1 Secrecy Capacity of Two-Hop Relay Assisted Wiretap Channels Meysam Mirzaee, Soroush Akhlaghi Shahed University, Tehran, Iran Emails: me.mirzaee,[email protected] Abstract—Incorporating the physical layer characteristics to andGaussian wiretapchannelsare investigatedin [3] and[4], 3 secure communications has received considerable attention in respectively. 1 recentyears. Moreover, cooperation withsomenodesofnetwork In [5], Csisza´r and Ko¨rner generalized Wyner’s approach 0 can give benefits of multiple-antenna systems, increasing the to broadcast channels which are not necessarily degraded. It secrecy capacity of such channels. In this paper, we consider 2 cooperative wiretap channel with the help of an Amplify and is assumed the source wishes to transmit a common message n Forward(AF)relaytotransmitconfidentialmessagesfromsource to both legitimate receiver and eavesdropper in addition to a to legitimate receiver in thepresenceof an eavesdropper. In this sendinga confidentialmessagetothe legitimatereceiver.This J regard, the secrecy capacity of AF relying is derived, assuming channel is termed as Broadcast Channel with Confidential 8 the relay is subject to a peak power constraint. To this end, an message (BCC). Accordingly, both the capacity-equivocation achievable secrecy rate for Gaussian input is evaluated through ] solving a non-convex optimization problem. Then, it is proved regionandthesecrecycapacityregionofBCCareestablished T thatanyratesgreaterthanthissecrecyrateisnotachievable.To in [5]. Moreover, it is shown that in the lack of a common I dothis,thecapacityofagenie-aidedchannelasanupperbound message, the secrecy capacity can be computed as . s for the secrecy capacity of the underlying channel is derived, c showing this upper bound is equal to the computed achievable [ secrecyratewithGaussianinput.Accordingly,thecorresponding Cs =max I(X;Y)−I(X;Z) (1) secrecy capacity is compared to the Decode and Forward (DF) 1 strategy whichisserved asthebenchmarkinthecurrent work. where X, Y and Z are, respectively, the source input, the v channel outputs at the legitimate receiver, and the eavesdrop- 1 Index Terms—Secrecy capacity, achievable secrecy rate, phys- 0 ical layer security, cooperative wiretap channel. per’s received signal where the maximization is taken over 7 the distribution of channel input signal. Note that the secrecy 1 capacitycanbeaffectedbychannelconditions.Forinstance,if 1. I. INTRODUCTION source-destinationchannelisweakerthansource-eavesdropper 0 SECURITY has been regarded as one of the important channel the secrecy capacity will be zero meaning no confi- 3 issues in wireless communication networks as it may dential message can be transmitted. To overcome this issue, 1 happen an illegitimate receiver to hear transmitted signal. multiple antenna systems can be employed [6]–[12]. : v As a result, enhancing security has attracted a great deal Due to the cost andsize limitation,using multipleantennas i X attentions in recent years in both of academia and industry. ateachnodemaynotbepracticallyfeasible.Cooperativecom- Information theoretic security is first proposed by Shannon munications, however, is an effective way to get advantages r a in his landmark paper [1] in which it is assumed both the of multi-antenna systems while incorporating single antenna legitimate receiver and eavesdropper (wiretapper) have direct nodes [13]–[18]. In cooperative communication, some nodes access to the transmitted signal. Accordingly, using crypto- can act as intermediate nodes, dubbed relays, to facilitate graphicapproachesandthenotionofequivocation,thelevelof the transmission between two nodesof network. Accordingly, uncertaintyaboutthemessageandthekeyattheeavesdropper there are some strategies to be employed at the relay nodes, side is measured.However,this approachmay notbe feasible among them, the Amplify and Forward (AF), and Decode for some of wireless technologies [2]. This motivated Wyner and Forward (DF) are mostly addressed in the literature. In in his pioneering work in [3] to