1 Multiple Access Wiretap Channel with Noiseless Feedback Bin Dai and Zheng Ma 7 Abstract 1 0 Thephysicallayersecurityintheup-linkofthewirelesscommunicationsystemsisoftenmodeledasthemultiple 2 accesswiretapchannel(MAC-WT),andrecentlyithasreceivedalotattention.Inthispaper,theMAC-WThasbeen n a re-visited by considering the situation that the legitimate receiver feeds his received channel output back to the J transmitters via two noiseless channels, respectively. This model is called the MAC-WT with noiseless feedback. 5 Inner and outer bounds on the secrecy capacity region of this feedback model are provided. To be specific, we first 1 presentadecode-and-forward(DF)innerboundonthesecrecycapacityregionofthisfeedbackmodel,andthisbound ] T is constructed by allowing each transmitter to decode the other one’s transmitted message from the feedback, and I then each transmitter uses the decoded message to re-encode his own messages, i.e., this DF inner bound allows the . s c independenttransmitterstoco-operatewitheachother.Then,weprovideahybridinnerboundwhichisstrictlylarger [ than the DF inner bound, and it is constructed by using the feedback as a tool not only to allow the independent 1 transmitters to co-operate with each other, but also to generate two secret keys respectively shared between the v 2 legitimate receiver and the two transmitters. Finally, we give a sato-type outer bound on the secrecy capacity region 5 of this feedback model. The results of this paper are further explained via a Gaussian example. 0 4 0 Index Terms . 1 0 Multiple-access wiretap channel, noiseless feedback, secrecy capacity region. 7 1 : v I. INTRODUCTION i X The physical layer security (PLS) was first investigated by Wyner in his landmark paper on the degraded wiretap r a channel [1]. Wyner’s degraded wiretap channel model consists of one transmitter and two receivers (a legitimate receiver and an eavesdropper). The transmitter sends a private message to the legitimate receiver via a discrete memoryless main channel, and an eavesdropper eavesdrops the output of the main channel via another discrete memoryless wiretap channel. We say that the perfect secrecy is achieved if no information about the private B. Dai is with the School of Information Science and Technology, Southwest JiaoTong University, Chengdu 610031, China, and with the StateKeyLaboratoryofIntegratedServicesNetworks,XidianUniversity,Xi(cid:48)an,Shaanxi710071,China,e-mail:[email protected]. Z. Ma is with the School of Information Science and Technology, Southwest JiaoTong University, Chengdu 610031, China, e-mail: [email protected]. 2 message is leaked to the eavesdropper. The secrecy capacity C , which is the maximum reliable transmission rate s with perfect secrecy constraint, was characterized by Wyner [1], and it is given by C =max(I(X;Y)−I(X;Z)), (1.1) s p(x) whereX,Y andZ aretheinputofthemainchannel,outputofthemainchannelandoutputofthewiretapchannel, respectively, and they satisfy the Markov chain X → Y → Z. Here note that (1.1) holds under the degradedness assumption X →Y →Z, and the secrecy capacity of the general wiretap channel (the wiretap channel without the degradedness assumption) was determined by Csisza´r and Ko¨rner [2]. The work of [1] and [2] lays a foundation for the PLS of the practical communication systems. SinceWozencraftetal.