t g , g , . . . , g L u , z l, cannot be written as g p x , ` β x , q ` n , . Hence, the remaining signals arenot aligned by the intended one as in (8). Although Evecollocated with R l has the Alice-Bob’s CSI, it cannot proceedsimilar to Bob to decode x , and x , because the remainingsymbols are not aligned by the intended one at the relays.Moreover, each symbol is aligned with m ´ symbols onthe same channel in z , , i.e., received over the same channelas m ´ symbols. Hence, Eve cannot use real interferencealignment [15]–[17] to separate the signals. In z , , β isunknown and Eve cannot use this signal to extract any of thetwo symbols. This heuristic interpretation suggests, from aninformation theoretic perspective, that the proposed techniqueachieves very low rate at Eve, which is proved in the followingLemma. Lemma 2.
At high SNR, the achievable rate associated to thesymbol pair t x , , x , u on the Alice ´ R l ( l P r , L s ) channelis given by: R R l p x , , x , q » log ¨˚˝ ` | g l | ř Lj “ j ‰ l | g j | ˛‹‚ . (14) Proof.
See Appendix B.The expression of the achievable rate in (14) is independentof the index of the transmitted pair of symbols. It depends onlyon the considered relays. Therefore, this expression is valid forall other symbol pairs. Consequently, the achievable rate perchannel use at high SNR at a given selected relay is upper bounded as R R l » m p m ` q log ˜ ` | g l | ř j “ ,j ‰ l | g j | ¸ »
12 log ˜ ` | g l | ř j “ ,j ‰ l | g j | ¸ ď
12 max l Pr ,L s log ¨˚˝ ` | g l | ř Lj “ j ‰ l | g j | ˛‹‚ . (15) C. The Achievable Rate for the Alice-Non Selected RelaysChannels
Let us now consider a non selected relay belonging to theset t R L ` , R L ` , . . . , R K u . We adopt the same strategy as inthe previous section where we first consider the symbol pair t x , , x , u . A non selected relay R l l P r L ` , K s can takeadvantage of the signal transmitted by Alice during the firstand third channel uses, in order to decode t x , , x , u , whichcan be written in vector form as „ z l, z l, “ « ř Ll “ h l,j ř mi “ x l,i ř Ll “ h l,j ̺ p g x , ´ β g x l, q ff ` „ n j, n j, , (16)where n l, and n l, are both AWGN with zero mean and vari-ance σ . In the second line of (16), we use h l,j to denote h l b Tj } b j } .We hereafter use h l to denote the vector p h l, , h l, , . . . , h l,L q T . Lemma 3.
At high SNR, the achievable rate associated to thesymbol pair p x , , x , q on the Alice ´ R l ( l P r L ` , K s )channel, does not scale with power for almost all channelrealizations and it can be written as R R l p x , , x , q » log ˜ } h l } } g } } h l } } g } ´ x h l , g y ¸ . (17) Proof.
See Appendix C.Given that the expression in (17) does not depend on thesymbol pair index, the achievable rate at high SNR at a givennon-selected relay is given as R R l » m p m ` q log ˜ } h l } } g } } h l } } g } ´ x h l , g y ¸ »
12 log ˜ } h l } } g } } h l } } g } ´ x h l , g y ¸ . (18)Now that we have obtained expressions for the achievable rateon all sub-channels, we proceed in the next section where wemake use of these results to determine the achievable secrecydiversity order.IV. A CHIEVABLE S ECRECY D IVERSITY O RDER
Given that the underlying channel is Rayleigh fading,there is a relationship between the outage probability andthe achievable secrecy rate. Specifically, P out p SNR , γ q “ P r r R s p SNR q ă γ s where SNR is defined as SNR ∆ “ mPσ and γ is a secrecy rate threshold [20]. Consequently, the achievablesecrecy diversity order is defined as d ∆ “ lim SNR Ñ8 ´ log P out p SNR , γ q log SNR . In Section III, we provided a lower bound on the achievablerate on the Alice-Bob channel. Moreover, we provided anupper bound on the achievable rate on the Alice-relay channel(selected and non selected). These give a lower bound onthe achievable secrecy rate. Since the diversity order is bydefinition computed asymptotically at high SNR, we make useof the secrecy rate lower bound to provide an upper bound onthe outage probability at high SNR, which is used later toprovide a lower bound on the achievable diversity order. Thesecrecy rate is given (9) and can be lower bounded at highSNR as R s “ max ˆ , R Bob ´ max l Pr ,K s R R l ˙ ě max $’’&’’% ,
12 log ¨˚˚˝ ` mP ř Ll “ | g l | σ ˆ ` ř Ll “ | g l | | h l,l | ˙ ˛‹‹‚ ´ ´
12 max »—– max l Pr ,L s log ¨˚˝ ` | g l | ř Lj “ j ‰ l | g j | ˛‹‚ , max l Pr L ` ,K s log ˜ } h l } } g } } h l } } g } ´ x h l , g y ¸ff+ . (19)On the other hand, the outage probability can be written as P out p SNR , γ q “ P r „ max ˆ , R Bob ´ max l Pr ,K s R R l ˙ ă γ “ P r r ă γ s loooomoooon “ P r „ˆ R Bob ´ max l Pr ,K s R R l ˙ ă γ “ P r „ˆ R Bob ´ max l Pr ,K s R R l ˙ ă γ . (20)Next, we show that for a given positive real constant ǫ ! ,which can be as small as desired, there exists a SNR th suchthat max l Pr ,K s R l can be upper bounded by log p SNR ǫ q @ SNR ě SNR th , i.e., lim SNR Ñ8 P r „ max l Pr ,K s R l ă log p SNR ǫ q “ . Thisbound is used later to provide an upper bound on the outageprobability.In (21) on the next page, we provide an expression forthe outage probability as a function of P out p SNR ǫ q “ P r „ max l Pr ,K s R l ă log p SNR ǫ q . Lemma 4. P out p SNR ǫ q Ñ SNR Ñ8 Proof.