investigate the possibility AF strategy, the relay sends an scaled version of its received of incorporating physical layer characteristics to secure the signal to the destination without any more changes, while in wireless communication networks. DF, the relay attempts to decode the information, re-encodes Wyner introduced the wiretap channel in which a source again and transmits a coded version of information to the wishes to send confidential message to a legitimate receiver destination. As a result, the AF strategy is more simpler than whilekeepingtheeavesdropperasignorantofthisinformation DF. Furthermore, in some applications, the relay nodes may as possible when the broadcast channel between the source have low security level, thus it is desirable that transmitted and the legitimate receiver and eavesdropper is a degraded messages to be confidential for the relays. These relays are one. Moreover, in [3] the maximum achievable secrecy rate, called untrusted relays [19], [20]. In such scenarios, the AF the rate below which the message can not be decoded at the strategy is the prominent choice as the relay nodes do not eavesdropper,is defined as the secrecy capacity. Accordingly, needtohaveanaccesstotheinformationbits,hence,theyare the secrecy capacity of discrete memoryless wiretap channels unable to eavesdrop the information bits. 2 More recently, a great deal of attentions are devoted to II. SYSTEMMODEL the physical layer security issues in cooperative communica- We consider a wireless communication network consisting tion networks, where it is shown that relaying can improve of a sourcenode S, a relay node R, a destinationnodeD, and the achievable secrecy rate of such networks [21]–[24]. For apassiveeavesdropperE(seeFig.1).Moreover,itisassumed instance, in [25] the secure communication for a source to all nodes are equipped with single antenna and operate in destination with the help of multiple cooperating relays in half duplex mode. Also, it is assumed that there is not a the presence of one or more eavesdroppers is investigated by direct link from S to D and E, and the communication is considering three cooperative strategies: (i) DF, (ii) AF and carried out in two hops through the use of a relay in the (iii) CooperativeJamming (CJ). In [26], the AF beamforming middle of transmission. We consider a quasi-static flat fading under total and individual relay power constraints is studied environmentwhere all channel coefficients are assumed to be where the goal is maximizing the secrecy rate when perfect statistically independent. Moreover, in addition to the source- channel state information (CSI) is available. Moreover, the to-relay channel gain, the channel gains from the relay to the idea of relay selection for secure cooperative networks is destination and eavesdropper are assumed to be completely considered in [27]–[29]. Also, there are some related works known at the relay. This is in accordance to what is assumed on this issue [25], [30], [31]. in some of related works including [32]. Accordingto the modeldepictedin Fig. 1, the communica- In this paper, we derive the secrecy capacity of a simple tion is occurred in two hops. During the first hop, S sends cooperativewiretap channelin which a source wishes to send the message W, which is uniformly taken from the index a confidential message to a legitimated destination with the set = 1,2,...,2nR , to the relay over a transmission helpofanuntrustedrelayincorporatingtheAFstrategy,where W { } interval of length n, where R and nR indicate, respectively, it is desirable to keep the information bits confidential from thetransmissionrateofsourceinunitsofbitsperchanneluse aneavesdropper.ReferringtoFig.1,itisassumedthereisnot and the message entropy. The mapping of each message W direct link from source to destination and eavesdropper and to a codeword xn χn is done by an encoder f : χn, the communication is occurred in two hops with the help of s ∈ s n W → s whereχn is thetransmittedvectorspace.Eachsourcesymbol an AF relay in the middle