[3]showedthatthetime-variantnoisytwo-waychannelscanbeusedtoprovidenoiseless feedback, whether this noiseless feedback helps to enhance the capacities of various communication channels motivates the researchers to study the channels with noiseless feedback. Shannon first proved that the noiseless feedback does not increase the capacity of a point-to-point discrete memoryless channel (DMC) [4]. After that, Cover et al. [5], [6] and Bross et al. [7] showed that the capacity regions of several multi-user channels, such as multiple-access channel (MAC) and relay channel, can be enhanced by feeding back the receiver’s channel output tothetransmitteroveranoiselesschannel.Then,itisnaturaltoask:doesthenoiselessfeedbackfromthelegitimate receiver to the transmitter also help to enhance the secrecy capacity of the wiretap channel? Ahlswede and Cai [8] answered this question by considering the wiretap channel with noiseless feedback. Since the noiseless feedback is known by the legitimate receiver and the transmitter, and it is not available for the eavesdropper, Ahlswede and Cai pointed out that the noiseless feedback can be used to generate a secret key shared only between the transmitter and the legitimate receiver, and we can use this key to encrypt the transmitted messages. Combining the idea of generating a secret key from the noiseless feedback with Wyner’s random binning technique used in the achievability proof of (1.1), Ahlswede and Cai showed that the secrecy capacity C of the degraded wiretap sf channel with noiseless feedback is given by C =maxmin{I(X;Y),I(X;Y)−I(X;Z)+H(Y|X,Z)}, (1.2) sf p(x) where X, Y and Z are defined the same as those in (1.1), and X → Y → Z forms a Markov chain. Comparing (1.2) with (1.1), it is easy to see that the noiseless feedback increases the secrecy capacity of the degraded wiretap channel. Other related works on the wiretap channel with noiseless feedback are in [9]-[11]. In recent years, the PLS in the up-link of wireless communication system receives a lot attention, see [12]-[16]. These work extends Wyner’s wiretap channel to a multiple access situation: the multiple-access wiretap channel (MAC-WT). Bounds on the secrecy capacity region of MAC-WT are provided in [12]-[16]. In order to investigate whetherthenoiselessfeedbackfromthelegitimatereceivertothetransmittershelpstoenhancethesecrecycapacity regionoftheMAC-WT,inthispaper,westudytheMAC-WTwithnoiselessfeedback,seeFigure1.Wefirstpresent a DF inner bound on the secrecy capacity region of the model of Figure 1, and this bound is constructed by using the DF strategy of the MAC-WT with noisy feedback [17], where each transmitter of the MAC decodes the other 3 one’s transmitted message from the noisy feedback and then uses it to re-encode his own messages. Second, note that the noiseless feedback can not only be used to