A Class of Three-Receiver Broadcast Channels with Degraded Message Sets and Side Information

Abstract

We study a class of three-receiver discrete memoryless broadcast channel (DM-BC) with three degraded message sets and side information. We derive achievable rate region for this three-receive DM-BC with side information noncausally available at the transmitter. When the receivers follow a degradedness order, we determine the perfect secrecy capacity region of this class of three-receiver DM-BC with three degraded message sets and side information noncausally available at both the transmitter and the receiver. The achievable secrecy region of this paper subsumes Steinberg’s rate region for two-receiver degraded BC with the side information and the secrecy capacity region for one-receiver two-eavesdropper DM-BC with no side information as its special cases.

1. Introduction

Wyner introduced wiretap channel in his seminal paper [1]. In his communication model, a transmitter wishes to send confidential messages to a legitimate receiver while keeping the messages secret from the eavesdropper. He assumed that the channel to the eavesdropper is a physical degraded version of the channel to the legitimate receiver. He characterized the trade-off between the information rate to the receiver and the level of secrecy, as measured by equivocation. Furthermore, Wyner proposed a basic principle coding strategy to obtain secure communication for wiretap channels. Subsequently, Csiszár and Körner extended Wyner’s results to general broadcast channel with confidential messages in [2], where no degradation order exists between the legitimate receiver and the eavesdropper. They derived a capacity-equivocation region of this general, not necessarily degraded, wiretap channel.

Körner and Marton considered a general two-receiver DM-BC with degraded message sets in [3]. In their model, a common message 𝑀₀ is to be transmitted to all receivers; a private message 𝑀₁ is to be sent only to the first receiver. They determined the capacity region by using superposition coding. Information theoretic study of the 𝑘-receiver (𝑘≥3) discrete memoryless broadcast channel (DM-BC) was initiated by Borade et al. in [4]. They introduced multilevel broadcast channels, which combined some aspects of degraded message sets and established an achievable rate region (named BZT region [5]) for these channels. They simply surmised that the straightforward extension of the Körner-Marton region for two-receiver BC with degraded message sets to 𝑘-receiver BC is in general optimal. Nair and El Gamal [5] showed that the direct extension of the Körner-Marton region to more than two receivers is not optimal in general. They established the capacity region for the multilevel BC and then extended the results of the multilevel BC to determine inner and outer bounds on the capacity region of the general three-receiver BC with two-degraded message sets. They proved that the capacity region for a class of three-receiver BC with two-degraded message sets can be strictly larger than the straightforward extension of the Körner-Marton region. One of the contributions of [5] is that it introduces a new decoding scheme named indirect decoding. Chia and El Gamal [6] considered two scenarios of three-receiver BC with common and confidential messages, two-receiver one-eavesdropper BC and one-receiver two-eavesdropper BC. In the two-receiver one-eavesdropper BC model, the confidential message is to be sent to the first and the second receivers while kept secret from the third receiver. In the second paradigm, one-receiver two-eavesdropper BC, the confidential message is to be sent to one receiver and kept secret from the other two receivers. They established inner bounds on the secrecy capacity region for these two scenarios by using Wyner channel coding, Nair-Gamal indirect decoding and the new ideal of generating secrecy from a public available superposition codebook. They also showed that the inner bound was tight for several special cases. Recently, Salehkalaibar et al. [7] studied one-receiver two-eavesdropper BC with three degraded message sets and proposed a more general network compared to [6].

Although all the aforementioned papers discussed the class of three-receiver BC with or without secrecy constraints, they did not consider the case where there is side information available at the transmitter or at both the transmitter and the receiver. Shannon first studied the channels with side information causally available at the transmitter in [8], where he determined the capacity region of the SISO channel. Following Shannon’s research, Gel’fand-Pinsker investigated the single-user channel with side information noncausally available at the transmitter and determined its capacity in [9]. Cover and Chiang [10] extended Gel’fand and Pinsker’s result to the case where the side information was available at both the transmitter and the receiver. After that, Steinberg [11] studied two paradigms where side information on the random parameter was provided to the transmitter in a noncausal or causal manner. He derived the inner and outer bounds on the capacity region with noncausal coding. For causal coding, he derived a single-letter characterization of the capacity region. Chia and El Gamal [12] established a lower bound on the secrecy capacity of the wiretap channel with state information available causally at both the encoder and the decoder. They showed that the lower bound is strictly larger than that for the noncausal case by Liu and Chen [13]. Recently, Hajizadeh and Hodtani [14] investigated two classes of three-receiver BC with side information available at the transmitter or at both ends, multilevel BC and three-receiver less noisy BC. For the multilevel BC with side information, they assumed that two degraded message sets are to be sent to three receivers, one for all receivers, the other only for the first receiver. They found the achievable rate region of multilevel BC with side information noncausally available at the transmitter. For the three-receiver less noisy BC with side information available at the transmitter, they assumed that three degraded message sets are to be sent to three receivers, respectively, and that the first receiver is less noisy than the second one and the second receiver is less noisy than the third one. When the side information is available at both the transmitter and the receiver, they established the capacity region of the three-receiver less noisy BC.

Different from [14], our paper considers another class of three-receiver, which is called one-receive two-eavesdropper, DM-BC with three degraded message sets and noncausally side information (see Figure 1). In this class of three-receiver BC, a common message 𝑀₀ is to be sent to three receivers (𝑌₁, 𝑍₂, 𝑍₃), a private message 𝑀₁ is to be sent to the first and the second receiver (𝑌₁, 𝑍₂), and meanwhile unrevealed to the third receiver (𝑍₃, viewed as one eavesdropper); then a second private message 𝑀₂ is to be sent to the first receiver (𝑌₁) while kept secret from the second and the third receiver (𝑍₂, 𝑍₃, viewed as two eavesdroppers). The objective of this paper is to study this class of three-receive BC with side information and three degraded message sets with secrecy constraints. When the side information is noncausally available only at the transmitter, we establish the achievable rates by using a combination of superposition coding, Gel’fand-Pinsker coding scheme, and Nair-El Gamal indirect decoding. The achievable secrecy region of this paper subsumes Steinberg’s rate region for two-receiver degraded BC with the side information and the secrecy capacity region for one-receiver two-eavesdropper DM-BC with no side information as its special cases. When the side information is noncausally available at both the transmitter and the three receivers, we determine the perfect secrecy capacity region of the one-receiver two-eavesdropper BC with three degraded message sets and side information when the receivers follow a degradedness order.

