Capacity-Equivocation Regions of the DMBCs with Noiseless Feedback

Abstract

The discrete memoryless broadcast channels (DMBCs) with noiseless feedback are studied. The entire capacity-equivocation regions of two models of the DMBCs with noiseless feedback are obtained. One is the degraded DMBCs with rate-limited feedback; the other is the less and reversely less noisy DMBCs with causal feedback. In both models, two kinds of messages are transmitted. The common message is to be decoded by both the legitimate receiver and the eavesdropper, while the confidential message is only for the legitimate receiver. Our results generalize the secrecy capacity of the degraded wiretap channel with rate-limited feedback (Ardestanizadeh et al., 2009) and the restricted wiretap channel with noiseless feedback (Dai et al., 2012). Furthermore, we use a simpler and more intuitive deduction to get the single-letter characterization of the capacity-equivocation region, instead of relying on the recursive argument which is complex and not intuitive.

1. Introduction

Secure data transmission is an important requirement in wireless communication. Wyner first studied the degraded (the wiretap channel is said to be (physically) degraded if 𝑋→𝑌→𝑍 form a Markov chain, where 𝑋 is the channel input and 𝑌 and 𝑍 are the channel outputs of the legitimate receiver and wiretapper, resp.) wiretap channel in [1], where the output 𝑍^𝑁 of the channel to the wiretapper is degraded to the output 𝑌^𝑁 of the channel to the legitimate receiver. In Wyner’s model, the transmitter aimed to send a confidential message 𝑆 to the legitimate receiver and keep the wiretapper as ignorant of the message as possible. Wyner obtained the secrecy capacity (the secrecy capacity is the best data transmission rate under perfect secrecy; i.e., the equivocation at the wiretapper 𝐻(𝑆∣𝑍^𝑁)=0. The formal definition of the secrecy capacity is given in Remark 3) and demonstrated that provable secure communication could be implemented by using information theoretic methods. This model was extended to a more general case by Csiszár and Körner [2], where broadcast channel with confidential messages was studied; see Figure 1. They considered transmitting not only the confidential messages 𝑆 to the legitimate receiver, but also the common messages 𝑊 to both the legitimate receiver and the eavesdropper. The capacity-equivocation region for the extended model was determined in [2]. This region contains all the achievable rate triples (𝑅₀,𝑅₁,𝑅_𝑒), where 𝑅₀ and 𝑅₁ are the rates of the common and confidential messages and 𝑅_𝑒 is the rate of the confidential message’s equivocation. Nevertheless, neither Wyner’s model nor Csiszár’s model considered feedback.

Figure 1 

Broadcast channel with confidential messages.

To explore more ways in achieving secure data transmission, [3–5] studied the effects of the feedback on the capacities of several channel models. They all showed that feedback could help enhance the secrecy in wireless transmission. In [3], Ahlswede and Cai presented both the inner and outer bounds on the secrecy capacity of the wiretap channel with secure causal feedback from the decoder and showed that the outer bound was tight for the degraded case. It was proved that, by using feedback, the secrecy capacity of the (degraded) wiretap channel was increased. After Ahlswede’s exploration, Ardestanizadeh et al. studied the wiretap channel with secure rate-limited feedback [4]. The main difference between Ardestanizadeh’s model and Ahlswede’s model is that the feedback in [4] is independent of the channel outputs, while the feedback in [3] is originated causally from the outputs of the channel to the legitimate receiver. In [4], the authors got an outer bound on the wiretap channel with rate-limited feedback through a recursive argument which was effective but not intuitive. They also showed the outer bound was tight for the degraded case. In addition, Dai et al. investigated the secrecy capacity of the restricted wiretap channel with noiseless causal feedback under the assumption that the main channel is independent of the wiretap channel [5].

However, all of these explorations [3–5] focused on sending only the confidential messages. They did not consider sending both the common and confidential messages. In fact, transmitting the two kinds of messages can be seen in many systems with feedback. For example, in the satellite television service, some channels are available to all users for free, but some other channels are only for those who have paid for them. Recently, [6] studied the problem of transmitting both the common and confidential messages in the degraded broadcast channels with feedback. Note that, like [3], the feedback in [6] was originated causally from the legitimate receiver’s channel outputs and not rate-limited. Besides, [7–9] studied the broadcast channel with feedback where no secure constraints were imposed.

To further investigate the secure data transmission with both common and confidential messages and noiseless feedback, this paper determines the capacity-equivocation regions of the following two DMBCs with both common and confidential messages. They are unsolved in the previous exploration.(i)Degraded DMBCs with rate-limited feedback, where the feedback rate is limited by 𝑅_𝑓 and the feedback is independent of the channel outputs; see Figure 2.(ii)Less and reversely less noisy (let 𝑋 be the input of the DMBC, 𝑌 the legitimate receiver’s channel output, and 𝑍 the eavesdropper’s channel output. A DMBC 𝑝(𝑦,𝑧∣𝑥) is said to be less noisy if 𝐼(𝑈;𝑌)≥𝐼(𝑈;𝑍) for all 𝑝(𝑢,𝑥); a DMBC 𝑝(𝑦,𝑧∣𝑥) is said to be reversely less noisy if 𝐼(𝑈;𝑌)≤𝐼(𝑈;𝑍) for all 𝑝(𝑢,𝑥), where 𝑢 is the value of the auxiliary random variable 𝑈) DMBCs with noiseless causal feedback, where the feedback is originated causally from the legitimate receiver’s channel outputs; see Figure 3.

Figure 2 

Degraded DMBCs with rate-limited feedback.

Figure 3 

Less and reversely less noisy DMBCs with noiseless causal feedback.

The two channel models are characterized in Section 2. The main results presented in Section 2 subsume some important previous findings about the secure data transmission with feedback. (1) By setting the auxiliary random variable 𝑈 to be constant in the secrecy capacity of the first model (see (9) in Remark 3), the secrecy capacity of the degraded wiretap with rate-limited feedback [4] is obtained. (2) By eliminating the common message in the second model, the capacity-equivocation region of restricted wiretap channel with noiseless feedback [5] is obtained. (3) We utilize a simpler and more intuitive deduction to get the single-letter characterization of the capacity-equivocation region, instead of relying on the recursive argument (see [4]) which is complex and not intuitive. (4) We find that even if the eavesdropper is in a better position than the legitimate receiver, provable secure communication could also be implemented in the DMBCs with both common and confidential messages.

The remainder of the paper is organized as follows. Section 2 gives the notations and main results, that is, the capacity-equivocation regions of the two channel models. Section 3 proves Theorem 2. Section 4 proves Theorems 4 and 5. Section 5 concludes the whole work.

2. Channel Models and Main Results

2.1. Notations

Throughout this paper, we use calligraphic letters, for example, 𝒳, 𝒴, to denote the finite sets and ‖𝒳‖ to denote the cardinality of the set 𝒳. Uppercase letters, for example, 𝑋, 𝑌, are used to denote random variables taking values from finite sets, for example, 𝒳, 𝒴. The value of the random variable 𝑋 is denoted by the lowercase letter 𝑥. We use 𝑍^𝑗_𝑖 to denote the (𝑗−𝑖+1)-vectors (𝑍_𝑖,𝑍_𝑖+1,…,𝑍_𝑗) of random variables for 1≤𝑖≤𝑗 and will always drop the subscript when 𝑖=1. Moreover, we use 𝑋∼𝑝(𝑥) to denote the probability mass function of the random variable 𝑋. For 𝑋∼𝑝(𝑥) and 0≤𝜖≤1, the set of the typical 𝑁-sequences 𝑥^𝑁 is defined as 𝒯^𝑁_𝑋(𝜖)={𝑥^𝑁∶|𝜋(𝑥∣𝑥^𝑁)−𝑝(𝑥)|≤𝜖𝑝(𝑥) for all 𝑥∈𝒳}, where 𝜋(𝑥∣𝑥^𝑁) denotes the frequency of occurrences of letter 𝑥 in the sequence 𝑥^𝑁 (for more details about typical sequences, please refer to [10, Chapter 2]). The set of the conditional typical sequences, for example, 𝒯^𝑁_𝑌∣𝑋(𝜖), follows similarly.

2.2. Channel Models and Main Results

This paper studies the secure data transmission for two subclasses of DMBCs with noiseless feedback. One is the case where the feedback is rate-limited and independent of the channel outputs (see Figure 2); the other is the case where the feedback is originated causally from the channel outputs (see Figure 3). Both models consist of a transmitter and two receivers, named receiver 1 (legitimate receiver) and receiver 2 (eavesdropper). The transmitter aims to convey a common message 𝑊 to both receivers in addition to a confidential message 𝑆 intended only for receiver 1. The confidential message 𝑆 should be kept secret from receiver 2 as much as possible. We use equivocation at receiver 2 to characterize the secrecy of the confidential message. 𝑊 and 𝑆 are mutually independent and uniformly distributed over 𝒲 and 𝒮.

2.2.1. Degraded DMBCs with Rate-Limited Feedback

The degraded DMBCs with rate-limited feedback (see Figure 2) are under the condition that the channel to receiver 2 is physically degraded from the channel to receiver 1; that is, 𝑝(𝑦,𝑧∣𝑥)=𝑝(𝑦∣𝑥)𝑝(𝑧∣𝑦) or 𝑋→𝑌→𝑍 form a Markov chain, where 𝑋 is the channel input and 𝑌,𝑍 are observations of receiver 1 and 2. In this model, the encoder encodes the messages (𝑊,𝑆) and feedback into codewords 𝑋^𝑁, where 𝑁 is the length of the codeword. They are transmitted over a discrete memoryless channel (DMC) with transition probability ∏^𝑁_𝑖=1𝑝(𝑦_𝑖,𝑧_𝑖∣𝑥_𝑖). Receiver 1 obtains 𝑌^𝑁 and decodes the common and confidential messages (𝑊,𝑆). Receiver 2 obtains 𝑍^𝑁 and decodes the common message 𝑊. More precisely, we define the encoder-decoder (𝑁,Δ,𝑃_𝑒1,𝑃_𝑒2) in Definition 1.

