Rui Ma,Haowei Wu,2,3,*,Liubin Wang,Jinglan Ou,3,Lian Yan

1 School of Microelectronics and Communication Engineering,Chongqing University,Chongqing 400044,China

2 Center of Communication and Tracking Telemetry Command,Chongqing University,Chongqing 400044,China

3 Chongqing Key Laboratory of Space Information Network and Intelligent Information Fusion,Chongqing 400044,China

Abstract: This work investigates the security issue of the energy-constrained untrusted relay network with imperfect channel reciprocity,where the relay without the built-in power supply can only scavenge energy from radio-frequency signals radiated by the source and destination.A hybrid power-splitting (PS)- and time-switching (TS)-based relaying (HPTR) protocol is presented to improve the degraded secrecy performance due to the incomplete self-interference cancellation at the destination.To evaluate the secrecy throughput (ST) of the studied system,the analytical expression of the secrecy outage probability under the delay-limited transmission mode and the closed-form lower bound of the ergodic secrecy capacity under delay-tolerant transmission mode are derived.Both linear and nonlinear models for the energy harvester at the relay are compared.The optimal PS and TS ratios are evaluated numerically.The theoretical derivations are validated by numerical results,revealing that the residual jamming has a negative effect on the secrecy performance of untrusted relay networks,which can be alleviated by the HPTR protocol.Besides,we compare the ST performance of the HPTR protocol with that of the PS and TS relaying schemes,and the results show that the HPTR protocol outperforms both PS and TS relaying protocols in terms of the ST.

Keywords: untrusted relay; physical layer security;secrecy outage probability; ergodic secrecy capacity;energy harvesting

I.INTRODUCTION

Cooperative relaying has been regarded as a powerful tool to expand communication coverage and enhance transmission reliability[1].In practice,however,relay nodes may be honest-and-curious when performing cooperative communication,because they are willing to comply with the designated protocol to forward the source’s information,while also have the potential to extract the secret message[2-6].Responding to this,the security issue of untrusted relay networks deserves more attention than that of traditional trusted relay networks[2].The destination-aided jamming method was first proposed to guard secret messages against the untrusted relay in[3,4],where the source radiated information signals while the destination sent the artificial noise (AN) to the relay.Subsequently,the source-and jammer-assisted interference strategies were also presented for untrusted relay systems in[5]and[6],respectively.

Another pivotal issue in untrusted relay networks is how to guarantee the continuous energy supply for relay nodes.At present,it is unrealistic to replenish energy by replacing batteries frequently or to ensure stable power output by collecting energy from natural resources.Fortunately,the simultaneous wireless information and power transfer (SWIPT) technique,which could transfer information and power concurrently,provides a feasible solution to the above problem[7,8].As such,two types of practical relaying protocols were proposed in[8],called timeswitching relaying (TSR) and power-splitting relaying (PSR).After that,the authors of[9]examined the secrecy metrics of the energy-limited amplify-andforward (AF) untrusted relay network,including the secrecy outage probability (SOP) and the ergodic secrecy capacity (ESC).In[10],the effects of the outdated channel state information (CSI) on the security of wireless-powered untrusted relay systems were revealed.Besides,a self-energy recycling full-duplex(FD) relay scheme was presented in[11].However,these studies only consider the PSR or TSR protocol,which may cause low coverage owing to power reduction or information loss due to time switching.

To settle this problem,a novel energy harvesting(EH)scheme,integrating the TSR and PSR protocols,was designed for traditional trusted relay networks in[12]and[13].Based on this,the authors of[14]put forward an improved hybrid relaying protocol for the decode-and-forward relay system with a direct link.In[15],a hybridized power-time splitting-based EH protocol was presented for the SWIPT-enabled communication networks,in which the analytical expression of the outage probability was derived.It can be found from[12-15]that the hybrid relaying protocol outperforms both the TSR and PSR protocols and is more generally applicable since it can be configured to work as any of the following three protocols,i.e.,PSR,TSR,or hybrid relaying.

