Enhancing Physical Layer Security in Internet of Things via Feedback: A General Framework

In this article, a general framework for enhancing the physical layer security (PLS) in the Internet of Things (IoT) systems via channel feedback is established. To be specific, first, we study the compound wiretap channel (WTC) with feedback, which can be viewed as an ideal model for enhancing the PLS in the downlink transmission of IoT systems via feedback. A novel feedback strategy is proposed and a corresponding lower bound on the secrecy capacity is constructed for this ideal model. Next, we generalize the ideal model (i.e., the compound WTC with feedback) by considering channel states and feedback delay, and this generalized model is called the finite state compound WTC with delayed feedback. The lower bounds on the secrecy capacities of this generalized model with or without delayed channel output feedback are provided, and they are constructed according to variations of the previously proposed feedback scheme for the ideal model. Finally, from a Gaussian fading example, we show that the delayed channel output feedback enhances the achievable secrecy rate of the finite state compound WTC with only delayed state feedback, which implies that feedback helps to enhance the PLS in the downlink transmission of the IoT systems.


I. INTRODUCTION
I NTERNET of Things (IoT) is taking the center stage of the upcoming 5G as the devices are expected to be a major component of 5G network. Due to the broadcasting nature of wireless communication, signals in the IoT systems are more vulnerable to eavesdropping, and hence the secure communication over the IoT systems is one of the most pressing problems needed to be solved. The study of the secure transmission over communication systems started from Wyner [1] in his groundbreaking work on the wiretap channel (WTC), where a transmitter broadcasts its message W over N channel uses to a legitimate receiver and an eavesdropper via a degraded broadcast channel (BC), and the perfect secrecy is guaranteed if the information leakage rate (1/N)I(W; Z N ), where Z N denotes the received signal at the eavesdropper, vanishes as the codeword length N tends to infinity. 1 The secrecy capacity, defined as the channel capacity with perfect secrecy constraint, was established in [1]. Subsequently, Csiszár and Körner [3] generalized the model studied in [1] by considering a general (not degraded) BC and the transmission of a common message which can be decoded by both legitimate receiver and eavesdropper. The follow-up studies of [1]- [3] include the Gaussian WTC [4], the BC with two secret messages [5], [6], and one transmitter broadcasts a secret message to multiple legitimate receivers and one eavesdropper (wiretap BC) [7], [8].
Here note that Wyner [1] further pointed out that the secrecy capacity is positive if the legitimate receiver's channel is less noisy than the eavesdropper's. Then it is natural to ask the following two questions.
1) How to achieve positive secrecy capacity when the eavesdropper's channel is less noisy than the legitimate receiver's?
2) When the legitimate receiver's channel is less noisy than the eavesdropper's, how to further enhance the secrecy capacity? The answer to both questions is artificial noise (AN) [9]- [16] and channel feedback (CF). However, we should notice that since the IoT devices (e.g., sensors and actuators) have significant energy constraint [17], [18], AN may not be suitable for IoT systems, and hence CF is of particular interest for enhancing the physical layer security (PLS) in IoT systems.
The study of the effect of CF on the PLS of communication channels started from [19], where the pioneering work [1] has been revisited by considering the situation that the legitimate receiver's received channel output is sent back to the transmitter through an additional noiseless feedback channel which is not known by the eavesdropper. Since the legitimate receiver's channel output is perfectly known by the transmitter and completely not known by the eavesdropper, we can generate the secret key from it, and this key helps to protect the transmitted message. Combining the above idea of generating the secret key from the CF with the random binning scheme for the WTC [1], Ahlswede and Cai [19] proposed a coding scheme which splits the transmitted message into two parts, where one submessage is encoded in the same way as that of the WTC [1], and the other is encrypted by the secret key. Compared with the secrecy capacity of the WTC [1], it is easy to see that encrypting part of the message by the key leads to the channel output feedback enhancing the secrecy capacity of the WTC.
