Journal of Applied Mathematics

Volume 2013 (2013), Article ID 423619, 17 pages

http://dx.doi.org/10.1155/2013/423619

Research Article

Wiretap Channel in the Presence of Action-Dependent States and Noiseless Feedback

Bin Dai,¹ A. J. Han Vinck,² and Yuan Luo³

¹School of Information Science and Technology, Southwest JiaoTong University, Chengdu 610031, China

²Institute for Experimental Mathematics, Duisburg-Essen University, Ellernstraße 29, 45326 Essen, Germany

³Computer Science and Engineering Department, Shanghai Jiao Tong University, Shanghai 200240, China

Received 29 November 2012; Accepted 3 January 2013

Academic Editor: Jin Liang

Copyright © 2013 Bin Dai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate the wiretap channel in the presence of action-dependent states and noiseless feedback. Given the message to be communicated, the transmitter chooses an action sequence that affects the formation of the channel states and then generates the channel input sequence based on the state sequence, the message, and the noiseless feedback, where the noiseless feedback is from the output of the main channel to the channel encoder. The main channel and the wiretap channel are two discrete memoryless channels (DMCs), and they are connected with the legitimate receiver and the wiretapper, respectively. The transition probability distribution of the main channel depends on the channel state. Measuring wiretapper’s uncertainty about the message by equivocation, the capacity equivocation regions are provided both for the case where the channel inputs are allowed to depend noncausally on the state sequence and the case where they are restricted to causal dependence. Furthermore, the secrecy capacities for both cases are formulated, which provide the best transmission rate with perfect secrecy. The result is further explained via a binary example.

1. Introduction

Equivocation was first introduced into channel coding by Wyner in his study of wiretap channel [1], see Figure 1. It is a kind of degraded broadcast channels. The object is to transmit messages to the legitimate receiver, while keeping the wiretapper as ignorant of the messages as possible.

Figure 1: Wiretap channel.

After the publication of Wyner’s work, Csiszár and Körner [2] investigated a more general situation: the broadcast channels with confidential messages, see Figure 2. The model of [2] is to transmit confidential messages to receiver 1 at rate and common messages to both receivers at rate , while keeping receiver 2 as ignorant of the confidential messages as possible. Measuring ignorance by equivocation, a single-letter characterization of all the achievable triples was provided in [2], where is the second receiver’s equivocation to the confidential messages. Note that the model of [2] is also a generalization of [3], where no confidentiality condition is imposed. In addition, Merhav [4] studied a specified wiretap channel and obtained the capacity region, where both the legitimate receiver and the wiretapper have access to some leaked symbols from the source, but the channels for the wiretapper are more noisy than the legitimate receiver, which shares a secret key with the encoder.

Figure 2: Broadcast channels with confidential messages.

In communication systems there is often a feedback link from the receiver to the transmitter, for example, the two-way channels for telephone connections. It is well known that feedback does not increase the capacity of discrete memoryless channel (DMC). However, does the feedback increase the capacity region of the wiretap channel? In order to solve this problem, Ahlswede and Cai studied the general wiretap channel (the wiretap channel does not need to be degraded) with noiseless feedback from the legitimate receiver to the channel encoder [5] (see Figure 3), and both upper and lower bounds of the secrecy capacity were provided. Specifically, for the degraded case, they showed that the secrecy capacity is larger than that of Wyner’s wiretap channel (without feedback). In the achievability proof, Ahlswede and Cai [5] used the noiseless feedback as a secret key shared by the transmitter and the legitimate receiver, while the wiretapper had no additional knowledge about the key except his own received symbols. Based on the work of [5], Dai et al. [6] studied a special case of the general wiretap channel with noiseless feedback and found that the noiseless feedback enhances the secrecy capacity of the nondegraded wiretap channel. Besides Ahlswede and Cai’s work, the wiretap channel with noisy feedback was studied in [7], and the wiretap channel with secure rate-limited feedback was studied in [8], and both of them focused on bounds of the secrecy capacity. Since the feedback in the model of wiretap channel is often used as a shared secret key, the techniques used in the secret sharing scheme play an important role in the construction of the practical secure communication systems, see [9–11].

Figure 3: The general wiretap channel with noiseless feedback.