The above literature[6-15]consider the hypothesis of the linear EH model,that is,the energy conversion efficiency is set as a constant irrelevant to the input power of the EH circuit.Boshkovska et al.recognized that in actual measures,the EH circuit exhibited the end-to-end nonlinear EH phenomena instead of the linear one[16].Thus,invoking the linear EH model in the SWIPT-enabled network would lead to imprecise analysis.To exactly delineate features of the real EH circuit,a logistic function-based nonlinear EH model was developed in[16].Based on this realistic model,the authors of[17]explored the dynamic FD protocol over the downlink orthogonal frequency division multiple access systems with a wireless-powered relay.In[18],the adaptive self-interference processing scheme with a higher degree of freedom in signal detection and energy recycling was proposed.Besides,a robust resource allocation scheme was devised in[19]to attain a trade-off between the transmission power and the collected energy over the SWIPT-enabled cognitive radio network.Under different CSI models,the transmission power minimization problem was formulated in[20]with the AN-assisted jamming scheme and a practical nonlinear EH model.Whereas,this nonlinear model is not mathematically tractable.To this end,a simplified nonlinear EH model,which catches distortion and saturation properties,was put forward in[21].This model is better suited for the derivation of the probability metrics and the solution of the capacityrelated problems[14,22].

Motivated by this,in the paper,a hybrid PS- and TS-based relaying (HPTR) protocol is presented for the energy-limited untrusted relay network,where the source sends the confidential message via an untrusted EH relay to the destination.Such networks have practical applications in military,government,and financial institutions,where all nodes within the institution may not have the same rights in accessing secret information[3-5].Moreover,given the fact that the channel reciprocity is not perfect in practice due to the transmission delay and the radio-frequency (RF)mismatch[10],the impacts of the imperfect channel reciprocity(ICR)on the secrecy performance are also taken into account.Although the ICR issues have been well studied in SWIPT-enabled networks with the PS or TS structure under the linear EH model[10,11,23],the untrusted hybrid relaying protocol has not been examined under the pragmatic nonlinear EH model.

Our work differs from the foregoing studies[12-15].Firstly,the untrusted relay scenario with the ICR is considered for the HPTR scheme,while the trusted relay network without security constraints was studied in these works.Secondly,the secrecy throughput(ST) in delay-limited transmission (DLT) and delaytolerant transmission (DTT) modes are assessed by the SOP and ESC,respectively,while only the system capacity in[13,14]or the outage probability in[15]was examined.Thirdly,both the classical linear and the simplified nonlinear EH models are employed for the proposed hybrid scheme,while only the linear EH model was cited in[12-15].Lastly,the closedform SOP and ESC for the untrusted relay network are derived,while only their integral-form expressions were given for the trusted relay network in[12].To our knowledge,the security studies of untrusted relay networks with the hybrid EH protocol and the ICR under the two modes are still missing.The main contributions can be outlined as follows:

·A three-phase secure HPTR protocol is presented for the untrusted relay network,where the hybrid EH structure,i.e.,TS and PS,is considered in a combining manner to power the relay.This protocol can make full use of time and power resources to enable the best performance.

·The exact integral and closed-form approximate expressions of the SOP under the DLT mode,and the accurate form and lower bound of the ESC under the DTT mode are derived to evaluate the ST performance of the studied system,respectively.

·Numerical results show that the untrusted HPTR protocol outperforms both TSR and PSR protocols and is equivalent to PSR protocol in the high power regime.Also,it reveals that the best deployment for the untrusted relay is close to the destination.

The rest of this paper is organized as follows.The system model and the untrusted HPTR protocol are first introduced in Section II.Then,the STs of the system under the DLT and DTT modes are derived in Section III.After that,numerical results are provided and discussed in Section IV.Finally,some conclusions are drawn in Section V.