Figure 1 

A class of three-receiver DM-BC with three degraded message sets and side information.

The rest of this paper is organized as follows. In Section 2, we describe the channel model and present some basic definitions. In Section 3, we will divide it into two subsections. In the first subsection, we derive achievable rate region for three-receiver DM-BC when the side information is noncausally available at the transmitter. In the second subsection, we determine the perfect secrecy capacity region of three-receiver degraded DM-BC when the side information is noncausally available at both the transmitter and the receiver. Conclusions and further discussions are given in Section 4.

2. Channel Model and Definitions

We denote discrete random variables and their realizations with uppercase and lowercase letters, respectively; for example, 𝑥 and 𝑦 are, respectively, the realization of 𝑋 and 𝑌. The random variables 𝑋, 𝑌₁, 𝑍₂, 𝑍₃, and 𝑆 may take values from finite sets 𝒳, 𝒴₁, 𝒵₂, 𝒵₃, and 𝒮. 𝑋^𝑛 denotes the 𝑛-sequence random variable, where the superscript is omitted when the choice is clear. The entropy of a discrete random variable is denoted by 𝐻(⋅) and the mutual information between two random variables is denoted by 𝐼(⋅;⋅). We denote by 𝒯_𝜖^(𝑛)(𝑋,𝑌) the set of 𝜖-strongly joint typical sequence of length 𝑛, and we assume that 𝑋^𝑛_𝑖 is the sequence of random variables (𝑋_𝑖, 𝑋_𝑖+1,…,𝑋_𝑛) throughout. Furthermore, 𝑝(𝑥) indicates the probability mass function of 𝑋 over 𝒳 and 𝑝(𝑥∣𝑦) indicates the conditional probability mass function of 𝑋 given 𝑌.

We consider a class of three-receiver discrete memoryless broadcast channels (DM-BC) with input alphabet 𝒳, three output alphabets 𝒴₁, 𝒵₂, 𝒵₃, set of states 𝑆_𝑖, each of them is finite, and a collection of conditional probability mass functions (pmfs) 𝑝(𝑦₁,𝑧₂,𝑧₃∣𝑥,𝑠) on 𝒴₁×𝒵₂×𝒵₃. 𝑆_𝑖 are independently and identically distributed (i.i.d.)∼𝑝(𝑠). The channel and state sequence are memoryless, for example, 𝑝𝑦^𝑛₁,𝑧^𝑛₂,𝑧^𝑛₃∣𝑥^𝑛,𝑠^𝑛=^𝑛_𝑖=1𝑝𝑦_1,𝑖,𝑧_2,𝑖,𝑧_3,𝑖∣𝑥_𝑖,𝑠_𝑖,𝑝_𝑆𝑠^𝑛=^𝑛_𝑖=1𝑝_𝑆𝑠_𝑖. (1) Denote the marginals of the channel by 𝑝_{𝑌1|𝑆,𝑋}(𝑦₁∣𝑠,𝑥) and 𝑝_{𝑍2|𝑆,𝑋}(𝑧₂∣𝑠,𝑥), 𝑝_{𝑍3|𝑆,𝑋}(𝑧₃∣𝑠,𝑥).

The states 𝑆 are known to be noncausally at the transmitter, or at both the transmitter and the receiver as side information. The transmitter wishes to send a common message 𝑀₀ to three receivers, 𝑀₁ to the first and the second receiver which needs to be kept secret from the third receiver and to send 𝑀₂ only to the fist receiver which needs to be kept secret from the second and the third receiver (viewed as two eavesdroppers) in 𝑛 uses of the channel.

A broadcast channel 𝑝(𝑦₁,𝑧₂,𝑧₃∣𝑠,𝑥) is said to be physically degraded if 𝑝𝑦₁,𝑧₂,𝑧₃∣𝑠,𝑥=𝑝_{𝑌1∣𝑆,𝑋}𝑦₁∣𝑠,𝑥𝑝_{𝑍2∣𝑌1}𝑧₂∣𝑦₁𝑝_{𝑍3∣𝑍2}𝑧₃∣𝑧₂ (2) in which case 𝑍₂ is a degraded version of 𝑌₁, and 𝑍₃ is a degraded version of 𝑍₂. According to this definition, we note that the side information parameter 𝑠 controls only the nondegraded channel, whereas 𝑝_{𝑍2∣𝑌1}(𝑧₂∣𝑦₁), 𝑝_{𝑍3∣𝑍2}(𝑧₃∣𝑧₂) is independent of the 𝑠.

Definition 1. A (2^𝑛𝑅0,2^𝑛𝑅1,2^𝑛𝑅2,𝑛)-code for the one-receiver two-eavesdropper DM-BC with three degraded message sets and side information available at the transmitter, as depicted in Figure 1, consists of an encoder map 𝑓∶ℳ₀×ℳ₁×ℳ₂×𝒮^𝑛→𝒳^𝑛,  ℳ₀={1,…,2^𝑛𝑅0}, (common message set for 𝑌₁,𝑍₂,𝑍₃), ℳ₁={1,…,2^𝑛𝑅1}, (private message set for 𝑌₁ and 𝑍₂), ℳ₂={1,…,2^𝑛𝑅2}, (private message set for 𝑌₁),
and a tuple of decoding maps 𝑔₁∶𝒴^𝑛₁→ℳ₀×ℳ₁×ℳ₂,  𝑔₂∶𝒵^𝑛₂→ℳ₀×ℳ₁,  𝑔₃∶𝒵^𝑛₃→ℳ₀. Applying the encoder to 𝑀₀, 𝑀₁, 𝑀₂, which are uniformly distributed over the message sets, yields channel input 𝑋=𝑋^𝑛, and the corresponding channel outputs are 𝑌₁ = 𝑌^𝑛₁, 𝑍₂ = 𝑍^𝑛₂, 𝑍₃ = 𝑍^𝑛₃. In particular, we have 𝑔₁(𝑌₁)=(𝑀₀₁,𝑀₁₁,𝑀₂₁)∈[1∶2^𝑛𝑅0]×[1∶2^𝑛𝑅1]×[1∶2^𝑛𝑅2], 𝑔₂(𝑍₂)=(𝑀₀₂,𝑀₁₂)∈[1∶2^𝑛𝑅0]×[1∶2^𝑛𝑅1], 𝑔₃(𝑍₃)=(𝑀₀₃)∈[1∶2^𝑛𝑅0], where the notation (⋅)̂ highlights that the decoded messages are estimate. The probability of error is defined as 𝑃^𝑛_𝑒=Pr𝑀_0𝑖≠𝑀₀,𝑖=1∶3or𝑀_1𝑗≠𝑀₁,𝑗=1∶2or𝑀₂₁≠𝑀₂. (3)
The rate tuples of the code are defined as follows: 𝑅₀,𝑅₁,𝑅₂=1𝑛log𝑀₀,log𝑀₁,log𝑀₂. (4)

