on the secrecy rate achievability in dual-hop amplify-and-forward relay networks

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IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. PP, NO. 99, AUGUST 2014 1

On the Secrecy Rate Achievability in Dual-hopAmplify-and-Forward Relay Networks

Auon Muhammad Akhtar, Aydin Behnad,Member, IEEE, and Xianbin Wang,Senior Member, IEEE

Abstract—The achievable secrecy rate of a dual-hop amplify-and-forward relaying system, in the presence of an eavesdropper,is investigated based on the different values of the wireless links’signal-to-noise ratios (SNRs). It is shown that when the directlink to the eavesdropper is in a better condition than the directlink to the destination, but the relaying link for the destination isbetter than the relaying link for the eavesdropper, under specificconditions for the links SNRs, it is possible to achieve the secrecyrate for a limited range of the relay power. On the other hand,when the direct link to the destination is better than the directlink to the eavesdropper, under certain conditions for the links’SNRs, it is not always beneficial to employ the relay, since thereexists a limited range of relaying power for which secrecy rateis not achievable. Finally, it is shown that the source transmitpower has no impact on the secrecy rate achievability. Simulationresults confirm the theoretical analysis carried out in thisletter.

Index Terms—Amplify and forward, cooperative communica-tion, physical layer security, secrecy rate.

I. I NTRODUCTION

I NFORMATION-THEORETIC secrecy was first introducedby Wyner in his seminal work on the wiretap channel

[1]. Wyner’s work was later extended to different scenariosand channel models [2]. Motivated by the growing trend forcooperative communications, recent works have focused onemploying decode-and-forward (DF) and amplify-and-forward(AF) relaying strategies in order to improve network security[3]–[6]. All of the aforementioned works develop differentschemes that increase the secrecy rates of the relay networks.However, none of the extant researches establish the existenceconstraints under which the secrecy rate is achievable.

In this letter, we investigate the secrecy rate achievabilityof cooperative diversity based dual-hop AF relaying. Specifi-cally, we are looking for the source-destination, source-relay,relay-destination, source-eavesdropper, and relay-eavesdropperlinks’ instantaneous signal-to-noise ratios (SNRs) for whichthe secrecy rate is achievable and, therefore, secure commu-nication is possible. As two non-trivial cases, it is shownthat when the source-destination link SNR is worse thanthe source-eavesdropper link SNR, but the relay-destinationlink SNR is better than relay-eavesdropper link SNR, theremay exist a limited power range for the relay for which thesecrecy rate is achievable. Conversely, it is shown that whenthe source-destination link SNR is better than the source-eavesdropper SNR, but the relay-eavesdropper link SNR isbetter than the relay-destination link SNR, there may exista

The authors are with the Department of Electrical and Computer Engineer-ing, Western Univeristy, London, ON, Canada, N6A 5B9, e-mail: ({aakhtar8,abehnad, xianbin.wang}@uwo.ca).

S

E

R

D

γsr

γsd

γse

γre

γrd

Fig. 1. A dual-hop amplify-and-forward relay network in thepresence of aneavesdropper.

finite range for the relay power for which the secrecy rateis not achievable. We derive the existence constraints and therelay power range for these two cases, separately. Furthermore,it is shown that the source transmit power has no effect onthe state of the secrecy rate, i.e., increasing or decreasing thesource power has no effect on the positivity or negativity ofthe secrecy rate. Hence, the analysis and derivations presentedin this work provide insight into the behavior of the achievablesecrecy rate, as a function of the source and the relay transmitpowers, under different network conditions.

The rest of this letter is organized as follows: Section IIdescribes the system model. Section III carries a detailed anal-ysis of the cooperative diversity based dual-hop amplify-and-forward relay networks. Simulation results are also presentedin the same section. Finally, Section IV draws the conclusions.