of transmission. In this case, the s x (t), which appearswithin one time slot, has zero mean and received signal at the destination is a degraded version of the s unitpower,i.e.,E x 2 =1.Inthiscase,thereceivedsignal relay’s. Thus, the DF strategy is optimal. However, we are | s| at the relay node can be written as, interested in cases in which we are dealing with an untrusted (cid:2) (cid:3) relaywhichisunawareofincorporatedcodebookatthesource y (t)= P h x (t)+z (t) for t=1,...,n , (2) r s r s r and the AF strategy is employed at this node. In this regard, where h is thpe channel fading coefficient from source to the secrecy capacity is fully characterized. r the relay, z is zero-mean Additive White Gaussian Noise r To this end, the achievable secrecy rate for Gaussian input (AWGN) at the destination which is of unit power, i.e., is derived. Then, it is demonstrated that any rate greater than E zr 2 =1. Finally, Ps is the transmit power per symbol. | | thisrateisnotachievable.Accordingly,thesecrecycapacityis Then, the relay depending on the incorporated strategy, (cid:2) (cid:3) compared to the capacity of DF relaying to get an indication broadcasts a variation of the received information to the regarding the capacity loss due to the use of AF relaying. destination as well as the eavesdropper. In the following sub- sections, two relaying strategies, AF and DF, are investigated. Thereminderofthispaperisorganizedasfollows.Thesys- tem model is discussed in Section II. Section III provides the A. Amplify and Forward problemformulationfollowedbythemainresults,wheresome In AF relaying, the relay transmits a scaled version of its of technical details are provided in the Appendix. Numerical received signal, i.e., y , to the destination as follows1, results are represented in Section IV. Finally, Section V r summarizes findings. x =ωy , (3) r r Thefollowingnotationsareusedthroughoutthispaper:We where x is the transmitted signal and ω is a scaling factor, r use bold upper and lower case characters for matrices and ensuring the peak power constraint at the relay is satisfied. vectors, respectively. Symbols h and I, respectively, denote As a result, the received signals at the nodes D and E can be differential entropy and mutual information. Rn is the set of written, respectively, as, all n-dimensional real-valued vectors. A 0 means that A (cid:23) y =h x +z = P h ωh x +h ωz +z , (4) is positive semi-definite matrix. Moreover, E[x], Var[x] and d d r d s d r s d r d Cov(x,y) denote the mean and variance of random variable ye =hexr+ze =pPsheωhrxs+heωzr+ze , (5) x and covariance of random variables x and y, respectively. where h and h are chanpnel fading coefficients from R to D The notations x , x and x refer to complex conjugate, d e ∗ ℜ{ } | | and E, respectively. Also, z (0,1) and z (0,1) real part and absolute value of complex variable x. Function d ∼CN e ∼ CN x +isequivalenttomax 0,x .Finallyxnand ∼(0,K) are additive white Gaussian noises at the nodes D and E, { } { } CN respectively. , respectively, denote a sequence of length n and a zero- meancircularlysymmetriccomplexGaussiandistributionwith 1Fornotationalconvenience, weignoretheindexofsymbolsintherestof covariance K. paper. 3 where p(x ) is the Probability Density Function (PDF) of x s s and ρ is the set of all possible PDFs associated with zero h d mean/unit variance random variables. Also, the factor 1 is 2 h due to the use of half-duplex nodes and the transmission is r done during two time slots. h (cid:7)(cid:135)(cid:149)(cid:150)(cid:139)(cid:144)(cid:131)(cid:150)(cid:139)(cid:145)(cid:144) e Evaluatingthesecrecycapacityofunderlyingchannelusing (10) may be computationally infeasible. This motivated us to (cid:22)(cid:145)(cid:151)(cid:148)(cid:133)(cid:135) (cid:21)(cid:135)(cid:142)(cid:131)(cid:155) propose the following theorem which aims at addressing this issue using an indirect approach. Theorem 1: The secrecy capacity of cooperative amplify (cid:8)(cid:131)(cid:152)(cid:135)(cid:149)(cid:134)(cid:148)(cid:145)(cid:146)(cid:146)(cid:135)(cid:148) and