Definition 2. A rate tuple (𝑅₀, 𝑅₁, 𝑅₂) with perfect secrecy is achievable if there exists a sequence of (2^𝑛𝑅0, 2^𝑛𝑅1, 2^𝑛𝑅2, 𝑛) codes as previously mentioned with lim_𝑛→∞𝑃^𝑛_𝑒=0,lim_𝑛→∞1sup𝑛𝐼𝑀₂;𝑍^𝑛₂=0,lim_𝑛→∞1sup𝑛𝐼𝑀₁,𝑀₂;𝑍^𝑛₃=0, (5) where sup denotes the supremum of an expression. The capacity region with secrecy is the closure of the set of all achievable rate tuples (𝑅₀, 𝑅₁, 𝑅₂) with secrecy.

3. One-Receiver Two-Eavesdropper BC with Side Information

In this section, we first consider the scenario that the side information is noncausally available at the transmitter. We derive achievable rate region for this three-receive DM-BC with side information noncausally available at the transmitter. The achievable secrecy region is established by using a combination of Gelfand-Pinsker binning scheme and rate splitting technique. In the achievability proof scheme, superposition coding scheme will be used to divide the available randomness into different levels so as to mislead the eavesdroppers. Achievability in this case also follows from the new idea of Nair-El Gamal indirect decoding. Subsequently, we consider the scenario that the side information is noncausally available at both the transmitter and the receiver. When the side information is available at both the transmitter and the receiver, we determine the perfect secrecy capacity region of this one-receive two-eavesdropper BC with three degraded message sets with secrecy constraints as the receivers form a degradedness order. Converse proof of Theorem 6 is provided in detail.

3.1. Side Information Noncausally Available at the Transmitter

Theorem 3. Rate tuples (𝑅₀,𝑅₁,𝑅₂) are achievable with perfect secrecy for one-receiver two-eavesdropper DM-BC with three degraded message sets and side information noncausally available at the transmitter when 𝑍₂ is a degraded version of   𝑌₁ and 𝑍₃ is a degraded version of 𝑍₂; that is, 𝑌₁→𝑍₂→𝑍₃ forms a Markov chain, provided that 𝑅₀≤𝐼𝑊;𝑍₃(),𝑅−𝐼𝑊;𝑆₁≤𝐼𝑉;𝑍₂∣𝑊−𝐼𝑉;𝑍₃(),𝑅∣𝑊−𝐼𝑉;𝑆∣𝑊₂≤𝐼𝑋;𝑌₁∣𝑈−𝐼𝑋;𝑍₂∣𝑈−𝐼(𝑋;𝑆∣𝑈) (6) for some distributions 𝑝(𝑤,𝑢,𝑣∣𝑠) and functions 𝑥(𝑤,𝑢,𝜐,𝑠).

Corollary 4. Setting 𝑆≡∅, 𝑈 = 𝑉 in achievable rates of Theorem 3, our achievable rate region with secrecy is reduced to the capacity region of the one-receiver two-eavesdropper DM-BC with three degraded message sets when 𝑌₁, 𝑍₂, 𝑍₃ forms a degradedness order as given in [7].

Corollary 5. Setting 𝑍₃=𝑌₁, 𝑈=𝑉=𝑊 in achievable rates of Theorem 3, the achievable rate region with secrecy is reduced to the capacity region of Steinberg’s rate region for two-receiver degraded DM-BC with noncausally side information [11].