Notations:Vectors and scalars are expressed in bold and italic lowercase letters,respectively.is the expectation measure.‖·‖indicates the Euclidean norm of a vector,|·|denotes the absolute value of a scalar,and (·)Hrepresents the conjugate transpose.is the combinatorial number.Ei(·) and Γ(·) are the functions of the exponential integral and gamma,respectively.Fv(·) andfv(·) stand for the cumulative distribution function and the probability density function of the variablev,respectively.is the Meijer-Gfunction.means thatXis a circularly symmetric complex Gaussian variable with meanκand varianceσ.

II.SYSTEM MODEL

As depicted in Figure 1,one multi-antenna sourceScommunicates with a destination nodeD,with the assistance of an untrusted AF relayRthat acts as both a helper to forward the source’s message and an eavesdropper to decode the confidential information.To guard the confidential information against being deciphered by the untrusted relay,Sradiates the message signal whileDemits the jamming signal to the relay.The sourceSis configured withMantennas,whileRandDare single-antenna nodes.There is no direct link betweenSandDdue to the deep shadowing[8-13,15].Besides,suppose thatRis an energylimited node with a rechargeable battery,whereasSandDhave a dedicated power supply[10].This scene happens in large wireless networks,e.g.,Internet-of-Things,heterogeneous networks,and ad hoc networks[4],in which the secret information is usually forwarded by multiple intermediate nodes that are not willing to consume their own energy.

Figure 1. System model of an untrusted relay network.

Figure 2. Transmission time-block structure for the untrusted relay network with the HPTR protocol.

The channels betweenSandR,RandD,DandRare represented asrespectively,and obey the zero-mean complex Gaussian distribution with variance{R,D},andm ∈{1,...,M}.As such,the channel power gain|hij|2is exponentially distributed with mean[22-24].Moreover,all channels are assumed to be quasi-static block-fading,i.e.,the fading amplitudes of channels keep constant over one block while vary independently for different blocks[5,6].We also suppose that instantaneous CSIs forandhRDare available atRandD,buthDRdiffers fromhRDowing to the ICR caused by the outdated CSI.Hence,hDRis modeled as[10,23]

where Φ is the channel correlation coefficient that measures the level of correlation betweenhDRandhRD.When Φ takes 1,the ideal channel reciprocity can be guaranteed such thathDRis exactly equal tohRD.With the decrease of Φ,the channel reciprocity becomes imperfect,resulting in a deviation betweenhDRandhRD[23].Also,hεdepicts the uncertain component ofhRDand is a complex Gaussian random variable with

For the considered system,the secure HPTR protocol is adopted at the untrusted relay,in which a complete transmission blockTis divided into three time slots,as illustrated in Figure 2.At the first time slot of durationαT,where 0≤α ＜1 is the TS factor,the relayRcollects energy from the received RF signals radiated bySandD.Then,in the second time slot that lasts(1−α)T/2,both the confidential signal fromSand the jamming signal fromDare received byRconcurrently.A fractionβof the collected power is used for the EH and the rest is split for the information processing(IP),where 0≤β ＜1 is the PS ratio.The last time slot of length (1−α)T/2 is used for the information transmission fromRtoD.

Figure 3. Structure diagram for the hybrid untrusted relay.

In terms of performance analysis,most existing works[6-13,15,23,25]cite the linear EH model due to its tractability.Given this,with the linear model,the total amount of energy collected atRbased on the hybrid relaying structure as shown in Figure 3,is linearly proportional to the RF input power and denoted as

In general,a fractionφEHof the total energy harvested at the relay is allocated to forward the observed signal,and the remaining(1−φ)EHis used to maintain the power consumption of the signal processing circuit[25].As a result,for the linear EH model,to sustain the energy balanced statistically in the last time slot,the average transmit power atRis

In practice,nonlinear elements at the harvester circuit cause a nonlinear input-output feature.Recent studies have indicated that the precise nonlinear EH model is hard to be directly utilized in performance analysis owing to its complex logistic function[14,26].To get around this,the rational function-based[21]and the piecewise function-based[27]nonlinear EH models are then presented.The sensitivity and saturation of the real energy harvester are integrated into both models.Compared with the piecewise EH model,the rational EH model is more mathematically tractable for the derivation of the probability-related metrics.Accordingly,the rational function-based nonlinear EH model in[21]is applied here to ensure the practicality and tractability of the derivation.As such,the relay’s average transmit power is given by

Besides,the signal observed atRis split for the IP and can be represented as

wherexSandxDare the unit-power information signal and the AN emitted from the source and destination,respectively;is the complex Gaussian noise at the relay.To enhance the transmission quality of the signal and energy,the maximum ratio transmission scheme with the optimal beamforming vectoris used atS[28].