forward wiretap channel is given by, Fig.1. SystemModel C (P )= s r 0 α β B. Decode and Forward ≤ InDFstrategy,therelayattemptstodecodethesourcemes- 21log2 ααββPPrr22++((ααµ++ββµ))PPrr++µµ α>β andPr ≤ αµβ sage and re-encodesthe estimated message W to a codeword 1log (cid:16)2√αβµ+αµ+β (cid:17) α>β andP >q µ , 2 2 2√αβµ+α+βµ r αβ ixnnrte∈rvaχlnrnb,yiannvoeknicnogdethregnch:aWnne→l cχonrd.inFgorthlaerogreemtra,ntshmeisrseiloany  (cid:16) (cid:17) q (11) can correctly decode the information signal as long as the where α= hd 2,β = he 2 and µ=1+Ps hr 2. | | | | | | transmission rate is not greater than the capacity of source- Proof: We prove the above theorem in two steps. First, relay link, which is given by, using (10), it is shown that (11) is achievable for Gaussian distribution. Next, for the converse part, we proposean upper CS−R =log2 1+Ps|hr|2 . (6) bound and show that any transmission rate greater than (11) is not achievable. After re-encoding, the relay br(cid:0)oadcasts a w(cid:1)eighted version of 1) The achievability of (11): For Gaussian input, the re-encodedsymbols, i.e., ωx , to D and E. Thus, the received r achievable secrecy rate can be computed as, signalsatthenodesD andEcanberespectivelyexpressedas, + 1 yd =hdωxr+zd , (7) Rs(Pr)= max I(xs;yd) I(xs;ye) . (12) ye =heωxr+ze . (8) (cid:26)E{|xr|2}≤Pr 2h − i(cid:27) Thus, referring to (4) and noting x (0,1), it follows, s In both AF and DF strategies, we assume that the relay is ∼N asunbdjeEct[|tωoxar|p2]ea≤k pPorweinr cDoFn.stTrahinuts,,it.hee. Esc[|axlirn|g2]f≤actPorr iant AthFe I(xs;yd)=log2(cid:18)1+ Ps1|h+d||2h|dω|2|2|ω|h|r2|2(cid:19) relay should satisfy the following constraints, 1+αµω 2 =log | | . (13) |ω|2 ≤ 1+|hPrr|2Ps for AF , Similarly, noting (5), one2c(cid:18)an1a+rrivαe|ωa|t2th(cid:19)e following, ω 2 P for DF , (9) r | | ≤ 1+βµω 2 where E[|xr|2]=1 is assumed in DF strategy. I(xs;ye)=log2 1+β ω| 2| . (14) Inthesequel,wearegoingtocomputethesecrecycapacity (cid:18) | | (cid:19) of this network. Substituting (13) and (14) into (12), it turns out that the achievable secrecy rate becomes, III. SECRECY CAPACITY OF CHANNEL R (P )= s r Thissectionaimstoaddressthesecrecycapacityofcooper- + ativewiretapchannelwhentherelaymakesuseofAFandDF 1 αβµω 4+(αµ+β)ω 2+1 max log | | | | . strategies which are addressed in subsections III.A and III.B, (|ω|2≤Pµr 2 2(cid:18)αβµ|ω|4+(α+βµ)|ω|2+1(cid:19)) respectively. (15) To address the optimal solution of (15), the following maxi- A. Amplify and Forward mization problem should be tackled, The Amplify and Forward cooperative wiretap single-input αβµω 4+(αµ+β)ω 2+1 single-outputchannel can be thoughtas a degraded broadcast max | | | | , (16) channel. Hence, the secrecy capacity of this channel can be |ω|2≤Pµr αβµ|ω|4+(α+βµ)|ω|2+1 computed as [33], which can be reformulated as, + αβµx2+(αµ+β)x+1 1 maxf(x)= , Cs(Pr)= max I(xs;yd) I(xs;ye) , (10) x αβµx2+(α+βµ)x+1 E{p|x(xrs|2)}∈≤ρPr 2h − i subject to 0≤x≤X (17)   4 where x=|ω|2 andX = Pµr. Althoughthe objectivefunction F(x,λ) F(x,λ) of (17) is the ratio of two convex quadratic functions, this functionisnotconvexingeneral[34],[35];hence,themethod ofLagrangeMultipliersdoesnotgivetheoptimalsolution.To find the optimal value of x, i.e., xˆ, we consider two possible cases of α β and α>β as the following. x x Caseα ≤β:Inthiscase,weshowthattheoptimalsolution X x x X ≤ of (17) is xˆ = 0. To this end, noting the definition of µ, (a) (b) indicating µ 1, it follows, ≥ α(µ 1) β(µ 1) , (18) Fig.2. Illustration offunction F(x,λ)fortwopossiblecases − ≤ − or equivalently, αµ+β α+βµ . (19) ≤ It should be noted that referring to (26), since 1 λ 0, − ≤ Thus, for 0 < x X and noting f(x) = αβµx2+(αµ+β)x+1, it turns out that F(x,λ) is a concave function of x and has ≤ αβµx2+(α+βµ)x+1 two positive roots2. As a result, depending on the value of λ, it turns out that the denominator of f(x) is greater than the F(x,λ) can be represented as one of the curves illustrated in nominator. Therefore, we have f(x) <1. On the other hand, Fig. 2. Assuming x˜ maximizes F(x,λ), if X is equal or less since f(0)=1, the optimal value of x becomes, than x˜, then x(λ) =X gives the maximum value of F(x,λ) xˆ=0 . (20) intheintervalx [0,X] see Fig.2(a) ;otherwise, x(λ) will ∈ be equal to x˜ see Fig. 2(b) . Thus, we have, Case α > β: In this case, we show that the optimal value (cid:0) (cid:1) of (17) can be computed as, (cid:0) (cid:1)X X x˜ x(λ)= ≤ (27) xˆ= Pµr Pr ≤ αµβ (21) (x˜ X >x˜ ,  1 P >q µ , where x˜ is computed by taking derivation of F(x,λ) with √αβµ r αβ respect to x and equating to zero as follows, q where xˆ is derived through using the following theorem.  