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

Authors: Bin Dai; A. J. Han Vinck; Yuan Luo
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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 R1 and common messages to both receivers at rate R0, 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 (R1,Re,R0) was provided in [2], where Re 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 Figure6. 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 Figure6, the transmitted message W is encoded as an action sequence AN, and AN is the input of a discrete memoryless channel (DMC). The output of this DMC is the channel state sequence SN. The main channel is a DMC with inputs XN and SN and output YN. The wiretap channel is also a DMC with input YN and output ZN. 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 W, the state sequence SN, and the noiseless feedback, while the output is XN. 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 Figure6 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 Figure4 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,UNdenotes a randomN-vector(U1,…,UN), anduN=(u1,…,uN)is a specific vector value in𝒰N, that is the,Nth Cartesian power of𝒰.UiNdenotes a randomN-i+1-vector(Ui,…,UN), anduiN=(ui,…,uN)is a specific vector value in𝒰iN. Let pV(v) denote the probability mass function Pr{V=v}. Throughout the paper, the logarithmic function is taken to the base 2.The remainder of this paper is organized as follows. In Section2, 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 Figure6 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 Figure6 with Noncausal Channel State Information

In this section, a description of the wiretap channel with noncausal action-dependent channel state information is given by Definition1 to Definition 5. The capacity-equivocation region 𝒞n composed of all achievable (R,Re) pairs is given in Theorem 6, where the achievable (R,Re) pair is defined in Definition 5.Definition 1 (action encoder).
The messageW takes values in 𝒲, and it is uniformly distributed over its range. The action encoder is a deterministic mapping:
(1)f1N:𝒲→𝒜N.
The input of the action encoder is W, while the output is AN.The channel state sequenceSN is generated by a DMC with input AN and output SN. The transition probability distribution is given by
(2)pSN∣AN(sN∣aN)=∏i=1N‍pSi∣Ai(si∣ai).
Note that the components of the state sequence SN may not be i.i.d. random variables, and this is due to the fact that AN is not i.i.d. generated. The transmission rate of the message is log∥𝒲∥/N.Definition 2 (channel encoder and the main channel).
The main channel is a DMC with finite input alphabet𝒳×𝒮, finite output alphabet 𝒴, and transition probability QM(y∣x,s), where x∈𝒳,s∈𝒮,y∈𝒴. QM(yN∣xN,sN)=∏n=1N‍QM(yn∣xn,sn). The inputs of the main channel are XN and SN, while the output is YN.There is a noiseless feedback from the output of the main channel to the channel encoder. At theith time, the feedback Yi-1 (where 2≤i≤N and Yi-1 takes values in 𝒴i-1) is the previous i-1 time output of the main channel. Since the channel encoder knows the state sequence sN in a noncausal manner, at the ith time, the inputs of the channel encoder are W, Yi-1, and SN, while the output is Xi; that is, the ith time channel encoder is a conditional probability f2,iN(xi∣w,sN,yi-1) that the message w, the feedback yi-1, and the channel state sequence sN are encoded as the ith time channel input xi.Definition 3 (wiretap channel).
The wiretap channel is also a DMC with finite input alphabet𝒴, finite output alphabet 𝒵, and transition probability QW(z∣y), where y∈𝒴,z∈𝒵. The input and output of the wiretap channel are YN and ZN, respectively. The equivocation to the wiretapper is defined as
(3)Δ=H(W∣ZN)N.
The cascade of the main channel and the wiretap channel is another DMC with transition probability:
(4)QMW(z∣x,s)=∑y∈𝒴‍QW(z∣y)QM(y∣x,s).Note thatW→AN→(XN,SN)→YN→ZN is a Markov chain in the model of Figure 6.Definition 4 (decoder).
The decoder for the legitimate receiver is a mappingfD:𝒴N→𝒲, with input YN and output W^. Let Pe be the error probability of the receiver, and it is defined as Pr{W≠W^}.Definition 5 (achievable(R,Re) pair in the model of Figure 6).
A pair(R,Re) (where R,Re>0) is called achievable if, for any ϵ>0 (where ϵ is an arbitrary small positive real number and ϵ→0), there exist channel encoders decoders (N,Pe) such that
(5)limN→∞log∥𝒲∥N=R,limN→∞Δ≥Re,Pe≤ϵ.The capacity-equivocation regionℛn is a set composed of all achievable (R,Re) 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 regionℛn is as follows:
(6)ℛ(n)={(R,Re):0≤Re≤R,R≤I(U;Y)-I(U;S∣A),Re≤H(Y∣Z),Re≤H(A∣Z)},
where pUASXYZ(u,a,s,x,y,z)=pZ∣Y(z∣y)pY∣X,S(y∣x,s)pUAXS(u,a,x,s), which implies that (A,U)→(X,S)→Y→Z.Remark 7.
There are some notes on Theorem6, see the following. (i)
The regionℛ(n) 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 variableU satisfies
(7)∥𝒰∥≤∥𝒳∥∥𝒜∥∥𝒮∥+1.The proof is in AppendixC.(iii)
Without the equivocation parameter, the capacity of the main channel with feedback is given by(8)CM=maxpX∣U,S(x∣u,s)pU∣A,S(u∣a,s)pA(a)(I(U;Y)-I(U;S∣A)).The formula (8) is proved by Weissman [16], and it is omitted here.(iv)
Secrecy capacity:  the points in ℛ(n) for which Re=R are of considerable interest, which imply the perfect secrecy H(W)=H(W∣ZN).Definition 8 (the secrecy capacityCs(n)).
The secrecy capacityCs(n) of the model of Figure 6 with noncausal channel state information is denoted by
(9)Cs(n)=max(R,Re=R)∈ℛ(n)R.
Clearly, we can easily determine the secrecy capacity Csn of the model of Figure 6 with noncausal channel state information by
(10)Csn=maxmin{I(U;Y)-I(U;S∣A),maxminH(Y∣Z),H(A∣Z)}.Proof of (10).
SubstitutingRe=R into the region ℛ(n) in Theorem 6, we have
(11)R≤H(Y∣Z),R≤I(U;Y)-I(U;S∣A),R≤H(A∣Z).
Therefore, the secrecy capacity Cs(n)=max(R,Re=R)∈ℛ(n)R=maxmin{I(U;Y)-I(U;S∣A),H(Y∣Z),H(A∣Z)}. Thus the proof is completed.