In the relaying phase,the scaled version of the received mixed-signal,xR=GoyR,is broadcasted by the relay to the destination,where the fixed-gain amplification coefficient is[29]

Next,in the information forwarding phase,the received signal at the destination can be given by

Through plugging (1) into (7),and subtracting the jamming part,can be rewritten as

where the residual jamming part is caused by the ICR.

As a consequence,substituting(6)into(8)and using some simple calculations,the received instantaneous signal-to-interference-plus-noise ratios (SINRs) atRandD,respectively,are calculated as

The achievable secrecy capacity for the untrusted relaying network is thus expressed as

where[x]+=max(x,0).

III.SECRECY THROUGHPUT ANALYSIS

In this section,the STs for the untrusted HPTR protocol with linear and nonlinear EH models are derived under both DLT and DTT modes.Based on these derived expressions,the optimal values of the TS factor and PS ratio to maximize STs can be found numerically with reduced computational complexity.

3.1 Delay-Limited Transmission Mode

For the DLT mode,the message should be sent within one transmission block time to guarantee the fast responding requirement at the receiver,and thus the signal can be decoded without buffering[8].Generally,the source sends data at a fixed rate,which may cause an outage owing to the random fading of wireless channels.Thus,it is suitable to select the outage-based metric for evaluating system performance.The ST of the system in the DLT mode is formulated as

Proposition 1.For the proposed hybrid relaying protocol,the closed-form approximate SOP of the untrusted relay system under the linear and nonlinear EH models can be uniformly expressed as

with

whereis the i-th root of the Laguerre polynomial Ln(u),and N is a parameter determining the tradeoff between complexity and accuracy.Proof.See Appendix.

Remark 1.It can be concluded from(14)that the SOP is proportional to the target secrecy rate.Instead,it is inversely proportional to the channel quality and correlation coefficient.This is intuitive as the higher Rth is set,the more it becomes hard to meet,and the likelihood of secure communication between S and D degrades.The improvement of channel quality and correlation will promote secrecy performance.

3.2 Delay-Tolerant Transmission Mode

In the DTT mode,the code length is greater than transmission block time,so that the code spans all the possible channel realizations during a codeword transmission[8].As a result,the destination can buffer the received information blocks and tolerate the delay when decoding the target signal.The ST in DTT mode is typically evaluated by the ESC[8]and defined as

where the ESC is formulated as.

Proposition 2.For the hybrid relaying protocol,the analytical ESC of the untrusted relay system with the linear and nonlinear EH models is unified as

where the instantaneous secrecy capacity is expressedas.

For all we know,the double integral (20) does not admit the closed-form solution,and it can be calculated by using numerical methods instead.Alternatively,we proceed to deduce a compact lower bound forunder the hybrid EH protocol.

Proposition 3.Under the linear and nonlinear EH models,the closed-form lower bound of the ESC for the untrusted relay system with the secure HPTR protocol is presented in a unified form as

with

whereandis the Meijer-G function.

Proof.See Appendix.

Remark 2.The lower bound shown in(21)is tight in the high power regime,which is presented in Figure 5.Besides,as P tends to ∞,we havegoes to ∞,andandΓR converge to constants,which means thatin the case of high power,the ESC will have a performance ceiling under the nonlinear EH model,but not under the linear EH model.This finding is significant as it reveals that the nonlinearity of the practical energy harvester induces variations to the design of wireless-powered networks.

Figure 4. SOP versus transmit power P for the HPTR scheme under linear and nonlinear EH models.

Figure 5. ESC versus transmit power P for the HPTR scheme under linear and nonlinear EH models.

