Rate splitting and sharing for degraded broadcast channel with secrecy outside a bounded range

A four-receiver degraded broadcast channel with secrecy outside a bounded range is studied, over which a transmitter sends four messages to four receivers. In the model considered, the channel quality gradually degrades from receiver 4 to receiver 1, and receiver k is required to decode the first k messages for k = 1, ..., 4. Furthermore, message 3 is required to be secured from receiver 1, and message 4 is required to be secured from receivers 1 and 2. The secrecy capacity region is established. The achievable scheme includes not only superposition, binning and embedded coding used in previous studies, but also rate splitting and sharing particularly designed for this model, which is shown to be critical to further enlarge the achievable region and enable the development of the converse proof.


Introduction
The broadcast channel models an important type of scenarios in which the transmitter's signal can simultaneously reach multiple receivers, and it has been widely used to model wireless communications. Within the communication range of the transmitter, some receivers are intended while some are non-intended or even eavesdroppers from which the messages should be kept secure. Due to this broadcast nature of wireless communications, security has arisen as an important issue. Various broadcast channel models with different transmission reliability constraints (i.e., legitimate receivers should decode messages destined for them) and different secrecy constraints (i.e., eavesdroppers should be kept ignorant of messages) have been intensively studied (see recent surveys [4][5][6][7]).
The basic broadcast channel with the secrecy constraint was the wiretap channel initiated by Wyner in [8], in which a transmitter has a message intended for a legitimate receiver and wishes to keep this message secured from an eavesdropper. Csiszár and Körner further generalized this model to the case with one more common message intended for both the legitimate receiver and the eavesdropper in [9].
These broadcast models were further generalized to the multi-receiver case in [10] and [11], in which a transmitter has a number of messages intended for a set of receivers, and all messages need to be secured from an eavesdropper. Another class of extension is degraded broadcast channel with layered decoding and layered secrecy [11][12][13], in which the transmitter has a number of messages intended for a set of receivers and as the channel quality of a receiver gets one level better, one more message is required to be decoded, and this message is required to be secured from all receivers with worse channel quality.
We note that for the model with layered decoding and layered secrecy, the additional message decoded by a better receiver needs to be kept secure from the receiver with only one level worse channel quality (layered secrecy, zero secrecy range). Although such a model is feasible for broadcast channels with discrete states (i.e., quality of receivers can be captured by discrete channel states), it cannot capture the scenarios with receivers' channel quality varying continuously. For such a case, it is more reasonable to require the message to be secured from the receivers with a certain amount of worse channel quality, instead of being secured from the receiver with one level worse channel quality, which is not even well defined for continuous channel quality. To be more explicit, we use an example to illustrate the motivation of such a model. Consider a degraded broadcast channel with infinite number of receivers, in which h denotes the amplitude of the channel gain (the larger h, the better the channel). In this case, it is impossible to require that the message intended for receivers with h ≥ h 0 to be secured from receivers with h < h 0 , because no positive secrecy rate can be achieved. Instead, it is more natural to require that the messages intended for receivers with h ≥ h 0 to be secured from receivers with h ≤ h 0 − ∆, where ∆ > 0. We refer to such a secrecy requirement as secrecy outside a bounded range.
In this paper, we consider the K-receiver degraded broadcast channel with secrecy outside a bounded range (see Fig. 1), in which a transmitter sends K messages to K receivers. The channel is assumed to satisfy the degradedness condition, i.e., the channel quality gradually degrades from receiver K to receiver 1. Furthermore, receiver k is required to decode the first k messages, W 1 , . . . , W k , for 1 ≤ k ≤ K, and to be kept secure of W k+2 , . . . , W K for k = 1, . . . , K − 2. Each message W k is required to be secured from the receiver k − 2, which has two level worse channel quality, for 3 ≤ k ≤ K. In this way, the secrecy is required outside a range of two level channel quality.
The main result of this paper lies in the complete characterization of the secrecy capacity region for the K-receiver degraded broadcast channel with secrecy outside a bounded range. To understand the challenges of the problem and the novelty of the paper, we first describe special cases, namely three-receiver and four-receiver models, studied by the authors in earlier conference versions [1,2]. For three-receiver model, we show in [1] that superposition of messages and joint random binning and embedded coding (using lower layer messages to protect higher layer messages) achieves the secrecy capacity. However, in [2] we show that a natural generalization of such a scheme does not provide the capacity region for the four-receiver model. A novel rate splitting and sharing scheme was proposed in [2], which is shown to be critical to further enlarge the achievable region and establish the secrecy capacity region for the four-receiver model. The idea is to first use lower-layer messages to serve as random sources to protect high-layer messages, and if the lower-layer messages are more than enough to protect high-layers messages, then further share remaining rate of lower-layer messages with upper-layer messages, as such part of lower layer messages can satisfy the same secrecy constraints as high-layer messages.
Further generalization of the capacity characterization for the above four-receiver model to the arbitrary K-user case becomes very challenging due to the following reasons. (1) Based on the understanding in the four-receiver model, each message as well as the random bin number at each layer can potentially serve as random source to protect all higher-layer messages (from lower layer receivers). The design of joint embedded coding and random binning is very complicated to handle. For example, consideration of whether to adopt random binning at layer k depends on whether embedded coding of layer k − 1 is sufficient to protect W k from receiver k −2, and whether embedded coding of layer k −2 and (possible) random binning in layer k − 1 are sufficient to protect W k−1 and W k from receiver k − 3, and so on. Incorporating all these considerations into the design of an achievable scheme is not feasible for arbitrary K-user model. (2) Due to rate splitting and sharing across adjacent layers, the rate region is expressed in terms of individual rate components. A typical technique to convert the rate region in terms of the (total) rate for each message is Fourier-Motzkin elimination. However, for the arbitrary K-user model, a large number of rate variables (more specifically, 2K) should be eliminated from the order of K 2 rate bounds. Such procedure is not analytically tractable in general. (3) Due to the reason that we employ joint embedded coding and random binning to secure multiple messages, the analysis of leakage rates is much more involved than the cases with only one or two messages secured by random binning.
Despite the challenges mentioned above, in this paper, we fully characterize the secrecy capacity region for the K-receiver model with secrecy outside a bounded range. Our solution of the problem includes the following new ingredients. (1) Our achievable scheme employs random binning in each layer, which avoids the complex consideration of whether or not it is necessary to employ random binning for each layer. We also make an important observation that rate sharing only between adjacent layers is sufficient. This observation is critical to keep the obtained rate region simple enough for further manipulation. (2) We design an induction algorithm to perform Fourier-Motzkin elimination. Instead of directly eliminating 2K variables from the order of K 2 rate bounds, we eliminate a pair of variables at a time. We then further show that the region after each elimination step possesses a common structure by induction. (3) In order to analyze the leakage rate for arbitrary K-user case, we generalize the analysis of the leakage rate provided in [14] for one confidential message using random binning to multiple confidential messages using joint embedded coding and random binning. (4) Our development of the converse proof involves recursive construction of rate upper bound on each message such that proper terms cancel out across adjacent messages, and manipulation of the terms by exploiting intuition in achievable schemes.
The remainder of this paper is organized as follows. In Section 2, we introduce our system model. In Section 3, we present two example models with three receivers and four receivers, respectively, which motivate the design of the achievable scheme for arbitrary K-receiver model. In Section 4, we present our main results for the model with arbitrary K receivers. Finally, in Section 5, we conclude our paper.