### 2.2.  The Model of Figure6 with Causal Channel State Information

The model of Figure6 with causal channel state information is similar to the model with noncausal channel state information in Section 2.1, except that the state sequence SN in Definition 1 is known to the channel encoder in a causal manner, that is, at the ith time (1≤i≤N), the output of the encoder xi=f2,i(w,si,yi-1), where si=(s1,s2,…,si) and f2,i is the probability that the message w, the feedback yi-1, and the state si are encoded as the channel input xi at time i.The capacity-equivocation regionℛc 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 regionℛc is as follows:
(12)ℛ(c)={(R,Re):0≤Re≤R,R≤I(U;Y),Re≤H(Y∣Z),Re≤H(A∣Z)},
where pUASXYZ(u,a,s,x,y,z)=pZ∣Y(z∣y)pY∣X,S(y∣x,s)pUAXS(u,a,x,s), which implies that (A,U)→(X,S)→Y→Z.Remark 10.
There are some notes on Theorem9, see the following. (i)
The regionℛ(c) is convex.(ii)
The range of the random variableU satisfies
(13)∥𝒰∥≤∥𝒳∥∥𝒜∥∥𝒮∥.The proof is similar to that in Theorem6, and it is omitted here.(iii)
Without the equivocation parameter, the capacity of the main channel with feedback is given by(14)CM*=maxpX∣U,S(x∣u,s)pU∣A,S(u∣a,s)pA(a)I(U;Y).The formula (14) is proved by Weissman [16], and it is omitted here.(iv)
Secrecy capacity:  the points in ℛ(c) for which Re=R are of considerable interest, which imply the perfect secrecy H(W)=H(W∣ZN).Definition 11 (the secrecy capacityCs(c)).
The secrecy capacityCs(c) of the model of Figure 6 with causal channel state information is denoted by
(15)Cs(c)=max(R,Re=R)∈ℛ(c)R.
Clearly, we can easily determine the secrecy capacity Cs(c) of the model of Figure 6 with causal channel state information by
(16)Cs(c)=maxmin{I(U;Y),H(Y∣Z),H(A∣Z)}.Proof of (16).
SubstitutingRe=R into the region ℛ(c) in Theorem 9, we have
(17)R≤H(Y∣Z),R≤I(U;Y),R≤H(A∣Z).
Therefore, the secrecy capacity Cs(c)=max(R,Re=R)∈ℛ(c)R=maxmin{I(U;Y),H(Y∣Z),H(A∣Z)}. Thus the proof is completed.