See Appendix D.Invoking the result in Lemma 4, the upper bound on the outage probability given in (21) becomes P out p γ, SNR qď P r „ R Bob ´
12 log p SNR ǫ q ă γ | max l Pr ,K s R l ă
12 log p SNR ǫ q . (22) Lemma 5.
At high SNR and for a given small constant ǫ , theoutage probability can be upper bounded as P out p γ, SNR q ď P r »—– L ÿ l “ l ‰ lm | g l | ă γ ` ` SNR ´ ǫ ´ γ ` SNR ǫ ˘ fiffifl , (23) where l m “ arg max l Pr ,L s p| g l | q .Proof. See Appendix E.Since the components of channel vector g follow theRayleigh distribution, ř Ll “ ,l ‰ lm | g l | follows a Chi-squaredistribution with order p L ´ q . From [20] and (19), weexpress the achievable secrecy diversity order as d “ lim SNR Ñ8 ´ log P r r R s ď γ s log SNR ě lim SNR Ñ8 ´ log P r „ř Ll “ l ‰ lm ` SNR ´ ǫ ´ γ ` SNR ǫ ˘ | g l | ă γ ` log SNR » lim SNR Ñ8 ´ log P r „ř Ll “ l ‰ lm SNR ´ ǫ | g l | ă γ ` log SNR “ p L ´ qp ´ ǫ q . (24)Since ǫ can be as small as desired, the proposed technique cantherefore achieve a secrecy diversity order equal to the numberof selected relays L . The only condition imposed on thenumber of selected relays is ď L ď min p N, K q , suggestingthat Alice can select up to p min p N, K q ´ q relays and thisyields a secrecy diversity order of up to min p N, K q ´ .V. N UMERICAL R ESULTS
A. System and Simulation Setup
In this section, we provide simulation and numerical resultsto validate the analytical results obtained in the previoussections. It is assumed that Alice is equipped with fourantennas, and there are relays. The number of selectedrelays varies from two to four. Bob is assumed to be equippedwith a single antenna. All channels are Rayleigh distributedwith variance one. We assume that each relay can provide anaverage SNR of mPσ which is independent from the number ofthe selected relays. The SNR considering in simulation resultsis the SNR per relay. This implies that the total used transmitpower increases as the number of the selected relays increase.However, increasing the total transmit power does not affectthe secrecy diversity order. P out p SNR , γ q“ P r „ R Bob ´ max l Pr ,K s R R l ă γ “ p ´ P out p SNR ǫ qq P r „ R Bob ´ max l Pr ,K s R R l ă γ | max l Pr ,K s R l ě
12 log p SNR ǫ q ` P out p SNR ǫ q P r „ R Bob ´ max l Pr ,K s R R l ă γ | max l Pr ,K s R l ă
12 log p SNR ǫ q ď p ´ P out p SNR ǫ qq P r „ R Bob ´ max l Pr ,K s R R l ă γ | max l Pr ,K s R l ě
12 log p SNR ǫ q ` P out p SNR ǫ q P r „ R Bob ´
12 log p SNR ǫ q ă γ | max l Pr ,K s R l ă
12 log p SNR ǫ q . (21) B. Achievable secrecy diversity order
We evaluate the achievable secrecy diversity order as afunction of the SNR in dB for the above system setup. Weplot the outage probability in Fig. 2 where the secrecy ratethreshold is set to γ “ . In the figure, we compare theexact outage probability with the corresponding upper bound.The exact expression for the achievable rate on the Alice-Bob channel is given by (25), where the exact expressionsof the denominator and nominator are given by (26) and (27),respectively. The other exact expressions of the achievable rateon the other channels can be found in Appendices B and C.The results corresponding to the upper bound on the outageprobability (UB) are obtained by considering the asymptoticlower bound on the secrecy rate given in (19). It is clear fromthe figure that the outage probability slope, for the proposedtechnique, is equal to L ´ and hence it matches the theoreticallower bound. The asymptotic lower bound is close to the exactoutage probability which proves the tightness of the providedsecrecy rate lower bound.We also compare in the same figure the performance ofthe proposed scheme with the distributed beamforming (DBF)scheme proposed in [12] which provides, to the best of ourknowledge, the best performance in terms of the achievablediversity order. As shown in the figure, the proposed techniqueoutperforms the DBF technique when L ą . Both schemeshave the same secrecy diversity order when L “ , howeverthe DBF outperforms the proposed scheme for this particularcase. The reason is that the results are plotted against the SNRin dB per relay, and this makes it more advantageous for theDBF since all relays are used for relaying, whereas only tworelays are used in the proposed scheme. −5 −4 −3 −2 −1 SNR (dB) O u t age p r obab ili t y ID, K=10, L=2ID,K=10, L=2, UBID, K=10, L=3ID,K=10, L=3, UBID, K=10, L=4ID,K=10, L=4, UBDBF, K=10 [12]
Fig. 2: Outage probability at the destination versus SNR(dB).