Channel Model
In this paper, we consider a K-receiver degraded broadcast channel model with secrecy outside a bounded range (see Fig. 1). A transmitter sends information to K receivers through a discrete memoryless channel. The channel transition probability function is P Y 1 ···Y K |X , where X ∈ X denotes the channel input, and Y k ∈ Y k denotes the channel output at receiver k, for 1 ≤ k ≤ K. The channel is assumed to be degraded, i.e., the following Markov chain condition holds: (1) Figure 1: The K-receiver broadcast channel with secrecy outside a bounded range Hence, the channel quality gradually degrades from receiver K to receiver 1. There are in total K messages W 1 , W 2 , . . . , W K intended for K receivers with the following decoding and secrecy requirements. Receiver k is required to decode messages W 1 , W 2 , . . . , W k , for k = 1, 2, . . . , K, and to be kept secure of W k+2 , . . . , W K , for k = 1, . . . , K − 2 (see Fig. 1).
A (2 nR 1 , · · · , 2 nR K , n) code for the channel consists of • K message sets: W k ∈ W k = {1, · · · , 2 nR k } for k = 1, · · · , K, which are independent from each other and each message is uniformly distributed over the corresponding message set; • A (possibly stochastic) encoder f n : W 1 × · · · × W K → X n that maps a message tuple to an input x n ; • K decoders g n k : Y n k → (W 1 , · · · , W k ) that maps an output y n k to a message tuple (ŵ 1 , . . . ,ŵ k ) for k = 1, · · · , K.
A rate tuple (R 1 , · · · , R K ) is said to be achievable, if there exists a sequence of (2 nR 1 , · · · , 2 nR K , n) codes such that both the average error probability and the leakage rate at each receiver k for k = 3, . . . , K approach zero as n goes to infinity.
Here, the asymptotically small error probability as in (2) implies that each receiver k is able to decode messages W 1 , . . . , W k , and asymptotically small leakage rate as in (3) for each receiver k implies that receiver k is kept ignorant of messages W k+2 , . . . , W K . Our goal is to characterize the secrecy capacity region that consists of all achievable rate tuples.