Definition 1. The encoder-decoder (𝑁,Δ,𝑃_𝑒1,𝑃_𝑒2) for the degraded DMBCs with rate-limited feedback (with rate limited by 𝑅_𝑓) is defined as follows.(i)The feedback alphabet 𝒦 satisfies lim_𝑁→∞(log‖𝒦‖/𝑁)≤𝑅_𝑓. The feedback is generated independent of the channel output symbols.(ii)The stochastic channel encoder 𝜑 is specified by a matrix of conditional probability distributions 𝜑(𝑥^𝑁∣𝑠,𝑤,𝑘) which denotes the probability that the message 𝑠,𝑤 and the feedback 𝑘 are encoded as the channel input 𝑥^𝑁, where 𝑥^𝑁∈𝒳^𝑁,𝑠∈𝒮,𝑤∈𝒲,𝑘∈𝒦, and ∑_𝑥𝑁𝜑(𝑥^𝑁∣𝑠,𝑤,𝑘)=1. Note that 𝒮 and 𝒲 are the confidential and common message sets.(iii)Decoder 1 is a mapping ℎ₁∶𝒴^𝑁→𝒮×𝒲. The input of decoder 1 is 𝑌^𝑁, and the output is 𝑆,𝑊. The decoding error probability of receiver 1 is defined as 𝑃_𝑒1=Pr{ℎ₁(𝑌^𝑁)≠(𝑆,𝑊)}. Similarly, Decoder 2 is defined as a mapping ℎ₂∶𝒵^𝑁→𝒲. The input of decoder 2 is 𝑍^𝑁, and the output is 𝑊. The decoding error probability of receiver 2 is defined as 𝑃_𝑒2=Pr{ℎ₂(𝑍^𝑁)≠𝑊}.(iv)The equivocation at receiver 2 is defined as 1Δ=𝑁𝐻𝑆∣𝑍^𝑁. (1)

A rate triple (𝑅₀,𝑅₁,𝑅_𝑒) is said to be achievable for the model in Figure 2 if there exists a channel encoder-decoder (𝑁,Δ,𝑃_𝑒1,𝑃_𝑒2) defined in Definition 1, such that lim_𝑁→∞log‖𝒲‖𝑁=𝑅₀,lim_𝑁→∞log‖𝒮‖𝑁=𝑅₁,lim_𝑁→∞log‖𝒦‖𝑁=𝑅^′_𝑓≤𝑅_𝑓,lim_𝑁→∞Δ≥𝑅_𝑒,𝑃_𝑒1≤𝜖,𝑃_𝑒2≤𝜖, (2)(3)(4)(5)(6) where 𝜖 is an arbitrary small positive real number, 𝑅₀,𝑅₁,𝑅^′_𝑓 are the rates of the common messages, confidential messages, and feedback, and 𝑅_𝑒 is the equivocation rate of the confidential messages. Note that the feedback rate is limited by 𝑅_𝑓. The capacity-equivocation region is defined as the convex closure of all achievable rate triples (𝑅₀,𝑅₁,𝑅_𝑒). The capacity-equivocation region of the degraded DMBCs with rate-limited feedback is shown in the following theorem.

Theorem 2. For the degraded DMBCs with limited feedback rate 𝑅_𝑓, the capacity-equivocation region is the set ℛ_𝑑=𝑅₀,𝑅₁,𝑅_𝑒∶0≤𝑅_𝑒≤𝑅₁,𝑅₀(),𝑅≤𝐼𝑈;𝑍₁(),𝑅≤𝐼𝑋;𝑌∣𝑈_𝑒≤𝐼(𝑋;𝑌∣𝑈)−𝐼(𝑋;𝑍∣𝑈)+𝑅_𝑓, (7) where 𝑈 is an auxiliary random variable and 𝑈→𝑋→𝑌→𝑍 form a Markov chain.

The proof of Theorem 2 is given in Section 3. The remark of Theorem 2 is shown below.

Remark 3. (i) The secrecy capacity of the model in Figure 2 is defined as the maximum rate at which confidential messages can be sent to receiver 1 in perfect secrecy; that is, 𝐶_𝑠=max_{(𝑅0=0,𝑅1,𝑅𝑒=𝑅1)∈ℛ}𝑅₁, (8) where ℛ is the capacity-equivocation region. Therefore, by the definition in (8), the secrecy capacity of the degraded DMBCs with limited feedback rate 𝑅_𝑓 is 𝐶_𝑠𝑑=maxmin𝐼(𝑋;𝑌∣𝑈),𝐼(𝑋;𝑌∣𝑈)−𝐼(𝑋;𝑍∣𝑈)+𝑅_𝑓. (9) This result subsumes the secrecy capacity of the degraded wiretap channel with rate-limited feedback (see [4]) by setting the auxiliary random variable 𝑈 to be constant in (9).
(ii) The capacity-equivocation region in (7) is bigger than that in [2] without feedback. This implies that feedback can be used to enhance the secrecy in the DMBCs. Note that this finding had already been verified in [3–6].

2.2.2. Less and Reversely Less Noisy DMBCs with Noiseless Causal Feedback

The model in Figure 3 is based on the assumption that the channel to receiver 1 is independent of the channel to receiver 2; that is, 𝑝(𝑦,𝑧∣𝑥)=𝑝(𝑦∣𝑥)𝑝(𝑧∣𝑥). The definition of the encoder-decoder for this model is similar to Definition 1 except for the feedback and the encoder. Different from the model in Figure 2, the feedback in Figure 3 is originated causally from the channel outputs of receiver 1 to the transmitter. The stochastic encoder for this model at time 𝑖, 1≤𝑖≤𝑁, is defined as 𝑓_𝑖(𝑥_𝑖∣𝑤_𝑖,𝑠_𝑖,𝑦^𝑖−1), where 𝑤_𝑖∈𝒲, 𝑠_𝑖∈𝒮, 𝑦^𝑖−1∈𝒴^𝑖−1 (the channel outputs of receiver 1 before time 𝑖) and ∑_{𝑥𝑖∈𝒳}𝑓_𝑖(𝑥_𝑖∣𝑤_𝑖,𝑠_𝑖,𝑦^𝑖−1)=1.

A rate triple (𝑅₀,𝑅₁,𝑅_𝑒) is said to be achievable for the model in Figure 3 if there exists a channel encoder-decoder (𝑁,Δ,𝑃_𝑒1,𝑃_𝑒2) such that (2), (3), (5), and (6) hold. Note that the definition of “achievable” here does not include (4) since the feedback in the model of Figure 3 is not rate limited. The definition of secrecy capacity is the same as that in Remark 3. Then, we present the capacity-equivocation regions of the less and reversely less noisy DMBCs with noiseless causal feedback in Theorems 4 and 5, respectively.

Theorem 4. For the less noisy DMBCs with noiseless causal feedback, the capacity-equivocation region is the set ℛ_𝑙=𝑅₀,𝑅₁,𝑅_𝑒∶0≤𝑅_𝑒≤𝑅₁,𝑅₀(),𝑅≤𝐼𝑈;𝑍₁(),𝑅≤𝐼𝑋;𝑌∣𝑈_𝑒≤𝐻(𝑌∣𝑍), (10) where 𝑈→𝑋→(𝑌,𝑍) form a Markov chain.

Theorem 5. For the reversely less noisy DMBCs with noiseless causal feedback, the capacity-equivocation region is the set ℛ_𝑟𝑙=𝑅₀,𝑅₁,𝑅_𝑒∶0≤𝑅_𝑒≤𝑅₁,𝑅₀(),𝑅≤𝐼𝑈;𝑌₁(),𝑅≤𝐼𝑋;𝑌∣𝑈_𝑒≤𝐻(𝑌∣𝑋), (11) where 𝑈→𝑋→(𝑌,𝑍) form a Markov chain.

The proof of Theorems 4 and 5 is given in Section 4. The remark of Theorems 4 and 5 is given below.

Remark 6. (i) By the definition in (8), the secrecy capacity of the less noisy DMBCs with noiseless causal feedback is 𝐶_𝑠𝑙=maxmin{𝐼(𝑋;𝑌∣𝑈),𝐻(𝑌∣𝑍)}. (12) The secrecy capacity of the reversely less noisy DMBCs with noiseless causal feedback is 𝐶_𝑠𝑟𝑙=maxmin{𝐼(𝑋;𝑌∣𝑈),𝐻(𝑌∣𝑋)}. (13) Setting the auxiliary random variable 𝑈 to be constant in (12) and (13), the capacity-equivocation region of the model in [5] is obtained.
(ii) In the model of Figure 3, it is assumed that the channel to receiver 1 is independent of the channel to receiver 2; that is, 𝑝(𝑦,𝑧∣𝑥)=𝑝(𝑦∣𝑥)𝑝(𝑧∣𝑥). This implies 𝑌→𝑋→𝑍. Therefore, it is easy to see 𝐻(𝑌∣𝑋)=𝐻(𝑌∣𝑋𝑍)≤𝐻(𝑌∣𝑍); that is, the upper bound on the equivocation rate 𝑅_𝑒 in (11) for the reversely less noisy case is smaller than that in (10) for the less noisy case. This tells that when the eavesdropper is in a better position than the legitimate receiver (see the reversely less noisy case), the uncertainty about the confidential messages at the eavesdropper is decreased. Besides, from (13), we see that even if the eavesdropper is in a better position, the secrecy capacity is a positive value, which means provable secure communication could also be implemented in such a bad condition.

3. Proof of Theorem 2

In this section, Theorem 2 is proved. The converse part of Theorem 2 gives the outer bound on the capacity-equivocation region of the degraded DMBCs with rate-limited feedback. The proof of the converse part is shown in Section 3.1. The key tools used in the proof include the identification of the random variables and Csiszár’s sum equality [2]. In Section 3.2, to prove the direct part of Theorem 2, a coding scheme is provided to achieve the achievable rate triples in ℛ_𝑑. The key ideas in the coding scheme are inspired by [4]. However, [4] only considers the transmission of the confidential messages. Our coding scheme considers both the confidential and common messages.