Note that in [19], the feedback channel is only used to send the legitimate receiver's channel output. What happens when the feedback channel can transmit anything as the legal receiver wishes? Ardestanizadeh et al. [20] studied this case and pointed out that for the legal receiver, the best way is to transmit randomly generated sequences (served as secret keys) over the feedback channel. Assumeing that the transmission rate of the feedback channel is up to R f , using a coding scheme in the same way as that in [19], Ardestanizadeh et al. [20] showed that sending the pure secret key is better than sending legal receiver's channel output if R f is larger than the key rate in [19], and vice versa. The works of [19] and [20] indicate that there is no difference between sending the pure secret key and sending legitimate receiver's channel output, and the main purpose of the feedback is to allow the legitimate receiver and the transmitter to share the secret key. In recent years, the above secret-key-based feedback coding scheme has been widely used in communication systems with feedback channel. To be specific, for the communication channels with legal receiver's channel output feedback, Dai et al. [23], [24] studied PLS of the channels with memory or memoryless states and legitimate receiver's channel output feedback, and proposed variations of the secret-key-based feedback coding scheme in [19]. For the communication systems with feedback channels directly transmitting pure secret keys, Schaefer et al. [21] extended the work of [20] to a broadcast situation, where two legitimate receivers of the BC independently send their secret keys to the transmitter via two noiseless feedback channels, and these keys help to increase the achievable secrecy rate region of the broadcast WTC [7]. Cohen and Cohen [22]  introduced memoryless channel state into the work of [20], and showed that the transmitted message can be protected by two keys, where one is from the feedback channel, and the other is generated by the channel state. Very recently, Dai and Luo [25] showed that for the general WTC with CF, a better choice of the transmitter is to produce not only the secret key but also the auxiliary message from CF, where the auxiliary message is used to improve the legal receiver's decoding performance. Dai and Luo [25] proved that for the WTC with channel output feedback, this new feedback scheme performs better than the widely used secret-key-based feedback scheme. Moreover, Li et al. [26] and Gundüz et al. [27] showed that the classical Schalkwijk-Kailath (SK) [28] feedback scheme for the Gaussian channel achieves the secrecy capacity of the Gaussian WTC with channel output feedback, and it equals the capacity of the same channel model without the secrecy constraint. However, we should notice that the results of [26] and [27] only work in the Gaussian case.
In IoT systems, the uplink transmission is from sensors to controllers, and the downlink transmission is from controllers to actuators. A classical scenario for the PLS in the downlink transmission of the IoT systems is depicted in Fig. 1, where a controller tries to send a secret message to several actuators in the presence of an eavesdropper. An ideal model characterizing this classical scenario is called the compound WTC, where the channels for all actuators and the eavesdropper are independent of one another. Achievable secrecy rates (lower bounds on the secrecy capacities) of various compound WTCs were provided in [29]- [31], and it is shown that if the eavesdropper's channel is less noisy than all actuators' (legitimate receivers') channels, the achievable secrecy rate equals zero. As we mentioned before, AN is not suitable for the scenario in Fig. 1. Moreover, we should notice that the already existing feedback schemes in [19]- [25] cannot be applied to the communication scenario in Fig. 1 due to the reason that each actuator does not know others' CF, and hence the feedback of all channels cannot be used to generate a common secret key shared between the transmitter and all the actuators.
In this article, we try to answer the following two fundamental questions.
1) How to increase the achievable secrecy rate of the compound WTC model by using CF? 2) Is there a more practical model for the scenario shown in Fig. 1? If so, can we apply the feedback scheme of 1) to this more practical model? This article provides the comprehensive answers to the aforementioned questions. Our main contributions are summarized as follows.
1) We study the compound WTC with feedback (see Fig. 2). An achievable secrecy rate, which is constructed according to a novel feedback strategy, is provided for the model of Fig. 2. 2) A more practical model for the scenario in Fig. 1 is provided (see Fig. 3), and an achievable secrecy rate for this new model is obtained according to a modified feedback scheme of the model of Fig. 2. From a Gaussian fading example, we show that the achievable secrecy rate of the new model is larger than that of the same model without channel output feedback, which implies that the proposed feedback scheme enhances the PLS in the downlink transmission of the IoT systems. In the remainder of this article, random variables (RVs), values, and alphabet are denoted by uppercase letters, lowercase letters, and calligraphic letters, respectively. The random vector and its value are written in a similar way. For example, suppose that X 1 is an RV, and x 1 is a real value in the alphabet X 1 . Similarly, suppose that X N 1,1 is a random vector (X 1,1 , . . . , X 1,N ), and x N 1,1 = (x 1,1 , . . . , x 1,N ) is a vector value in X N 1 (the Nth Cartesian power of X 1 ). Moreover, for simplicity, the probability Pr{X = x} is denoted by P(x), and in the remainder of this article, the base of the log function is 2.