II. SYSTEM MODEL

Consider a dual-hop amplify-and-forward relay network, asshown in Fig. 1, consisting of a source node S, relay nodeR, destination node D, and an eavesdropper E. When thesource transmits its data signal towards the destination, therelay receives and the eavesdropper overhears this transmissiondue to the broadcast nature of the wireless medium. Also,when the relay amplifies and forwards a copy of the receivedsignal towards the destination, this forward transmissionisagain overheard by the eavesdropper. It is assumed that thedestination and the eavesdropper use maximal-ratio combining(MRC) to combine the received direct and relaying signals.We also assume that the SNRs of S–R, R–D, S–D, S–E, andR–E links, for the reference realization of the system, aredenoted byγsr, γrd, γsd, γse, and γre, respectively. It is alsoassumed thatγsr, γsd, andγse are fixed, but R can changeits transmission power, which we model by a multiplying



2 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. PP, NO. 99, AUGUST 2014

factor α. Therefore, the received SNRs by D and E, due totransmission from R, areαγrd andαγre, respectively. Usingthis system model, the combined received signal-to-noise ratio(SNR) at the destination is given as [7]

γD = γsd+αγsrγrd

γsr + αγrd + 1. (1)

Similarly, at the eavesdropper side, the combined receivedSNR is given by

γE = γse+αγsrγre

γsr + αγre + 1. (2)

In the next section, we analyze the secrecy rate achievability ofthe aforementioned network. Note that the analysis presentedin this letter is based on the instantaneous SNRs, which areused to obtain different constraints for which the secrecy rateis achievable. Statistical analysis, e.g., those presented in [8],are considered as a future direction. Furthermore, it shouldbe noted that we are looking for the existence of the secrecyrate in terms of the links’ instantaneous SNRs and we are notinvestigating how these secrecy rates can be attained or howthe links’ SNRs can be estimated in practice.

III. SECRECY RATE IN DUAL -HOP NETWORKS

The achievable secrecy rate of the system model mentionedin Section II can be written as [5]

C = max{CD − CE, 0}, (3)

where [5]

CD =1

2log(1 + γD) (4)

is the total S–D end-to-end capacity and

CE =1

2log(1 + γE) (5)

denotes the total S–E end-to-end capacity. From (3), (4), and(5), it can be seen that the the secrecy rate is achievable whenγD > γE, i.e., using (1) and (2), when

γsd+αγsrγrd

γsr + αγrd + 1> γse+

αγsrγre

γsr + αγre + 1. (6)

Here, we analyze the secrecy rate achievability of thesystem under different SNR conditions. Based on the valuesof γrd, γsd, γse, andγre, we get four distinct cases, as follows1:

• Case 1:γsd > γse andγrd > γre.• Case 2:γsd < γse andγrd < γre.• Case 3:γsd < γse andγrd > γre.• Case 4:γsd > γse andγrd < γre.

In the following, we investigate these four cases, separately.

1We ignore the cases whereγsd = γse or γrd = γre, as it is impossible tohave these equalities in practice. However, the deriving results of this letteraccommodate these conditions with the assumptions thatγsd → γ

+se, γsd →

γ−

se, γrd → γ+re , or γrd → γ

−

re .

A. Case 1: γsd > γse and γrd > γre

In this case, it is trivial that the secrecy rate is achievable,as both the direct link and the relaying link to the destinationare better than those for the eavesdropper. Also, the secrecyrate holds for any value ofα > 0, i.e., it holds for any relaytransmit power. To show this analytically, we consider thefunction

f(x) =c1x

x+ c2

, (7)

wherec1 andc2 are two positive constants. The function f(x)is increasing forx > 0. This means that

αγsrγrd

γsr + αγrd + 1>

αγsrγre

γsr + αγre + 1(8)

holds as long asγrd > γre. Sinceγsd > γse, using (8), onefinds thatγD > γE and thereforeC > 0.

B. Case 2: γsd < γse and γrd < γre

Following similar steps as the ones for the previous case, itcan be shown that it is impossible to achieve the secrecy ratein this scenario.

C. Case 3: γsd < γse and γrd > γre

Case 3 reflects a scenario where the direct link for D isworse than that for E, but the relaying link for D is betterthan E. From (1) and (2), we obtain that

limα→0

γD = γsd.

limα→0

γE = γse.

limα→∞

γD = γsd+ γsr.

limα→∞

γE = γse+ γsr.