Motivating Examples
In this section, we study two simple cases with K = 3 and K = 4. The purpose is to motivate the development of the optimal achievable scheme for the case with arbitrary K receivers step by step. More specifically, we study the example with three receivers to introduce the technique of joint embedded coding and random binning. And we study the example with four receivers to introduce the technique of rate splitting and sharing. These schemes turn out to be necessary to achieve the secrecy capacity region for the case with arbitrary K receivers. We start with the case in which there are three receivers (see Fig. 2). In this case, receiver 1 is required to decode W 1 , receiver 2 is required to decode W 1 , W 2 , and receiver 3 is required to decode W 1 , W 2 , W 3 . The system is also required to satisfy the secrecy constraint that the message W 3 is kept secure from receiver 1. For such a model, a natural idea is to design superposition coding for encoding three messages W 1 , W 2 , W 3 into three layers, and then apply random binning in the third layer to protect W 3 from receiver 1. However, such a scheme is suboptimal because it ignores an important fact that the random message W 2 , which is not required to be decoded by receiver 1, can provide additional randomness to protect W 3 from receiver 1. This is referred to as embedded coding. In fact, if such a random source of W 2 is sufficient to protect W 3 from receiver 1, random binning is not necessary. If this is not sufficient to protect W 3 , we apply random binning in the third layer to further protect W 3 from receiver 1. The novelty of such an achievable scheme lies in exploiting the superposition layer of W 2 as embedded coding in addition to the random binning scheme to protect W 3 . Such a scheme turns out to achieve the secrecy capacity region as characterized in the following proposition. Proposition 1. Consider the three-receiver degraded broadcast channel with secrecy outside a bounded range as described in Section 2. The secrecy capacity region contains rate tuples

Lessons Learned from K = 3
for some P U 1 U 2 X such that the following Markov chain condition holds Proof. The proof can be found in [1].
The idea of the achievable scheme is also reflected in the expression of the capacity region in (4). The two bounds in "min" are corresponding to the two cases with the second layer of W 2 being sufficient and insufficient to protect W 3 , respectively. If I(U 2 ; Y 2 |U 1 ) > I(X; Y 1 |U 1 ), the randomness of W 2 is sufficient to exhaust receiver 1's decoding capability, and hence is good enough for protecting W 3 . Thus, in this case, no binning is required in layer 3, and R 3 ≤ I(X; Y 3 |U 2 ). On the other hand, if I(U 2 ; Y 2 |U 1 ) ≤ I(X; Y 1 |U 1 ), binning is required at layer 3 to protect W 3 in addition to randomness of W 2 , and hence, R 3 ≤ I(U 2 ; Y 2 |U 1 ) − I(X; Y 1 |U 1 ) + I(X; Y 3 |U 2 ).

