Multiple Access Wiretap Channel with Noiseless Feedback
11 Multiple Access Wiretap Channel withNoiseless Feedback
Bin Dai and Zheng Ma
Abstract
The physical layer security in the up-link of the wireless communication systems is often modeled as the multipleaccess wiretap channel (MAC-WT), and recently it has received a lot attention. In this paper, the MAC-WT has beenre-visited by considering the situation that the legitimate receiver feeds his received channel output back to thetransmitters via two noiseless channels, respectively. This model is called the MAC-WT with noiseless feedback.Inner and outer bounds on the secrecy capacity region of this feedback model are provided. To be specific, we firstpresent a decode-and-forward (DF) inner bound on the secrecy capacity region of this feedback model, and this boundis constructed by allowing each transmitter to decode the other one’s transmitted message from the feedback, andthen each transmitter uses the decoded message to re-encode his own messages, i.e., this DF inner bound allows theindependent transmitters to co-operate with each other. Then, we provide a hybrid inner bound which is strictly largerthan the DF inner bound, and it is constructed by using the feedback as a tool not only to allow the independenttransmitters to co-operate with each other, but also to generate two secret keys respectively shared between thelegitimate receiver and the two transmitters. Finally, we give a sato-type outer bound on the secrecy capacity regionof this feedback model. The results of this paper are further explained via a Gaussian example.
Index Terms
Multiple-access wiretap channel, noiseless feedback, secrecy capacity region.
I. I
NTRODUCTION
The physical layer security (PLS) was first investigated by Wyner in his landmark paper on the degraded wiretapchannel [1]. Wyner’s degraded wiretap channel model consists of one transmitter and two receivers (a legitimatereceiver and an eavesdropper). The transmitter sends a private message to the legitimate receiver via a discretememoryless main channel, and an eavesdropper eavesdrops the output of the main channel via another discretememoryless 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 theState Key Laboratory of Integrated Services Networks, Xidian University, Xi (cid:48) an, Shaanxi 710071, China, e-mail: [email protected]. Ma is with the School of Information Science and Technology, Southwest JiaoTong University, Chengdu 610031, China, e-mail:[email protected]. a r X i v : . [ c s . I T ] J a n message is leaked to the eavesdropper. The secrecy capacity C s , which is the maximum reliable transmission ratewith perfect secrecy constraint, was characterized by Wyner [1], and it is given by C s = max p ( x ) ( I ( X ; Y ) − I ( X ; Z )) , (1.1)where X , Y and Z are the input of the main channel, output of the main channel and output of the wiretap channel,respectively, and they satisfy the Markov chain X → Y → Z . Here note that (1.1) holds under the degradednessassumption X → Y → Z , and the secrecy capacity of the general wiretap channel (the wiretap channel without thedegradedness assumption) was determined by Csisz ´ a r and K¨orner [2]. The work of [1] and [2] lays a foundationfor the PLS of the practical communication systems.Since Wozencraft et al. [3] showed that the time-variant noisy two-way channels can be used to provide noiselessfeedback, whether this noiseless feedback helps to enhance the capacities of various communication channelsmotivates the researchers to study the channels with noiseless feedback. Shannon first proved that the noiselessfeedback 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 asmultiple-access channel (MAC) and relay channel, can be enhanced by feeding back the receiver’s channel outputto the transmitter over a noiseless channel. Then, it is natural to ask: does the noiseless feedback from the legitimatereceiver 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 feedbackis known by the legitimate receiver and the transmitter, and it is not available for the eavesdropper, Ahlswedeand Cai pointed out that the noiseless feedback can be used to generate a secret key shared only between thetransmitter and the legitimate receiver, and we can use this key to encrypt the transmitted messages. Combiningthe idea of generating a secret key from the noiseless feedback with Wyner’s random binning technique used inthe achievability proof of (1.1), Ahlswede and Cai showed that the secrecy capacity C sf of the degraded wiretapchannel with noiseless feedback is given by C sf = max p ( x ) min { I ( X ; Y ) , I ( X ; Y ) − I ( X ; Z ) + H ( Y | X, Z ) } , (1.2)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 wiretapchannel. 