re-encode the messages of the transmitters, but also be used to generate secret keys to encrypt the transmitted messages, thus we present a hybrid inner bound on the secrecy capacity region of the model of Figure 1 by combining Ahlswede and Cai’s idea of generating a secret key from the noiseless feedback [8] with the DF strategy used in [17], and we show that this hybrid inner bound is strictly larger than the DF inner bound. Third, we present a sato-type outer bound on the secrecy capacity region of the model of Figure 1. Finally, the results of this paper are further explained via a Gaussian example. The rest of this paper is organized as follows. In Section II, we show the definitions, notations and the main results of the model of Figure 1. An Gaussian example of the model of Figure 1 is provided in Section III. Final conclusions are presented in Section IV. Fig. 1: The multiple-access wiretap channel with noiseless feedback II. MODELDESCRIPTIONANDTHEMAINRESULT Basicnotations:Weusethenotationp (v)todenotetheprobabilitymassfunctionPr{V =v},whereV (capital V letter) denotes the random variable, v (lower case letter) denotes the real value of the random variable V. Denote the alphabet in which the random variable V takes values by V (calligraphic letter). Similarly, let UN be a random vector (U ,...,U ), and uN be a vector value (u ,...,u ). In the rest of this paper, the log function is taken to 1 N 1 N the base 2. Definitions of the model of Figure 1: LetW ,uniformlydistributedoverthefinitealphabetW ={1,2,...,M },bethemessagesentbythetransmitter 1 1 1 1. Similarly, let W , uniformly distributed over the finite alphabet W ={1,2,...,M }, be the message sent by the 2 2 2 transmitter 2. TheinputsofthechannelarexN andxN,whiletheoutputsareyN andzN.Thechannelisdiscretememoryless, 1 2 i.e., at the i-th time, the channel outputs Y and Z depend only on X and X , and thus we have i i 1,i 2,i P (yN,zN|xN,xN) YN,ZN|XN,XN 1 2 1 2 N (cid:89) = P (y ,z |x ,x ). (2.1) Y,Z|X1,X2 i i 1,i 2,i i=1 4 Since yN can be fed back to the transmitters via a noiseless feedback channel, at the i-th time, the channel input X (j =1,2) is given by j,i   f (W ), i=1 j,i j X = (2.2) j,i  f (W ,Yi−1), 2≤i≤N. j,i j Here note that the i-th time channel encoder f (j = 1,2) is a stochastic encoder, and the transmission rates of j,i the messages W and W are logM1 and logM2, respectively. 1 2 N N The decoder is a mapping ψ : YN →W ×W , with input YN and outputs Wˆ , Wˆ . The average probability 1 2 1 2 of error P is denoted by e 1 (cid:88)M1 (cid:88)M2 P = Pr{ψ(yN)(cid:54)=(i,j)|(i,j) sent}. (2.3) e M M 1 2 i=1j=1 The eavesdropper’s equivocation to the messages W and W is defined as 1 2 1 ∆= H(W ,W |ZN). (2.4) N 1 2 A positive rate pair (R ,R ) is called achievable with weak secrecy if, for any small positive (cid:15), there exists an 1 2 (M ,M ,N,P ) code such that 1 2 e logM logM 1 ≥R −(cid:15), 2 ≥R −(cid:15),∆≥R +R −(cid:15), P ≤(cid:15). (2.5) N 1 N 2 1 2 e Here we note that ∆ ≥ R +R −(cid:15) also ensures 1H(W |ZN) ≥ R −(cid:15) for t = 1,2, and the proof is in [17, 1 2 N t t p. 