3.1. The Converse Part of Theorem 2

In order to find the identification of the auxiliary random variables that satisfy the capacity-equivocation region characterized by ℛ_𝑑, we prove the converse part for the equivalent region (the fact that the two regions are equivalent follows similarly from [10, Chapter 5, problem 5.8]) containing all the rate triples (𝑅₀,𝑅₁,𝑅_𝑒) such that 0≤𝑅_𝑒≤𝑅₁,𝑅₀(),𝑅≤𝐼𝑈;𝑍₀+𝑅₁()(),𝑅≤𝐼𝑋;𝑌∣𝑈+𝐼𝑈;𝑍_𝑒≤𝐼(𝑋;𝑌∣𝑈)−𝐼(𝑋;𝑍∣𝑈)+𝑅_𝑓. (14)(15)(16)(17)

Now we show that all achievable triples (𝑅₀,𝑅₁,𝑅_𝑒) satisfy (14), (15), (16), and (17).

Condition (14) is proved as follows: 𝑅_𝑒≤lim_𝑁→∞Δ=lim_𝑁→∞𝐻𝑆∣𝑍^𝑁𝑁≤lim_𝑁→∞𝐻(𝑆)𝑁=𝑅₁. (18)

To prove condition (15), we calculate 𝐻(𝑊)=𝐼𝑊;𝑍^𝑁+𝐻𝑊∣𝑍^𝑁≤^(𝑎3.1)𝐼𝑊;𝑍^𝑁+𝜖₁=^𝑁_𝑖=1𝐼𝑊;𝑍_𝑖∣𝑍^𝑖−1+𝜖₁=^𝑁_𝑖=1𝐼𝑊;𝑍_𝑖∣𝑍^𝑁_𝑖+1+𝜖₁=^𝑁_𝑖=1𝐼𝑊𝑌^𝑖−1;𝑍_𝑖∣𝑍^𝑁_𝑖+1−𝐼𝑌^𝑖−1;𝑍_𝑖∣𝑍^𝑁_𝑖+1𝑊+𝜖₁≤^𝑁_𝑖=1𝐼𝑊𝑌^𝑖−1𝑍^𝑁_𝑖+1;𝑍_𝑖−𝐼𝑌^𝑖−1;𝑍_𝑖∣𝑍^𝑁_𝑖+1𝑊+𝜖₁≤^𝑁_𝑖=1𝐼𝑊𝐾^𝑁𝑌^𝑖−1𝑍^𝑁_𝑖+1;𝑍_𝑖+𝜖₁, (19) where (𝑎3.1) follows from Fano’s inequality and 𝜖₁ is a small positive number. Note that 𝐾^𝑁=(𝐾₁,𝐾₂,…,𝐾_𝑁), where 𝐾_𝑖 is the feedback symbol at time 𝑖, 1≤𝑖≤𝑁.

To prove condition (16), we consider 𝐻(𝑆)+𝐻(𝑊)=𝐻𝑆∣𝑊𝐾^𝑁+𝐻(𝑊)=𝐼𝑆;𝑌^𝑁∣𝑊𝐾^𝑁+𝐻𝑆∣𝑌^𝑁𝑊𝐾^𝑁+𝐼𝑊;𝑍^𝑁+𝐻𝑊∣𝑍^𝑁≤^(𝑎3.2)𝐼𝑆;𝑌^𝑁∣𝑊𝐾^𝑁+𝜖₂+𝐼𝑊𝐾^𝑁;𝑍^𝑁+𝜖₁=^𝑁_𝑖=1𝐼𝑆;𝑌_𝑖∣𝑌^𝑖−1𝑊𝐾^𝑁+^𝑁_𝑖=1𝐼𝑊𝐾^𝑁;𝑍_𝑖∣𝑍^𝑁_𝑖+1+𝜖₁+𝜖₂=^𝑁_𝑖=1𝐼𝑆𝑍^𝑁_𝑖+1;𝑌_𝑖∣𝑌^𝑖−1𝑊𝐾^𝑁−𝐼𝑍^𝑁_𝑖+1;𝑌_𝑖∣𝑌^𝑖−1𝑊𝐾^𝑁𝑆+^𝑁_𝑖=1𝐼𝑊𝐾^𝑁𝑌^𝑖−1;𝑍_𝑖∣𝑍^𝑁_𝑖+1−𝐼𝑌^𝑖−1;𝑍_𝑖∣𝑍^𝑁_𝑖+1𝑊𝐾^𝑁+𝜖₁+𝜖₂=^𝑁_𝑖=1𝐼𝑍^𝑁_𝑖+1;𝑌_𝑖∣𝑌^𝑖−1𝑊𝐾^𝑁+𝐼𝑆;𝑌_𝑖∣𝑍^𝑁_𝑖+1𝑌^𝑖−1𝑊𝐾^𝑁−𝐼𝑍^𝑁_𝑖+1;𝑌_𝑖∣𝑌^𝑖−1𝑊𝐾^𝑁𝑆+^𝑁_𝑖=1𝐼𝑊𝐾^𝑁𝑌^𝑖−1;𝑍_𝑖∣𝑍^𝑁_𝑖+1−𝐼𝑌^𝑖−1;𝑍_𝑖∣𝑍^𝑁_𝑖+1𝑊𝐾^𝑁+𝜖₁+𝜖₂≤^𝑁_𝑖=1𝐼𝑍^𝑁_𝑖+1;𝑌_𝑖∣𝑌^𝑖−1𝑊𝐾^𝑁+𝐼𝑆;𝑌_𝑖∣𝑍^𝑁_𝑖+1𝑌^𝑖−1𝑊𝐾^𝑁−𝐼𝑍^𝑁_𝑖+1;𝑌_𝑖∣𝑌^𝑖−1𝑊𝐾^𝑁𝑆+^𝑁_𝑖=1𝐼𝑊𝐾^𝑁𝑌^𝑖−1𝑍^𝑁_𝑖+1;𝑍_𝑖−𝐼𝑌^𝑖−1;𝑍_𝑖∣𝑍^𝑁_𝑖+1𝑊𝐾^𝑁+𝜖₁+𝜖₂=^{𝑁(𝑎3.3)}_𝑖=1𝐼𝑆;𝑌_𝑖∣𝑍^𝑁_𝑖+1𝑌^𝑖−1𝑊𝐾^𝑁−𝐼𝑍^𝑁_𝑖+1;𝑌_𝑖∣𝑌^𝑖−1𝑊𝐾^𝑁𝑆+^𝑁_𝑖=1𝐼𝑊𝐾^𝑁𝑌^𝑖−1𝑍^𝑁_𝑖+1;𝑍_𝑖+𝜖₁+𝜖₂≤^𝑁_𝑖=1𝐼𝑆;𝑌_𝑖∣𝑍^𝑁_𝑖+1𝑌^𝑖−1𝑊𝐾^𝑁+^𝑁_𝑖=1𝐼𝑊𝐾^𝑁𝑌^𝑖−1𝑍^𝑁_𝑖+1;𝑍_𝑖+𝜖₁+𝜖₂, (20) where 𝜖₂ is a small positive number and (𝑎3.2) and (𝑎3.3) follow from Fano’s inequality and Csiszár’s sum equality [2]; that is, ∑^𝑁_𝑖=1𝐼(𝑍^𝑁_𝑖+1;𝑌_𝑖∣𝑌^𝑖−1𝑊𝐾^𝑁)=∑^𝑁_𝑖=1𝐼(𝑌^𝑖−1;𝑍_𝑖∣𝑍^𝑁_𝑖+1𝑊𝐾^𝑁).

To prove condition (17), we calculate 𝐻𝑆∣𝑍^𝑁=𝐻𝑆∣𝑍^𝑁,𝑊+𝐼𝑆;𝑊∣𝑍^𝑁≤𝐻𝑆∣𝑍^𝑁,𝑊+𝐻𝑊∣𝑍^𝑁=𝐼𝑆;𝐾^𝑁,𝑌^𝑁∣𝑍^𝑁,𝑊+𝐻𝑆∣𝑍^𝑁,𝐾^𝑁,𝑌^𝑁,𝑊+𝐻𝑊∣𝑍^𝑁≤𝐼𝑆;𝐾^𝑁,𝑌^𝑁∣𝑍^𝑁,𝑊+𝐻𝑆∣𝐾^𝑁,𝑌^𝑁+𝐻𝑊∣𝑍^𝑁=𝐼𝑆;𝐾^𝑁∣𝑍^𝑁,𝑊+𝐼𝑆;𝑌^𝑁∣𝐾^𝑁,𝑍^𝑁,𝑊+𝐻𝑆∣𝐾^𝑁,𝑌^𝑁+𝐻𝑊∣𝑍^𝑁≤𝐻𝐾^𝑁+𝐼𝑆;𝑌^𝑁∣𝐾^𝑁,𝑍^𝑁,𝑊+𝐻𝑆∣𝐾^𝑁,𝑌^𝑁+𝐻𝑊∣𝑍^𝑁≤𝑁𝑅_𝑓+𝐼𝑆;𝑌^𝑁∣𝐾^𝑁,𝑍^𝑁,𝑊+𝜖₁+𝜖₂. (21)