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 investigatewhether the noiseless feedback from the legitimate receiver to the transmitters helps to enhance the secrecy capacityregion of the MAC-WT, in this paper, we study the MAC-WT with noiseless feedback, see Figure 1. We first presenta DF inner bound on the secrecy capacity region of the model of Figure 1, and this bound is constructed by usingthe DF strategy of the MAC-WT with noisy feedback [17], where each transmitter of the MAC decodes the other one’s transmitted message from the noisy feedback and then uses it to re-encode his own messages. Second, notethat the noiseless feedback can not only be used to re-encode the messages of the transmitters, but also be usedto generate secret keys to encrypt the transmitted messages, thus we present a hybrid inner bound on the secrecycapacity region of the model of Figure 1 by combining Ahlswede and Cai’s idea of generating a secret key fromthe noiseless feedback [8] with the DF strategy used in [17], and we show that this hybrid inner bound is strictlylarger than the DF inner bound. Third, we present a sato-type outer bound on the secrecy capacity region of themodel 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 mainresults of the model of Figure 1. An Gaussian example of the model of Figure 1 is provided in Section III. Finalconclusions are presented in Section IV.Fig. 1: The multiple-access wiretap channel with noiseless feedbackII. M ODEL DESCRIPTION AND THE MAIN RESULT
Basic notations:
We use the notation p V ( v ) to denote the probability mass function P r { V = v } , where V (capitalletter) denotes the random variable, v (lower case letter) denotes the real value of the random variable V . Denotethe alphabet in which the random variable V takes values by V (calligraphic letter). Similarly, let U N be a randomvector ( U , ..., U N ) , and u N be a vector value ( u , ..., u N ) . In the rest of this paper, the log function is taken tothe base 2. Definitions of the model of Figure 1:
Let W , uniformly distributed over the finite alphabet W = { , , ..., M } , be the message sent by the transmitter1. Similarly, let W , uniformly distributed over the finite alphabet W = { , , ..., M } , be the message sent by thetransmitter 2.The inputs of the channel are x N and x N , while the outputs are y N and z N . The channel is discrete memoryless,i.e., at the i -th time, the channel outputs Y i and Z i depend only on X ,i and X ,i , and thus we have P Y N ,Z N | X N ,X N ( y N , z N | x N , x N )= N (cid:89) i =1 P Y,Z | X ,X ( y i , z i | x ,i , x ,i ) . (2.1) Since y N can be fed back to the transmitters via a noiseless feedback channel, at the i -th time, the channel input X j,i ( j = 1 , ) is given by X j,i = f j,i ( W j ) , i = 1 f j,i ( W j , Y i − ) , ≤ i ≤ N. (2.2)Here note that the i -th time channel encoder f j,i ( j = 1 , ) is a stochastic encoder, and the transmission rates ofthe messages W and W are log M N and log M N , respectively.The decoder is a mapping ψ : Y N → W × W , with input Y N and outputs ˆ W , ˆ W . The average probabilityof error P e is denoted by P e = 1 M M M (cid:88) i =1 M (cid:88) j =1 P r { ψ ( y N ) (cid:54) = ( i, j ) | ( i, j ) sent } . (2.3)The eavesdropper’s equivocation to the messages W and W is defined as ∆ = 1 N H ( W , W | Z N ) . (2.4)A positive rate pair ( R , R ) is called achievable with weak secrecy if, for any small positive (cid:15) , there exists an ( M , M , N, P e ) code such that log M N ≥ R − (cid:15), log M N ≥ R − (cid:15), ∆ ≥ R + R − (cid:15), P e ≤ (cid:15). (2.5)Here we note that ∆ ≥ R + R − (cid:15) also ensures N H ( W t | Z N ) ≥ R t − (cid:15) for t = 1 , , and the proof is in [17,p. 609]. The secrecy capacity region C s of the model of Figure 1 is a set composed of all rate pairs ( R , R ) satisfying (2.5). The following Theorem 1 and Theorem 2 show two inner bounds on C s , and Theorem 3 shows anouter bound on C s . Theorem 1:
For the discrete memoryless MAC-WT with noiseless feedback, an inner bound C DFs on the secrecycapacity region C s is given by C DFs = { ( R ≥ , R ≥
0) : R ≤ I ( X ; Y | X , U ) R ≤ I ( X ; Y | X , U ) R + R ≤ min { I ( X , X ; Y ) , I ( X ; Y | X , U ) + I ( X ; Y | X , U ) } − I ( X , X ; Z ) } , for some distribution P Z,Y | X ,X ( z, y | x , x ) · P X | U ( x | u ) · P X | U ( x | u ) · P U ( u ) . (2.6) Proof:
In the MAC-WT with noisy feedback [17], the legitimate receiver’s channel output Y is sent to the transmittersvia two noisy feedback channels, and the outputs of the noisy feedback channel are Y and Y . Substituting Y = Y = Y (which implies the feedback channel is noiseless) into [17, Theorem 2], the DF inner bound C DFs for the model of Figure 1 is obtained, and the proof of C DFs is along the lines of that of [17, Theorem 2] (the fullDF inner bound on the secrecy capacity region of the MAC-WT with noisy feedback), and thus we omit the proofhere.
Remark 1:
In [17, Theorem 1], Tang et al. also provide a partial DF inner bound on the secrecy capacity regionof the MAC-WT with noisy feedback. Substituting Y = Y = Y into [17, Theorem 1], and using Fourier-Motzkinelimination (see, e.g., [18]) to eliminate R , R , R and R , it is not difficult to show that the partial DF innerbound C P DFs of the model of Figure 1 is exactly the same as the DF inner bound C DFs shown in Theorem 1.