Communication through state-dependent channels, with states known at the transmitter, was first investigated by Shannon [12] in 1958. In [12], the capacity of the discrete memoryless channel with causal (past and current) channel state information at the encoder was totally determined. After that, in order to solve the problem of coding for a computer memory with defective cells, Kuznecov and Cybakov [13] considered a channel in the presence of noncausal channel state information at the transmitter. They provided some coding techniques without the determination of the capacity. The capacity was found in 1980 by Gel’efand and Pinsker [14]. Furthermore, Costa [15] investigated a power-constrained additive noise channel, where part of the noise is known at the transmitter as side information. This channel is also called dirty paper channel. The assumption in these seminar papers, as well as in the work on communication with state-dependent channels that followed, is that the channel states are generated by nature and cannot be affected or controlled by the communication system.

In 2010, Weissman [16] revisited the above problem setting for the case where the transmitter can take actions that affect the formation of the states, see Figure 4. Specifically, Weissman considered a communication system where encoding is in two parts: given the message, an action sequence is created. The actions affect the formation of the channel states, which are accessible to the transmitter when producing the channel input sequence. The capacity of this model is totally determined both for the case where the channel inputs are allowed to depend noncausally on the state sequence and the case where they are restricted to causal dependence. Meanwhile, Weissman [16] found that the feedback from the channel output to the channel encoder cannot increase the channel capacity. This framework captures various new channel coding scenarios that may arise naturally in recording for magnetic storage devices or coding for computer memories with defects.

Figure 4: Channel with action-dependent states.

Inspired by the above works, Mitrpant et al. [17] studied the transmission of confidential messages through the channels with channel state information (CSI). In [17], an inner bound on the capacity-equivocation region was provided for the Gaussian wiretap channel with CSI. Furthermore, Chen and Vinck [18] investigated the discrete memoryless wiretap channel with noncausal CSI (see Figure 5) and also provided an inner bound on the capacity-equivocation region. Note that the coding scheme of [18] is a combination of those in [1, 14]. Based on the work of [18], Dai and Luo [19] provided an outer bound on the wiretap channel with noncausal CSI and determined the capacity-equivocation region for the memoryless case.

Figure 5: Wiretap channel with noncausal channel state information.

In this paper, we study the wiretap channel in the presence of action-dependent states and noiseless feedback, see Figure 6. This work is inspired by the wiretap channel with CSI [18], the channel with action-dependent CSI [16], and the wiretap channel with noiseless feedback [5]. The motivation of this work is to investigate the transmission of confidential messages through the channel with action-dependent CSI and noiseless feedback.

Figure 6: Wiretap channel with action-dependent channel state information and noiseless feedback.

More concretely, in Figure 6, the transmitted message is encoded as an action sequence , and is the input of a discrete memoryless channel (DMC). The output of this DMC is the channel state sequence . The main channel is a DMC with inputs and and output . The wiretap channel is also a DMC with input and output . Moreover, there exists a noiseless feedback from the output of the main channel to the channel encoder; that is, the inputs of the channel encoder are the transmitted message , the state sequence , and the noiseless feedback, while the output is . Since the action-dependent state captures various new coding scenarios for channels with a rewrite option that may arise naturally in storage for computer memories with defects or in magnetic recording, it is natural to ask the following two questions.

(i)How about the security of these channel models in the presence of a wiretapper?(ii)What can the noiseless feedback do in the model of wiretap channel with action-dependent CSI?

Measuring wiretapper’s uncertainty about the transmitted message by equivocation, the capacity-equivocation regions of the model of Figure 6 are provided both for the case where the channel input is allowed to depend noncausally on the state sequence and the case where it is restricted to causal dependence.

The contribution of this paper is as follows.

(i)Compared with the existing model of wiretap channel with side information [18] (see Figure 5), the channel state information in [18] is a special case of that in our new model; that is, the model of [18] is included in the model of Figure 6. Therefore, our result generalizes the result of [18].(ii)Our new model also extends the model of Figure 4 by considering an additional secrecy constraint, and therefore our result also generalizes the result of [16].

In this paper, random variab1es, sample values, and alphabets are denoted by capital letters, lowercase, letters and calligraphic letters, respectively. A similar convention is applied to the random vectors and their sample values. For example, denotes a random -vector , and is a specific vector value in , that is the, th Cartesian power of . denotes a random -vector , and is a specific vector value in . Let denote the probability mass function . Throughout the paper, the logarithmic function is taken to the base 2.