Lessons
Learned from K = 4 In this subsection, we study the model with four receivers (see Fig. 3). In this model, receiver k is required to decode messages W 1 , . . . , W k , for 1 ≤ k ≤ 4. Furthermore, the message W 3 is required to be secured from receiver 1, and the message W 4 is required to be secured from receivers 1 and 2.
Although this four-receiver model seems to be a straightforward generalization of the three-receiver model, our exploration turns out to show that the achievable scheme for the three-receiver model is not sufficient to establish the secrecy capacity region for the fourreceiver model. In order to understand this, we note that a direct generalization of the achievable scheme for the three-model involves first applying superposition coding to encode the four messages, and then use the random message W 3 as embedded coding together with the random binning in layer 4 (if necessary) to protect W 4 , and use the random message W 2 as embedded coding together with the random binning in layer 3 and layer 4 (if necessary) to protect W 3 and W 4 . Such a scheme then yields an achievable region with rate tuples (R 1 , R 2 , R 3 , R 4 ) satisfying It turns out to be very difficult to develop the converse proof for the bound R 4 ≤ I(X; Y 4 |U 3 ) in the above region. Thus, the optimality of the region (6) cannot be guaranteed.
The major novelty in our scheme for this four-receiver model lies in the design of rate splitting and sharing, which helps enlarge the achievable region and thus establish the secrecy capacity region. In order to further enlarge the above region, we incorporate the newly designed ingredient of rate splitting and sharing. More specifically, if W 3 is sufficient to protect W 4 , we further split W 3 into two parts, i.e., W 3,1 and W 3,2 , such that W 3,1 serves as a random source to secure both W 3,2 and W 4 from receiver 2. Thus,W 3,2 satisfies both the decoding and secrecy requirements for W 4 , and hence the rate of W 3,2 can be counted towards the rate of either W 3 or W 4 . In this way, the achievable region can be potentially enlarged. In fact, such an enlarged region can be shown to be the secrecy capacity region as characterized in the following proposition.
Proposition 2. Consider the four-receiver degraded broadcast channel with secrecy outside a bounded range as described in Section 2. The secrecy capacity region consists of rate tuples for some P U 1 U 2 U 3 X such that the following Markov chain condition holds Proof. The proof can be found in [2].
We note that by using rate splitting and sharing, the bound (7). Clearly, the region (7) is larger than the region (6) (for a given distribution of auxiliary random variables). Furthermore, the converse proof for the new bound on R 3 + R 4 in (7) can be derived, and thus establishes the region (7) as the secrecy capacity region.
Moreover, although we learn useful coding ingredients from the three-receiver and fourreceiver cases, direct generalization to arbitrary K-receiver model still gives rise to an analytically intractable achievable scheme. More specifically, the consideration of whether or not to use random binning in the higher layers and whether or not to split and share the rates will be complex. For example, when K = 5, whether to use random binning in the fifth layer depends on whether the embedded coding in the third layer and (possibly) random binning in the fourth layer are sufficient to protect W 4 , W 5 from receiver 3 and whether the embedded coding in the fourth layer is sufficient to protect W 5 from receiver 3. Such considerations become intractable when K is large. Thus, the major design issue for the arbitrary K-receiver case is to develop an achievable scheme that effectively incorporates the necessary coding ingredients as well as yielding a tractable rate region for analysis. This is the focus of the following section.

Main Results
In this section, we first present our main result of characterization of the secrecy capacity region for the K-receiver model, and then describe the idea of the design of the achievable scheme.

Secrecy Capacity Region
The following theorem states our main result. For simplicity of notation, we define U K = X. Theorem 1. Consider the K-receiver degraded broadcast channel with secrecy outside a bounded range as described in Section 2. The secrecy capacity region consists of rate tuples (R 1 , R 2 , . . . , R K ) satisfying for some P U 1 U 2 ...U K such that the following Markov chain condition holds: Proof. The proof of the achievability and the proof of converse are provided in Appendices A and D, respectively.
In the above capacity region, the bounds (9a) and (9b) are due to the decoding requirements, i.e., receiver k should decode messages W 1 , . . . , W k , for 1 ≤ k ≤ K. The sum rate bounds (9b) are due to the rate sharing scheme we design. The bounds (9c) are due to the secrecy requirements, i.e., messages W l , . . . , W k need to be kept secure from receiver l − 2 for 3 ≤ l ≤ k ≤ K. Furthermore, the bounds (9c) can be further written as is corresponding to the rate in layer j that can be secured from receiver l − 2 given the knowledge of layer j − 1. Those rates j=l R j in accordance to the secrecy requirement of keeping W l , . . . , W k secured from receiver l − 2.
If we set K = 3 and K = 4, the region in Theorem 1 reduces to equivalent but different forms from the regions in Proposition 1 and Proposition 2. The equivalence is justified by the converse proofs. In fact, the achievable schemes for the three-receiver model in Section 3.1 and the four-receiver model in Section 3.2 cannot be easily generalized to the arbitrary K-receiver model.
Our design of the achievable scheme for the general arbitrary K-receiver model is different from those for the three-receiver and four receiver models, and includes the following new ingredients. The scheme employs random binning in each layer, which avoids the complex consideration of whether or not it is necessary to employ random binning for each layer. And the rate sharing scheme is limited only between adjacent layers which captures the essence of the problem and helps simplify the obtained rate region. Furthermore, we design an induction algorithm to perform Fourier-Motzkin elimination, which makes the problem of eliminating 2K variables from the order of K 2 bounds analytically tractable. These ideas are described in more detail in Subsection 4.2.
The converse for the achievable region can be developed. The bounds (9a) and (9b) can be derived following standard steps. However, the proof for the bounds (9c) is more involved and requires careful recursive construction of the terms such that proper terms cancel out across adjacent messages.