3.3 Parameter Optimization

Through the above analysis,one can find that increasing the TS ratioαmeans that more energy can be collected by the relay,that is,the stronger forwarding power is promoted.As a result,the received SINR at the destination is increased and the ST is improved.However,the largerαalso signifies the shorter effective communication time.On another front,the increment of the PS ratioβcontributes to the amount of collected energy but reduces the strength of the information signal processed at the relay.Although the eavesdropping effect of the untrusted relay is weakened,the strength of the forwarded signal is also compromised.Thus,it is necessary to explore the reasonable design of the TS factor and PS ratio for maximizing the ST.

For the two transmission modes,the approximate optimal TS factor and PS ratio to maximize the ST can be gained by solving the following optimization issues

Note that the exact optimal values,α∗andβ∗,can be gained by the integral SOP in(A.3)and the integral ESC in (20),whileandare solved from the approximate SOP in(14)and the lower-bound ESC in(21),respectively.This degrades the computational dimension,making it easier and faster to search forandnumerically,compared to obtaining accurateα∗andβ∗.On the basis of the previous derivations,the pseudo-code for jointly optimizingαandβthat maximizes ST in a numerical manner is obtained,as shown in Algorithm 1.The core idea of the algorithm is to calculate the optimal TS and PS ratios alternately.

In each iteration of Algorithm 1,the optimalαandβare updated alternately until the fractional increase of the objective function ST is lower than a presupposed convergence tolerance,δ,which is sufficiently small.This means that the ST would monotonically increase at each iteration.Similarly,the optimal TS coefficientand PS factorin DTT mode can also be evaluated numerically.

Algorithm 1.ST maximization under DLT mode.1:Initialize the TS factor α(1),the error tolerance δ and system parameters such as P,M,μ,φ,η,etc.Given the objective function τoDLT in(12).I is the maximum number of iterations.2:for i from 1 to I,step=1,do 3:Calculate a solution β(i)of max images/BZ_130_2108_687_2129_733.pngα(i),βimages/BZ_130_2247_687_2268_733.pngusing the Golden section method.4:Calculate a solution α(i+1)of max 0≤β＜1 τoDLTimages/BZ_130_2123_821_2144_867.pngα,β(i)images/BZ_130_2247_821_2268_867.png0≤α＜1τoDLT using the Golden section method.5:ifimages/BZ_130_1440_952_1455_997.pngimages/BZ_130_1440_979_1455_1025.pngτoDLTimages/BZ_130_1546_953_1567_999.pngα(i+1),β(i)images/BZ_130_1753_953_1774_999.png−τoDLTimages/BZ_130_1906_953_1926_999.pngα(i),β(i)images/BZ_130_2069_953_2090_999.pngimages/BZ_130_2083_952_2098_997.pngimages/BZ_130_2083_979_2098_1025.png≤δ then 6:Exit.7:end if 8:end for 9:Output αappr DLT =α(i+1)and βapprDLT =β(i).

IV.NUMERICAL RESULTS

In this section,numerical results and Monte Carlo simulations with 106independent trials are provided to validate the theoretical analysis.Two classical counterparts,the PSR and TSR schemes in[9]are invoked as benchmarks to manifest the performance gain of the untrusted HPTR scheme.Without loss of generality,the channel power gain is defined aswheregijis the fading gain subject to the exponential distribution with meanλij,dijis the distance between nodes,andμis the path loss exponent[8-11,13].In what follows,the optimal ST represents the use of the best PS and TS factors,which are gained by Algorithm 1.Unless otherwise stated,the simulation parameters are set as Table 1,which are selected based on the works in[9]and[10].

Table 1. System parameters.