C. Achievable secrecy rate
Fig. 3 depicts the achievable secrecy rate as a functionof SNR for different numbers of untrusted relays, namely, K “ , , , , . The exact expression of the secrecy rateis considered in this figure. We set N “ L “ . The figureshows that the secrecy rate increases with the SNR, whichconfirms that the rates associated to the untrusted relays donot scale with power. The figure shows also that the secrecyrate decreases as the number of untrusted relays increases. Infact, the secrecy rate is equal to the difference between theachievable rate at Bob and the maximum of the achievablerates corresponding to the untrusted relays, which increasesas K increases. However, the loss in the secrecy rate is notconsiderable as the number of untrusted relays increases. Thesecrecy rate decreases by approximately a half bit as thenumber of untrusted relays goes from to . This provesthe robustness of the proposed technique to the number ofuntrusted relays. SNR (dB) R s i n b i t s pe r c hanne l u s e K=4K=6K=8K=10K=12
Fig. 3: Achievable secrecy rate versus SNR(dB).VI. C
ONCLUSION
We studied in this paper the achievable secrecy diversityorder of cooperative networks with untrusted relays. We con-sidered a two-hop network comprising an N -antenna Alice, K single-antenna relays and a single antenna Bob. We proposeda nonlinear PLS technique, based on a previously proposedinterference alignment scheme, that ensures secure communi-cation between Alice and Bob via the untrusted relays. Theproposed scheme was analyzed in terms the achievable secrecyrate and secrecy diversity order. It was shown that a secrecydiversity order of up to p min p N, K q ´ q is achievable. Thisis an important result because it is contrary to what has beenpublished so far on this subject. In particular, it has beenshown that the achievable secrecy diversity order is one for thecase when K may higher than N . Furthermore, the proposedmethod does not use artificial noise which is deemed theonly option to secure communications for the adopted systemmodel. The achieved performance is based on random relayselection. We believe that a proper relay selection method candramatically enhance the achievable diversity order.A PPENDIX
ASince the transmitted symbols and the noise componentsare Gaussian, the signals p y , y q tend to be Gaussian and theachievable rate associated to the symbol pair t x , , x , u atthe destination is written as R Bob p x , , x , q “ I p x , , x , ; y , y q“ H p y , y q ´ H p y , y | x , , x , q“ log ˆ | C p y , y q| E r| C p y , y | x , , x , q|s ˙ , (25) where C p y , y q and C p y , y | x , , x , q are the covariancesof p y , y q and p y , y q given p x , , x , q , respectively. Theirexplicit formulas are given in (26) and (27) on the next page,respectively. The first inequality in (27) comes from the factthat p ř Ll “ | g l |q ̺ “ p ř Ll “ | g l |q ř Ll “ | g l | ě and hence ` p ř Ll “ | g l |q ̺ ď p ř Ll “ | g l |q ̺ . Substituting (27) in (25), we obtain a lower bounded on theachievable rate as follows. R Bob p x , , x , q ě log ¨˝ ` mP ̺ σ ´ ` ř Ll “ | g l | α l ¯ ˛‚ ` log ˆ ˙ “ log ¨˝ ` mP ř Ll “ | g l | σ ´ ` ř Ll “ | g l | α l ¯ ˛‚ ` log ˆ ˙ “ log ¨˝ ` mP ř Ll “ | g l | σ ´ ` ř Ll “ | g l | α l ¯ ˛‚ ´ , (28)which proves (10). A PPENDIX