Achievable Scheme
In this subsection, we introduce the idea of the achievable scheme for the arbitrary K-receiver model, which is based on superposition coding, random binning, embedded coding, and rate splitting and sharing. We also sketch the novel induction idea to analyze Fourier-Motzkin elimination to characterize the achievable region.
Superposition, binning, embedded coding. We design one layer of codebook for each message, i.e., layer k corresponds to W k , for 1 ≤ k ≤ K. To avoid the complex consideration of whether to use random binning, we employ random binning in each layer. We divide the codewords in each layer into a number of bins, where the bin number contains the information of the corresponding message. We use joint embedded coding and random binning to provide randomness for secrecy.
Rate splitting and sharing. We design rate splitting and sharing to enlarge the achievable region. More specifically, within the k-th layer, we split the message W k into two parts W k,1 and W k,2 . The message W k,1 serves as embedded coding which is a random source in addition to the random binning to protect W k,2 and the higher layer messages from receiver Y k−1 , i.e., we require that (W k,2 , W k+1,1 , W k+1,2 , . . . , W K,1 , W K,2 ) are secured from receiver Y k−1 , for 2 ≤ k ≤ K − 1. Furthermore, the upstream receiver Y k+1 can also decode W k,2 because Y k+1 has a better channel quality than Y k . Thus, the message W k,2 satisfies both the decoding and secrecy requirements for message W k+1 , and hence, the rate of W k,2 can be counted towards the rate of either W k or W k+1 . By such a rate sharing strategy, the achievable region is enlarged.
We note that the rate can only be shared between adjacent receivers, which is an important observation of this problem, and is critical to reduce the complexity of the design of the rate splitting and sharing strategy. More specifically, the rate of W k,2 cannot be counted towards the rates of W k+2 , . . . , W K , because W k+2 , . . . , W K are required to be secured not only from receiver Y k−1 but also from Y k that are required to decode W k,2 .
Based on the above achievable scheme, we obtain the following achievable region: The above region are expressed in terms of component rates due to rate splitting. In order to express the above region in terms of total rate for each message, we introduce the technique of rate sharing. We define We then wish to project the region (11) onto the rate space (R 1 , . . . , R K ). This can be done by adding the above rate definitions to the achievable region (11) and then perform the Fourier-Motzkin elimination to eliminate R k,1 and R k,2 for 2 ≤ k ≤ K.
Fourier-Motzkin elimination via induction. The total number of bounds in the achievable region (11) is on the order of K 2 with 2K variables to be eliminated. Directly applying Fourier-Motzkin elimination is not analytically tractable. In order to solve such a problem, we design the following induction algorithm to perform Fourier Motzkin elimination. We eliminate the rate pairs R k−1,2 and R k,1 for 3 ≤ k ≤ K one at each step, and wish to show that the region R k after eliminating R k−1,2 and R k,1 possesses the following structure: Such a claim can be easily verified for the case when k = 3, 4, 5. If such a claim holds for R k , we then are able to show (see Appendix C for detailed proof) that the region R k+1 after eliminating R k,2 and R k+1,1 possesses the same structure given by The last step is to eliminate (R K−1,2 , R K,1 , R K,2 ). Thus, the above induction algorithm and arguments yield the achievable region in Theorem 1.