Proof. The most important part of the proof of the achievability is the code generation. We use superposition technique and Wyner’s wiretap coding [1] as the secrecy achievability method. We also employ indirect decoding scheme [5] where the first decoder finds 𝑀₁ from the satellite codewords instead of cloud centers and use superposition coding scheme to divide the available randomness into different levels, which are used to mislead the eavesdroppers.
Consider a DM-BC with (𝒳×𝒮,𝑝(𝑦₁,𝑧₂,𝑧₃∣𝑥,𝑠)𝑝(𝑠),𝒴₁,𝒵₂,𝒵₃). Fix 𝑛 and a joint distribution on 𝒫. The side information alphabet set 𝒮 is distributed i.i.d according to 𝑝_𝑆𝑠^𝑛=^𝑛_𝑖=1𝑝𝑠_𝑖. (7)
(1) Code Generation. Randomly and independently generate 2^{𝑛(𝑅0+𝑅′0)} sequences 𝑤^𝑛(𝑚₀,𝑚^′₀), 𝑚₀∈[1∶2^𝑛𝑅0], 𝑚^′₀∈[1∶2^𝑛𝑅′0], each one i.i.d according to ∏^𝑛_𝑖=1𝑝_𝑊(𝑤_𝑖) and randomly throw them into 2^𝑛𝑅0 bins. For each 𝑤^𝑛(𝑚₀,𝑚^′₀), randomly and independently generate 2^{𝑛(𝑅1+𝑅′1+}𝑅₁₂₎ sequences 𝜐^𝑛(𝑚₀,𝑚^′₀,𝑚₁,𝑚^′₁,𝑠₁₂), 𝑚₁∈[1∶2^𝑛𝑅1], 𝑚^′₁∈[1∶2^𝑛𝑅′1], 𝑠₁₂∈[1∶2^𝑛𝑅₁₂], each one i.i.d according to ∏^𝑛_𝑖=1𝑝_𝑉∣𝑊(𝜐_𝑖∣𝑤_𝑖(𝑚₀,𝑚^′₀)), and randomly throw them into 2^𝑛𝑅1 bins. For each 𝜐^𝑛(𝑚₀,𝑚^′₀,𝑚₁,𝑚^′₁,𝑠₁₂), randomly and independently generate 2^𝑛𝑅₁₁ sequences 𝑢^𝑛(𝑚₀,𝑚^′₀,𝑚₁,𝑚^′₁,𝑠₁₂,𝑠₁₁), 𝑠₁₁∈[1∶2^𝑛𝑅₁₁], each one i.i.d according to ∏^𝑛_𝑖=1𝑝_{𝑈|𝑊𝑉}(𝑢_𝑖|𝑤_𝑖,𝜐_𝑖), and randomly throw them into 2^𝑛𝑅1 bins. For each 𝑢^𝑛(𝑚₀,𝑚^′₀,𝑚₁,𝑚^′₁,𝑠₁₂,𝑠₁₁), randomly and independently generate 2^{𝑛(𝑅2+𝑅′2+}𝑅₂₎ sequences 𝑥^𝑛(𝑚₀,𝑚^′₀,𝑚₁,𝑚^′₁,𝑠₁₂,𝑠₁₁,𝑚₂,𝑚^′₂,𝑠₂), 𝑚₂∈[1∶2^𝑛𝑅2], 𝑚^′₂∈[1∶2^𝑛𝑅′2], 𝑠₂∈[1∶2^𝑛𝑅₂], each one i.i.d according to ∏^𝑛_𝑖=1𝑝_{𝑋∣𝑊𝑉𝑈}(𝑥_𝑖∣𝑤_𝑖,𝜐_𝑖,𝑢_𝑖), and randomly throw them into 2^𝑛𝑅2 bins. We define 𝑅₁=𝑅₁₁+𝑅₁₂. The randomness variables 𝑠₁₁, 𝑠₁₂, and 𝑠₂ are used to mislead the eavesdroppers.
(2) Encoding. Given the message (𝑚₀,𝑚₁,𝑚₂) and side information 𝑠^𝑛, the encoder finds (𝑚₀,𝑚₁,𝑚₂) according to the bin indices since the messages are in the indices. Then, encoder looks for a 𝑚^′₀ in the bin 𝑚₀ of 𝑤^𝑛 sequences such that (𝑤^𝑛(𝑚₀,𝑚^′₀),𝑠^𝑛)∈𝒯_𝜖^(𝑛). Then, the encoder finds 𝑚^′₁ such that (𝑤^𝑛(𝑚₀,𝑚^′₀),𝜐^𝑛(𝑚₀,𝑚^′₀,𝑚₁,𝑚^′₁,𝑠₁₂),𝑠^𝑛)∈𝒯_𝜖^(𝑛). The encoder next finds 𝑚^′₁ such that (𝑤^𝑛(𝑚₀,𝑚^′₀),𝑢^𝑛(𝑚₀,𝑚^′₀,𝑚₁,𝑚^′₁,𝑠₁₂,𝑠₁₁),𝑠^𝑛)∈𝒯_𝜖^(𝑛). Finally, the encoder finds 𝑚^′₂ in the 𝑥^𝑛 sequences such that (𝑤^𝑛(𝑚₀,𝑚^′₀), 𝜐^𝑛(𝑚₀,𝑚^′₀,𝑚₁,𝑚^′₁,𝑠₁₂), 𝑢^𝑛(𝑚₀,𝑚^′₀,𝑚₁,𝑚^′₁,𝑠₁₂,𝑠₁₁), 𝑥^𝑛(𝑚₀,𝑚^′₀,𝑚₁,𝑚^′₁,𝑠₁₂,𝑠₁₁,𝑚₂,𝑚^′₂,𝑠₂),𝑠^𝑛)∈𝒯_𝜖^(𝑛). We assume that the correct indices are found through the encoding procedure; that is, 𝑚^′₀=𝑀^′₀, 𝑚^′₁₁=𝑀^′₁, 𝑚^′₂=𝑀^′₂, and we send the found 𝑥^𝑛 sequence.
(3) Decoding and Analysis of the Probability of Error. Receiver 𝑍₃ finds 𝑚₀ by decoding 𝑊, 𝑍₂ finds 𝑚₁ by decoding 𝑉 and finds 𝑚₀ by decoding 𝑊, and 𝑌₁ finds 𝑚₀ by decoding 𝑊, finds 𝑚₁ by indirectly decoding 𝑈, and finds 𝑚₂ by decoding 𝑋. Without loss of generality, we assume that the tuple (𝑚₀,𝑚₁,𝑚₂)=(1,1,1) is sent.