609]. The secrecy capacity region C of the model of Figure 1 is a set composed of all rate pairs (R ,R ) s 1 2 satisfying (2.5). The following Theorem 1 and Theorem 2 show two inner bounds on C , and Theorem 3 shows an s outer bound on C . s Theorem 1: For the discrete memoryless MAC-WT with noiseless feedback, an inner bound CDF on the secrecy s capacity region C is given by s CDF ={(R ≥0,R ≥0):R ≤I(X ;Y|X ,U) s 1 2 1 1 2 R ≤I(X ;Y|X ,U) 2 2 1 R +R ≤min{I(X ,X ;Y),I(X ;Y|X ,U)+I(X ;Y|X ,U)}−I(X ,X ;Z)}, 1 2 1 2 1 2 2 1 1 2 for some distribution P (z,y|x ,x )·P (x |u)·P (x |u)·P (u). (2.6) Z,Y|X1,X2 1 2 X1|U 1 X2|U 2 U Proof: In the MAC-WT with noisy feedback [17], the legitimate receiver’s channel output Y is sent to the transmitters via two noisy feedback channels, and the outputs of the noisy feedback channel are Y and Y . Substituting 1 2 Y = Y = Y (which implies the feedback channel is noiseless) into [17, Theorem 2], the DF inner bound CDF 1 2 s for the model of Figure 1 is obtained, and the proof of CDF is along the lines of that of [17, Theorem 2] (the full s DF inner bound on the secrecy capacity region of the MAC-WT with noisy feedback), and thus we omit the proof here. 5 Remark 1: In [17, Theorem 1], Tang et al. also provide a partial DF inner bound on the secrecy capacity region of the MAC-WT with noisy feedback. Substituting Y =Y =Y into [17, Theorem 1], and using Fourier-Motzkin 1 2 elimination (see, e.g., [18]) to eliminate R , R , R and R , it is not difficult to show that the partial DF inner 10 12 20 21 bound CPDF of the model of Figure 1 is exactly the same as the DF inner bound CDF shown in Theorem 1. s s Theorem 2: For the discrete memoryless MAC-WT with noiseless feedback, an inner bound Cin on the secrecy s capacity region C is given by s Cin ={(R ≥0,R ≥0):R ≤I(X ;Y|X ,U) s 1 2 1 1 2 R ≤I(X ;Y|X ,U) 2 2 1 R +R ≤min{I(X ,X ;Y),I(X ;Y|X ,U) 1 2 1 2 1 2 +I(X ;Y|X ,U)}−I(X ,X ;Z) 2 1 1 2 +min{I(X ,X ;Z),H(Y|Z,X ,X )}}, 1 2 1 2 for some distribution satisfying (2.7). Proof: The hybrid inner bound Cin is constructed by combining Ahlswede and Cai’s idea of generating a secret key s from the noiseless feedback [8] with the DF strategy used in [17, Theorem 2], and it is achieved by the following key steps: • For the transmitter 1, split the transmitted message W1 into W1,0 and W1,1, and let W1∗ be a dummy message randomly generated by the transmitter 1, and it is used to confuse the eavesdropper. Analogously, for the transmitter2,splitthetransmittedmessageW intoW andW ,andletW∗ beadummymessagerandomly 2 2,0 2,1 2 generated by the transmitter 2, and it is used to confuse the eavesdropper. • ThemessagesW1 andW2 aretransmittedthroughnblocks,andinblocki(2≤i≤n),wheneachtransmitter receives the noiseless feedback, he tries to decode the other transmitter’s message (including the transmitted messageandthedummymessage)andusesittore-encodehisownmessage.Inaddition,thenoiselessfeedback isusedtogenerateapairofsecretkeys(K∗,K∗),andK∗ (j =1,2)isusedtoencryptthesub-messageW . 