(9)

Sinceγsd < γse, it can be seen from (9) that asα approaches0 or ∞, the secrecy rate is not achievable sinceγE is largerthanγD for both the extreme values ofα. The question here iswhether there is no secrecy rate for the whole range ofα > 0or there exist limited regions where the secrecy rate can beobtained. We show in Lemma 1 that under a constraint on thelinks’ SNRs, there exists one limited range ofα for whichthe secrecy rate is achievable, which means that under thisconstraint, there is a relay power range for which the secrecyrate is obtained.

Lemma 1: The secrecy rate is achievable if the relay poweris multiplied by α1 < α < α2, whereα1 andα2 are givenby (11a) and (11b) and the following holds for the referencescenario links’ SNRs:

γsr

γse− γsd>

γrd + γre + 2√γrdγre

γrd − γre. (10)

Proof: See Appendix A.In Lemma 1, equation (10) ensures thatα1 and α2 are

always positive and real. The right side of the inequality isrelated to the relaying links while the left side includes thedirect links’ SNRs in the reference realization. It is worthmentioning that (10) provides some useful insights about thebehavior of the secrecy rate achievability. First, the equationshows that if all other parameters are fixed, then an increase



AKHTAR et al.: ON THE SECRECY RATE ACHIEVABILITY IN DUAL-HOP AMPLIFY-AND-FORWARD RELAY NETWORKS 3

α1 = −γsr + 1

2γrdγre

(

γrd + γre + γsrγrd − γre

γsd− γse+

√

(


γsd− γse

)2

− 4γrdγre

)

, (11a)

α2 = −γsr + 1

2γrdγre

(


γsd− γse−√

(


γsd− γse

)2

− 4γrdγre

)

. (11b)

in γsr increases the left side of (10), which is desirable sincethe positive range ofα exists when (10) holds. Second, it canalso be seen that the difference betweenγse andγsd should besmall. This observation also relates to the fact that for Case3, γsd < γse and, therefore, the smaller the difference betweenthe two links’ SNRs, the better it is in terms of achieving thesecrecy rate. Lastly, we note that ifγrd ≫ γre, the right sideof (10) becomes approximately equal to 1. This implies thatif the R–D link is much better than the R–E link, the secrecyrate can be achieved as long asγsr > γse− γsd.

Fig. 2 depicts the behavior of the secrecy rate achievabilityas a function of the range ofα. The blue-colored regionsrepresent the valuesα for which there is no achievable secrecyrate while the white color represents the regions where thesecrecy rate can be obtained. It can be seen that the range ofα with secrecy rate increases as S–R link’s SNR improves.Note that for Case 3, transmission through the relay is theonly option, since the direct channel between the source andthe destination is worse than the channel between the sourceand the eavesdropper. Therefore, the relay must amplify itscopy of the received signal while ensuring thatα falls withinthe prescribed limits. Otherwise, there will be no secrecy rate.

D. Case 4: γsd > γse and γrd < γre

Case 4 reflects a scenario where the S–D link’s SNR isbetter than the S–E link’s SNR, while the R–D link’s SNR isworse than the R–E link’s SNR. In this case, (9) shows thatthe secrecy rate is achievable whenα goes to 0 or∞, sinceγD

is larger thanγE for both the extreme values ofα. Our goalhere is verify whether or not the secrecy rate is achievable forthe whole range ofα. In this context, Lemma 2 outlines thescenarios in which there is no secrecy rate.