At receiver 𝑍₃, the decoder finds the unique 𝑚₀ such that (𝑤^𝑛(𝑚₀,𝑚^′₀),𝑧^𝑛₃)∈𝒯_𝜖^(𝑛); otherwise, it declares an error. Thus, the possible error events are as the follows: 𝐸₃₁=𝑊^𝑛1,𝑀^′₀,𝑍₃∉𝒯_𝜖^(𝑛),𝐸₃₂=𝑊^𝑛𝑚₀,𝑚^′₀,𝑍₃∈𝒯_𝜖^(𝑛),forsome𝑚₀≠1,𝑚^′₀≠𝑀^′₀. (8) Thus, the probability of error for decoder 𝑍₃ is upper bounded as 𝑃𝐸₃≤𝑃𝐸₃₁+𝑃𝐸₃₂. (9) We now bound each term. By the law of large numbers (LLN) [15], 𝑃(𝐸₃₁) tends to zero as 𝑛→∞: 𝑃𝐸₃₂=_{𝑚0,𝑚′0𝒯𝜖(𝑛)}𝑃𝑤^𝑛𝑃𝑧^𝑛₃≤2^{𝑛(𝑅0+𝑅′0)}⋅2^{𝑛(𝐻(𝑊,𝑍3)+𝜖)}⋅2^{−𝑛(𝐻(𝑊)−𝜖)}⋅2^{−𝑛(𝐻(𝑍3)−𝜖)}=2^{𝑛(𝑅0+𝑅′0−𝐼(𝑊;𝑍3)+3𝜖)}. (10) Thus, 𝑃(𝐸₃₂)→0 as 𝑛→∞ if 𝑅₀+𝑅^′₀≤𝐼𝑊;𝑍₃−3𝜖. (11)
At receive 𝑍₂, we can similarly bound the probability of error for 𝑍₂. For 𝑍₂, we define the error events 𝐸₂₁=𝑊^𝑛𝑀^′₀,1,𝑉^𝑛𝑀^′₀,1,𝑀^′₁,1,𝑠₁₂,𝑍₂∉𝒯^𝑛_𝜖,𝐸₂₂=𝑊^𝑛𝑀^′₀,1,𝑉^𝑛𝑀^′₀,1,𝑚₁,𝑚^′₁,𝑠₁₂,𝑍₂∈𝒯^𝑛_𝜖forsome𝑚₁≠1,𝑚^′₁₂≠𝑀^′₁₂,𝑠₁₂≠𝑆₁₂,𝐸₂₃=𝑊^𝑛𝑚^′₀,𝑚₀,𝑉^𝑛𝑚^′₀,𝑚₀,𝑚₁,𝑚^′₁,𝑠₁₂,𝑍₂∈𝒯^𝑛_𝜖forsome𝑚₀≠1,𝑚^′₀≠𝑀^′₀,𝑚₁𝑚≠1,_′1≠𝑀^′₁,𝑠₁₂≠𝑆₁₂. (12) The probability of error for 𝑍₂ is then upper bounded by 𝑃𝐸₂≤𝑃𝐸₂₁+𝑃𝐸₂₂+𝑃𝐸₂₃. (13) By the LLN, 𝑃(𝐸₂₁) tends to zero as 𝑛→∞. Next, we consider the event 𝐸₂₂. For 𝑚₁≠1, 𝑉^𝑛(𝑀^′₀,1,𝑚₁,𝑚^′₁₂) is conditionally independent of (𝑉^𝑛(𝑀^′₀,1,𝑚₁,𝑚^′₁₂),𝑍₂) given 𝑊^𝑛(𝑀^′₀,1) and is distributed according to ∏^𝑛_𝑖=1𝑃_𝑉∣𝑊(𝜐_𝑖∣𝑢_𝑖). Thus, by the packing lemma [16], 𝑃(𝐸₂₂)→0 as 𝑛→∞ if 𝑅₁+𝑅^′₁+𝑅₁₂≤𝐼𝑉;𝑍₂∣𝑊−6𝜖. (14) Similarly, for event 𝐸₂₃, 𝑃(𝐸₂₃)→0 as 𝑛→∞ if 𝑅₀+𝑅^′₀+𝑅₁+𝑅^′₁+𝑅₁₂≤𝐼𝑊𝑉;𝑍₂−5𝜖. (15) We can see that this is a redundant inequality.
At receive 𝑌₁, we define the error events as 𝐸₁₁=𝑊^𝑛𝑀^′₀,1,𝑈^𝑛𝑀^′₀,1,𝑀^′₁,1,𝑠₁₂,𝑠₁₁,𝑋^𝑛𝑀^′₀,1,𝑀^′₁,1,𝑠₁₂,𝑠₁₁,𝑀^′₂,1,𝑠₂,𝑌^𝑛₁∉𝒯_𝜖^(𝑛),𝐸₁₂=𝑊^𝑛𝑀^′₀,1,𝑈^𝑛𝑀^′₀,1,𝑀^′₁,1,𝑠₁₂,𝑠₁₁,𝑋^𝑛𝑀^′₀,1,𝑀^′₁,1,𝑠₁₂,𝑠₁₁,𝑀^′₂,𝑚^′₂,𝑠₂,𝑌^𝑛₁∈𝒯_𝜖^(𝑛)forsome𝑚₂≠1,𝑚^′₂≠𝑀^′₂,𝑠₂≠𝑆₂,𝐸₁₃=𝑊^𝑛𝑀^′₀,1,𝑈^𝑛𝑀^′₀,1,𝑚^′₁,𝑚₁,𝑠₁₂,𝑠₁₁,𝑋^𝑛𝑀^′₀,1,𝑚^′₁,𝑚₁,𝑚₂,𝑚^′₂,𝑌^𝑛₁∈𝒯_𝜖^(𝑛)forsome𝑚₁₁≠1,𝑚^′₁≠𝑀^′₁,𝑚₂𝑚≠1,_′2≠𝑀^′₂,𝑠₁₂≠𝑆₁₂,𝑠₁₁≠𝑆₁₁,𝐸₁₄=𝑊^𝑛𝑚^′₀,𝑚₀,𝑈^𝑛𝑚^′₀,𝑚₀,𝑚^′₁,𝑚₁,𝑋^𝑛𝑚^′₀,𝑚₀,𝑚^′₁,𝑚₁,𝑚₂,𝑚^′₂,𝑌^𝑛₁∈𝒯_𝜖^(𝑛)forsome𝑚₀≠1,𝑚^′₀≠𝑀^′₀,𝑚₁≠1,𝑚^′₁≠𝑀^′₁,𝑚₂≠1,𝑚^′₂≠𝑀^′₂,𝑠₁₂≠𝑆₁₂,𝑠₁₁≠𝑆₁₁. (16) Therefore, the probability of error of the receiver 𝑌₁ can be made arbitrarily small if 𝑅₂+𝑅^′₂+𝑅₂≤𝐼𝑋,𝑌₁𝑅∣𝑈−6𝜖,₁+𝑅^′₁+𝑅₁₁+𝑅₂+𝑅^′₂+𝑅₁₂≤𝐼𝑋,𝑌₁𝑅∣𝑊−6𝜖,₀+𝑅^′₀+𝑅₀+𝑅₁+𝑅^′₁+𝑅₁₁+𝑅₂+𝑅^′₂+𝑅₁₂≤𝐼𝑋,𝑌₁−5𝜖. (17) Using a similar argument on analysis of information leakage rate as in [7], we have 𝐼(𝑀₁,𝑀₂;𝑍^𝑛₃∣𝐶)≤𝑛𝜖, if 𝑅₁+𝑅₂≥𝐼(𝑋;𝑍₃∣𝑊), also have 𝐼(𝑀₂;𝑍^𝑛₂∣𝐶)≤𝑛𝜖 if 𝑅₂≥𝐼(𝑋;𝑍₂∣𝑈), and have 𝐼(𝑀₁;𝑍^𝑛₃∣𝐶)≤𝑛𝜖 if 𝑅₁₂≥𝐼(𝑉;𝑍₃∣𝑊).
Subsequently, we use Gel’fand-Pinsker coding scheme and determine that the encoder can choose the proper 𝑚^′₀, 𝑚^′₁, 𝑚^′₂ indices to make the probability of error tend to zero proving that if for every 𝜖>0 and 𝑛→∞: 𝑅^′₀(),𝑅>𝐼𝑊;𝑆_′1(),𝑅>𝐼𝑉;𝑆∣𝑊_′2>𝐼(𝑋;𝑆∣𝑈). (18)(19)(20) Finally, combining all the terms and using Fourier-Motzkin Elimination, we obtain the expressions in Theorem 3 and complete the proof of achievability.