Theorem 2:
For the discrete memoryless MAC-WT with noiseless feedback, an inner bound C ins on the secrecycapacity region C s is given by C ins = { ( R ≥ , R ≥
0) : R ≤ I ( X ; Y | X , U ) R ≤ I ( X ; Y | X , U ) R + R ≤ min { I ( X , X ; Y ) , I ( X ; Y | X , U )+ I ( X ; Y | X , U ) } − I ( X , X ; Z )+ min { I ( X , X ; Z ) , H ( Y | Z, X , X ) }} , for some distribution satisfying (2.7). Proof:
The hybrid inner bound C ins is constructed by combining Ahlswede and Cai’s idea of generating a secret keyfrom the noiseless feedback [8] with the DF strategy used in [17, Theorem 2], and it is achieved by the followingkey steps: • For the transmitter 1, split the transmitted message W into W , and W , , and let W ∗ be a dummy messagerandomly generated by the transmitter 1, and it is used to confuse the eavesdropper. Analogously, for thetransmitter 2, split the transmitted message W into W , and W , , and let W ∗ be a dummy message randomlygenerated by the transmitter 2, and it is used to confuse the eavesdropper. • The messages W and W are transmitted through n blocks, and in block i ( ≤ i ≤ n ), when each transmitterreceives the noiseless feedback, he tries to decode the other transmitter’s message (including the transmittedmessage and the dummy message) and uses it to re-encode his own message. In addition, the noiseless feedbackis used to generate a pair of secret keys ( K ∗ , K ∗ ) , and K ∗ j ( j = 1 , ) is used to encrypt the sub-message W j, . • Comparing the above code construction of C ins with that of C DFs , the encoding and decoding schemes of thesetwo bounds are almost the same, except that the sub-message W j, ( j = 1 , ) is encrypted by a secret key K ∗ j .Thus the secrecy sum rate R + R is bounded by two part: the first part is the upper bound on the sum rateof C DFs , and the second part is the upper bound on the rate of the secret keys K ∗ and K ∗ . Using the balancedcoloring lemma introduced by Ahlswede and Cai [8], we conclude that the rate of the secret keys K ∗ and K ∗ is bounded by min { H ( Y | X , X , Z ) , I ( X , X ; Z ) } . Thus, the hybrid inner bound C ins is obtained.The details of the proof are in Appendix A. Remark 2:
Comparing the DF inner bound C DFs and the partial DF inner bound C P DFs with our hybrid newinner bound C ins , it is easy to see that our new inner bound C ins is strictly larger than C DFs and C P DFs . Theorem 3:
For the discrete memoryless MAC-WT with noiseless feedback, an outer bound C outs on the secrecycapacity region C s is given by C outs = { ( R ≥ , R ≥
0) : R + R ≤ H ( Y | Z ) } , for some distribution P Z,Y | X ,X ( z, y | x , x ) · P X X ( x , x ) . (2.7) Proof:
The outer bound C outs is a simple sato-type outer bound, and the proof is in Appendix B.III. G AUSSIAN E XAMPLE
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)where the channel noises N and N are independent and Gaussian distributed, i.e., N ∼ N (0 , σ ) , and N ∼N (0 , σ ) . The average power constraint of the transmitted signal X j ( j = 1 , ) is given by N N (cid:88) i =1 E [ X ji ] ≤ P j , j = 1 , . (3.2) The DF and partial DF inner bounds on the secrecy capacity region for the Gaussian case of the model ofFigure 1:
Theorem 4:
The DF inner bound C gdfs and the partial DF inner bound C gpdfs for the Gaussian case of the modelof Figure 1 are given by C gdfs = C gpdfs = { ( R ≥ , R ≥
0) : R ≤
12 log(1 + P σ ) ,R ≤
12 log(1 + P σ ) ,R + R ≤
12 log(1 + P + P σ ) −
12 log(1 + P + P σ ) } . (3.3) Proof:
In Remark 1, we have shown that for the model of Figure 1, the DF inner bound is the same as thepartial DF inner bound. Along the lines of [17, pp. 610-611], we have C gdfs = C gpdfs = { ( R ≥ , R ≥
0) : R ≤
12 log(1 + P σ ) ,R ≤
12 log(1 + P σ ) ,R + R ≤ min {
12 log(1 + P + P σ ) ,
12 log(1 + P σ ) + 12 log(1 + P σ ) }−