Lemma 2: The secrecy rate is not achievable if the relaypower is multiplied byα1 < α < α2, whereα1 and α2

are given by (11a) and (11b) and the following holds for thereference scenario links’ SNRs:

γsr

γsd− γse>

γrd + γre + 2√γrdγre

γre − γrd. (12)

Proof: The proof follows a similar procedure as that forLemma 1. Since we want to identify the range ofα with nosecrecy rate, the quadratic equation in (14) should be less than0, i.e.,

γrdγre(γsd− γse)α2 + (γsr + 1)[(γrd + γre)(γsd− γse)

+ γsr(γrd − γre)]α+ (γsd− γse)(γsr + 1)2 < 0 . (13)

Further,γsd − γse > 0, which implies that the coefficient ofα2 in (13) is positive, therefore, the quadratic inequality in

(13) holds if and only if the quadratic polynomial has two

−10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20

γsr=12dB

γsr=13dB

γsr=14dB

γsr=15dB

γsr=16dB

α (dB)

Fig. 2. The range ofα to have secrecy rate whenγrd = 18 dB, γsd = 6 dB,γse = 12 dB, γre = 0 dB, and for different values ofγsr.

real roots andα lies inside these two roots. The rest of theproof follows similar steps as those outlined for Lemma 1.

Case 4 is the exact opposite of Case 3. Noting that there isno secrecy rate whenα1 < α < α2, it is desirable to minimizethe left side of (12), since that minimizes the possibility of theexistence ofα with no secrecy rate. This implies that, when allother parameters are fixed, the probability of the existenceofα with no secrecy decreases with decreasingγsr. On the otherhand, the larger the difference betweenγsd andγse, the smallerthe probability thatα with no secrecy rate exists. Finally, itcan be seen that whenγre ≫ γrd, the right-hand side of (12)becomes roughly equal to 1, thus, there is no secrecy ratewhenγsr > γsd− γse.

Fig. 3 depicts the behavior of the achievable secrecy rateas a function of the range ofα. It can be seen that thereis no secrecy rate for a limited range ofα (blue regions).Further, the figure shows that with an increase in the differencebetweenγsd andγse, there is a decrease in the range ofα withno secrecy rate. This result confirms the discussion made inthe previous paragraph. Since the direct channel between thesource and the destination is better than the channel betweenthe source and the eavesdropper, the secrecy rate can beobtained either by avoiding relay transmission or by ensuringthat relay transmission power is within the derived limits.

As a final remark before concluding this section, we notethat the source transmit power has no impact on the state ofthe secrecy rate. The following Lemma further illustrates thisobservation:

Lemma 3: The state of secrecy rate, for a given scenario,remains unchanged when the source transmit power increases



4 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. PP, NO. 99, AUGUST 2014

−10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20

γsd=14dB

γsd=15dB

γsd=16dB

γsd=17dB

γsd=18dB

γse=4dB

γse=5dB

γse=6dB

γse=7dB

γse= 8dB

α (dB)

Fig. 3. The range ofα for which secrecy rate is not achievable whenγsr =

20dB, γrd = 10dB, γre = 23dB, and for different values ofγsd andγse.

or decreases.

Proof: Here, we prove that if we are in either one of thefour cases, we still remain in that case if the source transmitpower is increased or decreased. For Case 1 and Case 2, it isstraightforward to show that the source transmit power has noeffect on the state of the secrecy rate. For Case 3 and Case 4,it can be seen that if the source transmit power is increasedor decreased by a factorβ, then the left sides of the equations(10) and (12) remain unchanged, sinceβ in the numeratorcancels out with theβ in the denominator for both cases.

Since the source transmit power does not have any impacton the state of the secrecy rate, one can adjust the transmitpower at the source and the relay to maintain the data raterequirements at the destination while ensuring perfect secrecy.For example, in Case 3, the source can reduce its transmitpower such that the received SNR at the eavesdropper fallsbelow its minimum requirement for successful detection. Onthe other hand, the relay power is adjusted to compensate forthe drop in received SNR at the destination.

IV. CONCLUSIONS

The letter focused on the secrecy rate achievability of dual-hop amplify-and-forward relay networks. Based on the instan-taneous SNRs of various links within the network, differentscenarios were outlined and the state of the achievable secrecyrate, in each of the reference scenarios, was analyzed. We haveshown that increasing or decreasing the relay transmit powercan affect the state of the secrecy rate achievability. On theother hand, the source transmit power has no effect on thestate of the secrecy rate.