BThe rate associated to p x , , x , q given p z , , z , q is: R R l p x , , x , q “ I p x , , x , ; z l, , z l, q“ H p z l, , z l, q ´ H p z l, , z l, | x , , x , q“ log ˆ | C p z l, , z l, q| E r| C p z l, , z l, | x , , x , q|s ˙ . (29)Since only R receives a signal depending on t x , , x , u in the first channel use, the expressionof E r| C p z , , z , | x , , x , q|s differs slightly from theremaining covariance matrices. Explicit expressions of thecovariance matrices are given in (30), (31) and (32), wherethe expression in (32) is valid for all l P r , L s . At largevalues of m , we have m ´ » m and hence the covariancein (31) can be written as (32) by replacing l by one. Next,we consider the case of large values of m and we thus usethe general expression provided in (32). The expressions ofthe covariance matrix in (30) and (32) can be written in theform p P C ` σ qp P C ` σ q and σ p P C ` σ q ` P C ,respectively, where t C , C , C , C u are constants at highSNR. They depend only on the index of the considered relay.The achievable rate can be written as R R l p x , , x , q “ log ˆ p P C ` σ qp P C ` σ q σ p P C ` σ q ` P C ˙ . (33)We observe from (33) that, at high SNR, the denominatorscales with P . Given that the denominator also scales with P , the achievable rate becomes a constant at high SNR. Thatis, R R l p x , , x , q » log ˆ C C C ˙ » log ˜ ř Lj “ | g j | ř Lj “ ,j ‰ l | g j | ¸ » log ˜ ` | g l | ř Lj “ ,j ‰ l | g l | ¸ , (34)which proves (14). | C p y , y q| “ ˇˇˇˇ E r| y | s E r y p y q ˚ s E rp y q ˚ y s E r| y | s ˇˇˇˇ “ ˇˇˇˇˇˇ mP ̺ ` σ ´ ` ř Ll “ | g l | α l ¯ p ř Ll “ | g l |q ̺ mP ̺ ` σ ´ ` ř Ll “ | g l | α l ¯ ˇˇˇˇˇˇ “ ˜ mP ̺ ` σ ˜ ` L ÿ l “ | g l | α l ¸¸ ˜ mP p L ÿ l “ | g l |q ` σ ˜ ` L ÿ l “ | g l | α l ¸¸loooooooooooooooooooooooooomoooooooooooooooooooooooooon term . (26) E r| C p y , y | x , , x , q|s “ E „ˇˇˇˇ E r| y | | x , , x , s E r y y ˚ | x , , x , s E r y ˚ y | x , , x , s E r| y | | x , , x , s ˇˇˇˇ “ E »——–ˇˇˇˇˇˇˇˇ P ´ p m ´ q| g | ` m ř Ll “ | g l | ¯ ` σ ´ ` ř Ll “ | g l | α l ¯ ´ P c p ř Ll “ | g l |q ̺ ´ p m ´ q| g | ` m ř Ll “ | g l | ¯ ´ x , x , ¯ ˚ ´ P c p ř Ll “ | g l |q ̺ ´ p m ´ q P | g | ` m ř Ll “ | g l | ¯ x , x , P p ř Ll “ | g l |q ̺ ´ p m ´ q P | g | ` m ř ll “ | g l | ¯ x , x , ` σ ´ ` ř Ll “ | g l | α l ¯ ˇˇˇˇˇˇˇˇfiffiffifl “ ˜ p m ´ q P | g | ` mP L ÿ l “ | g l | ` σ ˜ ` L ÿ l “ | g l | α l ¸¸ ˜ p ř Ll “ | g l |q ̺ ˜ p m ´ q P | g | ` mP L ÿ l “ | g l | ¸ E „ x , x , ` σ ˜ ` L ÿ l “ | g l | α l ¸ ¸ ´ p ř Ll “ | g |q ̺ ˜ p m ´ q P | g | ` mP L ÿ l “ | g l | ¸ E „ x , x , “ σ ˜ ` L ÿ l “ | g l | α l ¸ « P ˜ ` p ř Ll “ | g |q ̺ ¸ ˜ p m ´ q| g | ` m L ÿ l “ | g l | ¸ ` σ ˜ ` L ÿ l “ | g l | α l ¸ff ď σ ˜ ` L ÿ l “ | g l | α l ¸ « P p ř Ll “ | g |q ̺ ˜ p m ´ q| g | ` m L ÿ l “ | g l | ¸ ` σ ˜ ` L ÿ l “ | g l | α l ¸ff ď σ ˜ ` L ÿ l “ | g l | α l ¸ « P p ř Ll “ | g |q ̺ ˜ m | g | ` m L ÿ l “ | g l | ¸ ` σ ˜ ` L ÿ l “ | g l | α l ¸ff “ σ ˜ ` L ÿ l “ | g l | α l ¸ »————– mP p L ÿ l “ | g |q ` σ ˜ ` L ÿ l “ | g l | α l ¸loooooooooooooooooooooooomoooooooooooooooooooooooon ˆ term fiffiffiffiffifl . (27) | C p z l, , z l, q| “ ˇˇˇˇˇ mP | h l,l | ` σ mP | h l,l | ̺ ̺ ` σ ˇˇˇˇˇ “ ´ mP | h l,l | ` σ ¯ ´ mP | h l,l | ` σ ¯ . (30) E r| C p z , , z , | x , , x , q|s “ E »—–ˇˇˇˇˇˇˇ P p m ´ q| h , | ` σ ´ ´ x , x , ¯ ˚ P p m ´ q | h , | ̺ p g q ˚ ´ x , x , P p m ´ q | h , | ̺ g | x , | | x , | P | h , | ̺ ´ p m ´ q| g | ` m ř Lj “ | g j | ¯ ` σ ˇˇˇˇˇˇˇfiffifl “ σ ˜ p m ´ q P | h , | ` E „ | x , | | x , | P | h , | ̺ ˜ p m ´ q| g | ` m L ÿ j “ | g j | ¸ ` σ ¸ ` E „ | x , | | x , | P m p m ´ q| h , | ř Lj “ | g j | ř Lj “ | g j | . (31) E r| C p z l, , z l, | x , , x , q|s “ E »—–ˇˇˇˇˇˇˇ P m | h l,l | ` σ ´ ´ x , x , ¯ ˚ P m | h l,l | ̺ p g l q ˚ ´ x , x , P m | h l,l | ̺ g l | x , | | x , | P | h l,l | ̺ ´ p m ´ q| g | ` m ř Lj “ | g j | ¯ ` σ ˇˇˇˇˇˇˇfiffifl “ σ ˜ mP | h l,l | ` E „ | x , | | x , | P | h l,l | ̺ ˜ p m ´ q| g | ` m L ÿ j “ | g j | ¸ ` σ ¸ ` E „ | x , | | x , | P | h l,l | ˜ m ř Lj “ ,j ‰ l | g j | ř Lj “ | g j | ` m p m ´ q | g | ř Lj “ | g j | ¸ . (32) A PPENDIX
CThe rate associated to p x , , x , q given p z l, , z l, q is ex-pressed as I p x , , x , ; z l, , z l, q “ log ˆ | C p z l, , z l, q| E r| C p z l, , z l, | x , , x , q|s ˙ . (35)Explicit expressions of the covariance matrices are given in(36) and (37) on the next page. These expressions can bewritten in the form p P C ` σ qp P C ` σ q and σ p P C ` σ q` P C , respectively, where t C , C , C , C u are constants athigh transmit power. The achievable rate can be written: R R l p x , , x , q “ log ˆ p P C ` σ qp P C ` σ q σ p P C ` σ q ` P C ˙ . (38)At high SNR and when C ‰ , we observe from (38) that thedenominator scales with P . Given that the denominator alsoscales with P , the achievable rate becomes a constant at highSNR. The achievable rate scales with the transmit power if andonly if C “ . By Cauchy Schwartz inequality, C is zeroif and only if p? m ´ h l, , ? mh l, , . . . , ? mh l,L q is parallelto p? m ´ g , ? mg , . . . , ? mg L q , i.e., p h l, , h l, , . . . , h l,L q is parallel to p g , g , . . . , g L q .The two channel vectors p h l, , h l, , . . . , h l,L q and p g , g , . . . , g L q are independent. The scenario whenthese two vectors are parallel is practically impossible andthus R R l p x , , x , q converges to a constant, i.e., does notscale with P at high SNR for almost all channel realizations.Therefore, for almost all channel realizations, the achievablerate, at high SNR and for high values of m , can be written as R R l p x , , x , q » log ˆ C C C ˙ » log ˆ } h l } } g } } h l } } g } ´ x h l , g y ˙ , (39)which proves (17) A PPENDIX
DNote that max l Pr ,K s R l ď log p SNR ǫ q implies that the rateon Alice-relay channel (selected or non selected) is less than log p SNR ǫ q . This is equivalent to $’&’% | g l | ř Lj “ j ‰ l | g j | ă SNR ǫ ´ , @ l P r , L s } h l } } g l } } h l } } g } ´x h l , g y ă SNR ǫ @ l P r L ` , K s (40) To get P out p SNR ǫ q Ñ SNR Ñ8 , the probability that the eventsin (40) occur should approach one as SNR tends to infinity.Now we have P r »—– | g l | ř Lj “ j ‰ l | g j | ă SNR ǫ ´ , @ l P r , L s fiffifl “ P r »—– max l Pr ,L s | g l | ř Lj “ j ‰ l | g j | ă SNR ǫ ´ fiffifl “ P r »—– | g lm | ă p SNR ǫ ´ q L ÿ j “ j ‰ lm | g j | fiffifl , (41)where l m “ arg max l Pr ,L s p| g l | q is the index of the maximumchannel gain among the vector