λ(α+βµ) (αµ+β) Theorem 2: We consider the following optimization prob- x˜= − , (28) 2αβµ(1 λ) lem, − As a result, using (28) and claim 1, (27) can be expressed as, xTQx+qTx+q0 where P and mQx∈aRaxnref(nx)=nxsTyPmxm+etpriTcxp+ospit0ive, semi-defi(n2i2te) x(λ)=(Xλ(α2+αββµµ)(−1(αλµ)+β) 122αα≤ββµµλXX≤++αα+µ22αα+ββββµµµXX<++λααµ+<+ββµααµ++ββµ . × − (29) matrices. To address the optimal solution, we define the Also, using (24) and (25), π(λ) can be obtained by, following function, F(x,λ)=xTQx+qTx+q0 λ(xTPx+pTx+p0), λ>0 . π(λ)=F x(λ),λ − (23) (cid:16)π (λ) 1(cid:17) λ 2αβµX+αµ+β Also, we define the functions, = 1 ≤ ≤ 2αβµX+α+βµ (30) (π2(λ) 22ααββµµXX++ααµ++ββµ <λ< ααµ++ββµ , x(λ)=argmaxF(x,λ) λ>0 , (24) x Rn ∀ where ∈ and π (λ)= αβµX2 α+βµ X 1 λ π(λ)= maxF(x,λ)=F(x(λ),λ) . (25) 1 − − − x∈Rn (cid:16) (cid:0)+αβµX(cid:1)2+ α(cid:17)µ+β X +1 , (31) If there exists λˆ > 0 for which π(λˆ) = 0, then ˆx x(λˆ) is ≡ and (cid:0) (cid:1) the optimal solution of (22). Proof: see [35]. 2 λ(α+βµ) (αµ+β) AccordingtotheTheorem2andreferringto(17),wedefine π (λ)= − λ+1 . (32) F(x,λ) as follows, 2 (cid:16) 4αβµ(λ 1) (cid:17) − − F(x,λ)=αβµ(1 λ)x2+ αµ+β λ(α+βµ) x+1 λ, Claim 2: λˆ can be written as, − − − (cid:16) (cid:17) (26) λ = αβµX2+(αµ+β)X+1 X 1 where we assume λ>0 and 0 x X. λˆ = 1 αβµX2+(α+βµ)X+1 ≤ √αβµ (33) Claim1:Theoptimalvalueo≤fλ, i≤.e.,λˆ, fallsintheinterval λ2 = 2(α+βµ)(2α(µα+ββ)µ−)82αβµ−√∆ X > √α1βµ , [1,αµ+β). − α+βµ Proof: see appendix I. 2Thenumberofpositiverootsofapolynomialwithrealcoefficientsordered intermsofascending powerofthevariable iseither equalto thenumberof Based on claim 1, it is sufficient to merely investigate variations in sign ofconsecutive non-zero coefficients or less than this bya F(x,λ) for 1 λ< αµ+β. multipleof2[36] ≤ α+βµ 5 where the first term in the right hand side of (39), i.e., h(y y ), d e | is maximized. On the other hand, we have, ∆=(8αβµ 2(α+βµ)(αµ+β))2 4(α βµ)2(αµ β)2 . − − − − (34) h(y y )=a h(y α y y ) d e d LMMSE e e | − | Proof: see appendix II. b Finally, substituting (33) into (29) yields, h(yd αLMMSEye) ≤ − log (πeλ ) , (41) X X 1 ≤ 2 LMMSE xˆ= ≤ √αβµ (35) (λ2(α2+αββµµ)(−1−(αλµ)+β) X > √α1βµ . wrahnedroem(av)acrioambleesdforoesmn’tthcehfaancgteththaeteandtdrionpgyaankdno(bw)nhvoaldlusesitnocae Moreover, comparing (35) with (27), one can arrive at the we always have h(y x) h(x). α is the corresponding LMMSE | ≤ following, coefficientof LinearMinimumMean SquareError(LMMSE) X X 1 estimation of yd by ye and λLMMSE is the error variance xˆ= ≤ √αβµ (36) conditional on knowing y , i.e., E[y α y 2 y ]. The ( 1 X > 1 , e | d− LMMSE e| | e √αβµ √αβµ last inequality in (41) is due to the fact that the maximum or equivalently, we have, differential entropy is achieved by Gaussian distribution. In the case that y and y are jointly Gaussian, the es- d e xˆ= Pµr Pr ≤ αµβ (37) tliimneaatriofnunecrtrioorn, oi.fe.,yyd[3−7],αtLhMuMsSEfyoer iGsaiunsdseiapnenidnepnutt owfeehvaevrye  1 P >q µ . e √αβµ r αβ h(yd αLMMSEye ye) = h(yd αLMMSEye). Noting, the maxi- − | − q mumdifferentialentropyisachievedbyGaussiandistribution, Asaresult,notingxˆ=|ωopt|2,itturnsoutthatifPr > αµβ, hence, the inequalities in (41) are held with equality for the relay doesn’t use all of its available power. This iqs due Gaussian input xs and I(xs;yd ye) is maximized. So, we can | to the fact that the relay sends a noisy version of x and rewrite (38) as, s additional relay’s transmit power may enhance the additive 1 noise, thereby decreasing the secrecy rate. C (P ) max I(x ;y y ) s r s d e Finally,using(20)and(37)andaftersomemathematics,one ≤ ω2 Pr 2 | | | ≤ µ can readily observe that (11) is the achievable secrecy rate of 1 = max log (πeλ ) AF relying for Gaussian input. In what follows, we are going ω2 Pr 2 2 LMMSE to show that any rate greater than (11) is not achievable (the | | ≤ µ 1 conversepart),thereby(11) isactuallythesecrecycapacityof h(hdωzr+zd heωzr+ze) , (42) − 2 | AF relaying. 