The remainder of this paper is organized as follows. In Section 2, we present the basic definitions and the main result on the capacity-equivocation region of wiretap channel with action-dependent channel state information and noiseless feedback. In Section 3, we provide a binary example of the model of Figure 6. Final conclusions are presented in Section 4.

2. Notations, Definitions, and the Main Results

In this section, the model of Figure 6 is considered into two parts. The model of Figure 6 with noncausal channel state information is described in Section 2.1, and the causal case is described in Section 2.2, see the following.

2.1. The Model of Figure 6 with Noncausal Channel State Information

In this section, a description of the wiretap channel with noncausal action-dependent channel state information is given by Definition 1 to Definition 5. The capacity-equivocation region composed of all achievable pairs is given in Theorem 6, where the achievable pair is defined in Definition 5.

Definition 1 (action encoder). The message takes values in , and it is uniformly distributed over its range. The action encoder is a deterministic mapping:

The input of the action encoder is , while the output is .

The channel state sequence is generated by a DMC with input and output . The transition probability distribution is given by

Note that the components of the state sequence may not be i.i.d. random variables, and this is due to the fact that is not i.i.d. generated. The transmission rate of the message is .

Definition 2 (channel encoder and the main channel). The main channel is a DMC with finite input alphabet , finite output alphabet , and transition probability , where . . The inputs of the main channel are and , while the output is .

There is a noiseless feedback from the output of the main channel to the channel encoder. At the th time, the feedback (where and takes values in ) is the previous time output of the main channel. Since the channel encoder knows the state sequence in a noncausal manner, at the th time, the inputs of the channel encoder are , , and , while the output is ; that is, the th time channel encoder is a conditional probability that the message , the feedback , and the channel state sequence are encoded as the th time channel input .

Definition 3 (wiretap channel). The wiretap channel is also a DMC with finite input alphabet , finite output alphabet , and transition probability , where . The input and output of the wiretap channel are and , respectively. The equivocation to the wiretapper is defined as

The cascade of the main channel and the wiretap channel is another DMC with transition probability:

Note that is a Markov chain in the model of Figure 6.

Definition 4 (decoder). The decoder for the legitimate receiver is a mapping , with input and output . Let be the error probability of the receiver, and it is defined as .

Definition 5 (achievable pair in the model of Figure 6). A pair (where ) is called achievable if, for any (where is an arbitrary small positive real number and ), there exist channel encoders decoders such that

The capacity-equivocation region is a set composed of all achievable pairs, and it is characterized by the following Theorem 6. The proof of Theorem 6 is in Appendices A and B.

Theorem 6. A single-letter characterization of the region is as follows:

where , which implies that .

Remark 7. There are some notes on Theorem 6, see the following.

(i) The region is convex, and the proof is directly obtained by introducing a time-sharing random variable into Theorem 6, and, therefore, we omit the proof here.(ii) The range of the random variable satisfies

 The proof is in Appendix C.(iii) Without the equivocation parameter, the capacity of the main channel with feedback is given by

 The formula (8) is proved by Weissman [16], and it is omitted here.(iv) Secrecy capacity:  the points in for which are of considerable interest, which imply the perfect secrecy .

Definition 8 (the secrecy capacity ). The secrecy capacity of the model of Figure 6 with noncausal channel state information is denoted by

Clearly, we can easily determine the secrecy capacity of the model of Figure 6 with noncausal channel state information by

Proof of (10). Substituting into the region in Theorem 6, we have

Therefore, the secrecy capacity . Thus the proof is completed.

2.2. The Model of Figure 6 with Causal Channel State Information

The model of Figure 6 with causal channel state information is similar to the model with noncausal channel state information in Section 2.1, except that the state sequence in Definition 1 is known to the channel encoder in a causal manner, that is, at the th time (), the output of the encoder , where and is the probability that the message , the feedback , and the state are encoded as the channel input at time .

The capacity-equivocation region for the model of Figure 6 with causal channel state information is characterized by the following Theorem 9, and it is proved in Appendices D and E.

Theorem 9. A single-letter characterization of the region is as follows:

where , which implies that .

Remark 10. There are some notes on Theorem 9, see the following.

(i) The region is convex.(ii) The range of the random variable satisfies

 The proof is similar to that in Theorem 6, and it is omitted here.(iii) Without the equivocation parameter, the capacity of the main channel with feedback is given by

 The formula (14) is proved by Weissman [16], and it is omitted here.(iv) Secrecy capacity:  the points in for which are of considerable interest, which imply the perfect secrecy .