p| g | , | g | , . . . , | g L | q . Theterm p SNR ǫ ´ q ř Lj “ j ‰ lm | g j | increases as SNR increases.Therefore, the probability in (41) approaches one as SNR tendsto infinity.Let us now analyze P r „ } h l } } g } } h l } } g } ´x h l , g y ă SNR ǫ . When-ever } h l } } g } } h l } } g } ´x h l , g y ă SNR ǫ , it means that ˇˇˇ x h l , g y ˇˇˇ } h l } } g } ă ´ SNR ǫ . Let us consider the normalized vector h l “ h l } h l } and itsorthogonal normalized vector p h l q K “ p h l q K p} h l }q K . The channelvector g can be decomposed into parallel and orthogonalcomponents as follows. g “ x g , h l y h l ` x g , p h l q K yp h l q K . We use g K “ x h l , p h l q K y and g k “ x g , h l y to denotethe perpendicular and parallel components, respectively. Since | C p z l, , z l, q| “ ˇˇˇˇˇˇ mP ř Lj “ | h l,j | ` σ
00 2 mP p ř Lj “ h l,j q ̺ ř Lj “ | g j | ` σ ˇˇˇˇˇˇ “ ˜ mP L ÿ j “ | h l,j | ` σ ¸ ˜ mP p ř Lj “ h l,j q ̺ L ÿ j “ | g j | ` σ ¸ “ ˜ mP L ÿ j “ | h l,j | ` σ ¸ ˜ mP p L ÿ j “ h l,j q ` σ ¸ . (36) E r| C p z l, , z l, | x , , x , q|s“ E »—–ˇˇˇˇˇˇˇ P pp m ´ q| h l, | ` m ř Ll “ | h l,j | q ` σ ´ p ř Lj “ h l,j q ˚ x ˚ , ̺x ˚ , P ´ p m ´ qp g q ˚ h l, ` m ř Lj “ p g j q ˚ h l,j ¯ p ř Lj “ h l,j q x , ̺x , P ´ p m ´ qp g qp h l, q ˚ ` m ř Lj “ p g j qp h l,j q ˚ ¯ | x , | | x , | P p ř lj “ h l,j q ̺ ´ p m ´ q| g | ` m ř Lj “ | g j | ¯ ` σ ˇˇˇˇˇˇˇfiffifl “ σ ˜ P pp m ´ q| h l, | ` m L ÿ j “ | h l,j | q ` P p ř Lj “ h l,j q ̺ ˜ p m ´ q| g | ` m L ÿ j “ | g j | ¸ ` σ ¸ ` P p ř Lj “ h l,j q ̺ «˜ p m ´ q| g | ` m L ÿ j “ | g j | ¸ ˜ p m ´ q| h l,j | ` m L ÿ j “ | h l,j | ¸ ´ ˇˇˇˇˇ p m ´ qp g q ˚ h l, ` m L ÿ j “ p g j q ˚ h l,j ˇˇˇˇˇ ff . (37) } g } “ } g K } ` } g k } , we can write ˇˇˇ x h l , g y ˇˇˇ } h l } } g } “ ˇˇˇˇ x h l } h l } , g y ˇˇˇˇ } g } “ ˇˇˇ x h l , g y ˇˇˇ } g } “ } g k } } g K } ` } g k } . (42)The magnitude of the elements of the channel vector g be-tween the selected relays and the destination follows Rayleighdistribution. From [21], } g k } „ Γ p , q and }p g q K } „ Γ p L ´ , q , where Γ p p, λ q denotes the Γ distribution withparameters p p, λ q . By applying this result and considering (42),we can obtain [22] ˇˇˇ x h l , g y ˇˇˇ } h l } } g } „ β p , L ´ q , where β p p, λ q denotes the Beta distribution with parameters p p, λ q . Furthermore, it is known that the cumulative distribu-tion function (CDF) of the Beta distribution is the regularized incomplete Beta function [22]. Consequently, we obtain, P r « } h l } } g } } h l } } g } ´ x h l , g y ă SNR ǫ , @ l P r L ` , K s ff “ K ź l “ L ` P r « } h l } } g } } h l } } g } ´ x h l , g y ă SNR ǫ ff “ « P r ˜ } h L ` } } g } } h L ` } } g } ´ x h L ` , g y ă SNR ǫ ¸ff K ´ L “ ¨˚˝ P r »—– ˇˇˇ x h L ` , g y ˇˇˇ } h L ` } } g } ă ´ SNR ǫ fiffifl˛‹‚ K ´ L “ ˜ β ` ´ SNR ǫ ; 1 , L ´ ˘ β p , L ´ q ¸ K ´ L “ ´ I ´ SNR ǫ p , L ´ q ¯ K ´ L “ ˜ ´ ˆ SNR ǫ ˙ L ´ ¸ K ´ L Ñ SNR Ñ8 , (43)where β ` ´ SNR ǫ ; 1 , L ´ ˘ and I ´ SNR ǫ p , L ´ q denotesthe incomplete Beta function and the regularized incompleteBeta function, respectively. The probabilities in (41) and (43)approach one as SNR tends to infinity. There product is equalto P out p SNR q Ñ
SNR Ñ8 . This proves (22). A PPENDIX E P out p γ, SNR qď P r „ R Bob ´