The last inequality in (21) follows from the Fano’s inequality and the fact that the feedback rate is limited by 𝑅_𝑓. Then, 𝐼(𝑆;𝑌^𝑁∣𝐾^𝑁,𝑍^𝑁,𝑊) will be calculated as follows: 𝐼𝑆;𝑌^𝑁∣𝐾^𝑁,𝑍^𝑁=,𝑊_𝑁𝑖=1𝐼𝑆;𝑌_𝑖∣𝑌^𝑖−1𝑍^𝑁𝑊𝐾^𝑁=^𝑁_𝑖=1𝐼𝑆;𝑌_𝑖∣𝑌^𝑖−1,𝑍^𝑖−1,𝑍_𝑖,𝑍^𝑁_𝑖+1,𝑊,𝐾^𝑁=^{𝑁(𝑎3.4)}_𝑖=1𝐼𝑆;𝑌_𝑖∣𝑌^𝑖−1,𝑍^𝑖−1,𝑍_𝑖,𝑍^𝑁_𝑖+1,𝑊,𝐾^𝑁+𝐼𝑍^𝑖−1;𝑌_𝑖∣𝑌^𝑖−1,𝑍^𝑁_𝑖,𝑊,𝐾^𝑁−𝐼𝑍^𝑖−1;𝑌_𝑖∣𝑌^𝑖−1,𝑍^𝑁_𝑖,𝑊,𝑆,𝐾^𝑁=^𝑁_𝑖=1𝐼𝑆,𝑍^𝑖−1;𝑌_𝑖∣𝑌^𝑖−1,𝑍_𝑖,𝑍^𝑁_𝑖+1,𝑊,𝐾^𝑁−𝐼𝑍^𝑖−1;𝑌_𝑖∣𝑌^𝑖−1,𝑍^𝑁_𝑖,𝑊,𝑆,𝐾^𝑁=^𝑁_𝑖=1𝐼𝑆;𝑌_𝑖∣𝑌^𝑖−1,𝑍_𝑖,𝑍^𝑁_𝑖+1,𝑊,𝐾^𝑁, (22) where (𝑎3.4) follows from the Markov chain 𝑌_𝑖→𝑌^𝑖−1𝑍^𝑁_𝑖𝑊𝐾^𝑁→𝑍^𝑖−1 and 𝑌_𝑖→𝑌^𝑖−1𝑍^𝑁_𝑖𝑊𝑆𝐾^𝑁→𝑍^𝑖−1. Then, we introduce a random variable 𝑄 which is independent of 𝑆𝑊𝐾^𝑁𝑋^𝑁𝑌^𝑁𝑍^𝑁 and uniformly distributed over {1,2,…,𝑁}. Set 𝑈=𝑍^𝑁_𝑄+1𝑌^𝑄−1𝑊𝐾^𝑁𝑄,𝑉=𝑈𝑆,𝑌=𝑌_𝑄,𝑋=𝑋_𝑄,𝑍=𝑍_𝑄. It is straightforward to see that 𝑈→𝑉→𝑋→𝑌→𝑍 form a Markov chain. After using the standard time sharing argument [10, Section 5.4], (19), (20), and (22) are simplified into 𝐻(𝑊)≤^𝑁_𝑖=1𝐼𝑊𝐾^𝑁𝑌^𝑖−1𝑍^𝑁_𝑖+1;𝑍_𝑖+𝜖₁=𝑁𝐼(𝑈;𝑍)+𝜖₁,𝐻(𝑆)+𝐻(𝑊)≤^𝑁_𝑖=1𝐼𝑆;𝑌_𝑖∣𝑍^𝑁_𝑖+1𝑌^𝑖−1𝑊𝐾^𝑁+^𝑁_𝑖=1𝐼𝑊𝐾^𝑁𝑌^𝑖−1𝑍^𝑁_𝑖+1;𝑍_𝑖+𝜖₁+𝜖₂=𝑁𝐼(𝑆;𝑌∣𝑈)+𝑁𝐼(𝑈;𝑍)+𝜖₁+𝜖₂=𝑁𝐼(𝑉;𝑌∣𝑈)+𝑁𝐼(𝑈;𝑍)+𝜖₁+𝜖₂,𝐼𝑆;𝑌^𝑁∣𝐾^𝑁,𝑍^𝑁,𝑊=^𝑁_𝑖=1𝐼𝑆;𝑌_𝑖∣𝑌^𝑖−1,𝑍_𝑖,𝑍^𝑁_𝑖+1,𝑊,𝐾^𝑁()()().=𝑁𝐼𝑆;𝑌∣𝑍,𝑈=𝑁𝐼𝑈𝑆;𝑌∣𝑍,𝑈=𝑁𝐼𝑉;𝑌∣𝑍,𝑈 (23)(24)(25) Substituting (25) into (21) and utilizing (5), we get 𝑅_𝑒≤lim_𝑁→∞Δ=lim_𝑁→∞𝐻𝑆∣𝑍^𝑁𝑁≤lim_𝑁→∞𝑁𝑅_𝑓+𝐼𝑆;𝑌^𝑁∣𝐾^𝑁,𝑍^𝑁,𝑊+𝜖₁+𝜖₂𝑁=𝐼(𝑉;𝑌∣𝑍,𝑈)+𝑅_𝑓=𝐼(𝑉;𝑌∣𝑈)−𝐼(𝑉;𝑍∣𝑈)+𝑅_𝑓. (26) The last equality in (26) follows from the Markov chain 𝑈→𝑉→𝑌→𝑍.

To finish the proof of (16) and (17), we need to show that 𝐼(𝑉;𝑌∣𝑈)≤𝐼(𝑋;𝑌∣𝑈) and 𝐼(𝑉;𝑌∣𝑈)−𝐼(𝑉;𝑍∣𝑈)≤𝐼(𝑋;𝑌∣𝑈)−𝐼(𝑋;𝑍∣𝑈). We first prove 𝐼(𝑉;𝑌∣𝑈,𝑋)=0 and 𝐼(𝑉;𝑍∣𝑈,𝑋)=0: 𝐼()()()=𝑉;𝑌∣𝑈,𝑋=𝐻𝑌∣𝑈,𝑋−𝐻𝑌∣𝑈,𝑉,𝑋_(𝑎3.5)𝐻()()𝐼()()()=𝑌∣𝑋−𝐻𝑌∣𝑋=0,𝑉;𝑍∣𝑈,𝑋=𝐻𝑍∣𝑈,𝑋−𝐻𝑍∣𝑈,𝑉,𝑋_(𝑎3.6)𝐻(𝑍∣𝑋)−𝐻(𝑍∣𝑋)=0, (27) where (𝑎3.5) follows from the Markov chains 𝑈→𝑋→𝑌 and (𝑈𝑉)→𝑋→𝑌 and (𝑎3.6) follows from the Markov chains 𝑈→𝑋→𝑍 and (𝑈𝑉)→𝑋→𝑍. Utilizing (27), we obtain 𝐼()()()()()()()(),𝐼()()()()()()()().𝑉;𝑌∣𝑈=𝐼𝑉,𝑋;𝑌∣𝑈−𝐼𝑋;𝑌∣𝑈,𝑉=𝐼𝑋;𝑌∣𝑈+𝐼𝑉;𝑌∣𝑈,𝑋−𝐼𝑋;𝑌∣𝑈,𝑉=𝐼𝑋;𝑌∣𝑈−𝐼𝑋;𝑌∣𝑈,𝑉𝑉;𝑍∣𝑈=𝐼𝑉,𝑋;𝑍∣𝑈−𝐼𝑋;𝑍∣𝑈,𝑉=𝐼𝑋;𝑍∣𝑈+𝐼𝑉;𝑍∣𝑈,𝑋−𝐼𝑋;𝑍∣𝑈,𝑉=𝐼𝑋;𝑍∣𝑈−𝐼𝑋;𝑍∣𝑈,𝑉 (28)(29)

From (28), it is straightforward to see that 𝐼(𝑉;𝑌∣𝑈)≤𝐼(𝑋;𝑌∣𝑈). This proves condition (16).

Then, we prove 𝐼(𝑉;𝑌∣𝑈)−𝐼(𝑉;𝑍∣𝑈)≤𝐼(𝑋;𝑌∣𝑈)−𝐼(𝑋;𝑍∣𝑈). Since the channel model in Figure 1 is (physically) degraded, 𝐼(𝑋;𝑌∣𝑈=𝑢,𝑉=𝑣)−𝐼(𝑋;𝑍∣𝑈=𝑢,𝑉=𝑣)≥0 holds for every (𝑢,𝑣), which implies 𝐼(𝑋;𝑌∣𝑈,𝑉)−𝐼(𝑋;𝑍∣𝑈,𝑉)≥0. (30)

Therefore, utilizing (28), (29), and (30), we get 𝐼()()()()−[𝐼()(()().𝑉;𝑌∣𝑈−𝐼𝑉;𝑍∣𝑈=𝐼𝑋;𝑌∣𝑈−𝐼𝑋;𝑍∣𝑈𝑋;𝑌∣𝑈,𝑉−𝐼𝑋;𝑍∣𝑈,𝑉)]≤𝐼𝑋;𝑌∣𝑈−𝐼𝑋;𝑍∣𝑈 (31) This proves condition (17).

The converse part of Theorem 2 is proved.

3.2. A Coding Scheme Achieving ℛ_𝑑

A coding scheme is provided to achieve the achievable triples (𝑅₀,𝑅₁,𝑅_𝑒)∈ℛ_𝑑. The key methods used in the scheme include the superposition coding, rate splitting, and random binning. The confidential message is split into two parts. One part is reliably transmitted using superposition coding and random binning; the other part is securely transmitted with the help of the feedback. Note that Section 3.1 has already given the outer bound on the capacity-equivocation region. When 𝑅_𝑓≥𝐼(𝑋;𝑍∣𝑈), it can be seen from (9) that the secrecy capacity for the degraded DMBCs with rate-limited feedback always equals to 𝐼(𝑋;𝑌∣𝑈). Therefore, in order to investigate the effects of the feedback, the feedback rate 𝑅_𝑓<𝐼(𝑋;𝑍∣𝑈) will only be considered in this subsection.

We need to prove that all the triples (𝑅₀,𝑅₁,𝑅_𝑒)∈ℛ_𝑑 for the model of Figure 2 with any feedback rate 𝑅^′_𝑓 limited by 𝑅_𝑓 are achievable (see Definition 1). This subsection is organized as follows. The codebook generation and encoding scheme is given in Section 3.2.1. The decoding scheme is given in Section 3.2.2. The analysis of error probability and equivocation are shown in Sections 3.2.3 and 3.2.4, respectively.