## 2.1. The Model of Figure6 with Noncausal Channel State Information

In this section, a description of the wiretap channel with noncausal action-dependent channel state information is given by Definition1 to Definition 5. The capacity-equivocation region 𝒞n composed of all achievable (R,Re) pairs is given in Theorem 6, where the achievable (R,Re) pair is defined in Definition 5.Definition 1 (action encoder).
The messageW takes values in 𝒲, and it is uniformly distributed over its range. The action encoder is a deterministic mapping:
(1)f1N:𝒲→𝒜N.
The input of the action encoder is W, while the output is AN.The channel state sequenceSN is generated by a DMC with input AN and output SN. The transition probability distribution is given by
(2)pSN∣AN(sN∣aN)=∏i=1N‍pSi∣Ai(si∣ai).
Note that the components of the state sequence SN may not be i.i.d. random variables, and this is due to the fact that AN is not i.i.d. generated. The transmission rate of the message is log∥𝒲∥/N.Definition 2 (channel encoder and the main channel).
The main channel is a DMC with finite input alphabet𝒳×𝒮, finite output alphabet 𝒴, and transition probability QM(y∣x,s), where x∈𝒳,s∈𝒮,y∈𝒴. QM(yN∣xN,sN)=∏n=1N‍QM(yn∣xn,sn). The inputs of the main channel are XN and SN, while the output is YN.There is a noiseless feedback from the output of the main channel to the channel encoder. At theith time, the feedback Yi-1 (where 2≤i≤N and Yi-1 takes values in 𝒴i-1) is the previous i-1 time output of the main channel. Since the channel encoder knows the state sequence sN in a noncausal manner, at the ith time, the inputs of the channel encoder are W, Yi-1, and SN, while the output is Xi; that is, the ith time channel encoder is a conditional probability f2,iN(xi∣w,sN,yi-1) that the message w, the feedback yi-1, and the channel state sequence sN are encoded as the ith time channel input xi.Definition 3 (wiretap channel).
The wiretap channel is also a DMC with finite input alphabet𝒴, finite output alphabet 𝒵, and transition probability QW(z∣y), where y∈𝒴,z∈𝒵. The input and output of the wiretap channel are YN and ZN, respectively. The equivocation to the wiretapper is defined as
(3)Δ=H(W∣ZN)N.
The cascade of the main channel and the wiretap channel is another DMC with transition probability:
(4)QMW(z∣x,s)=∑y∈𝒴‍QW(z∣y)QM(y∣x,s).Note thatW→AN→(XN,SN)→YN→ZN is a Markov chain in the model of Figure 6.Definition 4 (decoder).
The decoder for the legitimate receiver is a mappingfD:𝒴N→𝒲, with input YN and output W^. Let Pe be the error probability of the receiver, and it is defined as Pr{W≠W^}.Definition 5 (achievable(R,Re) pair in the model of Figure 6).
A pair(R,Re) (where R,Re>0) is called achievable if, for any ϵ>0 (where ϵ is an arbitrary small positive real number and ϵ→0), there exist channel encoders decoders (N,Pe) such that
(5)limN→∞log∥𝒲∥N=R,limN→∞Δ≥Re,Pe≤ϵ.The capacity-equivocation regionℛn is a set composed of all achievable (R,Re) 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 regionℛn is as follows:
(6)ℛ(n)={(R,Re):0≤Re≤R,R≤I(U;Y)-I(U;S∣A),Re≤H(Y∣Z),Re≤H(A∣Z)},
where pUASXYZ(u,a,s,x,y,z)=pZ∣Y(z∣y)pY∣X,S(y∣x,s)pUAXS(u,a,x,s), which implies that (A,U)→(X,S)→Y→Z.Remark 7.
There are some notes on Theorem6, see the following. (i)
The regionℛ(n) 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 variableU satisfies
(7)∥𝒰∥≤∥𝒳∥∥𝒜∥∥𝒮∥+1.The proof is in AppendixC.(iii)
Without the equivocation parameter, the capacity of the main channel with feedback is given by(8)CM=maxpX∣U,S(x∣u,s)pU∣A,S(u∣a,s)pA(a)(I(U;Y)-I(U;S∣A)).The formula (8) is proved by Weissman [16], and it is omitted here.(iv)
Secrecy capacity:  the points in ℛ(n) for which Re=R are of considerable interest, which imply the perfect secrecy H(W)=H(W∣ZN).Definition 8 (the secrecy capacityCs(n)).
The secrecy capacityCs(n) of the model of Figure 6 with noncausal channel state information is denoted by
(9)Cs(n)=max(R,Re=R)∈ℛ(n)R.
Clearly, we can easily determine the secrecy capacity Csn of the model of Figure 6 with noncausal channel state information by
(10)Csn=maxmin{I(U;Y)-I(U;S∣A),maxminH(Y∣Z),H(A∣Z)}.Proof of (10).
SubstitutingRe=R into the region ℛ(n) in Theorem 6, we have
(11)R≤H(Y∣Z),R≤I(U;Y)-I(U;S∣A),R≤H(A∣Z).
Therefore, the secrecy capacity Cs(n)=max(R,Re=R)∈ℛ(n)R=maxmin{I(U;Y)-I(U;S∣A),H(Y∣Z),H(A∣Z)}. Thus the proof is completed.