12 log p SNR ǫ q ă γ | max l Pr ,K s R l ă
12 log p SNR ǫ q “ P r »——–
12 log ¨˚˚˝ ` SNR ř Ll “ | g l | ˆ ` ř Ll “ | g l | | h l,l | ˙ ˛‹‹‚ ´ ´
12 log p SNR ǫ q ă γ | max l Pr ,K s R l ă
12 log p SNR ǫ q ff ď P r »–
12 log ¨˝ SNR ´ ǫ ř Ll “ | g l | ` ř Ll “ | g l | | h l,l | ˛‚ ă γ ` | max l Pr ,K s R l ă
12 log p SNR ǫ q ff . (44)whenever R Bob ´ log p SNR ǫ q ă γ and max l Pr ,K s R l ă log p SNR ǫ q yield $’’&’’% SNR ´ ǫ ř Ll “ | g l | ` ř Ll “ | gl | | h l,l | ă γ ` | g lm | ă p SNR ǫ ´ q ř Lj “ j ‰ lm | g j | . (45)This is equivalent to the following inequalities. $’&’% SNR ´ ǫ ř Ll “ | g l | ´ γ ` ř Ll “ | g l | | h l,l | ă γ ` | g lm | ă p SNR ǫ ´ q ř Lj “ j ‰ lm | g j | , (46)which is equivalent to $’&’% SNR ´ ǫ | g lm | ´ γ ` | g lm | | h lm,lm | ` ř Ll “ l ‰ lm | g l | p SNR ´ ǫ ´ γ ` | h l,l | q ă γ ` | g lm | ă p SNR ǫ ´ q ř Lj “ j ‰ lm | g j | . (47)To obtain an upper bound on the outage probability at highSNR, we provide a lower bound on the first inequality in(47). This can be obtained by subtracting the first term, inthe first inequality, and replacing | g lm | in the second term, inthe first inequality, by its upper bound provided by the secondinequality in (47). That is, ´ γ ` ř Ll “ l ‰ lm | g l | | h lm,lm | ` L ÿ l “ l ‰ lm | g l | p SNR ´ ǫ ´ γ ` | h l,l | q ă γ ` which can be rearranged as L ÿ l “ l ‰ lm | g l | ˜ SNR ´ ǫ ´ γ ` ˜ | h l,l | ` | h lm,lm | ¸¸ ă γ ` . (48)At high SNR, the outage probability can thus be outer boundedas given in (49) on the next page.The probability P out p SNR ǫ q “ P r „ max l Pr ,L s | h l,l | ă SNR ǫ approaches one as SNR tends to infinity. Indeed, the magnitude of the channel vector h l resulting from ZFBF follows Rayleighdistribution [18]. Therefore P out p SNR ǫ q can be written asfollows. P r « max l Pr ,L s | h l,l | ă SNR ǫ ff “ P r « | h l,l | ă SNR ǫ , @ l P r , L s ff “ ˆ P r „ | h | ă SNR ǫ ˙ L “ ˆ ´ P r „ | h | ą SNR ǫ ˙ L “ ´ ´ e SNR ǫ ¯ L Ñ SNR Ñ8 . (50)This yields at high SNR an upper bound on the outageprobability as P out p γ, SNR q ď P r »—– L ÿ l “ l ‰ lm | g l | ˜ SNR ´ ǫ ´ γ ` | h l,l | ` γ ` | h lm | ¸ ă γ ` | max l Pr ,L s | h l,l | ă SNR ǫ ff ď P r »—– L ÿ l “ l ‰ lm | g l | ` SNR ´ ǫ ´ γ ` p SNR ǫ q ˘ ă γ ` fiffifl “ P r »—– L ÿ l “ l ‰ lm | g l | ă γ ` ` SNR ´ ǫ ´ γ ` SNR ǫ ˘ fiffifl . (51) R EFERENCES[1] R. Pabst, B. H. Walke, D. C. Schultz, P. Herhold, H. Yanikomeroglu,S. Mukherjee, H. Viswanathan, M. Lott, W. Zirwas, M. Dohler, H. Agh-vami, D. D. Falconer, and G. P. Fettweis, “Relay-based deploymentconcepts for wireless and mobile broadband radio,” vol. 42, no. 9, pp.80–89, Sep. 2004.[2] B. Pabst, R.and Walke, D. Schultz, P. Herhold, H. Yanikomeroglu,S. Mukherjee, H. Viswanathan, M. Lott, W. Zirwas, M. Dohler, H. Agh-vami, D. Falconer, and G. Fettweis, “Relay-based deployment conceptsfor wireless and mobile broadband radio,” vol. 42, no. 9, pp. 80–89,Sept. 2004.[3] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity.part I & II,” vol. 51, no. 11, pp. 1939–1948, Nov. 2003.[4] X. He and A. Yener, “Cooperation with an untrusted relay: A secrecyperspective,” vol. 56, no. 8, pp. 3807–3827, Aug. 2010.[5] F. Armknecht, J. Girao, A. Matos, and R. L. Aguiar, “Who said that?privacy at link layer,” in