Figures 4 and 5 describe the SOP and ESC versus the transmit powerP,respectively,for the proposed HPTR scheme under linear and nonlinear EH models.It can be seen that the SOP obtained by the accurate integral expression (A.3) and the ESC calculated by the exact integral expression (20) match their respective simulation points well,which confirms the correctness of the theoretical analysis.Moreover,in the case ofN=10,the performance gap between the approximate SOP computed by(14)and the exact result gained by(A.3)is narrow,which manifests the validity of the approximate method.One can also note that the derived lower bound of the ESC in (21) is tight at the high power regime.The observation of interest is that the nonlinear EH model exhibits the higher SOP and the lower ESC than the linear EH model.Besides,for the nonlinear model,with the increase ofP,the SOP and ESC gradually approach the performance floor and ceiling respectively,due to the saturation of the EH circuit.This indicates that selecting a practical model for the energy harvester is very important.

Figure 6.ST versus channel correlation coefficient Φ under DTT and DLT modes,with different antenna numbers.

In Figure 6,the STs for both the DTT and DLT modes are plotted versus the channel correlation coefficient Φ,with the number of antennaM=3 and 5.As depicted in the figure,with augmentation of the antenna number,the ST of the system is improved due to the increased degrees of freedom.We can also observe that for the DTT mode,the increment in channel correlation has a more apparent impact on performance improvement compared to the DLT mode.The main reason is that for different delay requirements,the STs in the two transmission modes are respectively assessed by the ESC and SOP.In addition,as the channel correlation coefficient rises,the ST enhances as expected.

To investigate the convergence of Algorithm 1,we examine its performance under two different initial values,i.e.,α(1)= 0.5 and 0.9,and the convergence toleranceδis set to 10-4.The ST at each round of the iteration is demonstrated in Figure 7.It shows that algorithm 1 always converges after two iterations for different initial conditions.Additionally,we can also observe that under the given simulation settings,the iterative algorithm converges to an optimal value of 0.5033bps/Hz in the linear EH model and DTT mode,0.4495bps/Hz in the nonlinear EH model and DTT mode,0.1165bps/Hz in the linear EH model and DLT mode,and 0.1076bps/Hz in the nonlinear EH model and DLT mode.These optimal values are almost the same as 0.5029bps/Hz,0.4496bps/Hz,0.1165bps/Hz,0.1075bps/Hz obtained by ergodic search,which verifies that the numerical algorithm is effective.

Figure 7. Convergence of Algorithm 1.

Figure 8. Effects of target secrecy rate Rth on the optimal ST under DLT mode,with perfect channel correlation coefficient and different energy conversion efficiencies.

Under the DLT mode,Figure 8 shows the variation in the optimal ST against the target secrecy rate under linear and nonlinear EH models.It can be observed that the HPTR scheme works as the PSR in lowRth,and it operates as the TSR in highRth,while the HPTR has a higher ST than TSR and PSR schemes in the intermediate transition state.This is because a smallRthimplies weak security constraint and thus does not require additional EH to ensure the system’s outage requirements.As the required secrecy rate constraint becomes tighter,the more it becomes difficult to satisfy,and the likelihood of secure communication between the source and the destination is lower.Thus,the extra time slot is used for the EH,which leads to the raise of the transmit power at the relay and the improvement of received SINR at the destination.We can also see that differentηinduces the discrepancy in performance between the two EH models.

Figure 9. Optimal ST versus P for Φ=0.9.

Figure 10. Optimal ST versus P for Φ=1.0.

The ST performance comparison between the three schemes under DLT and DTT modes is drawn in Figures 9 and 10 for different Φ.It can be seen that as the transmit powerPincreases,the optimal ST improved for all schemes.We also observe that the HPTR protocol outperforms both TSR and PSR protocols.It can be explained by the fact that optimizing the ST under the PSR and TSR protocols can only achieve a local optimum since only a single constraint(time or power)is involved.The difference is the optimized hybrid protocol can make full use of time and power to realize better performance.Moreover,the ST is improved as the coefficient Φ increases,because the strength of the residual jamming part is weakened as the reciprocity error reduces.On top of that,under the linear EH model,the presence of residual interference makes the ST performance in the DTT scenario not continue to be improved with the increase ofP.In contrast,the ST is nearly saturated in the DLT scenario as the target secrecy rate is pre-set.