The outline of this article is organized as follows. Section II investigates the compound WTC with feedback (see Fig. 2), and provides bounds on the secrecy capacity of this ideal model. Section III studies a generalized version of the model of Fig. 2, called the finite state compound WTC with delayed feedback (see Fig. 3), and also provides bounds on the secrecy capacity of this generalized model. In Section IV, the capacity results on the model of Fig. 3 are further illustrated by a Gaussian fading example. The final conclusion is given in Section V.

II. COMPOUND WIRETAP CHANNEL WITH FEEDBACK
The discrete memoryless compound WTC with feedback is shown in Fig. 2, where a transmitter wishes to broadcast his/her secret message to L legitimate receivers and an eavesdropper attempts to eavesdrop this secret message via a WTC. The overall channel transition probability of the model of Fig. 2 is given by where x i ∈ X , y j,i ∈ Y j , and z i ∈ Z. Here note that z N (x N ) is an abbreviation of z N 1 (x N 1 ), and a similar convention is applied to Z N 1 (X N 1 ). The transmitted message W is uniformly distributed over W = {1, 2, . . . , |W|}. Since all legitimate receivers send their received channel outputs back to the transmitter via feedback channels, the ith (i ∈ {1, 2, . . . , N}) channel input X i is given by where f i is a stochastic encoding function, and Y i−1 j,1 (j ∈ {1, 2, . . . , L}) is the jth legitimate receiver's channel output feedback at time i.
For the jth (j ∈ {1, 2, . . . , L}) legitimate receiver, after receiving Y N j,1 , he/she produces an estimationŴ (j) = ψ j (Y N j,1 ) (ψ j is the jth legitimate receiver's decoding function), and the average decoding error probability equals The secrecy level of the transmitted message W at the eavesdropper is formulated as Given a non negative number R, if for any > 0, there exist encoder and decoders such that R is achievable under weak perfect secrecy constraint. The secrecy capacity C f s is composed of all such achievable R defined in (5), and bounds on C f s are given in the remainder of this section.
[a] + = a for a ≥ 0, [a] + = 0 for a < 0, and the joint probability is defined as P u j |v j , y j P y j |x .

(7)
Proof Sketch: The lower bound R f s is constructed according to a block Markov coding scheme, and the encodingdecoding procedure of each block is briefly explained in Fig. 4. From Fig. 4, we see that in each block, after receiving the CF of receiver j (j ∈ {1, 2, . . . , L}), the transmitter encodes the transmitted message of current block and the feedback of receiver j as a codeword v N j,1 , and the channel input of current block is generated via an auxiliary discrete memoryless channel with inputs v N 1,1 , . . . , v N L,1 , and output x N . For receiver j, after receiving the channel outputs of all blocks, he/she uses the backward and jointly typical decoding scheme to decode the transmitted v N j,1 for all blocks. If v N j,1 is decoded without error, the messages of all blocks can be obtained by receiver j. The details about the encoding-decoding scheme of Theorem 1 are in Appendix A.
where the joint distribution is denoted by P(z, y 1 , . . . , y L , x, u) = P(z|x)P(y 1 , . . . , y L |x)P(x|u)P(u) In general, we do not know whether R f s is larger than R s or not. In Section IV, from a Gaussian fading example, we show that R f s is larger than R s , which indicates that CF may increase the achievable secrecy rate of the compound WTC.