Definition 11 (the secrecy capacity ). The secrecy capacity of the model of Figure 6 with causal channel state information is denoted by

Clearly, we can easily determine the secrecy capacity of the model of Figure 6 with causal channel state information by

Proof of (16). Substituting into the region in Theorem 9, we have

Therefore, the secrecy capacity . Thus the proof is completed.

3. A Binary Example for the Model of Figure 6 with Causal Channel State Information

In this section, we calculate the secrecy capacity of a special case of the model of Figure 6 with causal channel state information.

Suppose that the channel state information is available at the channel encoder in a casual manner. Meanwhile, the random variables , , , , , and take values in , and the transition probability of the main channel is defined as follows.

When ,

When ,

Note that here .

The wiretap channel is a (binary symmetric channel) BSC with crossover probability , that is,

The channel for generating the state sequence is a BSC with crossover probability (), that is,

From Remark 10 we know that the secrecy capacity for the causal case is given by

Note that , , and are achieved if is a function of and is a function of and , and this is similar to the argument in [16]. Define and , then (22) can be written as

and this is because the joint probability distribution can be calculated by

Since is a function of , we have

Then, it is easy to see that

Now it remains to calculate the characters , , and ; see the remaining of this section.

Let take values in . The probability of is defined as follows: and .

In addition, there are 16 kinds of and 4 kinds of . Define the following:

The character depends on the joint probability mass functions , and we have

The character depends on the joint probability mass functions , and we have

The character depends on the joint probability mass functions , and we have

where is from .

By choosing the above , , and , we find that

where . Moreover, is achieved when , and .

On the other hand,

and “=” is achieved if , and .

Moreover,

and “=” is achieved if , and .

Therefore, the secrecy capacity for the causal case is given by

Figures 7, 8, and 9 give the secrecy capacity of the model of Figure 6 with causal channel state information for several values of . It is easy to see that the secrecy capacity is increasing while is getting larger.

Figure 7: When , the secrecy capacity of the model of Figure 6 is with causal channel state information.

Figure 8: When , the secrecy capacity of the model of Figure 6 is with causal channel state information.

Figure 9: When , the secrecy capacity of the model of Figure 6 is with causal channel state information.

4. Conclusion

In this paper, we study the model of the wiretap channel with action-dependent channel state information and noiseless feedback. The capacity-equivocation regions are provided both for the case where the channel inputs are allowed to depend noncausally on the state sequence and the case where they are restricted to causal dependence. Furthermore, the secrecy capacities for both cases are formulated, which provide the best transmission rate with perfect secrecy. The result is further explained via a binary example.

The contribution of this paper is as follows.

(i)Compared with the existing model of wiretap channel with side information [18] (see Figure 5), the channel state information in [18] is a special case of that in our new model; that is, the model of [18] is included in the model of Figure 6. Therefore, our result generalizes the result of [18].(ii)Our new model also extends the model of Figure 4 by considering an additional secrecy constraint, and therefore our result also generalizes the result of [16].

Appendices

A. Proof of the Direct Part of Theorem 6

In this section, we will show that any pair is achievable. Gel’efand-Pinsker’s binning, block Markov coding, and Ahlswede-Cai’s secret key on the feedback system are used in the construction of the code book.

Now the remainder of this section is organized as follows. The code construction is in Appendix  A.1. The proof of achievability is given in Appendix  A.2.

A.1. Code Construction

Since , and , it is sufficient to show that the pair is achievable, and note that this implies that .

The construction of the code and the proof of achievability are considered into two cases.

(i)Case  1:  .(ii)Case  2:  .

We use the block Markov coding method. The random vectors , , , , , and consist of blocks of length . The message for blocks is , where () are i.i.d. random variables uniformly distributed over . Note that in the first block, there is no .

Let () be the output of the wiretap channel for block , ,   . Similarly, , and () is the output of the main channel for block . The specific values of the above random vectors are denoted by lowercase letters.

(i) Code Construction for Case  1. Given a pair , choose a joint probability mass function such that

The message set satisfies the following condition:

where is a fixed positive real numbers and

Note that is from and (A.2). Let .

Code-book generation:

(a) Construction of and . In the first block, generate a i.i.d. sequence according to the probability mass function , and choose it as the output of the action encoder. Let be the state sequence generated in response to the chosen action sequence .