Proc. IEEE INFOCOM , May 2007.[6] K. Zeng, K. Govindan, and P. Mohapatra, “Non-cryptographic authen-tication and identification in wireless networks [security and privacy inemerging wireless networks],” vol. 17, no. 5, pp. 56–62, Oct. 2010.[7] J. Pang, B. Greenstein, R. Gummadi, S. Seshan, and D. Wetherall,“802.11 user fingerprinting,” in
Proc. of the 13th Annual ACM Inter-national Conference on Mobile Computing and Networking , 2007.[8] A. D. Wyner, “The wire-tap channel,”
The Bell System TechnicalJournal , vol. 54, no. 8, pp. 1355–1387, Oct. 1975.[9] C. Jeong, I. M. Kim, and D. I. Kim, “Joint secure beamforming designat the source and the relay for an amplify-and-forward mimo untrustedrelay system,” vol. 60, no. 1, pp. 310–325, Jan. 2012.[10] L. Sun, T. Zhang, Y. Li, and H. Niu, “Performance study of two-hopamplify-and-forward systems with untrustworthy relay nodes,” vol. 61,no. 8, pp. 3801–3807, Oct. 2012.[11] L. Sun, P. Ren, Q. Du, Y. Wang, and Z. Gao, “Security-aware relayingscheme for cooperative networks with untrusted relay nodes,” vol. 19,no. 3, pp. 463–466, March 2015. P out p SNR , γ qď P r »—– L ÿ l “ l ‰ lm | g l | ˜ SNR ´ ǫ ´ γ ` ˜ | h l,l | ` | h lm | ¸¸ ă γ ` fiffifl “ ˜ ´ P r « max l Pr ,L s | h l,l | ă SNR ǫ ff¸ P r »—– L ÿ l “ l ‰ lm | g l | ˜ SNR ´ ǫ ´ γ ` ˜ | h l,l | ` | h lm | ¸¸ ă γ ` | max l Pr ,L s | h l,l | ě SNR ǫ fiffifl ` P r « max l Pr ,L s | h l,l | ă SNR ǫ ff P r »—– L ÿ l “ l ‰ lm | g l | ˜ SNR ´ ǫ ´ γ ` ˜ | h l,l | ` | h lm | ¸¸ ă γ ` | max l Pr ,L s | h l,l | ă SNR ǫ fiffifl (49) [12] J. B. Kim, J. Lim, and J. M. Cioffi, “Capacity scaling and diversity orderfor secure cooperative relaying with untrustworthy relays,” vol. 14, no. 7,pp. 3866–3876, July 2015.[13] W. Wang, K. C. Teh, and K. H. Li, “Relay selection for secure successiveaf relaying networks with untrusted nodes,” vol. 11, no. 11, pp. 2466–2476, Nov. 2016.[14] M. Chraiti, A. Ghrayeb, and C. Assi, “Nonlinear interference alignmentin a one-dimensional space,” http://arxiv.org/abs/1606.06021 , 2016.[15] A. Motahari, A. Khandani, and S. Gharan, “On the degrees of freedomof the 3-user Gaussian interference channel: The symmetric case,” in Proc. IEEE ISIT , June 2009.[16] R. Etkin and E. Ordentlich, “On the degrees-of-freedom of the K-userGaussian interference channel,” in
Proc. IEEE ISIT , June 2009.[17] A. Motahari, S. Oveis-Gharan, M.-A. Maddah-Ali, and A. Khandani,“Real interference alignment: Exploiting the potential of single antennasystems,” vol. 60, no. 8, pp. 4799–4810, Aug. 2014.[18] R. de Francisco, M. Kountouris, D. T. M. Slock, and D. Gesbert,“Orthogonal linear beamforming in MIMO broadcast channels,” in
Proc.IEEE WCNC , Mar. 2007.[19] M. R. Bloch and J. N. Laneman, “Strong secrecy from channel resolv-ability,” vol. 59, no. 12, pp. 8077–8098, Dec. 2013.[20] R. U. Nabar, H. Bolcskei, and A. J. Paulraj, “Diversity and outageperformance in space-time block coded ricean mimo channels,” vol. 4,no. 5, pp. 2519–2532, Sep. 2005.[21] M. Maleki and H. R. Bahrami, “On the distribution of norm ofvector projection and rejection of two complex normal random vectors,”
Mathematical Problems in Engineering , vol. 2015, no. 8, p. 4, 2015.[22] A. Jeffrey, D. Zwillinger, I. Gradshteyn, and I. Ryzhik,