2) For the compound WTC with noiseless feedback, it is natural to ask: When not directly using the noiseless feedback channels for secure communication, is the total secrecy rate of the original compound WTC and the noiseless feedback channels larger than R f s given in Theorem 1? To answer this question, a binary symmetric case of the model of Fig. 2 is investigated (see Fig. 5), and we show that for this special case, directly using noiseless feedback channels for secure communication may not always be the best choice. In Fig. 5, one transmitter wishes to send a message W to two legitimate receivers via two BSCs with crossover probabilities p 1 and p 2 , respectively, and an eavesdropper tries to eavesdrop W via another BSC with crossover probability q. In addition, the legitimate receivers send their received signals back to the transmitter via two noiseless feedback channels. From Theorem 1, we see that an achievable secrecy rate R f s for the model of Fig. 5 is given by Then defining P( where Next, if the noiseless feedback In Fig. 6, two messages W 1 and W 2 are transmitted, where W 1 is transmitted through the binary symmetric compound WTC in the model of Fig. 5 without feedback, and W 2 is transmitted through the noiseless feedback channels and due to the broadcast nature of wireless communication, W 2 can also be eavesdropped by the eavesdropper via a binary symmetric WTC with crossover probability q. From [29, Th. 1], an achievable secrecy rate R * 1 of W 1 is given by where (a) is from defining P(X = 0) = α, P(X = 1) = 1 − α, and using the fact that the maximum is achieved when α = (1/2). Analogously, an achievable secrecy rate R * 2 of W 2 is given by where (b) follows from substituting p 1 = p 2 = 0 into (12). Hence the total secrecy rate R * = R * 1 + R * 2 is given by   of Fig. 6 for p 1 = 0.0001, p 2 = 0.5, and several values of q. From this figure, we see that the feedback scheme proposed in Theorem 1 performs no better than directly using the feedback channels for secure transmission when one legitimate receiver's channel is completely noisy (i.e., p 2 = 0.5). Fig. 8 plots R f s and R * for p 1 = 0.0001, p 2 = 0.3, and several values of q. From this figure, we see that the feedback scheme proposed in Theorem 1 performs better than directly using the feedback channel for secure transmission when q is very small. From the above figures, we see that directly using the feedback channels for secure transmission may not always be the best choice, and sometimes the feedback scheme of Theorem 1 may perform better. (15) and the joint probability is defined as Proof: This outer bound can be directly obtained by using the fact that the secrecy capacity cannot exceed the capacity of each channel and feedback does not increase the capacity of a discrete memoryless channel. Hence the secrecy capacity C f s is upper bounded by the minimum of each channel's capacity [here note that the capacity of channel j is max P(x) I(X; Y j )], and the proof is completed.
Here note that the compound WTC with feedback investigated in this section is only an ideal model for the PLS in the downlink transmission of the IoT system. In the next section, we will study a more practical model, which we call the finite state compound WTC with delayed feedback. The lower bound on the secrecy capacity of this more practical model is constructed according to a variation of the feedback strategy in Theorem 1 (see the remainder of this article).

III. FINITE STATE COMPOUND WIRETAP CHANNEL WITH DELAYED FEEDBACK
The practical IoT systems often consist of time-varying and fading channels, and the states of these channels are often obtained by the transmitter via receivers's delayed feedback. In [33], the time-varying fading channel in the presence of one transmitter, one legitimate receiver, one eavesdropper and delayed CF is modeled as the finite state Markov WTC (FSM-WTC) with delayed feedback. In this section, we extend the FSM-WTC with delayed feedback to a more general case, i.e., the finite state compound WTC with delayed feedback (see Fig. 3). In Fig. 3, the channel consists of multiple legitimate receivers and one eavesdropper, and each legitimate receiver sends his/her received signal back to the transmitter via a corresponding feedback channel with different delayed feedback time. In the remainder of this section, we first give formal definition of the model of Fig. 3, and then we show bounds on the secrecy capacity of this new model.
Model Formulation: 1) The overall channel transition probability of the model of Fig. 3 is given by . . , L}) are supposed to be stationary irreducible aperiodic ergodic Markov chains. The state processes are independent of one another, and they are independent of the transmitted message. Moreover, the state process {S j,i } is independent of the channel input and outputs given the previous states, i.e., Define the one-step transition probability matrix of the state process {S j,i } as K j . Denote the steady state probabilities of {S j,i } and {S e,i } by π j and π e , respectively.
Note that For the case that all legitimate receivers only send their received channel states back to the transmitter via feedback channels with delay times For the case that all legitimate receivers send their received channel outputs and channel states back to the transmitter via feedback channels with delay times d 1 , . . . , d L , the ith (i ∈ {1, 2, . . . , N}) channel input X i is given by Here note that f i in (21) and (22) is a stochastic encoding function. 4) For receiver j (j ∈ {1, 2, . . . , L}), after receiving Y N j,1 and S N j,1 , he/she produces an estimationŴ (j) = ψ j (Y N j,1 , S N j,1 ), and his/her average decoding error probability is defined as The secrecy level of the transmitted message W at the eavesdropper is formulated as The definition of a non-negative number R achieving weak perfect secrecy is the same as that in (5). The secrecy capacity of the model of Fig. 3 with delayed channel output feedback is denoted by C f −dy s , and without delayed channel output feedback is denoted by C are given in the following theorems.