Figure 11. Optimal ST versus distance dSR or dRD for three schemes under DTT and DLT modes with Φ=1.0.

For the DTT and DLT modes,to study the impacts ofdSRanddRDon the optimal ST,we fixed one of them to 200 m,while the other changed from 100 to 300 m,shown in Figure 11.One can observe that the optimal ST reduces as the distancesdSRanddRDenlarge.This is because the longer the distance,the faster the strength of the information-and energybearing signals attenuate,which not only impairs the interception of the relay but also worsens the reception performance of the destination.Besides,it can be seen thatdRDhas a more significant impact on ST performance thandSRsince the interference effect and forwarding performance deteriorated at the same time in the second hop.This reveals that the best deployment for the untrusted relay is close to the destination.

V.CONCLUSION

The hybrid EH protocol for energy-constrained untrusted relay networks with imperfect channel reciprocity was studied under the DLT and DTT modes.The analytical expressions of the SOP and ESC were respectively given for two transmission modes to assess the ST of the system.The linear and nonlinear EH models for the untrusted relay were examined.Simulation results showed that the ICR deteriorated the secrecy performance of the untrusted relay network,and the HPTR protocol generally yielded a better ST performance than both PSR and TSR protocols.These results provide useful guidance on the design of the EH relaying protocol for untrusted relay systems with different delay requirements.For future work,more realistic multi-relay multi-user scenarios with different security requirements would be considered to further reveal the advantages of the hybrid EH protocol.

ACKNOWLEDGEMENT

This work was supported by Special Foundation for Chongqing Science and Technology Talent(cstc2019y szx-jcyjX0006 and cstc2020yszx-jscxX0004) and Advanced Research Project of Civil Aerospace Technologies in 13th Five-year Plan(D010201).

APPENDIX

PROOF OF PROPOSITION 1

In line with the definition of the SOP in(13),denotingwe can obtain

Based on the basic probability theory,and along with some straightforward manipulations,yields

where Δ is obtained by solving the equationg(x2)=0 and the solution less than 0 is deleted,as(18).

Note thatX1is a gamma-distributed variable with the shape parameterMand scale parameteras well asX2follows the exponential distribution with meanHence,lett=x2−Δ,yielding

Observing (A.3),it can be found that the integralis difficult to be solved numerically in a closed form.The famous Gauss-Laguerre quadrature is an effective approach to calculate the value of the integral of the given functionf(x) over the interval (0,∞)and expressed aswhere the variablexiis thei-th root of Laguerre polynomialLn(x) and the weightAi=Therefore,an approximation ofcan be computed as

PROOF OF PROPOSITION 3

For the untrusted relay system,the exact and lowerbound ESC can be given by

where (c) is obtained based on the general fact.

Next,according to the basic operations of the logarithm function,Cχ1can be attained as

Hereby,we invoke the identity[31,Eq.(8.4.6.5)]

With the help of[32,Eq.(7.813.1)],and again using the Gauss-Laguerre quadrature method,we can obtain

where (d) is gained by makingwithAnd then,can be computed as

where Δ1=ξ5¯λRD,and Ei(x)indicates the exponential integral function[32,Eq.(8.211.1)].

Inserting(A.9)and(A.8)into(A.7),one can get

After that,through applying[32,Eq.(4.337.2)],we can easily get

Moreover,by defining(ξ1X1+ξ2X2)/ξ3=Z,Cχ4can be rewritten as

The variableZcontains two diverse fading distributions,and thus,its PDF is mathematically intractable.Invoking the Taylor expansion of the function ln(1+x) with respect tothe secondorder analytical expression ofcan be sufficiently approached as[33]

Based on the assumptions of the channel model,X1andX2are independent random variables that obey gamma distribution and exponential distribution,respectively.Hence,the first-and second-order statistical moments ofZare,respectively,calculated as

Substituting (A.15) and (A.16) into (A.14),after some simple algebraic manipulations,(A.13) can be further expressed as

where Ω=MΔ3+Δ2.

In the end,inserting(A.6)-(A.13)into(A.5),can be attained as(21).