## 2.2.  The Model of Figure6 with Causal Channel State Information

The model of Figure6 with causal channel state information is similar to the model with noncausal channel state information in Section 2.1, except that the state sequence SN in Definition 1 is known to the channel encoder in a causal manner, that is, at the ith time (1≤i≤N), the output of the encoder xi=f2,i(w,si,yi-1), where si=(s1,s2,…,si) and f2,i is the probability that the message w, the feedback yi-1, and the state si are encoded as the channel input xi at time i.The capacity-equivocation regionℛc 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 regionℛc is as follows:
(12)ℛ(c)={(R,Re):0≤Re≤R,R≤I(U;Y),Re≤H(Y∣Z),Re≤H(A∣Z)},
where pUASXYZ(u,a,s,x,y,z)=pZ∣Y(z∣y)pY∣X,S(y∣x,s)pUAXS(u,a,x,s), which implies that (A,U)→(X,S)→Y→Z.Remark 10.
There are some notes on Theorem9, see the following. (i)
The regionℛ(c) is convex.(ii)
The range of the random variableU satisfies
(13)∥𝒰∥≤∥𝒳∥∥𝒜∥∥𝒮∥.The proof is similar to that in Theorem6, and it is omitted here.(iii)
Without the equivocation parameter, the capacity of the main channel with feedback is given by(14)CM*=maxpX∣U,S(x∣u,s)pU∣A,S(u∣a,s)pA(a)I(U;Y).The formula (14) is proved by Weissman [16], and it is omitted here.(iv)
Secrecy capacity:  the points in ℛ(c) for which Re=R are of considerable interest, which imply the perfect secrecy H(W)=H(W∣ZN).Definition 11 (the secrecy capacityCs(c)).
The secrecy capacityCs(c) of the model of Figure 6 with causal channel state information is denoted by
(15)Cs(c)=max(R,Re=R)∈ℛ(c)R.
Clearly, we can easily determine the secrecy capacity Cs(c) of the model of Figure 6 with causal channel state information by
(16)Cs(c)=maxmin{I(U;Y),H(Y∣Z),H(A∣Z)}.Proof of (16).
SubstitutingRe=R into the region ℛ(c) in Theorem 9, we have
(17)R≤H(Y∣Z),R≤I(U;Y),R≤H(A∣Z).
Therefore, the secrecy capacity Cs(c)=max(R,Re=R)∈ℛ(c)R=maxmin{I(U;Y),H(Y∣Z),H(A∣Z)}. Thus the proof is completed.