Theorem 3 (Lower Bound on
the auxiliary RVS j represents S j,i−d L , S j represents S j,i , and the joint probability is defined as Proof Sketch: The lower bound R f −dy s is constructed by combining the coding scheme of Theorem 1 with the multiplexing encoding-decoding scheme for the FSM-WTC with delayed feedback [33], and the encoding-decoding procedure is briefly explained in Fig. 9. From Fig. 9, we see where k j is the size of the alphabet S j , i.e., k j = |S j |. Moreover, in block i, after receiving the delayed feedback channel output and state of receiver j, the transmitter encodes each submessage W i,l (l ∈ {1, 2, . . . , k j }) and the delayed feedback as a subcodeword v N l j,i,1 (similar to the coding scheme in the proof of Theorem 1, see Appendix A), where N l is the subcodeword length for W i,l , and . Similar to the coding scheme of Theorem 1, the channel input of block i is generated via an auxiliary discrete memoryless channel In the decoding procedure, for receiver j, after receiving the channel outputs and states of all blocks, he/she uses the backward, de-multiplexing and jointly typical decoding scheme to decode the transmitted v n j, 1 is decoded without error, the messages for all blocks can be obtained by receiver j. The details about the encoding-decoding scheme of Theorem 3 are in Appendix B.
Theorem 4 (Lower Bound on C the auxiliary RVS j represents S j,i−d L , S j represents S j,i , and the joint probability is defined as Proof: First, recall that in the proof of Theorems 1 and 3, the auxiliary RV U 1 , . . . , U L are generated by the channel output feedback and they are used to improve the legitimate receivers' decoding performance. Then, note that in Theorem 4, there is no channel output feedback, which indicates that U 1 , . . . , U L are useless. Finally, substituting  cannot exceed the capacity of each channel without secrecy constraint. To be specific, first, note that in the model of Fig. 3, the channel j (j ∈ {1, 2, . . . , L}) with delayed feedback and without eavesdropper has already been investigated by Viswanathan [34]. It has been shown in [34] that the capacity C f −d of each channel with only delayed state feedback equals the capacity C f −dy of the same channel with delayed both state and channel output feedback, and they are given by Then, using the fact that C

IV. GAUSSIAN FADING EXAMPLE OF THE FINITE STATE COMPOUND WIRETAP CHANNEL WITH DELAYED FEEDBACK
In this section, we compute the capacity bounds in Section III via a Gaussian fading example, and we would like to know how the delayed feedback time affects the capacity bounds. The remainder of this section is organized as follows. In Section IV-A, we show bounds on the secrecy capacities of the Gaussian fading case of the model of Fig. 3 with or without delayed channel output feedback. In Section IV-B, the bounds in Section IV-A are further explained via numerical results. Fig. 3 For the Gaussian fading case of the model of Fig. 3, at time i (1 ≤ i ≤ N), the channel inputs and outputs are given by

A. Gaussian Fading Case of the Model of
where [x] + = x for x ≥ 0, [x] + = 0 for x < 0, and P(s j ) (j ∈ {1, 2, . . . , L}) is the transmitter's power allocated to the states j . Proof: First, for j ∈ {1, 2, . . . , L}, define Here note that V 1 , . . . , V L are independent of one another. From the definition (35), it is easy to check that the power constraint (33) holds. Next, define where P(s j ) is the transmitter's power allocated to the states j , and it satisfies s j π j s j P s j = s j π j s j E X 2 |s j = E X 2 ≤ P. (37) Further define From (37) and (38), it is easy to check that Finally, note that for j ∈ {1, 2, . . . , L}, U j is generated from the feedback Y j and the transmitted codeword V j , define  is given by Proof: Substituting the definitions (32) and (36) into Theorem 5, C f −out * s is obtained. The proof of Corollary 3 is completed.