3.2.1. Codebook Generation and Encoding

Split the confidential message into two parts; that is, 𝒮=(ℳ₁,ℳ₂). The corresponding variables 𝑀₁, 𝑀₂ are uniformly distributed over {1,2,3,…,2^𝑁𝑅′} and {1,2,3,…,2^{𝑁𝑅′𝑓}}, where (when 𝑅_𝑓≥𝑅₁, the confidential message 𝑆 can be totally protected by using part of the feedback (as the shared key between the transmitter and receiver 1). The remaining part of the feedback is redundant. Therefore, in order to study the effects of the feedback on the capacity region, only 𝑅_𝑓<𝑅₁ comes into our consideration) 0≤𝑅^′_𝑓≤𝑅_𝑓,𝑅^′=𝑅₁−𝑅^′_𝑓>0. (32)

It is important to notice that 𝑅₁ is the rate of the private message 𝒮, which consists of ℳ₁ and ℳ₂. This means that 𝑅₁=lim_𝑁→∞logℳ₁ℳ₂𝑁=lim_𝑁→∞logℳ₁𝑁+logℳ₂𝑁. (33)

Define the index sets 𝒥_𝑁, ℒ_𝑁, ℱ_𝑁, and ℳ_𝑁 satisfying lim_𝑁→∞1𝑁log𝒥_𝑁=𝐼(𝑋;𝑍∣𝑈)−𝑅^′_𝑓,lim_𝑁→∞1𝑁logℒ_𝑁=𝐼(𝑋;𝑌∣𝑈)−𝐼(𝑋;𝑍∣𝑈),lim_𝑁→∞1𝑁logℱ_𝑁=𝑅^′_𝑓,lim_𝑁→∞1𝑁logℳ_𝑁=𝐼(𝑈;𝑍). (34)

We use 𝑗∈𝒥_𝑁, 𝑙∈ℒ_𝑁, 𝑓∈ℱ_𝑁, 𝑚∈ℳ_𝑁 to index the codeword 𝑥^𝑁. Take 𝒲⊂ℳ_𝑁 such that (2) holds. Since 𝑅₁≤𝐼(𝑋;𝑌∣𝑈), it is easy to see ‖𝒥_𝑁×ℒ_𝑁×ℱ_𝑁‖≥2^𝑁𝑅1. Therefore, let ℳ₁=𝒟_𝑁×ℒ_𝑁, ℳ₂=ℱ_𝑁, where 𝒟_𝑁 is an arbitrary set such that (3) holds. Let 𝑔_𝑗 be a mapping of 𝒥_𝑁 into 𝒟_𝑁 partitioning 𝒥_𝑁 into subsets of size ‖𝒥_𝑁‖/‖𝒟_𝑁‖; that is, 𝑔_𝑗∶𝒥_𝑁⟶𝒟_𝑁, (35) where 𝑔_𝑗(𝑗)=𝑑,𝑗∈𝒥_𝑁,𝑑∈𝒟_𝑁.

For each 𝑤∈𝒲, we generate a codeword 𝑢^𝑁(𝑤) according to ∏^𝑁_𝑖=1𝑝(𝑢_𝑖). Then, for each 𝑢^𝑁(𝑤), a codebook 𝒞ℬ_𝑤 (see Figure 4) containing ‖𝒥_𝑁‖⋅‖ℒ_𝑁‖⋅‖ℱ_𝑁‖ codewords 𝑥^𝑁_{𝑗𝑙𝑓𝑚} is constructed according to ∏^𝑁_𝑖=1𝑝(𝑥_𝑖∣𝑢_𝑖), where 𝑗∈𝒥_𝑁,𝑙∈ℒ_𝑁,𝑓∈ℱ_𝑁,𝑚=𝑤∈ℳ. Those 𝑥^𝑁 are put into ‖ℒ_𝑁‖⋅‖ℱ_𝑁‖ bins so that each bin contains ‖𝒥_𝑁‖ codewords. Each bin is indexed by (𝑙,𝑓), where 𝑙∈ℒ_𝑁, 𝑓∈ℱ_𝑁. Then, we divide each bin into ‖𝒟_𝑁‖ subbins such that each subbin contains ‖𝒥_𝑁‖/‖𝒟_𝑁‖ codewords. The codebook structure is presented in Figure 4.

Figure 4 

The codebook 𝒞ℬ𝑤 for each 𝑢𝑁(𝑤).

Let 𝒦={1,2,3,…,2^{𝑁𝑅′𝑓}}, where 𝑘∈𝒦 is the key sent to the transmitter from receiver 1 through the secure feedback link. It is kept secret from receiver 2. The corresponding variable 𝐾 is uniformly distributed over 𝒦 and independent of 𝑆 and 𝑊.

In order to send 𝑠=(𝑑,𝑙,𝑚₂)∈𝒟_𝑁×ℒ_𝑁×ℱ_𝑁 and 𝑤∈𝒲, a codeword 𝑥^𝑁_{𝑗𝑙𝑓𝑚} is chosen as follows. According to the common message 𝑤, we first find the sequence 𝑢^𝑁(𝑤). For the determined 𝑢^𝑁(𝑤), there is a corresponding codebook 𝒞ℬ_𝑤; see Figure 4. Then, the corresponding codeword 𝑥^𝑁_{𝑗𝑙𝑓𝑚} is sent into the channel, where 𝑗 is chosen randomly from the set 𝑔_𝑗⁻¹(𝑑), 𝑓=𝑘⊕𝑚₂, and 𝑚=𝑤 (here ⊕ is modulo addition over ℱ_𝑁). Figure 4 shows how to select 𝑥^𝑁_{𝑗𝑙𝑓𝑚} in detail. According to 𝑢^𝑁(𝑤), we can find the corresponding codebook 𝒞ℬ_𝑤. In the codebook 𝒞ℬ_𝑤, we choose the corresponding bin according to 𝑓 and 𝑙. Then, in that bin, the subbin is found according to 𝑑. Finally, a codeword 𝑥^𝑁 (which is denoted by 𝑥^𝑁_{𝑗𝑙𝑓𝑚}) is randomly chosen from that subbin.

3.2.2. Decoding

Receiver 2 tries to find a unique sequence 𝑢^𝑁(𝑤) such that (𝑢^𝑁(𝑤),𝑧^𝑁)∈𝑇^𝑁_𝑈𝑍(𝜖₁). If there exists such a unique sequence, decoder 2 outputs 𝑤; otherwise, an error is declared. Since the size of 𝒲 is smaller than 2^{𝑁𝐼(𝑈;𝑍)}, the decoding error probability for receiver 2 approaches zero.

For receiver 1, he can also decode the common message 𝑤 since the output of channel 2 is a degraded version of the output of channel 1. Then, receiver 1 tries to find a unique codeword 𝑥^𝑁̂𝑗𝑙𝑓̂𝑚 indexed by 𝑗̂, 𝑙̂, 𝑓, 𝑚, such that (𝑥^𝑁̂𝑗𝑙𝑓̂𝑚,𝑦^𝑁)∈𝑇^𝑁_{𝑋𝑌∣𝑈}(𝜖₂). If there exists such a unique codeword 𝑥^𝑁̂𝑗𝑙𝑓̂𝑚, receiver 1 calculates 𝑓⊖𝑘 as 𝑚₂ (here ⊖ is modulo subtraction over ℱ_𝑁, and 𝑚=𝑤) and finds 𝑑 according to 𝑔_𝑗(𝑗)̂. Note that receiver 1 knows the secret key 𝑘. Decoder 1 outputs 𝑠̂=(𝑑,𝑙,̂𝑚₂) and 𝑤. If no such 𝑥^𝑁̂𝑗𝑙𝑓̂𝑚 or more than one such 𝑥^𝑁̂𝑗𝑙𝑓̂𝑚 exist, an error is declared.

3.2.3. Analysis of Error Probability

Since the number of 𝑢^𝑁(𝑤) is upper bound by 2^{𝑁𝐼(𝑈;𝑍)} and the DMBCs under discussion are degraded, both receivers can decode the common message 𝑤 with error probability approaching zero by applying the standard channel coding theorem [11, Theorem 7.7.1]. Moreover, it can be calculated that given the codeword 𝑢^𝑁(𝑤), the number of 𝑥^𝑁 is ℱ_𝑁⋅𝒥_𝑁⋅ℒ_𝑁=2^{𝑁𝐼(𝑋;𝑌∣𝑈)}. (36)

So, after determining the codeword 𝑢^𝑁(𝑤), receiver 1 can decode the codeword 𝑥^𝑁 with error probability approaching zero by applying the standard channel coding theorem [11, Theorem 7.7.1]. This proves (6).

3.2.4. Analysis of Equivocation

The proof of (5) is given below: 𝐻𝑆∣𝑍^𝑁=𝐻𝑀₁,𝑀₂∣𝑍^𝑁=𝐻𝑀₁∣𝑍^𝑁+𝐻𝑀₂∣𝑍^𝑁,𝑀₁≥𝐻𝑀₁∣𝑍^𝑁+𝐻𝑀₂∣𝑍^𝑁,𝑀₁,𝐾⊕𝑀₂=^(𝑏3.1)𝐻𝑀₁∣𝑍^𝑁+𝐻𝑀₂∣𝐾⊕𝑀₂=^(𝑏3.2)𝐻𝑀₁∣𝑍^𝑁+𝐻𝑀₂=^(𝑏3.3)𝐻𝑀₁∣𝑍^𝑁+𝑁𝑅^′_𝑓, (37) where (𝑏3.1) follows from the Markov chain 𝑀₂→𝑀₂⊕𝐾→(𝑍^𝑁,𝑀₁), (𝑏3.2) follows from the fact that 𝑀₂ is independent of 𝑀₂⊕𝐾, and (𝑏3.3) follows from that 𝑀₂ is uniformly distributed over {1,2,3,…,2^{𝑁𝑅′𝑓}}. The proof of the fact that 𝑀₂ is independent of 𝑀₂⊕𝐾 is shown as follows (the proof can also be seen in [6]): 𝑝𝑀₂=⊕𝐾=𝑎_𝑘𝑝𝑀₂𝑝()=⊕𝐾=𝑎∣𝐾=𝑘𝐾=𝑘_{(𝑏3.4)𝑘}𝑝𝑀₂1⊕𝐾=𝑎∣𝐾=𝑘ℱ_𝑁=1ℱ_𝑁𝑘𝑝𝑀₂=1⊕𝐾=𝑎∣𝐾=𝑘ℱ_𝑁𝑘𝑝𝑀₂==𝑎⊖𝑘∣𝐾=𝑘_(𝑏3.5)1ℱ_𝑁𝑘𝑝𝑀₂=1=𝑎⊖𝑘ℱ_𝑁𝑝𝑀₂⊕𝐾=𝑎,𝑀₂=𝑚₂=𝑝𝐾=𝑎⊖𝑚₂,𝑀₂=𝑚₂=^(𝑏3.6)𝑝𝐾=𝑎⊖𝑚₂𝑝𝑀₂=𝑚₂=^(𝑏3.7)1ℱ_𝑁⋅1ℱ_𝑁, (38) where (𝑏3.5) and (𝑏3.6) follow from that 𝑀₂ is independent of 𝐾, and (𝑏3.4) and (𝑏3.7) follow from that 𝑀₂ and 𝐾 are both uniformly distributed over ℱ_𝑁. According to (38), 𝑝𝑀₂⊕𝐾=𝑎,𝑀₂=𝑚₂=𝑝𝑀₂⊕𝐾=𝑎𝑝𝑀₂=𝑚₂. (39) Therefore, 𝑀₂ is independent of 𝑀₂⊕𝐾.