1 2 j j,1 • Comparing the above code construction of Csin with that of CsDF, the encoding and decoding schemes of these two bounds are almost the same, except that the sub-message W (j =1,2) is encrypted by a secret key K∗. j,1 j Thus the secrecy sum rate R +R is bounded by two part: the first part is the upper bound on the sum rate 1 2 of CDF, and the second part is the upper bound on the rate of the secret keys K∗ and K∗. Using the balanced s 1 2 coloring lemma introduced by Ahlswede and Cai [8], we conclude that the rate of the secret keys K∗ and K∗ 1 2 is bounded by min{H(Y|X ,X ,Z),I(X ,X ;Z)}. Thus, the hybrid inner bound Cin is obtained. 1 2 1 2 s The details of the proof are in Appendix A. Remark 2: Comparing the DF inner bound CDF and the partial DF inner bound CPDF with our hybrid new s s inner bound Cin, it is easy to see that our new inner bound Cin is strictly larger than CDF and CPDF. s s s s 6 Theorem 3: For the discrete memoryless MAC-WT with noiseless feedback, an outer bound Cout on the secrecy s capacity region C is given by s Cout ={(R ≥0,R ≥0):R +R ≤H(Y|Z)}, s 1 2 1 2 for some distribution P (z,y|x ,x )·P (x ,x ). (2.7) Z,Y|X1,X2 1 2 X1X2 1 2 Proof: The outer bound Cout is a simple sato-type outer bound, and the proof is in Appendix B. s III. GAUSSIANEXAMPLE A. Capacity Results on the Gaussian MAC-WT with Noiseless Feedback For the Gaussian case of the model of Figure 1, the channel inputs and outputs satisfy Y =X +X +N Z =X +X +N , (3.1) 1 2 1 1 2 2 where the channel noises N and N are independent and Gaussian distributed, i.e., N ∼ N(0,σ2), and N ∼ 1 2 1 1 2 N(0,σ2). The average power constraint of the transmitted signal X (j =1,2) is given by 2 j N 1 (cid:88) E[X2]≤P , j =1,2. (3.2) N ji j i=1 The DF and partial DF inner bounds on the secrecy capacity region for the Gaussian case of the model of Figure 1: Theorem 4: The DF inner bound Cgdf and the partial DF inner bound Cgpdf for the Gaussian case of the model s s of Figure 1 are given by 1 P Cgdf =Cgpdf ={(R ≥0,R ≥0):R ≤ log(1+ 1), s s 1 2 1 2 σ2 1 1 P R ≤ log(1+ 2), 2 2 σ2 1 1 P +P 1 P +P R +R ≤ log(1+ 1 2)− log(1+ 1 2)}. (3.3) 1 2 2 σ2 2 σ2 1 2 Proof: In Remark 1, we have shown that for the model of Figure 1, the DF inner bound is the same as the partial DF inner bound. Along the lines of [17, pp. 610-611], we have 1 P Cgdf =Cgpdf ={(R ≥0,R ≥0):R ≤ log(1+ 1), s s 1 2 1 2 σ2 1 1 P R ≤ log(1+ 2), 2 2 σ2 1 1 P +P 1 P 1 P R +R ≤min{ log(1+ 1 2), log(1+ 1)+ log(1+ 2)} 1 2 2 σ2 2 σ2 2 σ2 1 1 1 1 P +P − log(1+ 1 2)}. (3.4) 2 σ2 2 Note that in (3.4), 1log(1+ P1+P2) ≤ 1log(1+ P1)+ 1log(1+ P2), and thus (3.3) is obtained. The proof is 2 σ2 2 σ2 2 σ2 1 1 1 completed. 7 The hybrid inner bound on the secrecy capacity region for the Gaussian case of the model of Figure 1: Theorem 5: The hybrid inner bound Cgi for the Gaussian case of the model of Figure 1 is given by s 1 P Cgi ={(R ≥0,R ≥0):R ≤ log(1+ 1), s 1 2 1 2 σ2 1 1 P R ≤ log(1+ 2), 2 2 σ2 1 1 P +P 1 P +P R +R ≤ log(1+ 1 2)− log(1+ 1 2) 1 2 2 σ2 2 σ2 1 2 1 1 P +P +min{ log(2πeσ2), log(1+ 1 2)}}. (3.5) 2 1 2 σ2 2 √ (cid:112) Proof: Similar to the corresponding proof in [17, pp. 610-611], substituting X = (1−α)P U + αP U 1 1 1 1 √ (cid:112) (0 ≤ α ≤ 1) and X = (1−β)P U + βP U (0 ≤ β ≤ 1) into (3.1), and using the fact that U, U 2 2 2 2 1 and U2 are independent and Gaussian distributed with zero mean and unit variance, and 12log(1 + P1σ+2P2) ≤ 1 1log(1+ P1)+ 1log(1+ P2), (3.5) is directly obtained. Here note that (3.5) is achieved when α=1 and β =1. 