3.2. Side Information Noncausally Available at Both the Transmitter and the Receivers

Theorem 6. The secrecy capacity region of one-receiver two-eavesdropper DM-BC with three degraded message sets and side information noncausally available both at the transmitter and at the receiver when 𝑍₂ is a degraded version of 𝑌₁ and 𝑍₃ is a degraded version 𝑍₂, that is, 𝑌₁→𝑍₂→𝑍₃ forms a Markov chain, is given by the union rate tuples (𝑅₀,𝑅₁,𝑅₂) such that 𝑅₀≤𝐼𝑊;𝑍₃,𝑅∣𝑆₁≤𝐼𝑉;𝑍₂∣𝑊,𝑆−𝐼𝑉;𝑍₃,𝑅∣𝑊,𝑆₂≤𝐼𝑋;𝑌₁∣𝑉,𝑆−𝐼𝑋;𝑍₂∣𝑉,𝑆. (21)

Proof

(1) Achievability. Setting 𝑌₁=(𝑌₁,𝑆), 𝑍_𝑘=(𝑍_𝑘,𝑆), 𝑘=2, 3, 𝑈=𝑉, and substituting it into (6), we can achieve the direct part of the proof.

(2) Converse. We define 𝑊_𝑖∶=𝑀₀,𝑆^𝑖−1,𝑆^𝑛_𝑖+1,𝑍^𝑛_3,𝑖+1,𝑉_𝑖∶=𝑀₀,𝑀₁,𝑆^𝑖−1,𝑆^𝑛_𝑖+1,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1. (22)

First, we consider the rate 𝑅₀𝑛𝑅₀=𝐻_0(a)𝑀≤𝐼𝑀₀;𝑍₃∣𝑆^𝑛+𝑛𝜖_𝑛=^𝑛_𝑖=1𝐼𝑀₀;𝑍_3,𝑖∣𝑆^𝑛,𝑍^𝑛_3,𝑖+1+𝑛𝜖_𝑛=^𝑛_𝑖=1𝐼𝑀₀,𝑍^𝑛_3,𝑖+1;𝑍_3,𝑖∣𝑆^𝑛−^𝑛_𝑖=1𝐼𝑍^𝑛_3,𝑖+1;𝑍_3,𝑖∣𝑆^𝑛+𝑛𝜖_𝑛≤^𝑛_𝑖=1𝐼𝑀₀,𝑍^𝑛_3,𝑖+1;𝑍_3,𝑖∣𝑆^𝑛+𝑛𝜖_𝑛≤^𝑛_𝑖=1𝐼𝑀₀,𝑍^𝑛_3,𝑖+1,𝑆^𝑖−1,𝑆^𝑛_𝑖+1;𝑍_3,𝑖∣𝑆^𝑛+𝑛𝜖_𝑛=^𝑛_𝑖=1𝐼𝑊_𝑖,𝑍_3,𝑖∣𝑆_𝑖+𝑛𝜖_𝑛, (23)

where (a) follows from Fano’s inequality.