For the th block (), generate i.i.d. sequences , according to the probability mass function . Index each sequence by . For a given message (), choose a corresponding as the output of the action encoder. Let be the state sequence generated in response to the action sequence .

(b) Construction of the Secret Key. For the th block (), firstly we generate a mapping (note that ). Define a random variable (), which is uniformly distributed over , and is independent of  . Reveal the mapping to both receivers and the transmitter.

Then, when the transmitter receives the output of the -1th block, he computes as the secret key used in the th block.

(c) Construction of . In the first block, for the transmitted action sequence and the corresponding state sequence , generate a i.i.d. sequence according to the probability mass function . Choose as a realization of for the first block.

For the th block (), given the transmitted action sequence and the corresponding state sequence , generate ( as ) i.i.d. sequences , according to the probability mass function . Distribute these sequences at random into bins such that each bin contains sequences. Index each bin by .

For a given , , , and a secret key , the transmitter chooses a sequence from the bin (where is the modulo addition over ) such that . If such multiple sequences in bin exist, choose the one with the smallest index in the bin. If no such sequence exists, declare an encoding error.

(d) Construction of .  For each block, the is generated according to a new discrete memoryless channel (DMC) with inputs , , and output . The transition probability of this new DMC is , which is obtained from the joint probability mass function .

The probability is calculated as follows:

Decoding. For the th block (), given a vector and a secret key ( is known by the receiver), try to find a sequence such that . If there exist sequences with the same , by using the secret key , put out the corresponding . Otherwise, that is, if no such sequence exists or multiple sequences have different message indices, declare a decoding error.

(ii) Code Construction for Case  2.  Given a pair , choose a joint probability mass function such that

The message set satisfies the following condition:

Let .

Code-book generation:

(a) Construction of and .  In the first block, generate a i.i.d.  sequence according to the probability mass function , and choose it as the output of the action encoder. Let be the state sequence generated in response to the chosen action sequence .

For the th block (), generate i.i.d. sequences , according to the probability mass function . Index each sequence by . For a given message (), choose a corresponding as the output of the action encoder. Let be the state sequence generated in response to the action sequence .

(b) Construction of the Secret Key.  For the th block (), firstly we generate a mapping (note that ). Define a random variable (),  which is uniformly distributed over , and is independent of  . Reveal the mapping to both receivers and the transmitter.

Then, when the transmitter receives the output of the -1th block, he computes as the secret key used in the th block.

(c) Construction of .  In the first block, for the transmitted action sequence and the corresponding state sequence , generate a i.i.d. sequence according to the probability mass function . Choose as a realization of for the first block.

For the th block (), given the transmitted action sequence and the corresponding state sequence , generate ( as ) i.i.d. sequences , according to the probability mass function . Distribute these sequences at random into bins such that each bin contains sequences. Index each bin by .

For a given , , , and a secret key , the transmitter chooses a sequence from the bin (where is the modulo addition over ) such that . If such multiple sequences in bin exist, choose the one with the smallest index in the bin. If no such sequence exists, declare an encoding error.

(d) Construction of .  The is generated the same as that for the case 1, and it is omitted here.

Decoding. For the th block (), given a vector and a secret key ( is known by the receiver), try to find a sequence such that . If there exist sequences with the same , by using the secret key , put out the corresponding . Otherwise, that is, if no such sequence exists or multiple sequences have different message indices, declare a decoding error.

A.2. Proof of Achievability

The rate of the message is defined as

and it satisfies

In addition, note that the encoding and decoding scheme for Theorem 6 is exactly the same as that in [16], except that the transmitted message for the legitimate receiver is . Since the legitimate receiver knows , the decoding scheme for Theorem 6 is in fact the same as that in [16]. Hence, we omit the proof of here.

It remains to show that , see the following.

Since the message is encrypted by , the equivocation about is equivalent to the equivocation about the secret key . There are two ways for the wiretapper to obtain the secret key . One way is that he tries to guess the from its alphabet . The other way is that he tries to guess the feedback ( is the output of the main channel for the previous block, and ) from the conditional typical set , and this is because for a given and sufficiently large , . Note that there are sequences when and . Therefore, the equivocation about is , and note that and , and then is obtained.

The details about the proof are as follows.

First, we will show that is independent of and , and this is used in the proof of .