2 σ2 2 σ2 1 1 The proof is completed. The outer bound on the secrecy capacity region for the Gaussian case of the model of Figure 1: Theorem 6: For the case that σ2 ≥σ2, the outer bound Cgo for the Gaussian case of the model of Figure 1 is 1 2 s given by 1 Cgo ={(R ≥0,R ≥0):R +R ≤ log(2πe(σ2−σ2))}. (3.6) s 1 2 1 2 2 1 2 For the case that σ2 ≤σ2, the outer bound Cgo is given by 1 2 s 1 1 P +P +σ2 Cgo ={(R ≥0,R ≥0):R +R ≤ log(2πe(σ2−σ2))+ log 1 2 1}. (3.7) s 1 2 1 2 2 2 1 2 P +P +σ2 1 2 2 Proof: • For the case that σ12 ≥σ22, (3.1) can be re-written as Y =X +X +N +N −N Z =X +X +N . (3.8) 1 2 2 1 2 1 2 2 Substituting (3.8) into Theorem 3, we have R +R ≤h(Y|Z)=h(X +X +N +N −N |X +X +N ) 1 2 1 2 2 1 2 1 2 2 1 =h(N −N |X +X +N )≤h(N −N )= log(2πe(σ2−σ2)). (3.9) 1 2 1 2 2 1 2 2 1 2 • For the case that σ12 ≤σ22, (3.1) can be re-written as Y =X +X +N Z =X +X +N +N −N . (3.10) 1 2 1 1 2 1 2 1 8 Substituting (3.10) into Theorem 3, we have R +R ≤h(Y|Z)=h(Y,Z)−h(Z)=h(Z|Y)+h(Y)−h(Z) 1 2 =h(X +X +N +N −N |X +X +N )+h(Y)−h(Y +N −N ) 1 2 1 2 1 1 2 1 2 1 =h(N −N |X +X +N )+h(Y)−h(Y +N −N ) 2 1 1 2 1 2 1 ≤h(N −N )+h(Y)−h(Y +N −N ) 2 1 2 1 (a) 1 ≤ h(N −N )+h(Y)− log(22h(Y)+22h(N2−N1)) 2 1 2 (b) 1 1 ≤ h(N −N )+ log(2πe(P +P +σ2))− log(2πe(P +P +σ2)+2πe(σ2−σ2)) 2 1 2 1 2 1 2 1 2 1 2 1 1 1 1 = log(2πe(σ2−σ2))+ log(2πe(P +P +σ2))− log(2πe(P +P +σ2)+2πe(σ2−σ2)) 2 2 1 2 1 2 1 2 1 2 1 2 1 1 1 P +P +σ2 = log(2πe(σ2−σ2))+ log 1 2 1, (3.11) 2 2 1 2 P +P +σ2 1 2 2 where(a)isfromtheentropypowerinequality,i.e.,22h(Y+N2−N1) ≥22h(Y)+22h(N2−N1),and(b)isfromthe factthath(Y)−12log(22h(Y)+22h(N2−N1))isincreasingwhileh(Y)isincreasing,h(Y)=h(X1+X2+N1)≤ 1log(2πe(P +P +σ2)) and h(N −N )= 1log(2πe(σ2−σ2)). 2 1 2 1 2 1 2 2 1 The proof is completed. Finally, recall that Tekin and Yener [12] have shown that for the Gaussian MAC-WT without feedback, an inner bound Cgmac−wt is given by s 1 P 1 P Cgmac−wt ={(R ≥0,R ≥0):R ≤ log(1+ 1)− log(1+ 1 ), s 1 2 1 2 σ2 2 σ2+P 1 2 2 1 P 1 P R ≤ log(1+ 2)− log(1+ 2 ), 2 2 σ2 2 σ2+P 1 2 1 1 P +P 1 P +P R +R ≤ log(1+ 1 2)− log(1+ 1 2)}. (3.12) 1 2 2 σ2 2 σ2 1 2 For the case that σ2 ≤σ2, the following Figure 2 shows the inner bound Cgi, the partial (Cgpdf) and full (Cgdf) 1 2 s s s DF inner bounds for the Gaussian case of Figure 1, the outer bound Cgo and Tekin-Yener’s inner bound Cgmac−wt s s of the Gaussian MAC-WT [12] for P = P = 1, σ2 = 1 and σ2 = 10. From Figure 2, it is easy to see that 1 2 1 2 our new inner bound Cgi is larger than the DF inner bounds Cgpdf and Cgdf, and the noiseless feedback helps to s s s enhance the secrecy rate region Cgmac−wt of the Gaussian MAC-WT. s For the case that σ2 ≥ σ2, the DF bounds Cgpdf, Cgdf and Tekin-Yener’s inner bound Cgmac−wt reduce to 1 2 s s s the point (R = 0,R = 0). The