B. Numerical Results
In this section, we investigate a two-state example, i.e., for j ∈ {1, 2, . . . , L}, S j consists of two elements G j (good state) and B j (bad state), and S e also consists of two elements G e and B e . The state process of {S j } is given by and the steady probabilities of G j and B j are given by In addition, the state process of {S e } is given by For the noise N s j of the channel from the transmitter to receiver j, its variance σ 2 s j in state G j is σ 2 G j , and in state B j is σ 2 B j . Similarly, the noise N s e of the channel from the transmitter to the eavesdropper, its variance σ 2 s e in state G e is σ 2 G e , and in state B e is σ 2 B e . Fig. 10. Comparison of the bounds in Theorems 1-3 for σ 2  For L = 3, which indicates that there are three legitimate receivers in the model of Fig. 3, Fig. 10 plots the lower and upper bounds on the secrecy capacities of the Gaussian fading case of the model of Fig. 3 with delayed state feedback, and with or without delayed channel output feedback for σ 2 , which indicates that the secrecy capacity of the Gaussian fading case of the model of Fig. 3 with delayed both states and channel output feedback is determined. Moreover, we see that channel output feedback helps to enhance the achievable secrecy rate of the Gaussian fading case of the model of Fig. 3 with only delayed state feedback. Fig. 11 plots the bounds for the same values of the parameters given in Fig. 10 except that σ 2 G e = 1 and σ 2 B e = 2.5. As depicted in this figure, if the eavesdropper's channel noise variance (σ 2 G e = 1, σ 2 B e = 2.5) is decreasing, the gap between the lower and upper bounds on C f −dy * s is increasing. In addition, we see that channel output feedback still helps to enhance the achievable secrecy rate R f −d * s of the Gaussian fading case of the model of Fig. 3 with only delayed state feedback.    Fig. 3 with only delayed state feedback equals 0, which implies that the perfect secrecy cannot be guaranteed for this case. Using the channel output feedback, the positive achievable secrecy rate R f −dy * s is derived, and hence the PLS of the Gaussian fading case of the model of Fig. 3 with only delayed state feedback is enhanced. In addition, we should notice that there still exists a huge gap between the lower and upper bounds on C f −dy * s . To investigate how the delayed feedback time d j (j ∈ {1, 2, . . . , L}) affects the secrecy rates of the model of Fig. 3, Fig. 13 plots the lower bounds in Theorems 1 and 2 for the case that L = 3 (three legitimate receivers) and d 1 = d 2 = d 3 = d, which implies that the delayed feedback times of the three legitimate receivers are the same and equal d. As depicted in this figure, the achievable secrecy rates of the Gaussian fading case of the model of Fig. 3

V. CONCLUSION
This article established a general framework for enhancing the PLS in the downlink transmission of IoT systems via feedback. Two models, including the compound WTC with feedback and the finite state compound WTC with delayed feedback, were studied, and bounds on the secrecy capacities of the two models were given. From a Gaussian fading example, we see that the delayed channel output feedback enhances the lower bound on the secrecy capacity of the finite state compound WTC with only delayed state feedback, and the corresponding feedback strategy may achieve the secrecy capacity if the eavesdropper's channel noise variance is sufficiently large. Moreover, numerical results indicate that the secrecy rates are decreasing while the feedback delay time is increasing, and the secrecy rates are approaching their infinite asymptotes while the feedback delay time is sufficiently large. However, we should notice that all the capacity results given in this article only work well under the perfect weak secrecy condition, and how to design the corresponding encodingdecoding schemes under the strong perfect secrecy condition is of further interest to us.

APPENDIX A PROOF OF THEOREM 1
The messages are conveyed to the receivers via n blocks. The blocklength of block i ∈ {1, 2, . . . , n − 1} is N, and for block n (the last block), its blocklength is γ N, where γ is a positive real number and will be determined later. In block i (i ∈ {1, 2, . . . , n − 1}), the random sequences X N , Z N ,   1 , are denoted byX n ,Z n ,Ȳ 1,n ,Ȳ 2,n , . . . ,Ȳ L,n , V 1,n , andV 2,n , . . . ,V L,n , respectively. In addition, the value of the random vector is written in lower case letter.
Code-Book Construction: 1) The message W is sent to all legitimate receivers via n blocks, i.e., the message W is composed of n components (W = (W 1 , . . . , W n )), and each component the channel inputx i is i.i.d. generated according to P(x|v 1 , . . . , v L ). For convenience, Fig. 15 provides some important notations in the proof of Theorem 1.