Next, we consider the rate 𝑅₁𝑛𝑅₁=𝐻_1(a)𝑀≤𝐼𝑀₁;𝑍^𝑛₂∣𝑀₀,𝑆^𝑛+𝑛𝜖_𝑛(b)≤𝐼𝑀₁;𝑍^𝑛₂∣𝑀₀,𝑆^𝑛−𝐼𝑀₁;𝑍^𝑛₃∣𝑆^𝑛+2𝑛𝜖_𝑛=𝐼𝑀₁;𝑍^𝑛₂∣𝑀₀,𝑆^𝑛−𝐼𝑀₁;𝑍^𝑛₃,𝑀₀∣𝑆^𝑛+𝐼𝑀₁;𝑀₀∣𝑍₃,𝑆^𝑛+2𝑛𝜖_𝑛=𝐼𝑀₁;𝑍^𝑛₂∣𝑀₀,𝑆^𝑛−𝐼𝑀₁;𝑍^𝑛₃∣𝑀₀,𝑆^𝑛−𝐼𝑀₁;𝑀₀∣𝑆^𝑛+𝐼𝑀₁;𝑀₀∣𝑍^𝑛₃,𝑆^𝑛+2𝑛𝜖_𝑛=𝐼𝑀₁;𝑍^𝑛₂∣𝑀₀,𝑆^𝑛−𝐼𝑀₁;𝑍^𝑛₃∣𝑀₀,𝑆^𝑛+𝐼𝑀₁;𝑀₀∣𝑍^𝑛₃,𝑆^𝑛𝑀+2𝑛𝜖=𝐼₁;𝑍^𝑛₂∣𝑀₀,𝑆^𝑛−𝐼𝑀₁;𝑍^𝑛₃∣𝑀₀,𝑆^𝑛+𝐻𝑀₀∣𝑍^𝑛₃,𝑆^𝑛−𝐻𝑀₀∣𝑍^𝑛₃,𝑆^𝑛,𝑀₁+2𝑛𝜖_𝑛≤𝐼𝑀₁;𝑍^𝑛₂∣𝑀₀,𝑆^𝑛−𝐼𝑀₁;𝑍^𝑛₃∣𝑀₀,𝑆^𝑛+𝐻𝑀₀∣𝑍^𝑛₃,𝑆^𝑛+2𝑛𝜖_𝑛(c)≤𝐼𝑀₁;𝑍^𝑛₂∣𝑀₀,𝑆^𝑛−𝐼𝑀₁;𝑍^𝑛₃∣𝑀₀,𝑆^𝑛+3𝑛𝜖_𝑛=^𝑛_𝑖=1𝐼𝑀₁;𝑍_2,𝑖∣𝑀₀,𝑍₂^𝑖−1,𝑆^𝑛−^𝑛_𝑖=1𝐼𝑀₁;𝑍_3,𝑖∣𝑀₀,𝑆^𝑛,𝑍^𝑛_3,𝑖+1+3𝑛𝜖_𝑛=^𝑛_𝑖=1𝐼𝑀₁,𝑍^𝑛_3,𝑖+1;𝑍_2,𝑖∣𝑀₀,𝑍₂^𝑖−1,𝑆^𝑛−^𝑛_𝑖=1𝐼𝑍^𝑛_3,𝑖+1;𝑍_2,𝑖∣𝑀₀,𝑀₁,𝑍₂^𝑖−1,𝑆^𝑛−^𝑛_𝑖=1𝐼𝑀₁,𝑍₂^𝑖−1;𝑍_3,𝑖∣𝑀₀,𝑆^𝑛,𝑍^𝑛_3,𝑖+1+^𝑛_𝑖=1𝐼𝑍₂^𝑖−1;𝑍_3,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍^𝑛_3,𝑖+1+3𝑛𝜖_𝑛(d)=^𝑛_𝑖=1𝐼𝑀₁,𝑍^𝑛_3,𝑖;𝑍_2,𝑖∣𝑀₀,𝑆^𝑛,𝑍₂^𝑖−1−^𝑛_𝑖=1𝐼𝑀₁,𝑍₂^𝑖−1;𝑍_3,𝑖∣𝑀₀,𝑆^𝑛,𝑍^𝑛_3,𝑖+1+3𝑛𝜖_𝑛=^𝑛_𝑖=1𝐼𝑀₁;𝑍_2,𝑖∣𝑀₀,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1+^𝑛_𝑖=1𝐼𝑍^𝑛_3,𝑖+1;𝑍_2,𝑖∣𝑀₀,𝑆^𝑛,𝑍₂^𝑖−1−^𝑛_𝑖=1𝐼𝑀₁;𝑍_3,𝑖∣𝑀₀,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1−^𝑛_𝑖=1𝐼𝑍₂^𝑖−1;𝑍_3,𝑖∣𝑀₀,𝑆^𝑛,𝑍^𝑛_3,𝑖+1+3𝑛𝜖_𝑛(e)=^𝑛_𝑖=1𝐼𝑀₁;𝑍_2,𝑖∣𝑀₀,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1−^𝑛_𝑖=1𝐼𝑀₁;𝑍_3,𝑖∣𝑀₀,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1+3𝑛𝜖_𝑛=^𝑛_𝑖=1𝐼𝑀₁,𝑍₂^𝑖−1;𝑍_2,𝑖∣𝑀₀,𝑆^𝑛,𝑍^𝑛_3,𝑖+1−^𝑛_𝑖=1𝐼𝑍₂^𝑖−1;𝑍_2,𝑖∣𝑀₀,𝑆^𝑛,𝑍^𝑛_3,𝑖+1−^𝑛_𝑖=1𝐼𝑀₁,𝑍₂^𝑖−1;𝑍_3,𝑖∣𝑀₀,𝑆^𝑛,𝑍^𝑛_3,𝑖+1+^𝑛_𝑖=1𝐼𝑍₂^𝑖−1;𝑍_3,𝑖∣𝑀₀,𝑆^𝑛,𝑍^𝑛_3,𝑖+1+3𝑛𝜖_𝑛(f)≤^𝑛_𝑖=1𝐼𝑀₁,𝑍₂^𝑖−1;𝑍_2,𝑖∣𝑀₀,𝑆^𝑛,𝑍^𝑛_3,𝑖+1−^𝑛_𝑖=1𝐼𝑀₁,𝑍₂^𝑖−1;𝑍_3,𝑖∣𝑀₀,𝑆^𝑛,𝑍^𝑛_3,𝑖+1+3𝑛𝜖_𝑛=^𝑛_𝑖=1𝐼𝑀₁,𝑍₂^𝑖−1;𝑍_2,𝑖∣𝑀₀,𝑆^𝑖−1,𝑆_𝑖,𝑆^𝑛_𝑖+1,𝑍^𝑛_3,𝑖+1−^𝑛_𝑖=1𝐼𝑀₁,𝑍₂^𝑖−1;𝑍_3,𝑖∣𝑀₀,𝑆^𝑖−1,𝑆_𝑖,𝑆^𝑛_𝑖+1,𝑍^𝑛_3,𝑖+1+3𝑛𝜖_𝑛=^𝑛_𝑖=1𝐼𝑉_𝑖;𝑍_2,𝑖∣𝑊_𝑖,𝑆_𝑖−^𝑛_𝑖=1𝐼𝑉_𝑖;𝑍_3,𝑖∣𝑊_𝑖,𝑆_𝑖+3𝑛𝜖_𝑛, (24)

where (a) and (c) follow from Fano’s inequality, (b) follows from the secrecy condition, and (d) and (e) follow from the Csiszár sum identity; thus, we have ^𝑛_𝑖=1𝐼𝑍^𝑛_3,𝑖+1;𝑍_2,𝑖∣𝑀₀,𝑀₁,𝑍₂^𝑖−1,𝑆^𝑛=^𝑛_𝑖=1𝐼𝑍₂^𝑖−1;𝑍_3,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍^𝑛_3,𝑖+1, (25)

(f) follows the fact ^𝑛_𝑖=1𝐼𝑍₂^𝑖−1;𝑍_3,𝑖∣𝑀₀,𝑆^𝑛,𝑍^𝑛_3,𝑖+1≤^𝑛_𝑖=1𝐼𝑍₂^𝑖−1;𝑍_2,𝑖∣𝑀₀,𝑆^𝑛,𝑍^𝑛_3,𝑖+1 (26)

which is due to the fact that 𝑍₃ is a degraded version of 𝑍₂.