following Figure 3 shows the inner bound Cgi and the outer bound Cgo for 1 2 s s P =P =10, σ2 =5, σ2 =2. It is easy to see that when σ2 ≥σ2, our hybrid inner bound still provides positive 1 2 1 2 1 2 secrecy rates, while there is no positive secrecy rate in the partial and full DF inner bounds. B. Power Control for the Maximum Secrecy Sum Rate of Cgi s In this subsection, we assume that the average power constraints of the transmitters satisfy 0≤P ,P ≤P, (3.13) 1 2 9 Fig. 2: The bounds Cgi, Cgpdf, Cgdf, Cgo, and Cgmac−wt for P =P =1, σ2 =1, σ2 =10 s s s s s 1 2 1 2 and define the maximum secrecy sum rate R∗ of Cgi as sum s 1 P +P 1 P +P R∗ = max log(1+ 1 2)− log(1+ 1 2) sum P1,P2 2 σ12 2 σ22 1 1 P +P +min{ log(2πeσ2), log(1+ 1 2)}. (3.14) 2 1 2 σ2 2 In the remainder of this subsection, we calculate the maximum secrecy sum rate R∗ of Cgi, and show the sum s optimum power control (the optimum of P and P is denoted by P∗ and P∗, respectively) for R∗ . 1 2 1 2 sum Theorem 7: If σ2 >σ2, the maximum secrecy sum rate R∗ of Cgi is given by 1 2 sum s   1log(1+ 2P), 0≤P ≤ (2πeσ12−1)σ22 R∗ = 2 σ12 2 (3.15) sum  1log(1+ (2πeσ12−1)σ22), P ≥ (2πeσ12−1)σ22, 2 σ2 2 1 and the optimum power control is given by   (P,P), 0≤P ≤ (2πeσ12−1)σ22 (P∗,P∗)= 2 (3.16) 1 2  ((2πeσ12−1)σ22,(2πeσ12−1)σ22), P ≥ (2πeσ12−1)σ22. 2 2 2 If σ2 ≤σ2, the maximum secrecy sum rate R∗ of Cgi is given by 1 2 sum s   1log(1+ 2P), 0≤P ≤ (2πeσ12−1)σ22 R∗ = 2 σ12 2 (3.17) sum  1log(2πeσ2)+ 1log(1+ 2P)− 1log(1+ 2P), P ≥ (2πeσ12−1)σ22, 2 1 2 σ2 2 σ2 2 1 2 and the optimum power control is given by   (P,P), 0≤P ≤ (2πeσ12−1)σ22 (P∗,P∗)= 2 (3.18) 1 2  (P,P), P ≥ (2πeσ12−1)σ22. 2 10 Fig. 3: The bounds Cgi and Cgo for P =P =10, σ2 =5, σ2 =2 s s 1 2 1 2 Proof: From Theorem 5, it is easy to see that the secrecy sum rate R of Cgi is given by sum s 1 P +P 1 P +P 1 1 P +P R = log(1+ 1 2)− log(1+ 1 2)+min{ log(2πeσ2), log(1+ 1 2)},(3.19) sum 2 σ2 2 σ2 2 1 2 σ2 1 2 2 and (3.19) can be re-written as  Rsum = 12log(1+ P1σ+12P2), 0≤P1+P2 ≤(2πeσ12−1)σ22 (3.20)  21log(1+ P1σ+2P2)− 12log(1+ P1σ+2P2)+ 12log(2πeσ12), P1+P2 >(2πeσ12−1)σ22. 1 2 Since 0≤P +P ≤2P, the secrecy sum rate R in (3.20) can be considered into the following three cases: 1 2 sum • (Case 1:) If 0 ≤ P ≤ (2πeσ122−1)σ22, it is easy to see that Rsum is increasing while P1 and P2 are increasing, and thus we have R∗ = 1log(1+ 2P), and the corresponding optimum P∗ and P∗ equal to P. sum 2 σ2 1 2 1 • (Case 2:) If P > (2πeσ122−1)σ22 and σ12 ≤σ22, (3.20) is re-written as  Rsum = 12log(1+ P1σ+12P2), 0≤P1+P2 ≤(2πeσ12−1)σ22  21log(1+ P1σ+2P2)− 12log(1+ P1σ+2P2)+ 21log(2πeσ12), (2πeσ12−1)σ22 <P1+P2 ≤2P. 1 2 (3.21) It is not difficult to show that for this case, R∗ = 1log(1+ 2P)− 1log(1+ 2P)+ 1log(2πeσ2), and the sum 2 σ2 2 σ2 2 1 1 2 corresponding optimum P∗ and P∗ equal to P. 1 2 • (Case 3:) If P > (2πeσ122−1)σ22 and σ12 > σ22, it is not difficult to show that for this case, Rs∗um = 12log(1+ (2πeσ12−1)σ22), and the corresponding optimum P∗ and P∗ equal to (2πeσ12−1)σ22. σ2 1 2 2 1 Combining the above three cases, Theorem 7 is obtained, and the proof is completed.