2) TheConverseapproach: Fortheconversepart,weshow where λLMMSE can be computed as [37], that any rate greater than R (P ) defined in (15) is not s r λ =Var(y α y y ) achievable. To do this, we investigate the capacity of genie- LMMSE d− LMMSE e| e aided channel as an upper bound on the secrecy capacity of =Var(y ) |Cov(yd,ye)|2 . (43) d underlying channel. Then, we show that this upper bound is − Var(y ) e tight for Gaussian distribution. The following lemma estab- We assume that the receivednoises at nodesD and E, i.e., z lishes the capacity of corresponding genie-aided channel, d and z , are jointly Gaussian with covariance matrix K , i.e., Lemma 1 [7]: An upper bound on the secrecy capacity of e φ cooperative wiretap channel is, z 1 φ d (0,K ), K = ∗ . (44) 1 ze ∼CN φ φ φ 1 C (P ) max I(x ;y y ) . (38) (cid:20) (cid:21) (cid:20) (cid:21) s r s d e ≤ p(xs)∈ρ 2 | Using (44) and after some mathematics, it turns out that (43) ω2 Pr | | ≤ µ can be computed as, In what follows, we show that for AF relaying, the Gaussian 1+(α+β)µx φ2 2 µxh h φ distribution maximizes I(xs;yd|ye). To this end, we have, λLMMSE = −1+| β|µ−x ℜ{ d ∗e } . (45) I(x ;y y )=h(y y ) h(y x ,y ) . (39) s d e d e d s e | | − | The proof is provided in Appendix III. The second term in the right hand side of (39) can be Moreover, the second term in (42) can be computed as, expressed, using (4) and (5), as, h(h ωz +z h ωz +z ) d r d e r e | h(yd|xs,ye)=h( Pshdωhrxs+hdωzr+zd|xs,ye) =h(hdωzr+zd,heωzr+ze)−h(heωzr+ze) =h(phdωzr+zd xs,ye) πe 1+(α+β)x φ2 2 xh h φ | =log −| | − ℜ{ d ∗e } . =h(hdωzr+zd heωzr+ze) . (40) 2 1+βx | (cid:0) (cid:1) (46) One can readily observe that (40) does not depend on the distribution of x , thus, p(x ) should be chosen such that The proof is given in appendix IV. s s 6 Plugging (45) and (46) into (42) and after some manipula- Referring to (15), it turns out that (56) is actually an tions, it follows, achievable rate for the underlying channel. Thus, we have, Cs(Pr) 1 αβµx2+(αµ+β)x+1 ≤ C (P )= max log . (57) 1 1+βx s r 0 x X 2 2 αβµx2+(α+βµ)x+1 max log ≤ ≤ 0 x X2 2(1+βµx Considering the obtained results in (52) and (57), Theorem 1 ≤ ≤ is proved. 1+(α+β)µx φ2 2 µxh h φ −| | − ℜ{ d ∗e } . × 1+(α+β)x−|φ|2−2ℜ{xhdh∗eφ} ) (47) B. Decode and Forward Note that the covariance matrix Kφ should be positive semi- For DF relaying, using max-flow min-cut theorem, it turns definite, i.e., Kφ 0. This results in, out that the secrecy capacity can be computed as, (cid:23) φ 1 . (48) 1 | |≤ C (P )= min C ,C , (58) s r 2 S−R sR−D Thus, (47) yields an upper bound just for values of φ which (cid:8) (cid:9) satisfy (48). where as mentioned earlier C is the capacity of source- S R Proceeding,weagainconsidertwocases α≤β andα>β. to-relay and CsR−D is the secre−cy capacity of the second hop Foreachcase,anupperboundofsecrecycapacityiscomputed operating at full power which is given by [7], with a special value of φ. + Case α β: In this case, we choose, ≤ φ= h∗d . (49) CsR−D = ( |pωm(x|2ra≤)x∈PρrhI(xr;yd)−I(xr;ye)i) h∗e 1+αP + = log r . (59) Noting, 2 1+βP α (cid:26) (cid:18) r(cid:19)(cid:27) φ2 = 1 , (50) | | β ≤ If C C , then we have C (P ) = 1C , thus substituting (49) into (47) gives the following upper otherswRi−sDe, th≤e miSn−imRum value of CsR−Dsandr CS−R2isseRq−uDal bound, toCS R andinthiscase,therelaydoesnotneedtouseallof − its available power, i.e., P . In other words, when the secrecy r Cs(Pr) capacity associated with the second hop is greater than the ≤ 1 (1+βx)(1 α +(β α)µx) available information at the relay, the relay can simply adjust max log − β − 0 x X 2 2 (1+βµx)(1 α +(β α)x) itspowersothatnottowasteanymorepower.Inthiscase,we ≤ ≤ − β − have C = C , where referring to (9), one can arrive = max 1log βµ(β−α)x2+(β−α)(µ+1)x+1− αβ at the fsoRll−oDwing 3S,−R 0 x X 2 2 βµ(β α)x2+(β α)(µ+1)x+1 α ≤ ≤ − − − β 1+αω 2 =0 . (51) C =log | | . (60) S−R 2 1+β ω 2 (cid:18) | | (cid:19) This results in, By noting (6) and the definition of µ, we get, C (P )=0. (52) s r µ 1 Case α>β: In this case, φ is set to, ω 2 = − . (61) | | α βµ h − φ= e , (53) As a result, the secrecy capacity of DF relaying is given by h d where we should note the following, 0 α β ≤ |φ|2 = αβ <1 . (54) Cs(Pr)=121lloogg2(cid:16)µ11++αβPPrr(cid:17) αα>>ββ aanndd 111+++ααβPPPrrr ≤>µµ , Substituting (54) into (47), we arrive at the following, 2 2 1+βPr (62) (1+βx) 1 β +(α β)µx and the optimum relay’s power can be written as, C (P ) max 1log − α − s r ≤ 0 x X 2 2 (1+βµx(cid:16)) 1 β +(α β)x(cid:17) 0 α β ≤ ≤ 1 1+βx (cid:16) 1−+ααµx − (cid:17) |ωopt|2 =Pr α≤>β and 11++αβPPrr ≤µ (63) = 0m≤xa≤xX 2log2(cid:18)1+βµx × 1+αx (cid:19) (55) αµ−−β1µ α>β and 11++αβPPrr >µ . 1 αβµx2+(αµ+β)x+1  = 0mxaxX 2log2 αβµx2+(α+βµ)x+1 , (56) 3Itisworthmentioningthat 11++αβ||ωω||22 isanincreasingfunctionwithrespect ≤ ≤ to|ω|forα>β,thusdecreasing |ω|reducesthesecrecyrateofthesecond where (55) is proved in Appendix V. hop. 7 0.7 1.2 σ2=1 σ2=1 hd hd σ2=2 σ2=2 0.6 hd hd σ2=4 1 σ2=4 hd hd mission)0.5 σ2hd=8 mission)0.8 σ2hd=8 Capacity (bits/trans00..34 Capacity (bits/trans0.6 Secrecy 0.2 Secrecy 0.4 0.2 0.1 0 0 −20 −15 −10 −5 0 5 10 15 20 25 30 −20 −15 −10 −5 0 5 10 15 20 25 30 Power Budget (dBW) Power Budget (dBW) Fig.3. Secrecycapacity ofAFrelaying versuspowerbudget Fig.4. Secrecycapacity ofDFrelaying versuspowerbudget IV. SIMULATIONRESULTS 30 σ2=1 hd This section aims to provide some numerical results to σ2hd=2 20 σ2=4 illustrate the secrecy capacity versus the power budget at the hd σ2=8 relayforcooperativewire-taprelaychannelemployingthe AF hd eafnfidcDieFntsstoraftseoguiersc.eT-rherlaoyug(hhor)u,trtehleays-imdeusltaintiaotniosn,t(hhedc)haanndnreellcaoy-- Power (dBW) 100 ADFF Aealvseos,dtrhoeppreecrei(vheed)naorieseasssautmtheedretolayb,ethReadyelsetiignhatidoinstrainbduttehde. Relay’s −10 eavesdropperareassumedtobecircularlysymmetriccomplex Gaussian randomvariables with zero mean and unit variance. −20 Moreover,the results are derivedfordifferentvaluesof relay- destination channel strengths σ2 = 1,2,4 and 8, while it is −30 assumed σ2 = σ2 = 1 throhudghout the simulations. Also, −20 −15 −10 −5 P0ower Bud5get (dB1W0) 15 20 25 30 hr he source transmit power is set to P =10dBW 4. s Figs. 3 and 4, respectively, show the secrecy capacity of Fig.5. Relay’s powerforAFandDFstrategies versuspowerbudget AFandDFcooperativewiretapchannelsversuspowerbudget for various relay-destination channel strengths, implying the VI. APPENDIXI:PROOF OF CLAIM 1 secrecycapacityofDFisgreaterthanthatofAFstrategy.This Note thatλˆ should satisfy the equality π(λˆ)=0, wherewe isduetothefactthatthereceivedsignalatthedestinationisa have f(xˆ) = λˆ. Thus, the value of λˆ resides between lower degradedversionoftherelay’s,thustheDFstrategyisoptimal. bound and upper bound of f(x). In the case of α > β, we Moreover,itisdemonstratedthatastherelay-destinationchan- have, nel strength is increased, the secrecy capacity is consistently αµ+β >α+βµ , (64) increased. Moreover, the secrecy capacity approaches to a constant value as the relay’s power tends to infinity. This and therefore, is due to the fact that the capacity of the first hop acts as αβµx2+(αµ+β)x+1 αβµx2+(α+βµ)x+1 , (65) bottleneck. Also, Fig. 5 depicts the consumed relay’s power ≥ versusavailable powerforAF and DF strategies, showing the where the equality is satisfied for x = 0. Thus, we have AF strategy saves more power as compared to DF strategy f(x) 1. ≥ when the power budget at the relay increases. On the other hand, we know that for positive values a, b and c, when b<a, we have, V. CONCLUSION a+c a < . (66) This paper aimed at exploring the secrecy capacity of b+c b AF and DF