Next, we focus on the first term in (37). The method of the equivocation analysis in [2] will be used: 𝐻𝑀₁∣𝑍^𝑁≥𝐻𝑀₁∣𝑍^𝑁𝑀,𝑊=𝐻₁,𝑍^𝑁∣𝑊−𝐻𝑍^𝑁𝑀∣𝑊=𝐻₁,𝑍^𝑁,𝑋^𝑁∣𝑊−𝐻𝑋^𝑁∣𝑀₁,𝑍^𝑁𝑍,𝑊−𝐻^𝑁𝑀∣𝑊=𝐻₁,𝑋^𝑁∣𝑊+𝐻𝑍^𝑁∣𝑀₁,𝑋^𝑁𝑋,𝑊−𝐻^𝑁∣𝑀₁,𝑍^𝑁,𝑊−𝐻𝑍^𝑁𝑋∣𝑊≥𝐻^𝑁∣𝑊+𝐻𝑍^𝑁∣𝑀₁,𝑋^𝑁𝑋,𝑊−𝐻^𝑁∣𝑀₁,𝑍^𝑁,𝑊−𝐻𝑍^𝑁∣𝑊. (40) Note that 𝑊 in inequality (40) is the random variable of the common message 𝒲. The four terms 𝐻(𝑋^𝑁∣𝑊), 𝐻(𝑍^𝑁∣𝑀₁,𝑋^𝑁,𝑊), 𝐻(𝑋^𝑁∣𝑀₁,𝑍^𝑁,𝑊), and 𝐻(𝑍^𝑁∣𝑊) will be bounded as follows.

Given 𝑤∈𝒲, the number of 𝑥^𝑁 is ‖𝒥_𝑁‖⋅‖ℒ_𝑁‖⋅‖ℱ_𝑁‖. By applying [12, Lemma 2.5], we obtain 𝐻𝑋^𝑁∣𝑊≥log𝒥_𝑁⋅ℒ_𝑁⋅ℱ_𝑁()−1=𝑁𝐼𝑋;𝑌∣𝑈−1. (41)

Since (𝑀₁,𝑊)→𝑋^𝑁→𝑍^𝑁 and the channel to receiver 2 is discrete memoryless, it is easy to get 𝐻𝑍^𝑁∣𝑀₁,𝑋^𝑁,𝑊=𝐻𝑍^𝑁∣𝑋^𝑁=𝑁𝐻(𝑍∣𝑋). (42)

With the knowledge of (𝑑,𝑙)∈ℳ₁ and 𝑤∈𝒲, the number of 𝑥^𝑁 is 2^{𝑁𝑅′𝑓}⋅𝒥_𝑁𝒟_𝑁<2^{𝑁𝑅′𝑓}⋅𝒥_𝑁=2^{𝑁𝑅′𝑓}⋅2^{𝑁(𝐼(𝑋;𝑍∣𝑈)−𝑅′𝑓)}=2^{𝑁𝐼(𝑋;𝑍∣𝑈)}. (43)

So, receiver 2 can decode the codeword 𝑥^𝑁 with error probability approaching zero by using the standard channel coding theorem [11, Theorem 7.7.1]. Therefore, using Fano’s inequality, we get 𝐻𝑋^𝑁∣𝑀₁,𝑍^𝑁,𝑊⟶0. (44)

Moreover, using the similar deduction in [2, Section 4], we get 𝐻𝑍^𝑁∣𝑊≤log𝑇^𝑁_𝑍∣𝑈𝜖₁≤𝑁𝐻(𝑍∣𝑈). (45)

Substituting (41), (42), (44), and (45) into (40), we get 𝐻𝑀₁∣𝑍^𝑁()()()()(),≥𝑁𝐼𝑋;𝑌∣𝑈+𝑁𝐻𝑍∣𝑋−𝑁𝐻𝑍∣𝑈=𝑁𝐼𝑋;𝑌∣𝑈−𝑁𝐼𝑋;𝑍∣𝑈 (46) where the equality in (46) follows from the Markov chain 𝑈→𝑋→𝑍.

Finally, (5) is verified by substituting (46) into (37): lim_𝑁→∞Δ=lim_𝑁→∞𝐻𝑆∣𝑍^𝑁𝑁≥lim_𝑁→∞𝐻𝑀₁∣𝑍^𝑁𝑁+𝑅^′_𝑓≥lim_𝑁→∞𝑁𝐼(𝑋;𝑌∣𝑈)−𝑁𝐼(𝑋;𝑍∣𝑈)𝑁+𝑅^′_𝑓=𝐼(𝑋;𝑌∣𝑈)−𝐼(𝑋;𝑍∣𝑈)+𝑅^′_𝑓≥𝑅_𝑒. (47) This completes the proof of Theorem 2.

4. Proof of Theorems 4 and 5

In this section, Theorems 4 and 5 are proved. In the model of Figure 3, it is assumed that the channel to receiver 1 is independent of the channel to receiver 2; that is, 𝑝(𝑦,𝑧∣𝑥)=𝑝(𝑦∣𝑥)𝑝(𝑧∣𝑥). To prove Theorem 4, we first give the outer bound on the capacity-equivocation region of the less noisy DMBCs with noiseless causal feedback in Section 4.1. Then, a coding scheme is provided to achieve the outer bound. Similarly, to prove Theorem 5, the outer bound on the capacity-equivocation region of the reversely less noisy DMBCs with noiseless causal feedback is given in Section 4.2. Moreover, we also provide a coding scheme to achieve the outer bound. The methods used to prove the converse parts of the two theorems are from [5]. The coding schemes are inspired by [3, 5].

4.1. Less Noisy DMBCs with Noiseless Causal Feedback

We first show the converse part of Theorem 4, and then we prove the direct part of Theorem 4 by providing a coding scheme.

In order to find the identification of the auxiliary random variables that satisfy the capacity-equivocation region characterized by ℛ_𝑙, we prove the converse part for the equivalent region containing all the rate triples (𝑅₀,𝑅₁,𝑅_𝑒) such that 0≤𝑅_𝑒≤𝑅₁,𝑅₀(),𝑅≤𝐼𝑈;𝑍₀+𝑅₁()(),𝑅≤𝐼𝑋;𝑌∣𝑈+𝐼𝑈;𝑍_𝑒≤𝐻(𝑌∣𝑍). (48)(49)(50)(51)

The proof of (48), (49), and (50) follows exactly the same line of proving (14), (15), and (16) in Section 3 except for the identification of the auxiliary random variable 𝑈,𝑉 (which will be given subsequently) and therefore is omitted. We focus on proving (51): 𝐻𝑆∣𝑍^𝑁≤𝐻𝑆∣𝑍^𝑁+𝐼𝑆;𝑍^𝑁∣𝑌^𝑁=𝐻𝑆∣𝑍^𝑁+𝐻𝑆∣𝑌^𝑁−𝐻𝑆∣𝑌^𝑁,𝑍^𝑁=𝐼𝑆;𝑌^𝑁∣𝑍^𝑁+𝐻𝑆∣𝑌^𝑁≤𝐻𝑌^𝑁∣𝑍^𝑁+𝐻𝑆∣𝑌^𝑁=^𝑁_𝑖=1𝐻𝑌_𝑖∣𝑌^𝑖−1,𝑍^𝑁+𝐻𝑆∣𝑌^𝑁≤^𝑁_𝑖=1𝐻𝑌_𝑖∣𝑍_𝑖+𝜖₃, (52) where 𝜖₃ is a small positive number. The last inequality in (52) follows from the fact that conditioning does not increase entropy and Fano’s inequality. To complete the proof of (51), define a time-sharing random variable 𝑄 which is uniformly distributed over 1,2,…,𝑁 and independent of 𝑆𝑊𝑋^𝑁𝑌^𝑁𝑍^𝑁. Set 𝑈=𝑍^𝑁_𝑄+1𝑌^𝑄−1𝑊𝑄, 𝑉=𝑈𝑆, 𝑋=𝑋_𝑄, 𝑌=𝑌_𝑄, 𝑍=𝑍_𝑄. It is easy to see 𝑈→𝑉→𝑋→(𝑌,𝑍) form a Markov chain. After using the standard time-sharing argument [10, Section 5.4], (52) simplifies to 𝐻𝑆∣𝑍^𝑁≤𝑁𝐻(𝑌∣𝑍)+𝜖₃. (53)

Finally, utilizing lim_𝑁→∞Δ≥𝑅_𝑒 in the definition of “achievable” and (53), we obtain (51). This completes the proof of the converse part of Theorem 4.

Next, a coding scheme is presented to achieve the rate triple (𝑅₀,𝑅₁,𝑅_𝑒)∈ℛ_𝑙. We should prove that all triples (𝑅₀,𝑅₁,𝑅_𝑒)∈ℛ_𝑙 are achievable. Note that the noiseless feedback for the less noisy DMBCs is causally transmitted from receiver 1 to the transmitter. The scheme includes codebook generation and encoding scheme in Section 4.1.1, decoding scheme in Section 4.1.2, analysis of error probability in Section 4.1.3, and equivocation analysis in Section 4.1.4. Techniques like block Markov coding, superposition coding, and random binning are used.