12 log(1 + P + P σ ) } . (3.4)Note that in (3.4), log(1 + P + P σ ) ≤ log(1 + P σ ) + log(1 + P σ ) , and thus (3.3) is obtained. The proof iscompleted. 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 C gis for the Gaussian case of the model of Figure 1 is given by C gis = { ( R ≥ , R ≥
0) : R ≤
12 log(1 + P σ ) ,R ≤
12 log(1 + P σ ) ,R + R ≤
12 log(1 + P + P σ ) −
12 log(1 + P + P σ )+ min {
12 log(2 πeσ ) ,
12 log(1 + P + P σ ) }} . (3.5) Proof:
Similar to the corresponding proof in [17, pp. 610-611], substituting X = (cid:112) (1 − α ) P U + √ αP U ( ≤ α ≤ ) and X = (cid:112) (1 − β ) P U + √ βP U ( ≤ β ≤ ) into (3.1), and using the fact that U , U and U are independent and Gaussian distributed with zero mean and unit variance, and log(1 + P + P σ ) ≤ log(1 + P σ ) + log(1 + P σ ) , (3.5) is directly obtained. Here note that (3.5) is achieved when α = 1 and β = 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 σ ≥ σ , the outer bound C gos for the Gaussian case of the model of Figure 1 isgiven by C gos = { ( R ≥ , R ≥
0) : R + R ≤
12 log(2 πe ( σ − σ )) } . (3.6)For the case that σ ≤ σ , the outer bound C gos is given by C gos = { ( R ≥ , R ≥
0) : R + R ≤
12 log(2 πe ( σ − σ )) + 12 log P + P + σ P + P + σ } . (3.7) Proof: • For the case that σ ≥ σ , (3.1) can be re-written as Y = X + X + N + N − N Z = X + X + N . (3.8)Substituting (3.8) into Theorem 3, we have R + R ≤ h ( Y | Z ) = h ( X + X + N + N − N | X + X + N )= h ( N − N | X + X + N ) ≤ h ( N − N ) = 12 log(2 πe ( σ − σ )) . (3.9) • For the case that σ ≤ σ , (3.1) can be re-written as Y = X + X + N Z = X + X + N + N − N . (3.10) 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 )= h ( X + X + N + N − N | X + X + N ) + h ( Y ) − h ( Y + N − N )= h ( N − N | X + X + N ) + h ( Y ) − h ( Y + N − N ) ≤ h ( N − N ) + h ( Y ) − h ( Y + N − N ) ( a ) ≤ h ( N − N ) + h ( Y ) −
12 log(2 h ( Y ) + 2 h ( N − N ) ) ( b ) ≤ h ( N − N ) + 12 log(2 πe ( P + P + σ )) −
12 log(2 πe ( P + P + σ ) + 2 πe ( σ − σ ))= 12 log(2 πe ( σ − σ )) + 12 log(2 πe ( P + P + σ )) −
12 log(2 πe ( P + P + σ ) + 2 πe ( σ − σ ))= 12 log(2 πe ( σ − σ )) + 12 log P + P + σ P + P + σ , (3.11)where (a) is from the entropy power inequality, i.e., h ( Y + N − N ) ≥ h ( Y ) + 2 h ( N − N ) , and (b) is from thefact that h ( Y ) − log(2 h ( Y ) +2 h ( N − N ) ) is increasing while h ( Y ) is increasing, h ( Y ) = h ( X + X + N ) ≤ log(2 πe ( P + P + σ )) and h ( N − N ) = log(2 πe ( σ − σ )) .The proof is completed.Finally, recall that Tekin and Yener [12] have shown that for the Gaussian MAC-WT without feedback, an innerbound C gmac − wts is given by C gmac − wts = { ( R ≥ , R ≥
0) : R ≤
12 log(1 + P σ ) −
12 log(1 + P σ + P ) ,R ≤
12 log(1 + P σ ) −
12 log(1 + P σ + P ) ,R + R ≤
12 log(1 + P + P σ ) −
12 log(1 + P + P σ ) } . (3.12)For the case that σ ≤ σ , the following Figure 2 shows the inner bound C gis , the partial ( C gpdfs ) and full ( C gdfs )DF inner bounds for the Gaussian case of Figure 1, the outer bound C gos and Tekin-Yener’s inner bound C gmac − wts of the Gaussian MAC-WT [12] for P = P = 1 , σ = 1 and σ = 10 . From Figure 2, it is easy to see thatour new inner bound C gis is larger than the DF inner bounds C gpdfs and C gdfs , and the noiseless feedback helps toenhance the secrecy rate region C gmac − wts of the Gaussian MAC-WT.For the case that σ ≥ σ , the DF bounds C gpdfs , C gdfs and Tekin-Yener’s inner bound C gmac − wts reduce tothe point ( R = 0 , R = 0) . The following Figure 3 shows the inner bound C gis and the outer bound C gos for P = P = 10 , σ = 5 , σ = 2 . It is easy to see that when σ ≥ σ , our hybrid inner bound still provides positivesecrecy 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 C gis In this subsection, we assume that the average power constraints of the transmitters satisfy ≤ P , P ≤ P, (3.13) Fig. 2: The bounds C gis , C gpdfs , C gdfs , C gos , and C gmac − wts for P = P = 1 , σ = 1 , σ = 10 and define the maximum secrecy sum rate R ∗ sum of C gis as R ∗ sum = max P ,P