Finally, we consider the 𝑅₂𝑛𝑅₂=𝐻𝑀₂=𝐼𝑀₂;𝑌^𝑛₁∣𝑀₀,𝑀₁,𝑆^𝑛+𝐻𝑀₂∣𝑀₀,𝑀₂,𝑆^𝑛,𝑌^𝑛_1(a)≤𝐼𝑀₂;𝑌^𝑛₁∣𝑀₀,𝑀₁,𝑆^𝑛+𝑛𝜖_𝑛(b)≤𝐼𝑀₂;𝑌^𝑛₁∣𝑀₀,𝑀₁,𝑆^𝑛−𝐼𝑀₂;𝑍^𝑛₂∣𝑆^𝑛+2𝑛𝜖_𝑛=𝐼𝑀₂;𝑌^𝑛₁∣𝑀₀,𝑀₁,𝑆^𝑛−𝐼𝑀₂;𝑍^𝑛₂,𝑀₀,𝑀₁∣𝑆^𝑛+𝐼𝑀₂;𝑀₀,𝑀₁∣𝑍^𝑛₂,𝑆^𝑛+2𝑛𝜖_𝑛=𝐼𝑀₂;𝑌^𝑛₁∣𝑀₀,𝑀₁,𝑆^𝑛−𝐼𝑀₂;𝑀₀,𝑀₁∣𝑆^𝑛−𝐼𝑀₂;𝑍^𝑛₂∣𝑀₀,𝑀₁,𝑆^𝑛+𝐼𝑀₂;𝑀₀,𝑀₁∣𝑍^𝑛₂,𝑆^𝑛+2𝑛𝜖_𝑛=𝐼𝑀₂;𝑌^𝑛₁∣𝑀₀,𝑀₁,𝑆^𝑛−𝐼𝑀₂;𝑍^𝑛₂∣𝑀₀,𝑀₁,𝑆^𝑛+𝐼𝑀₂;𝑀₀,𝑀₁∣𝑍^𝑛₂,𝑆^𝑛+2𝑛𝜖_𝑛(c)≤𝐼𝑀₂;𝑌^𝑛₁∣𝑀₀,𝑀₁,𝑆^𝑛−𝐼𝑀₂;𝑍^𝑛₂∣𝑀₀,𝑀₁,𝑆^𝑛+3𝑛𝜖_𝑛=^𝑛_𝑖=1𝐼𝑀₂;𝑌_1,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑌^𝑛_1,𝑖+1−^𝑛_𝑖=1𝐼𝑀₂;𝑍_2,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1+3𝑛𝜖_𝑛≤^𝑛_𝑖=1𝐼𝑀₂,𝑌^𝑛_1,𝑖+1;𝑌_1,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1−^𝑛_𝑖=1𝐼𝑀₂,𝑌^𝑛_1,𝑖+1;𝑍_2,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1+3𝑛𝜖_𝑛=^𝑛_𝑖=1𝐼𝑀₂,𝑌^𝑛_1,𝑖+1,𝑋_𝑖;𝑌_1,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1−^𝑛_𝑖=1𝐼𝑋_𝑖;𝑌_1,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1,𝑌^𝑛_1,𝑖+1,𝑀₂−^𝑛_𝑖=1𝐼𝑀₂,𝑌^𝑛_1,𝑖+1,𝑋_𝑖;𝑍_2,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1+^𝑛_𝑖=1𝐼𝑋_𝑖;𝑍_2,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1,𝑌^𝑛_1,𝑖+1,𝑀₂+3𝑛𝜖_𝑛(d)≤^𝑛_𝑖=1𝐼𝑀₂,𝑌^𝑛_1,𝑖+1,𝑋_𝑖;𝑌_1,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1−^𝑛_𝑖=1𝐼𝑀₂,𝑌^𝑛_1,𝑖+1,𝑋_𝑖;𝑍_2,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1+3𝑛𝜖_𝑛=ⁿ_𝑖=1𝐼𝑋_𝑖;𝑌_1,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1+^𝑛_𝑖=1𝐼𝑀₂,𝑌^𝑛_1,𝑖+1;𝑌_1,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1,𝑋_𝑖−^𝑛_𝑖=1𝐼𝑋_𝑖;𝑍_2,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1−^𝑛_𝑖=1𝐼𝑀₂,𝑌^𝑛_1,𝑖+1;𝑍_2,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1,𝑋_𝑖+3𝑛𝜖_𝑛(e)=^𝑛_𝑖=1𝐼𝑋_𝑖;𝑌_1,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1−^𝑛_𝑖=1𝐼𝑋_𝑖;𝑍_2,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1+3𝑛𝜖_𝑛=^𝑛_𝑖=1𝐼𝑋_𝑖;𝑌_1,𝑖∣𝑀₀,𝑀₁,𝑆^𝑖−1,𝑆_𝑖,𝑆^𝑛_𝑖+1,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1−^𝑛_𝑖=1𝐼𝑋_𝑖;𝑍_2,𝑖∣𝑀₀,𝑀₁,𝑆^𝑖−1,𝑆_𝑖,𝑆^𝑛_𝑖+1,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1+3𝑛𝜖_𝑛=^𝑛_𝑖=1𝐼𝑋_𝑖;𝑌_1,𝑖∣𝑉_𝑖,𝑆_𝑖−^𝑛_𝑖=1𝐼𝑋_𝑖;𝑍_2,𝑖∣𝑉_𝑖,𝑆_𝑖+3𝑛𝜖_𝑛, (27)

where (a) and (c) follow Fano’s inequality, (b) follows from the secrecy condition, (d) follows the fact that ^𝑛_𝑖=1𝐼𝑋_𝑖;𝑍_2,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1,𝑌^𝑛_1,𝑛+1,𝑀₂≤^𝑛_𝑖=1𝐼𝑋_𝑖;𝑌_1,𝑖∣𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1,𝑌^𝑛_1,𝑖+1,𝑀₂ (28)

which is due to the degradation condition, and (e) is due to the following Markov Chain: 𝑀₂,𝑌^𝑛_1,𝑖+1,𝑀₀,𝑀₁,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1⟶𝑋_𝑖,𝑀₀,𝑀₁,𝑆^𝑛,𝑍₂^𝑖−1,𝑍^𝑛_3,𝑖+1⟶𝑌_1,𝑖,𝑍_2,𝑖. (29)

Finally, let 𝑄∈{1,2,…,𝑛} be a uniformly distributed random variable independent of all other random variables and define 𝑊 = (𝑊_𝑄,𝑄), 𝑉=(𝑉_𝑄,𝑄), 𝑋=𝑋_𝑄, 𝑌₁ = 𝑌_1𝑄, 𝑍₂ = 𝑍_2𝑄, 𝑍₃ = 𝑍_3𝑄 that completes the converse proof in the standard way [15].

4. Conclusion

A class of three-receiver, one-receiver two-eavesdropper, discrete memoryless broadcast channel (DM-BC) with three degraded message sets and noncausally side information is considered in this paper. We study this class of three-receiver BC with three degraded message sets and side information with secrecy constraints. We establish the achievable rates using a combination of superposition coding, Gel’fand-Pinsker binning scheme, and Nair-El Gamal indirect decoding when the side information is noncausally available at the transmitter. The achievable secrecy region of this paper subsumes Steinberg’s rate region for two-receiver degraded BC with the noncausally side information and the secrecy capacity region for one-receiver two-eavesdropper DM-BC with no side information as its special cases. As the side information is noncausally available at both the transmitter and the three receivers, we determine the perfect secrecy capacity region of the one-receive two-eavesdropper BC with three degraded message sets when the receivers follow a degradedness order; that is, 𝑌₁→𝑍₂→𝑍₃ forms a Markov chain. In our future work, we will consider more general three-receiver BC with two or three degraded message sets and side information causally or noncausally available at the transmitter and the receiver (the receivers without following the degradedness order).

Conflict of Interests

The authors declare that they have no financial and personal relationships with other people or organizations that can inappropriately influence their work, and there is no professional or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of this paper.

Acknowledgment

This work was supported in part by the National Science Foundation of China under Grants no. 60932003 and no. 61271220.