To serve the block Markov coding, let random vectors 𝑈^𝑁, 𝑋^𝑁, 𝑌^𝑁, and 𝑍^𝑁 consist of 𝑛 blocks of length 𝑁. Let 𝑊^𝑛≜(𝑊₁,…,𝑊_𝑛) stand for the common messages of 𝑛 blocks, where 𝑊₁,…,𝑊_𝑛 are independent and identically distributed random variables over 𝒲. Let 𝑆^𝑛≜(𝑆₂,…,𝑆_𝑛) stand for the confidential messages of 𝑛 blocks, where 𝑆₂,…,𝑆_𝑛 are independent and identically distributed random variables over 𝒮. Note that in the first block, there is no 𝑆₁. Let 𝑍^𝑛=(𝑍₁,𝑍₂,…,𝑍_𝑛), 𝑍^𝑏=(𝑍₁,𝑍₂,…,𝑍_𝑏−1,𝑍_𝑏+1,…,𝑍_𝑛), where 𝑍_𝑏 is the output vector at receiver 2 at the end of the 𝑏th block, where 1≤𝑏≤𝑛. Similarly, 𝑌_𝑏 denotes the output vector at receiver 1 at the end of the 𝑏th block, and 𝑋_𝑏 denotes the input vector of the channel in the 𝑏th block. These notations coincide with [6].

4.1.1. Codebook Generation and Encoding

Let the common message set 𝒲 and the confidential message set 𝒮 satisfy lim_𝑁→∞log‖𝒲‖𝑁=𝑅₀,lim_𝑁→∞log‖𝒮‖𝑁=𝑅₁, (54) where 𝑅₀ and 𝑅₁ satisfy (10).

Fix 𝑝(𝑢) and 𝑝(𝑥∣𝑢). In the 𝑏th block, 1≤𝑏≤𝑛, we generate 2^𝑁𝑅0 independent and identically distributed (i.i.d) sequences 𝑢^𝑁(𝑤_𝑏) according to ∏^𝑁_𝑖=1𝑝(𝑢_𝑖), where 𝑤_𝑏∈𝒲 is the common message to be sent in the 𝑏th block. For each 𝑢^𝑁(𝑤_𝑏), generate 2^{𝑁𝐼(𝑋;𝑌∣𝑈)} codewords 𝑥^𝑁(𝑢^𝑁(𝑤_𝑏)) according to ∏^𝑁_𝑖=1𝑝(𝑥_𝑖∣𝑢_𝑖). Put the 2^{𝑁𝐼(𝑋;𝑌∣𝑈)} codewords into 2^𝑁𝑅1 bins, so each bin contains 2^{𝑁(𝐼(𝑋;𝑌∣𝑈)−𝑅1)} codewords. The 2^𝑁𝑅1 bins are denoted by 𝑄₁,𝑄₂,…,𝑄_‖𝒮‖, where ‖𝒮‖=2^𝑁𝑅1. The codebook structure is shown in Figure 5. Reveal all the codebooks to the transmitter, receiver 1, and receiver 2.

Figure 5 

The codebook structure.

Let 𝑔 be a mapping from 𝒴^𝑁 into 𝒮. Reveal the mapping 𝑔 to the transmitter, receiver 1, and receiver 2. Define a random variable 𝑆^′=𝑔(𝑌^𝑁) uniformly distributed over 𝒮 and independent of the confidential message 𝑆. It can be similarly proved from (39) that 𝑆⊕𝑆^′ is independent of 𝑆. In the first block, that is, 𝑏=1, to send the common message 𝑤₁ (note that there is no confidential message to be sent in the first block), the transmitter tries to find 𝑢^𝑁(𝑤₁) and randomly choose a codeword 𝑥^𝑁(𝑢^𝑁(𝑤₁)) from the corresponding 2^{𝑁𝐼(𝑋;𝑌|𝑈)} codewords. In the 𝑏th block (𝑏=2,3,…,𝑛), to send the common message 𝑤_𝑏 and confidential message 𝑠_𝑏, the transmitter calculates 𝑠^′_𝑏=𝑔(𝑦_𝑏−1) and randomly chooses a codeword 𝑥^𝑁(𝑢^𝑁(𝑤_𝑏),𝑠_𝑏) from the bin 𝑄_{𝑠𝑏⊕𝑠′𝑏}. Here, 𝑦_𝑏−1 is the output vector of the (𝑏−1)th block at receiver 1, and ⊕ is the modulo addition over 𝒮.

4.1.2. Decoding

In the first block, as there is no confidential message, only the common message needs to be decoded for both receivers. For receiver 2, he tries to find a unique sequence 𝑢^𝑁(𝑤₁) such that (𝑢^𝑁(𝑤₁),𝑧₁)∈𝑇^𝑁_𝑈𝑍(𝜖^′₁), where 𝜖^′₁ is a small positive number. If there exists such a unique sequence, decoder 2 outputs 𝑤₁; otherwise, an error is declared. For receiver 1, he tries to find a unique sequence 𝑢^𝑁(𝑤₁) such that (𝑢^𝑁(𝑤₁),𝑦₁)∈𝑇^𝑁_𝑈𝑌(𝜖₁^′′), where 𝜖₁^′′ is a small positive number. If there exists such a unique sequence, output is 𝑤₁; otherwise, declare an error.

In the 𝑏th block, 2≤𝑏≤𝑛, receiver 2 aims to decode the common message, and receiver 1 aims to decode both confidential and common messages. The method of decoding the common message 𝑤_𝑏 for both receivers follows the same as that in the first block. Then, receiver 1 tries to find a unique sequence 𝑥^𝑁(𝑢^𝑁(𝑤_𝑏),𝑠_𝑏̂) such that (𝑥^𝑁(𝑢^𝑁(𝑤_𝑏),𝑠_𝑏̂),𝑦_𝑏)∈𝑇^𝑁_{𝑋𝑌∣𝑈}(𝜖^′₂), where 𝜖^′₂ is a small positive number. If there exists such a unique sequence in one bin, denoting the corresponding index of that bin by 𝑠_𝑏^′′, receiver 1 calculates 𝑠_𝑏^′′⊖𝑠^′_𝑏 as 𝑠_𝑏̂ (here ⊖ is modulo subtraction over 𝒮, and receiver 1 knows 𝑠^′_𝑏=𝑔(𝑦_𝑏−1)); otherwise, declare an error.

4.1.3. Analysis of Error Probability

Since the number of 𝑢^𝑁(𝑤_𝑏) is upper bounded by 2^{𝑁𝐼(𝑈;𝑍)}, receiver 2 can decode the common message 𝑤_𝑏 with error probability approaching zero by applying the standard channel coding theorem [11, Theorem 7.7.1]. Moreover, since the DMBCs under discussion in Section 4.1 are less noisy, receiver 1 can also decode the common message with error probability approaching zero. It can be calculated that given the codeword 𝑢^𝑁(𝑤_𝑏), the number of 𝑥^𝑁 is 2^{𝑁𝐼(𝑋;𝑌∣𝑈)}. So, after determining the codeword 𝑢^𝑁(𝑤_𝑏), receiver 1 can decode the codeword 𝑥^𝑁 with error probability approaching zero by applying the standard channel coding theorem [11, Theorem 7.7.1] and obtain the confidential message with the help of the feedback.

4.1.4. Analysis of Equivocation

In this part, lim_𝑁→∞Δ≥𝑅_𝑒 is proved by utilizing the methods in [5, 6]: lim_𝑁→∞Δ=lim_{𝑁,𝑛→∞}𝐻𝑆^𝑛∣𝑍^𝑛𝑛𝑁=lim^𝑛_{𝑁,𝑛→∞𝑖=2}𝐻𝑆_𝑖∣𝑆^𝑖−1,𝑍^𝑛=𝑛𝑁_(𝑎4.1)lim_{𝑁,𝑛→∞}∑^𝑛_𝑖=2𝐻𝑆_𝑖∣𝑍_𝑖𝑛𝑁≥lim_{𝑁,𝑛→∞}∑^𝑛_𝑖=2𝐻𝑆_𝑖∣𝑍_𝑖,𝑍_𝑖−1,𝑆_𝑖⊕𝑆^′_𝑖=𝑛𝑁_(𝑎4.2)lim_{𝑁,𝑛→∞}∑^𝑛_𝑖=2𝐻𝑆_𝑖∣𝑍_𝑖−1,𝑆_𝑖⊕𝑆^′_𝑖=𝑛𝑁_(𝑎4.3)lim_{𝑁,𝑛→∞}∑^𝑛_𝑖=2min𝑁𝐻(𝑌∣𝑍),log‖𝒮‖𝑛𝑁=lim_𝑛→∞∑^𝑛_𝑖=2min𝐻(𝑌∣𝑍),𝑅₁𝑛=min𝐻(𝑌∣𝑍),𝑅₁≥^(𝑎4.4)𝑅_𝑒. (55)

In the above deduction, (𝑎4.1) follows from 𝑆_𝑖→𝑍_𝑖→(𝑆^𝑖−1,𝑍^𝑖). (𝑎4.2) follows from 𝑆_𝑖→(𝑆_𝑖⊕𝑆^′_𝑖,𝑍_𝑖−1)→𝑍_𝑖. (𝑎4.3) follows from the fact that receiver 2 can choose a better way to intercept the secret key at will and 𝑆_𝑖⊕𝑆^′_𝑖 is independent of 𝑆_𝑖 and uniformly distributed over 𝒮. (𝑎4.4) follows from (10).

This completes the proof of Theorem 4.

4.2. Reversely Less Noisy DMBCs with Noiseless Causal Feedback

In this subsection, Theorem 5 will be proved. The converse part will be shown first, and then a coding scheme is given for proving the direct part.