## 3. A Binary Example for the Model of Figure6 with Causal Channel State Information

In this section, we calculate the secrecy capacity of a special case of the model of Figure6 with causal channel state information.Suppose that the channel state informationSN is available at the channel encoder in a casual manner. Meanwhile, the random variables U, A, X, S, Y, and Z take values in {0,1}, and the transition probability of the main channel is defined as follows.Whens=0,
(18)pY∣X,S(y∣x,s=0)={1-p,ify=x,p,otherwise.Whens=1,
(19)pY∣X,S(y∣x,s=1)={p,ify=x,1-p,otherwise.
Note that here 0≤p≤1.The wiretap channel is a (binary symmetric channel) BSC with crossover probabilityq(0≤q≤1/2), that is,
(20)pZ∣Y(z∣y)={1-q,ify=x,q,otherwise.The channel for generating the state sequenceSN is a BSC with crossover probability r (0≤r≤1), that is,
(21)pS∣A(s∣a)={1-r,ify=x,r,otherwise.From Remark10 we know that the secrecy capacity for the causal case is given by
(22)Csc=maxmin{I(U;Y),H(Y∣Z),H(A∣Z)}.
Note that maxI(U;Y), maxH(A∣Z), and maxH(Y∣Z) are achieved if A is a function of U and X is a function of U and S, and this is similar to the argument in [16]. Define a=g(u) and x=f(u,s), then (22) can be written as
(23)Csc=maxf,g,pU(u)min{I(U;Y),H(A∣Z),H(Y∣Z)},
and this is because the joint probability distribution pAUSXYZ(a,u,s,x,y,z) can be calculated by
(24)pAUSXYZ(a,u,s,x,y,z)=pZ∣Y(z∣y)pY∣X,S(y∣x,s)1x=f(u,s)pS∣A×(s∣a)1a=g(u)pU(u).SinceA is a function of U, we have
(25)H(A∣Z)=H(U∣Z).
Then, it is easy to see that
(26)maxf,g,pU(u)min{I(U;Y),H(A∣Z),H(Y∣Z)}=maxf,g,pU(u)maxf,g,pU(u)min{I(U;Y),H(U∣Z),H(Y∣Z)}.Now it remains to calculate the charactersmaxf,g,pU(u)H(Y∣Z), maxf,g,pU(u)H(U∣Z), and maxf,g,pU(u)I(U;Y); see the remaining of this section.LetU take values in {0,1}. The probability of U is defined as follows: pU(0)=α and pU(1)=1-α.In addition, there are 16 kinds off and 4 kinds of g. Define the following:
(27)f(1)(u,s):{00⟶0,01⟶0,10⟶0,11⟶0,f(2)(u,s):{00⟶0,01⟶0,10⟶0,11⟶1,f(3)(u,s):{00⟶0,01⟶0,10⟶1,11⟶0,f(4)(u,s):{00⟶0,01⟶0,10⟶1,11⟶1,f(5)(u,s):{00⟶0,01⟶1,10⟶0,11⟶0,f(6)(u,s):{00⟶0,01⟶1,10⟶0,11⟶1,f(7)(u,s):{00⟶0,01⟶1,10⟶1,11⟶0,f(8)(u,s):{00⟶0,01⟶1,10⟶1,11⟶1,f(9)(u,s):{00⟶1,01⟶0,10⟶0,11⟶0,f(10)(u,s):{00⟶1,01⟶0,10⟶0,11⟶1,f(11)(u,s):{00⟶1,01⟶0,10⟶1,11⟶0,f(12)(u,s):{00⟶1,01⟶0,10⟶1,11⟶1,f(13)(u,s):{00⟶1,01⟶1,10⟶0,11⟶0,f(14)(u,s):{00⟶1,01⟶1,10⟶0,11⟶1,f(15)(u,s):{00⟶1,01⟶1,10⟶1,11⟶0,f(16)(u,s):{00⟶1,01⟶1,10⟶1,11⟶1,g(1)(u):{0⟶0,1⟶0,g(2)(u):{0⟶0,1⟶1,g(3)(u):{0⟶1,1⟶0,g(4)(u):{0⟶1,1⟶1.
The character I(U;Y) depends on the joint probability mass functions pUY(u,y), and we have
(28)pUY(u,y)=∑x,s,a‍pUYXSA(u,y,x,s,a)=∑x,s,a‍pY∣XS(y∣x,s)pX∣U,S(x∣u,s)pU(u)×pA∣U(a∣u)pS∣A(s∣a).
The character H(U∣Z) depends on the joint probability mass functions pUZ(u,z), and we have
(29)pUZ(u,z)=∑y‍pUYZ(u,y,z)=∑y‍pZ∣Y(z∣y)pU,Y(u,y).The characterH(Y∣Z) depends on the joint probability mass functions pYZ(u,z), and we have
(30)pYZ(y,z)=∑u‍pUYZ(u,y,z)=(a)∑u‍pZ∣Y(z∣y)pU,Y(u,y),
where (a) is from U→Y→Z.By choosing the abovef, g, and α, we find that
(31)maxf,g,pU(u)H(U∣Z)=h(p+q-2pq),
where h(p+q-2pq)=(p+q-2pq)log(1/(p+q-2pq))+(1-p-q+2pq)log(1/(1-p-q+2pq)). Moreover, h(p+q-2pq) is achieved when f=f(7), g=g(2) and α=1/2.On the other hand,(32)maxf,g,pU(u)I(U;Y)=1-h(p),
and “=” is  achieved if f=f(7), g=g(2) and α=1/2.Moreover,(33)maxf,g,pU(u)H(Y∣Z)=h(q),
and “=” is achieved if f=f(7), g=g(2) and α=1/2.Therefore, the secrecy capacity for the causal case is given by(34)Csc=min{h(p+q-2pq),1-h(p),h(q)}.Figures7, 8, and 9 give the secrecy capacity of the model of Figure 6 with causal channel state information for several values of q. It is easy to see that the secrecy capacity Csc is increasing while q is getting larger.Figure 7
Whenq=0.2, the secrecy capacity of the model of Figure 6 is with causal channel state information.Figure 8
Whenq=0.1, the secrecy capacity of the model of Figure 6 is with causal channel state information.Figure 9
Whenq=0.02, 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 Figure4 by considering an additional secrecy constraint, and therefore our result also generalizes the result of [16].



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*Source: 423619-2013-03-10.xml*