12 log(1 + P + P σ ) −
12 log(1 + P + P σ )+ min {
12 log(2 πeσ ) ,
12 log(1 + P + P σ ) } . (3.14)In the remainder of this subsection, we calculate the maximum secrecy sum rate R ∗ sum of C gis , and show theoptimum power control (the optimum of P and P is denoted by P ∗ and P ∗ , respectively) for R ∗ sum . Theorem 7: If σ > σ , the maximum secrecy sum rate R ∗ sum of C gis is given by R ∗ sum = log(1 + Pσ ) , ≤ P ≤ (2 πeσ − σ log(1 + (2 πeσ − σ σ ) , P ≥ (2 πeσ − σ , (3.15)and the optimum power control is given by ( P ∗ , P ∗ ) = ( P, P ) , ≤ P ≤ (2 πeσ − σ ( (2 πeσ − σ , (2 πeσ − σ ) , P ≥ (2 πeσ − σ . (3.16)If σ ≤ σ , the maximum secrecy sum rate R ∗ sum of C gis is given by R ∗ sum = log(1 + Pσ ) , ≤ P ≤ (2 πeσ − σ log(2 πeσ ) + log(1 + Pσ ) − log(1 + Pσ ) , P ≥ (2 πeσ − σ , (3.17)and the optimum power control is given by ( P ∗ , P ∗ ) = ( P, P ) , ≤ P ≤ (2 πeσ − σ ( P, P ) , P ≥ (2 πeσ − σ . (3.18) Fig. 3: The bounds C gis and C gos for P = P = 10 , σ = 5 , σ = 2 Proof:
From Theorem 5, it is easy to see that the secrecy sum rate R sum of C gis is given by R sum = 12 log(1 + P + P σ ) −
12 log(1 + P + P σ ) + min {
12 log(2 πeσ ) ,
12 log(1 + P + P σ ) } , (3.19)and (3.19) can be re-written as R sum = log(1 + P + P σ ) , ≤ P + P ≤ (2 πeσ − σ log(1 + P + P σ ) − log(1 + P + P σ ) + log(2 πeσ ) , P + P > (2 πeσ − σ . (3.20)Since ≤ P + P ≤ P , the secrecy sum rate R sum in (3.20) can be considered into the following three cases: • (Case 1:) If ≤ P ≤ (2 πeσ − σ , it is easy to see that R sum is increasing while P and P are increasing,and thus we have R ∗ sum = log(1 + Pσ ) , and the corresponding optimum P ∗ and P ∗ equal to P . • (Case 2:) If P > (2 πeσ − σ and σ ≤ σ , (3.20) is re-written as R sum = log(1 + P + P σ ) , ≤ P + P ≤ (2 πeσ − σ log(1 + P + P σ ) − log(1 + P + P σ ) + log(2 πeσ ) , (2 πeσ − σ < P + P ≤ P. (3.21)It is not difficult to show that for this case, R ∗ sum = log(1 + Pσ ) − log(1 + Pσ ) + log(2 πeσ ) , and thecorresponding optimum P ∗ and P ∗ equal to P . • (Case 3:) If P > (2 πeσ − σ and σ > σ , it is not difficult to show that for this case, R ∗ sum = log(1 + (2 πeσ − σ σ ) , and the corresponding optimum P ∗ and P ∗ equal to (2 πeσ − σ .Combining the above three cases, Theorem 7 is obtained, and the proof is completed. The following Figure 4 and Figure 5 show the maximum secrecy sum rate R ∗ sum and the corresponding optimumpower control for σ > σ and σ ≤ σ , respectively. It is easy to see that for both cases, R ∗ sum tends to a constantwhile P tends to infinity.Fig. 4: The maximum secrecy sum rate R ∗ sum and the corresponding optimum power control for σ = 5 and σ = 2 IV. C
ONCLUSIONS
In this paper, we present two inner bounds and one outer bound on the secrecy capacity region of the MAC-WTwith noiseless feedback. To be specific, the first inner bound is constructed by using the DF strategy, where eachtransmitter decodes the other one’s transmitted message from the noiseless feedback and then uses the decodedmessage to re-encode his own messages. The second inner bound is constructed by combining Ahlswede and Cai’sidea of generating a secret key from the noiseless feedback [8] with the DF strategy used in the first inner bound.The outer bound is a simple sato-type bound. We show that the second inner bound is strictly larger than the firstone, and the capacity results are further explained via a Gaussian example.A
CKNOWLEDGEMENT