APPENDIX APROOF OFLEMMA 1

Starting from (6), after some simplifications and rearrange-ments of terms, we obtain

γrdγre(γsd− γse)α2 + (γsr + 1)[(γrd + γre)(γsd− γse)

+ γsr(γrd − γre)]α+ (γsd− γse)(γsr + 1)2 > 0 . (14)

The left side of the inequality in (14) is a quadratic poly-nomial ofα. Sinceγsd− γse< 0 and therefore the coefficientof α

2 is negative, the quadratic inequality in (14) holds ifand only if the quadratic polynomial has two real roots andα

lies between these two roots. The roots ofα for the quadraticpolynomial in (14) are obtained as (11a) and (11b), after somesimplifications and rearrangements. One immediate constraintto have distinct real values forα1 andα2 is

(


γsd− γse

)2

− 4γrdγre > 0. (15)

Also, from (11), one obtains

α1α2 =(γsr + 1)2

γrdγre, (16a)

α1 + α2 = −γsr + 1

γrdγre

(


γsd− γse

)

. (16b)

Eq. (16a) means that both roots are either positive ornegative. As the relay power is a positive value, thenα shouldbe positive and, therefore,α1 andα2 should be positive. Using(16b), the roots are positive if


γsd− γse< 0 . (17)

Hence, using (15) and (17), the secrecy rate is achievable forα1 < α < α2 if the following hold:

{∣

∣γrd + γre + γsrγrd−γreγsd−γse

∣

∣ > 2√γrdγre ,

γrd + γre + γsrγrd−γreγsd−γse

< 0 .(18)

The constraint in (18) is equivalent to{

−(

γrd + γre + γsrγrd−γre

γsd−γse

)

> 2√γrdγre;

γrd + γre + γsrγrd−γre

γsd−γse< 0,

(19)

which can be rearranged as

γsr

γse−γsd>

γrd+γre+2√γrdγre

γrd−γre;

γsr

γse−γsd>

γrd+γre

γrd−γre,

(20)

and it is further simplified to (10).

REFERENCES

[1] A. D. Wyner, “The wire-tap channel,”Bell Syst. Tech. J., vol. 54, no. 8,pp. 1355–1387, Oct. 1975.

[2] I. Csiszar and J. Korner, “Broadcast channels with confidential messages,”IEEE Trans. Inf. Theory, vol. IT-24, no. 3, pp. 339–348, May 1978.

[3] L. Lai and H. El Gomel, “The relay-eavesdropper channel:cooperationfor secrecy,” IEEE Trans. Inf. Theory, vol. 54, no. 9, pp. 4005–4019,Sept. 2008.

[4] G. Chen, Z. Tian, Y. Gong, Z. Chen, and J. A. Chambers, “Max-ratio relayselection in secure buffer-aided cooperative wireless networks,” IEEETrans. Inf. Forensics Security, vol. 9, no. 4, pp. 719–729, 2014.

[5] V. N. Q. Bao, N. Linh-Trung, and M. Debbah, “Relay selection schemesfor dual-hop networks under security constraints with multiple eavesdrop-pers,” IEEE Trans. Wireless Commun., vol. 12, no. 12, pp. 6076–6085,Dec. 2013.

[6] J. Huang, A. Mukherjee, and A. L. Swindlehurst, “Secure communicationvia an untrusted non-regenerative relay in fading channels,” IEEE Trans.Signal Process., vol. 61, no. 10, pp. 2536–2550, May 2013.

[7] A. Behnad, A. M. Rabiei, and N. C. Beaulieu, “Performanceanalysis ofopportunistic relaying in a poisson field of amplify-and-forward relays,”IEEE Trans. Commun., vol. 61, no. 1, pp. 97–107, Jan. 2013.

[8] Z. Ding, M. Zheng, and F. Pingzhi, “Asymptotic studies for the impactof antenna selection on secure two-way relaying communications witharticial noise,”IEEE Trans. Wireless Commun., vol. 13, no. 4, pp. 2189–2203, April 2014.

on the secrecy rate achievability in dual-hop amplify-and-forward relay networks

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