Since is independent of (), and all of them are uniformly distributed over , the fact that is independent of and is proved by the following (A.9):

Then, for both cases is proved by the following (A.10):

where is from that is a Markov chain, is from that is a Markov chain, follows from the fact that is independent of and , and is from the fact that the wiretapper can guess the specific vector (corresponding to the key ) from the conditional typical set , and is uniformly distributed over ( is the key used in block ).

On the other hand,

where (1) is from the fact that , () is the state sequence for block , and is a function of , (2) is from and that are i.i.d. generated random vectors, and the channels are discrete memoryless.

Therefore, it is easy to see that, for the case 1, is proved by (A.10) and (A.11). For the case 2, is proved by the formula (A.10).

Thus, for both cases is proved. The proof of Theorem 6 is completed.

B. Proof of the Converse Part of Theorem 6

In this section, we prove the converse part of Theorem 6: all the achievable pairs are contained in the set . Suppose that is achievable; that is, for any given , there exists a channel encoder-decoder such that

Then we will show the existence of random variables such that

Since is uniformly distributed over , we have . The formulas (B.3), (B.4), and (B.5) are proved by Lemma  B.1; see the following.

Lemma  B.1. The random vectors and and the random variables , , , , , , and of Theorem 6 satisfy

where . Note that .

Substituting and (5) into (B.6), (B.7), and (B.8) and using the fact that , the formulas (B.3), (B.4), and (B.5) are obtained. The formula (B.2) is from

It remains to prove Lemma  B.1; see the following.

Proof of Lemma  B.1.  The formula (B.6) follows from (B.10), (B.13), and (B.21). The formula (B.7) is from (B.11), (B.17) and (B.22). The formula (B.8) is proved by (B.12), (B.18), and (B.23).

Part (i). We begin with the left parts of the inequalities (B.6), (B.7), and (B.8); see the following.

Since is a Markov chain, for the message , we have

For the equivocation to the wiretapper, we have

Moreover,

Note that and follow from Fano’s inequality, and (B.12) is from the fact that is a deterministic function of .

Part (ii). By using chain rule, the character in formulas (B.10) and (B.11) can be bounded as follows:

where formula (1) follows from that , and formula (2) follows from that

and formula (3) follows from that .

Proof of (B.14).  The left part of (B.14) can be rewritten as

where (1) is from the fact that is a deterministic function of .

The right part of (B.14) can be rewritten as

where (2) is from the fact that is a deterministic function of .

The formula (B.14) is proved by (B.15) and (B.16). The proof is completed.

Part (iii). The character in formula (B.11) can be rewritten as follows:

Part (iv). The character in formula (B.12) can be rewritten as follows:

Part (v) (single letter). To complete the proof, we introduce a random variable , which is independent of , , , , and . Furthermore, is uniformly distributed over . Define

Part (vi). Then (B.13) can be rewritten as

Analogously, (B.17) is rewritten as follows:

Similarly, (B.18) can be rewritten as follows:

Substituting (B.21), (B.22), (B.23) into (B.10), (B.11), and (B.12), Lemma  B.1 is proved.

The proof of Theorem 6 is completed.

C. Size Constraint of the Auxiliary Random Variable in Theorem 6

By using the support lemma (see [20, page 310]), it suffices to show that the random variable can be replaced by new one, preserving the Markovity and the mutual information , , , and furthermore, the range of the new satisfies, . The proof is in the reminder of this section.

Let

Define the following continuous scalar functions of :

Since there are functions of , the total number of the continuous scalar functions of is .

Let . With these distributions , we have

According to the support lemma ([20, page 310]), the random variable can be replaced by new ones such that the new takes at most different values and the expressions (C.3), (C.4), and (C.5) are preserved.

D. Proof of the Direct Part of Theorem 9

In this section, we will show that any pair is achievable. Block Markov coding and Ahlswede-Cai’s secret key on the feedback system are used in the construction of the code-book.

Now the remainder of this section is organized as follows. The code construction is in Appendix D.1. The proof of achievability is given in Appendix D.2.

D.1. Code Construction

 Since , , and , it is sufficient to show that the pair is achievable, and note that this implies that .

We use the block Markov coding method. The random vectors , , , , , and consist of blocks of length . The message for blocks is , where () are i.i.d. random variables uniformly distributed over . Note that, in the first block, there is no .