This work was supported by the National Natural Science Foundation of China under Grants 61671391, 61301121and 61571373, the fundamental research funds for the Central universities (No. 2682016ZDPY06), and the OpenResearch Fund of the State Key Laboratory of Integrated Services Networks, Xidian University (No. ISN17-13). Fig. 5: The maximum secrecy sum rate R ∗ sum and the corresponding optimum power control for σ = 1 and σ = 10 A PPENDIX AP ROOF OF T HEOREM W = ( W , , ..., W ,n ) and W = ( W , , ..., W ,n ) are transmitted through n blocks. In block i ( ≤ i ≤ n ), the transmitted message w j,i ( j = 1 , ) is denoted by w j,i = ( w j,i, , w j,i, ) , where w j,i, ∈{ , , ..., NR j } , w j,i, ∈ { , , ..., NR j } , w j,i ∈ { , , ..., NR j } and R j = R j + R j . Here note that in block ,the transmitted message w j, = ( w j, , , const ) ( j = 1 , ), which implies that the sub-message w j, , is a constant.For block i ( ≤ i ≤ n ), let w ∗ ,i and w ∗ ,i be the randomly generated dummy messages for transmitters 1 and 2,respectively. Here w ∗ j,i ∈ { , , ..., NR ∗ j } ( j = 1 , ).For ≤ i ≤ n , let (cid:101) X j,i ( j = 1 , ), (cid:101) U i , (cid:101) Y i and (cid:101) Z i be the random vectors with length N for block i . Thespecific values of the above random vectors are denoted by lower case letters. Moreover, let X nj = ( (cid:101) X j, , ..., (cid:101) X j,n ) , U n = ( (cid:101) U , ..., (cid:101) U n ) , Y n = ( (cid:101) Y , ..., (cid:101) Y n ) and Z n = ( (cid:101) Z , ..., (cid:101) Z n ) . Construction of the code-books : In each block i ( ≤ i ≤ n ), for a fixed joint probability P Z,Y | X ,X ( z, y | x , x ) P X | U ( x | u ) P X | U ( x | u ) P U ( u ) , randomly generate N ( R + R + R ∗ + R + R + R ∗ ) i.i.d. sequences (cid:101) u i according to P U ( u ) , and index these sequences as (cid:101) u i ( w (cid:48) ,i ) , where ≤ w (cid:48) ,i ≤ N ( R + R + R ∗ + R + R + R ∗ ) .For each w (cid:48) ,i , randomly generate N ( R j + R j + R ∗ j ) ( j = 1 , ) i.i.d. sequences (cid:101) x j,i according to P X j | U ( x j | u ) , andindex these sequences as (cid:101) x j,i ( w (cid:48) j,i ) , where ≤ w (cid:48) j,i ≤ N ( R j + R j + R ∗ j ) . Encoding scheme : In block , both the transmitters choose w (cid:48) , = 1 as the index of the transmitted (cid:101) u , and send (cid:101) u (1) . Furthermore, the transmitter j ( j = 1 , ) chooses w (cid:48) j, = ( w j, , , w j, , = const, w ∗ j, ) as the index of the transmitted codeword (cid:101) x j, .In block i ( ≤ i ≤ n ), suppose that transmitter 1 has already obtained w (cid:48) ,i − and w (cid:48) ,i − = ( w ,i − , , w ,i − , , w ∗ ,i − ) .Since the transmitter 1 receives the feedback (cid:101) y i − , he tries to find a unique sequence (cid:101) x ,i − ( ˇ w (cid:48) ,i − , w (cid:48) ,i − ) suchthat ( (cid:101) x ,i − ( ˇ w (cid:48) ,i − , w (cid:48) ,i − ) , (cid:101) x ,i − ( w (cid:48) ,i − , w (cid:48) ,i − ) , (cid:101) u i − ( w (cid:48) ,i − ) , (cid:101) y i − ) are jointly typical sequences. From AEP, it is easy to see that the error probability P r { ˇ w (cid:48) ,i − (cid:54) = w (cid:48) ,i − } goes to if R + R + R ∗ ≤ I ( X ; Y | X , U ) . (A1)Thus in block i , the transmitter sends (cid:101) u i with the index w (cid:48) ,i = ( w (cid:48) ,i − , ˇ w (cid:48) ,i − ) .Analogously, since the transmitter 2 receives the feedback (cid:101) y i − , he tries to find a unique sequence (cid:101) x ,i − ( ˜ w (cid:48) ,i − , w (cid:48) ,i − ) such that ( (cid:101) x ,i − ( ˜ w (cid:48) ,i − , w (cid:48) ,i − ) , (cid:101) x ,i − ( w (cid:48) ,i − , w (cid:48) ,i − ) , (cid:101) u i − ( w (cid:48) ,i − ) , (cid:101) y i − ) are jointly typical sequences. From AEP, it is easy to see that the error probability P r { ˜ w (cid:48) ,i − (cid:54) = w (cid:48) ,i − } goes to if R + R + R ∗ ≤ I ( X ; Y | X , U ) . (A2)Thus in block i , the transmitter sends (cid:101) u i with the index w (cid:48) ,i = ( ˜ w (cid:48) ,i − , w (cid:48) ,i − ) .In block i ( ≤ i ≤ n ), before choosing the transmitted codewords (cid:101) x ,i and (cid:101) x ,i , we generate a mapping g i : (cid:101) y i − → { , , ..., N ( R + R ) } . Furthermore, we define K ∗ i = ( K ∗ i, , K ∗ i, ) = g i ( (cid:101) Y i − ) as a random variableuniformly distributed over { , , ..., N ( R + R ) } , and it is independent of (cid:101) X ,i , (cid:101) X ,i , (cid:101) Y i , (cid:101) Z