After decodingv j,n , receiver j picks out w * j,n−1 from it. Then he/she tries to choose a uniqueū j,n−1 such that given w * j,n−1 , u j,n−1 andȳ j,n−1 are jointly typical. If multiple or no such u j,n−1 exists, an error occurs. From the packing lemma [32], this error vanishes ifR Once such uniqueū j,n−1 is obtained, receiver j seeks a uniquē v j,n−1 such that (v j,n−1 ,ȳ j,n−1 ,ū j,n−1 ) are jointly typical. From the packing lemma [32], this error vanishes if then applying Fano's lemma, H(V j,1 |W 1 ,Z 1 ) ≤ N 4 is obtained, where 4 → 0 while N → ∞, and analogously, for i ∈ {2, . . . , n − 1} given w i ,z i , the eavesdropper attempts to find a uniquev j,i jointly typical with his/her receivedz i , and from the packing lemma [32], this decoding error vanishes if Here note that (53) is included in (54), and thus we only need to use (54) to derive the final region.
The bound (52) implies that if ≥ R − is satisfied by choosing sufficiently large n and N. Now it remains to use the above conditions (47), (50), (51), (54), and (55) to derive the lower bound in Theorem 1, as follows.
First, note that from (47) and (50), we have Next, substituting (56) into (51), we get Then, note that from (54) and (55), we can conclude that Now substituting (58) into (51), we have From the above (57) and (59), we have Next, note that if I(V j ; U j , Y j ) ≤ I(V j ; Z), from (51), we have Combining (61) with (58), and observing that R ≥ 0, we can . Hence (60) should be rewritten as Note that (62) should be satisfied for all j ∈ {1, 2, . . . , L}, hence we have Finally, note that the effective transmission rate is which indicates that the effective transmission rate approaches R as the number of blocks n → ∞, then maximizing the bound in (63), Theorem 1 is proved, and the proof is completed.

APPENDIX B PROOF OF THEOREM 3
The encoding-decoding scheme of Theorem 3 combines that of Theorem 1 with the multiplexing encoding-decoding scheme for the finite state Markov channel with delayed feedback [34]. The detail about the coding scheme is given below.
Definition: Similar to the definitions in the proof of Theorem 1, the messages are transmitted via n blocks. Since The dummy message W also consists of n components (W = (W 1 , . . . , W n ) For convenience, Fig. 18 and w * * j,i,s j ∈ {1, 2, . . . , 2 Once such uniqueū After decodingv where (c) follows from the fact that H(W i |V j,i ) = 0, (d) follows from the fact thatS e,i is independent ofV j,i , (e) follows from the construction ofV j,i and a similar argument in [3, eqs. (16) and (23)], i.e., H(V j,i ) ≥ N(R + R + R * j − 1 ), where 1 → 0 as N → ∞, (f ) follows from a similar argument in [5, Lemma 3], i.e., I(V j,i ;Z i |S e,i ) ≤ N(I(V j ; Z|S e ) + 2 ), where 2 → 0 as N → ∞, and (g) follows from that given w i , z i , the eavesdropper attempts to find a uniquev j,i jointly typical with his/her receivedz i , and from the packing lemma [32], this error vanishes if then applying Fano's lemma, H(V j,i |W i ,Z i ,S e,i ) ≤ N 3 is obtained, where 3 → 0 while N → ∞. Substituting (76) into (75), we have The bound (78) implies that if ≥ R − is satisfied by choosing sufficiently large n and N. Now it remains to use the above conditions (71), (73), (74), (77), and (79) to derive the lower bound in Theorem 3, as follows.
Next, substituting (83) into (82), we get Then, note that from (77) and (79), we can conclude that Now substituting (85) into (82), we have From the above (84) and (86), we have Next, note that if I(V j ; U j , Y j |S j ,S j ) ≤ I(V j ; Z|S e ), from (82), we have Combining (88) with (85), and observing that R ≥ 0, we can conclude that R = 0 if I(V j ; U j , Y j |S j ,S j ) ≤ I(V j ; Z|S e ). Hence (87) should be rewritten as R ≤ min [I V j ; U j , Y j |S j ,S j − I V j ; Z|S e ] + , I V j ; Y j |S j ,S j .
Note that (89) should be satisfied for all j ∈ {1, 2, . . . , L}, hence we have Finally, note that the effective transmission rate is which indicates that the effective transmission rate approaches R as the number of blocks n → ∞, then maximizing the bound in (90), Theorem 3 is proved, and the proof is completed.