relay-assisted wire-tap channel. Accordingly, the Based on this, by choosing a = αµ+β, b = α+βµ and secrecy capacity of the aforementioned strategies are derived c=αβµx2+1, we arrive at, and numerically compared for Rayleigh channels. Although αβµx2+(αµ+β)x+1 αµ+β the secrecy capacity of DF relying outperforms that of AF, f(x)= < . (67) αβµx2+(α+βµ)x+1 α+βµ less power is consumed when relying on AF strategy. Therefore, we conclude that, 4PleasenotethathereitisassumedthetransmitSNRatthesourceis10dB. αµ+β Thus,notingthereceivednoiseatthisnodeisofunitpower,thusthetransmit 1 λˆ < . (68) poweratthesourcebecomes10dBW. ≤ α+βµ 8 VII. APPENDIXII:PROOF OF CLAIM2 π(λ) π(λ) Using(26)and(29),onecanreadilyverifythatF(x,λ)and π1(λ) π1(λ) x(λ) are continuous functions of x and λ. Therefore, π(λ) am+b will also be a continuous function of λ. Furthermore, π(λ) is λ* λ π a+bm λ a decreasing convex function of λ [35]. Moreover, π(λ) has π 1 λ1 aam++bbm 1 λ* λ2 λ3 π2(λ) π2(λ) positive and negative values, respectively, at the start and end (a) (b) points of interval λ [1,αµ+β], since from (31) we have the ∈ α+βµ following for λ=1, Fig.6. Illustration offunction π(λ)fortwopossiblecases π(1)=π (1)=(αµ+β) (α+βµ)>0 , (69) 1 − It is clear that λ is the desirable root of π (λ). Thus, we and for λ= ααµ++ββµ, using (32), it follows, have, 2 2 αµ+β αµ+β αµ+β αβµX2+(αµ+β)X+1 X 1 π(α+βµ)=π2(α+βµ)=1− α+βµ <0 . (70) λˆ =α2(βαµ+Xβ2µ+)((2αα(µ+α+ββµβ))µ−X)82+α1βµ−√∆ X >≤ √√αα1ββµµ . (78) − Thus, noting π(λ) is strictly decreasing function, it has one root in the interval λ [1,αµ+β], where this root should  ∈ α+βµ VIII. APPENDIXIII either reside in the region in which π(λ)=π (λ) or π(λ) = 1 According to (4), we have, π (λ) as respectively illustrated in Fig.6 (a) or Fig.6 (b). To 2 determine which of these conditions is occurred, we should Var(y )=P ω 2 h 2 h 2+ ω 2 h 2+1 d s d r d | | | | | | | | | | compute π(λ) at the point in which these curves meet each =1+ ω 2 h 2(1+P h 2) d s r other, i.e., at the point λ = 2αβµX+αµ+β as illustrated in | | | | | | ∗ 2αβµX+α+βµ =1+αµx . (79) Fig.6, Similarly, using (5), it follows, 2αβµX +αµ+β π¯ =π(λ∗)=π1(2αβµX +α+βµ) Var(ye)=1+βµx . (80) (αµ+β (α+βµ))(αβµX2 1) Also, the cross covariance of y and y can be computed as, d e = − − . (71) 2αβµX +(α+βµ) Cov(y ,y )=E P h ωh x +h ωz +z d e s d r s d r d This implies that for the case X ≤ √α1βµ, π¯ has negative (cid:20)(cid:16)p (cid:17) value, hence, π(λ) can be represented as Fig.6 (a). Thus, the P h ωh x +h ωz +z ∗ s e r s e r e arosoftoollfowπs(,λ) can be computed through setting π1(λ) to zero = 1×+(cid:16)Pps|hr|2 |ω|2hdh∗e+φ∗ (cid:17) (cid:21) π (λ)=( αβµX2 (α+βµ)X 1)λ+ =µ(cid:0)xhdh∗e +φ∗(cid:1). (81) 1 − − − Using (79), (80) and (81), (43) can be written as, αβµX2+(αµ+β)X +1=0 , (72) αβµ2x2+ φ2+2 µxh h φ which gives, λLMMSE =1+αµx− |1|+βµℜx{ d ∗e } 1+(α+β)µx φ2 2 µxh h φ αβµX2+(αµ+β)X +1 = −| | − ℜ{ d ∗e } . (82) λ1 = αβµX2+(α+βµ)X +1 . (73) 1+βµx IX. APPENDIXIV Alternatively, Fig.6 (b) corresponds to the case that we have We begin with the following definition, X > 1 . Consequently, the root of π(λ) is derived by √αβµ setting π2(λ) to zero as follows, K ,Cov hdωzr+zd z h ωz +z e r e (λ(α+βµ) (αµ+β))2 (cid:18)(cid:20) (cid:21)(cid:19) π2(λ)= 4αβµ(−λ2 1) −λ+1=0 , (74) = |ωω|22h|hdh|2++φ1 |ωω|22hdhh∗e2++φ1∗ . (83) − (cid:20)| | ∗d e | | | e| (cid:21) or equivalently, we have, We know that the following holds, πeK (α βµ)2λ2+(8αβµ 2(α+βµ)(αµ+β))λ+(αµ β)2 =0, h(h ωz +z h ωz +z )=log | z| , (84) − − − d r d| e r e 2 Var(h ωz +z ) (75) e r e which has the following roots, where 2(α+βµ)(αµ+β) 8αβµ √∆ |Kz|=1+(α+β)x−|φ|2−2ℜ{xhdh∗eφ} , (85) λ = − − , (76) 2 2(α βµ)2 and − 2(α+βµ)(αµ+β) 8αβµ+√∆ Var(heωzr+ze)=βx+1 . (86) λ = − . (77) 3 2(α βµ)2 So, we arrive at (46). − 9 X. APPENDIXV [20] X.He and A.Yener, “Cooperation with an untrusted relay: A secrecy perspective,” IEEE Trans. Inf. Theory, vol. 56, no. 8, pp. 3807–3827, To prove (55), we use the following equality, Aug.2010. [21] E. Tekin and A. 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