i , W ,i , W ,i , W ∗ ,i and W ∗ ,i . Here note that K ∗ i,j ( j = 1 , ) is used as a secret key shared by the transmitter j and the receiver,and k ∗ i,j ∈ { , , ..., NR j } is a specific value of K ∗ i,j . Reveal the mapping g i to the transmitters, receiver andthe eavesdropper. After the generation of the secret key, the transmitter j ( j = 1 , ) sends (cid:101) x j,i with the index w (cid:48) j,i = ( w j,i, , w j,i, ⊕ k ∗ i,j , w ∗ j,i ) . Decoding scheme for the receiver : The intended receiver does backward decoding after the transmission of all n blocks is completed, and the receiver’s decoding scheme is exactly the same as that of the classical MAC withfeedback [5, pp. 295-296]. Following similar steps of error probability analysis for MAC with feedback [5, pp.295-296], we have R + R + R ∗ + R + R + R ∗ ≤ I ( X , X ; Y ) . (A3) Equivocation analysis (1): For block ≤ i ≤ n , a lower bound on H ( K ∗ i | (cid:101) X ,i − , (cid:101) X ,i − , (cid:101) Z i − ) : Given (cid:101) X ,i − , (cid:101) X ,i − and (cid:101) Z i − , the eavesdropper’s equivocation about the secret key k ∗ i can be bounded by Ahlswede and Cai’sbalanced coloring lemma [8, p. 260], see the followings. Lemma 1: (Balanced coloring lemma)
For arbitrary (cid:15), δ > , sufficiently large N , all N -type P X X Y ( x , x , y ) and all (cid:101) x ,i − , (cid:101) x ,i − ∈ T NX X ( ≤ i ≤ n ), there exists a γ - coloring c : T NY | X ,X ( (cid:101) x ,i − , (cid:101) x ,i − ) → { , , .., γ } of T NY | X ,X ( (cid:101) x ,i − , (cid:101) x ,i − ) such that for all joint N -type P X X Y Z ( x , x , y, z ) with marginal distribution P X X Z ( x , x , z ) and | T NY | X ,X ,Z ( (cid:101) x ,i − , (cid:101) x ,i − , (cid:101) z i − ) | γ ≥ N(cid:15) , (cid:101) x ,i − , (cid:101) x ,i − , (cid:101) z i − ∈ T X X Z , | c − ( k ) | ≤ | T NY | X ,X ,Z ( (cid:101) x ,i − , (cid:101) x ,i − , (cid:101) z i − ) | (1 + δ ) γ , (A4)for k = 1 , , ..., γ , where c − is the inverse image of c .From Lemma 1, we see that the typical set T NY | X ,X ,Z ( (cid:101) x ,i − , (cid:101) x ,i − , (cid:101) z i − ) maps into at least | T NY | X ,X ,Z ( (cid:101) x ,i − , (cid:101) x ,i − , (cid:101) z i − )) | | T NY | X ,X ,Z ( (cid:101) x ,i − , (cid:101) x ,i − , (cid:101) z i − ) | (1+ δ ) γ = γ δ (A5)colors. On the other hand, the typical set T NY | X ,X ,Z ( (cid:101) x ,i − , (cid:101) x ,i − , (cid:101) z i − ) maps into at most γ colors. From (A5),we can conclude that H ( K ∗ i | (cid:101) X ,i − , (cid:101) X ,i − , (cid:101) Z i − ) ≥ log γ δ . (A6)Here note that | T NY | X ,X ,Z ( (cid:101) x ,i − , (cid:101) x ,i − , (cid:101) z i − ) | γ ≥ N(cid:15) implies that γ ≤ | T NY | X ,X ,Z ( (cid:101) x ,i − , (cid:101) x ,i − , (cid:101) z i − ) | . Choosing γ = | T NY | X ,X ,Z ( (cid:101) x ,i − , (cid:101) x ,i − , (cid:101) z i − ) | and noticing that | T NY | X ,X ,Z ( (cid:101) x ,i − , (cid:101) x ,i − , (cid:101) z i − ) | ≥ (1 − (cid:15) )2 N (1 − (cid:15) ) H ( Y | X ,X ,Z ) , (A7)where (cid:15) and (cid:15) tend to as N tends to infinity, (A6) can be further bounded by H ( K ∗ i | (cid:101) X ,i − , (cid:101) X ,i − , (cid:101) Z i − ) ≥ log 1 − (cid:15) δ + N (1 − (cid:15) ) H ( Y | X , X , Z ) . (A8) Equivocation analysis (2): Bound on eavesdropper’s equivocation ∆ : For all blocks, the equivocation ∆ isbounded by ∆ = 1 nN H ( W , W | Z n ) ( a ) = 1 nN ( H ( W (cid:48) , , W (cid:48) , | Z n )+ H ( W (cid:48) , , W (cid:48) , | Z n , W (cid:48) , , W (cid:48) , )) , (A9)where (a) is from the definitions W (cid:48) j, = ( W j, , , ..., W j,n, ) and W (cid:48) j, = ( W j, , , ..., W j,n, ) for j = 1 , . Theconditional entropy H ( W (cid:48) , , W (cid:48) , | Z n ) of (A9) is bounded by H ( W (cid:48) , , W (cid:48) , | Z n ) = H ( W (cid:48) , , W (cid:48) , , Z n ) − H ( Z n )= H ( W (cid:48) , , W (cid:48) , , Z n , X n , X n ) − H ( X n , X n | W (cid:48) , , W (cid:48) , , Z n ) − H ( Z n ) ( b ) = H ( Z n | X n , X n ) + H ( X n , X n ) − H ( X n , X n | W (cid:48) , , W (cid:48) , , Z n ) − H ( Z n ) ( c ) = nN ( R + R + R ∗ + R + R + R ∗ ) − nN I ( X , X ; Z ) − H ( X n , X n | W (cid:48) , , W (cid:48) , , Z n ) ( d ) ≥ nN ( R + R + R ∗ + R + R + R ∗ ) − nN I ( X , X ; Z ) − nN (cid:15) , (A10)where (b) is from H ( W (cid:48) , | X n ) = 0 and H ( W (cid:48) , | X n ) = 0 , (c) is from the code constructions of X n , X n and thefact that the channel is memoryless, and (d) is from the fact that given w (cid:48) , , w (cid:48) , and z n , the eavesdropper tries tofind unique w (cid:48) , , w (cid:48) , , w ∗ = ( w ∗ , , ..., w ∗ ,n ) and w ∗ = ( w ∗ , , ..., w ∗ ,n ) such that ( x n , x n , z n ) are jointly typical,and from the properties of AEP, we see that the eavesdropper’s decoding error probability tends to if R , + R , + R ∗ + R ∗ ≤ I ( X , X ; Z ) , (A11) then by using Fano’s inequality, we have nN H ( X n , X n | W (cid:48) , , W (cid:48) , , Z n ) ≤ (cid:15) , where (cid:15) → as n, N → ∞ .Moreover, the conditional entropy H ( W (cid:48) , , W (cid:48) , | Z n , W (cid:48) , , W (cid:48) , ) of (A9) is bounded by H ( W (cid:48) , , W (cid:48) , | Z n , W (cid:48) , , W (cid:48) , ) ≥ n (cid:88) i =2 H ( W ,i, , W ,i, | Z n , W (cid:48) , , W (cid:48) , , W , , , W , , ,..., W ,i − , , W ,i − , , W ,i, ⊕ K ∗ i, , W ,i, ⊕ K ∗ i, ) ( e ) = n (cid:88) i =2 H ( W ,i, , W ,i, | (cid:101) Z i − , W ,i, ⊕ K ∗ i, , W ,i, ⊕ K ∗ i, ) ≥ n (cid:88) i =2 H ( W ,i, , W ,i, | (cid:101) Z i − , (cid:101) X ,i − , (cid:101) X ,i − , W ,i, ⊕ K ∗ i, , W ,i, ⊕ K ∗ i, )= n (cid:88) i =2 H ( K ∗ i, , K ∗ i, | (cid:101) Z i − , (cid:101) X ,i − , (cid:101) X ,i − , W ,i, ⊕ K ∗ i, , W ,i, ⊕ K ∗ i, ) ( f ) = n (cid:88) i =2 H ( K ∗ i | (cid:101) Z i − , (cid:101) X ,i − , (cid:101) X ,i − ) ( g ) ≥ ( n − − (cid:15) δ + N (1 − (cid:15) ) H ( Y | X , X , Z )) , (A12)where (e) is from the Markov chain ( W ,i, , W ,i, ) → ( (cid:101) Z i − , W ,i, ⊕ K ∗ i, , W ,i, ⊕ K ∗ i, ) → ( W (cid:48) , , W (cid:48) , , W , , , W , , , ..., W ,i − , , W ,i − , , (cid:101) Z , ..., (cid:101) Z i − , (cid:101) Z i , ..., (cid:101) Z n ) , (f) is from the definition K ∗ i = ( K ∗ i, , K ∗ i, ) and the Markov chain K ∗ i → ( (cid:101) Z i − , (cid:101) X ,i − , (cid:101) X ,i − ) → ( W ,i, ⊕ K ∗ i, , W ,i, ⊕ K ∗ i, ) , and (g) is from (A8).Substituting (A10) and (A12) into (A9), we have ∆ ≥ R + R + R ∗ + R + R + R ∗ − I ( X , X ; Z ) − (cid:15) + n − nN log 1 − (cid:15) δ + n − n (1 − (cid:15) ) H ( Y | X , X , Z ) . (A13)The bound (A13) implies that if R ∗ + R ∗ ≥ I ( X , X ; Z ) − H ( Y | X , X , Z ) (A14)we can prove that ∆ ≥ R + R + R + R − (cid:15) by choosing sufficiently large n and N .Finally, applying Fourier-Motzkin elimination (see, e.g., [18]) on (A1), (A2), (A3), (A11) and (A14), Theorem2 is obtained. The proof of Theorem 2 is completed.A PPENDIX BP ROOF OF T HEOREM R + R − (cid:15) (1) ≤ H ( W , W | Z N ) N = 1 N ( H ( W , W | Z N ) − H ( W , W | Z N , Y N ) + H ( W , W | Z N , Y N )) (2) ≤ N ( I ( W , W ; Y N | Z N ) + δ ( P e )) ≤ N ( H ( Y N | Z N ) + δ ( P e ))= 1 N N (cid:88) i =1 H ( Y i | Y i − , Z N ) + δ ( P e ) N ≤ N N (cid:88) i =1 H ( Y i | Z i ) + δ ( P e ) N (3) = 1 N N (cid:88) i =1 H ( Y i | Z i , J = i ) + δ ( P e ) N (4) = H ( Y J | Z J , J ) + δ ( P e ) N (5) ≤ H ( Y J | Z J ) + δ ( (cid:15) ) N (6) = H ( Y | Z ) + δ ( (cid:15) ) N , (A15)where (1) is from (2.5), (2) is from Fano’s inequality, (3) and (4) are from the fact that J is a random variable(uniformly distributed over { , , ..., N } ), and it is independent of Y N , Z N , W and W , (5) is from P e ≤ (cid:15) and δ ( P e ) is increasing while P e is increasing, and (6) is from the definitions Y (cid:44) Y J and Z (cid:44) Z J . Letting (cid:15) → , R + R ≤ H ( Y | Z ) is proved. 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