Let () be the output of the wiretap channel for block , ,   . Similarly, and () are the output of the main channel for block . The specific values of the above random vectors are denoted by lowercase letters.

Given a pair , choose a joint probability mass function such that

The message set satisfies the following condition:

where is a fixed positive real numbers.

Code-book generation:

(i) Construction of and .  In the first block, generate a i.i.d. sequence according to the probability mass function , and choose it as the output of the action encoder. Let be the state sequence generated in response to the chosen action sequence .

For the th block (), generate i.i.d. sequences , according to the probability mass function . Index each sequence by . For a given message (), choose a corresponding as the output of the action encoder. Let be the state sequence generated in response to the action sequence .

(ii) Construction of the Secret Key.  For the th block (), firstly we generate a mapping (note that ). Define a random variable (), which is uniformly distributed over , and is independent of . Reveal the mapping to both receivers and the transmitter.

Then, when the transmitter receives the output of the -1th block, he computes as the secret key used in the th block.

(iii) Construction of .  In the first block, for the transmitted action sequence and the corresponding state sequence , generate a i.i.d. sequence according to the probability mass function . Choose as a realization of for the first block.

For the th block (), given the transmitted action sequence and the corresponding state sequence , generate i.i.d. sequences , according to the probability mass function . Index each sequence by .

For a given and a secret key , the transmitter chooses a sequence (where is the modulo addition over ) to transmit.

(iv) Construction of . For each block, the is generated according to a new discrete memoryless channel (DMC) with inputs , , and output . The transition probability of this new DMC is , which is obtained from the joint probability mass function .

The probability is calculated as follows:

Decoding. For the th block (), given a vector and a secret key ( is known by the receiver), try to find a sequence such that . If there exist sequences with the same , by using the secret key , put out the corresponding . Otherwise, that is, if no such sequence exists or multiple sequences have different message indices, declare a decoding error.

D.2. Proof of Achievability

 The proof of achievability for Theorem 9 is along the lines of that for Theorem 6, and, therefore, we omit the proof here.

The proof of Theorem 9 is completed.

E. Proof of the Converse Part of Theorem 9

In this section, we prove Theorem 9: all the achievable pairs are contained in the set . Suppose that is achievable; that is, for any given , there exists a channel encoder-decoder such that

Then we will show the existence of random variables such that

Since is uniformly distributed over , we have . The formulas (E.3), (E.4), and (E.5) are proved by Lemma  E.1; see the following.

Lemma  E.1.  The random vectors and and the random variables , , , , , , and of Theorem 9 satisfy

where . Note that .

Substituting and (5) into (E.6), (E.7), and (E.8) and using the fact that , the formulas (E.3), (E.4), and (E.5) are obtained. The formula (E.2) is from

It remains to prove Lemma  E.1; see the following.

Proof of Lemma  E.1. The formula (E.6) follows from (E.10), (E.13), and (E.17). The formula (E.7) is from (E.11), (E.14), and (E.18). The formula (E.8) is proved by (E.12), (E.15), and (E.19).

Part (i). We begin with the left parts of the inequalities (E.6), (E.7), and (E.8); see the following.

Since is a Markov chain, for the message , we have

For the equivocation to the wiretapper, we have

Note that and follow from Fano’s inequality, and (E.12) is from the fact that is a deterministic function of .

Part (ii). By using chain rule, the character in formulas (E.10) and (E.11) can be bounded as follows:

Part (iii). Similar to (E.13), the character in formula (E.11) can be rewritten as follows:

Part (iv). The character in formula (E.12) can be rewritten as follows:

Part (v) (single letter). To complete the proof, we introduce a random variable , which is independent of , , , , , and . Furthermore, is uniformly distributed over . Define

Part (vi). Then (E.13) can be rewritten as

Analogously, (E.14) is rewritten as follows:

Similarly, (E.15) can be rewritten as follows,

Substituting (E.17), (E.18), and (E.19) into (E.10), (E.11), and (E.12), Lemma  E.1 is proved.

The proof of Theorem 9 is completed.

Acknowledgments

The authors would like to thank Professor N. Cai for his help to improve this paper. This work was supported by a subproject in National Basic Research Program of China under Grant 2012CB316100 on Broadband Mobile Communications at High Speeds, the National Natural Science Foundation of China under Grant 61271222, and the Research Fund for the Doctoral Program of Higher Education of China (no. 20100073110016).

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