In order to find the identification of the auxiliary random variables that satisfy the capacity-equivocation region characterized by ℛ_𝑟𝑙, we prove the converse part for the equivalent region containing all the rate triples (𝑅₀,𝑅₁,𝑅_𝑒) such that 0≤𝑅_𝑒≤𝑅₁,𝑅₀(),𝑅≤𝐼𝑈;𝑌₀+𝑅₁()(),𝑅≤𝐼𝑋;𝑌∣𝑈+𝐼𝑈;𝑌_𝑒≤𝐻(𝑌∣𝑋). (56)(57)(58)(59)

The inequalities (56), (57), and (58) can be proved using similar deduction of the converse part of Theorem 2 in Section 3 except for the identification of the auxiliary random variables. We focus on (59): 𝐻𝑆∣𝑍^𝑁=𝐻𝑆∣𝑋^𝑁,𝑍^𝑁+𝐼𝑆;𝑋^𝑁∣𝑍^𝑁=^(𝑏4.1)𝐻𝑆∣𝑋^𝑁+𝐼𝑋^𝑁;𝑆∣𝑍^𝑁=𝐻𝑆,𝑌^𝑁∣𝑋^𝑁−𝐻𝑌^𝑁∣𝑋^𝑁𝑋,𝑆+𝐼^𝑁;𝑆∣𝑍^𝑁=𝐻𝑌^𝑁∣𝑋^𝑁+𝐻𝑆∣𝑌^𝑁,𝑋^𝑁−𝐻𝑌^𝑁∣𝑋^𝑁,𝑆+𝐼𝑋^𝑁;𝑆∣𝑍^𝑁≤𝐻𝑌^𝑁∣𝑋^𝑁+𝐻𝑆∣𝑌^𝑁,𝑋^𝑁+𝐼𝑋^𝑁;𝑆∣𝑍^𝑁≤𝐻𝑌^𝑁∣𝑋^𝑁+𝐻𝑆∣𝑌^𝑁,𝑋^𝑁+𝐻𝑋^𝑁∣𝑍^𝑁=^(𝑏4.2)𝐻𝑌^𝑁∣𝑋^𝑁+𝐻𝑆∣𝑌^𝑁,𝑋^𝑁+𝐻𝑋^𝑁∣𝑌^𝑁=𝐻𝑌^𝑁∣𝑋^𝑁+𝐻𝑆,𝑋^𝑁∣𝑌^𝑁=^(𝑏4.3)𝐻𝑌^𝑁∣𝑋^𝑁+𝐻𝑆∣𝑌^𝑁=^𝑁_𝑖=1𝐻𝑌_𝑖∣𝑌^𝑖−1,𝑋^𝑁+𝐻𝑆∣𝑌^𝑁≤^{𝑁(𝑏4.4)}_𝑖=1𝐻𝑌_𝑖∣𝑋_𝑖+𝜖₃, (60) where (𝑏4.1) from the Markov chain 𝑆→𝑋^𝑁→𝑍^𝑁, (𝑏4.2) from the assumption that the channel is reversely less noisy (by setting 𝑈=𝑋), (𝑏4.3) from that 𝑋^𝑁 is a function of 𝑆,𝑌^𝑁, and (𝑏4.4) from the fact that conditioning does not increase entropy and Fano’s inequality. To complete the proof of (59), define a time-sharing random variable 𝑄 which is uniformly distributed over {1,2,…,𝑁} and independent of 𝑆𝑊𝑋^𝑁𝑌^𝑁𝑍^𝑁. Set 𝑈=𝑍^𝑁_𝑄+1𝑌^𝑄−1𝑊𝑄, 𝑉=𝑈𝑆, 𝑋=𝑋_𝑄, 𝑌=𝑌_𝑄, 𝑍=𝑍_𝑄. It is easy to see 𝑈→𝑉→𝑋→(𝑌,𝑍) form a Markov chain. After using the standard time-sharing argument [10, Section 5.4], (60) simplifies to 𝐻𝑆∣𝑍^𝑁≤𝑁𝐻(𝑌∣𝑋)+𝜖₃. (61)

Finally, utilizing lim_𝑁→∞Δ≥𝑅_𝑒 in the definition of “achievable” and (61), we obtain (59). This completes the proof of the converse part of Theorem 5.

Next, a coding scheme will be provided for achieving the triple (𝑅₀,𝑅₁,𝑅_𝑒)∈ℛ_𝑟𝑙. We should prove that all triples (𝑅₀,𝑅₁,𝑅_𝑒)∈ℛ_𝑟𝑙 are achievable. The codebook generation, encoding, and decoding follow exactly the lines of the coding scheme for the less noisy case in Section 4.1. We present the analysis of error probability and equivocation as follows.

4.2.1. Analysis of Error Probability

Since the number of 𝑢^𝑁(𝑤_𝑏) is upper bounded by 2^{𝑁𝐼(𝑈;𝑌)}, receiver 1 can decode the common message 𝑤_𝑏 with error probability approaching zero by applying the standard channel coding theorem [11, Theorem 7.7.1]. Moreover, since the DMBCs under discussion in Section 4.2 are reversely less noisy, receiver 2 can also decode the common message with error probability approaching zero. It can be calculated that given the codeword 𝑢^𝑁(𝑤_𝑏), the number of 𝑥^𝑁 is 2^{𝑁𝐼(𝑋;𝑌∣𝑈)}. So, after determining the codeword 𝑢^𝑁(𝑤_𝑏), receiver 1 can decode the codeword 𝑥^𝑁 with error probability approaching zero by applying the standard channel coding theorem [11, Theorem 7.7.1] and obtain the confidential message with the help of the feedback.

4.2.2. Analysis of Equivocation

In this part, lim_𝑁→∞Δ≥𝑅_𝑒 will be proved. Special attention should be paid to receiver 2 since the DMBCs are reversely less noisy; that is, 𝐼(𝑈;𝑍)≥𝐼(𝑈;𝑌) for all 𝑝(𝑢,𝑥), which implies 2^{𝑁𝐼(𝑋;𝑍∣𝑈)}≥2^{𝑁𝐼(𝑋;𝑌∣𝑈)}. Therefore, receiver 2 can also decode the codeword 𝑥^𝑁. With the knowledge of 𝑥^𝑁 and 𝑧^𝑁, receiver 2 can guess receiver 1’s channel output 𝑦^𝑁 from the conditional typical set 𝒯^𝑁_{𝑌∣𝑋𝑍}(𝜖₃). Note that receiver 2 can intercept the confidential messages in two ways. One is guessing the secret key 𝑠^′_𝑏 from 𝒮 directly; the other is guessing the channel output 𝑦_𝑏−1 and finding 𝑠^′_𝑏 through 𝑔(𝑦_𝑏−1) indirectly. Intuitively, receiver 2 will always choose a better way to implement eavesdropping. More formally, lim_𝑁→∞Δ=lim_{𝑁,𝑛→∞}𝐻𝑆^𝑛∣𝑍^𝑛𝑛𝑁=lim^𝑛_{𝑁,𝑛→∞𝑖=2}𝐻𝑆_𝑖∣𝑆^𝑖−1,𝑍^𝑛=𝑛𝑁_(𝑏4.5)lim_{𝑁,𝑛→∞}∑^𝑛_𝑖=2𝐻𝑆_𝑖∣𝑍_𝑖,𝑍_𝑖−1𝑛𝑁≥lim_{𝑁,𝑛→∞}∑^𝑛_𝑖=2𝐻𝑆_𝑖∣𝑍_𝑖,𝑍_𝑖−1,𝑋_𝑖−1,𝑆_𝑖⊕𝑆^′_𝑖=𝑛𝑁_(𝑏4.6)lim_{𝑁,𝑛→∞}∑^𝑛_𝑖=2𝐻𝑆_𝑖∣𝑍_𝑖−1,𝑋_𝑖−1,𝑆_𝑖⊕𝑆^′_𝑖=𝑛𝑁_(𝑏4.7)lim_{𝑁,𝑛→∞}∑^𝑛_𝑖=2min𝑁𝐻(𝑌∣𝑋𝑍),log‖𝒮‖=𝑛𝑁_(𝑏4.8)lim_{𝑁,𝑛→∞}∑^𝑛_𝑖=2min𝑁𝐻(𝑌∣𝑋),log‖𝒮‖𝑛𝑁=lim_𝑛→∞∑^𝑛_𝑖=2min𝐻(𝑌∣𝑋),𝑅₁𝑛=min𝐻(𝑌∣𝑋),𝑅₁≥^(𝑏4.9)𝑅_𝑒. (62)

In the above deduction, (𝑏4.5) follows from 𝑆_𝑖→(𝑍_𝑖,𝑍_𝑖−1)→(𝑆^𝑖−1,𝑍^𝑖−2,𝑍^𝑛_𝑖+1).(𝑏4.6) follows from 𝑆_𝑖→(𝑆_𝑖⊕𝑆^′_𝑖,𝑍_𝑖−1,𝑋_𝑖−1)→𝑍_𝑖. (𝑏4.7) follows from the fact that receiver 2 can choose a better way to intercept the secret key at will, and 𝑆_𝑖⊕𝑆^′_𝑖 is independent of 𝑆_𝑖 and uniformly distributed over 𝒮. Note that the number of 𝑦^𝑁∈𝑇^𝑁_{𝑌|𝑋𝑍}(𝜖₃) is about 2^{𝑁𝐻(𝑌∣𝑋𝑍)} based on the property of strong typical sequence [10]. (𝑏4.8) follows from the fact that 𝑌 is independent of 𝑍 conditioning on 𝑋, which is obtained from the assumption 𝑝(𝑦,𝑧∣𝑥)=𝑝(𝑦∣𝑥)𝑝(𝑧∣𝑥). (𝑏4.9) follows from (11).

This completes the proof of Theorem 5.

5. Conclusion

This paper studies two models of the DMBCs with noiseless feedback. One is the degraded DMBCs with rate-limited feedback; the other is the less and reversely less noisy DMBCs with feedback. The difference between them is that the feedback in the first model is independent of the channel outputs and rate limited, while the feedback in the second model is originated causally from the channel outputs. The capacity-equivocation regions of the two models are obtained in this paper. We should point out that the second model studied in this paper is under the assumption that the channel to receiver 1 (the legitimate receiver) is independent of the channel to receiver 2 (the eavesdropper); that is, the channel output 𝑌^𝑁 is independent of 𝑍^𝑁 given the channel input 𝑋^𝑁. However, without this assumption, the capacity-equivocation region remains unknown for the general DMBCs with noiseless feedback.

Acknowledgments

This work was supported in part by the National Science Foundation of China under Grant no. 60932003 and Grant no. 61271220. The authors also would like to thank the anonymous reviewers for helpful comments.