Conclusion
In this paper, we have studied the K-receiver degraded broadcast channel with secrecy outside a bounded range. We have proposed a novel achievable scheme based on superposition coding, joint embedded coding and random binning, and rate splitting and sharing. The combination of embedded coding and random binning to achieve secrecy captures the fact that lower-layer message can serve as embedded coding to protect higher-layer messages.
And rate splitting and sharing are critical to enlarge the achievable region for which the converse proof can be established. Moreover, our design exploits an important property that the rate sharing should be only between adjacent receivers, which significantly reduces the complexity of the achievable scheme. We have further proposed a novel induction algorithm to perform Fourier-Motzkin elimination on the achievable region with 2K variables to be eliminated from the order of K 2 bounds. We have also constructed a converse proof, which involves careful recursive construction of rate bounds, and exploits the intuition gained from embedded coding in the achievable scheme. By the converse proof, we have demonstrated the optimality of our achievable scheme and established the secrecy capacity region.
Several directions are interesting to explore in the future. The current results can be further generalized to the model with continuously changing channel state parameters, e.g., Gaussian fading channel [15], and Gaussian multiple input multiple output (MIMO) channel [16]. This paper has focused on characterizing the information theoretic performance limits which is based on random coding arguments. It is of further interest to design practical coding schemes such as low density parity check (LDPC) codes [17] and polar codes [18,19] to achieve the secrecy capacity region.

Appendix A Achievability Proof of Theorem 1
The achievability proof is based on superposition coding, embedded coding, random binning, rate splitting and sharing. We use random codes and fix a distribution P U 1 U 2 ...,U K−1 X P Y 1 ...Y K |X satisfying the Markov chain condition in (10). We let T n (P U 1 ...U K−1 XY 1 ...Y K ) denote the strongly jointly -typical set based on the fixed distribution. We design the achievable scheme as follows: Random codebook generation: For simplicity, we define U K = X in the following proof, i.e., P U 1 ···U K−1 X = P U 1 ···U K .
Otherwise, it declares an error.
Otherwise, it declares an error.
Analysis of error probability: By the law of large numbers and packing lemma, it can be shown that receiver k decodes the message (w 1 , . . . , w k,1 , w k,2 ) for 2 ≤ k ≤ K and receiver 1 decodes the message w 1 with asymptotically small error probabilities if the following inequalities are satisfied: Analysis of leakage rate: We require that W k−1,2 , W k,1 , W k,2 , . . . , W K,1 , W K,2 be secured from receiver Y k−2 for 3 ≤ k ≤ K. It suffices to show that the average of the leakage rate over the random codebook ensemble converges to zero, as n → ∞, i.e., for 3 ≤ k ≤ K. This implies the existence of one codebook that guarantees secrecy.
We note that l k in random codebook generation is a realization of the random variable L k here. And we let L K 3 = (L 3 , . . . , L K ). We generalize the analysis in [14] for one secure message via random binning to multiple secure messages via joint embedded coding and random binning. For any 3 ≤ k ≤ K, we derive the following bound: = I(W 1 , W 2,1 , W 2,2 , . . . , W K,1 , W K,2 , L K 3 ; Y n k−2 |C) − I(W 1 , W 2,1 , W 2,2 , . . . , W k−1,1 , L K 3 ; Y n k−2 |W k−1,2 , . . . , W K,2 , C) where (a) follows from the chain rule of mutual information, and (b) is due to the Markov chain condition (W 1 , . . . , We then bound the three terms in (22) one by one. We bound the first term as follows: where (a) is due to the Markov chain condition U n k−2 → U n K → Y n k−2 , and (b) is due to the fact that H(Y k−2,i |Y n k−2,i+1 , U n k−2 , C) ≤ H(Y k−2,i |U k−2,i ) and the Markov chain condition (U i−1 K , U n K,i+1 , Y n k−2,i+1 , C) → U K,i → Y k−2,i . For the second term, due to the independence of the messages W 1 , W 2,1 , W 2,2 , . . . , W K,1 , W K,2 and L 3 , . . . , L K , we have We bound the third term as follows: where (a) is due to Fano's inequality [20], (b) is due to the fact that U n k−2 is a function of (C, W 1 , . . . , W k−2,2 , L k−2 3 ), and (c) follows from Lemma 1 given below if the conditions (26) are satisfied. Lemma 1. For 3 ≤ k ≤ K, if the following conditions are satisfied: then the following inequality holds: Proof. See Appendix B.
Combining the bounds in (20) and (26), we conclude that the rate tuple (R 1 , R 2,1 , R 2,2 , . . . , R K,1 , R K,2 ) is achievable if Rate Sharing: We note that our achievable scheme guarantees W k−1,2 , W k,1 , W k,2 , . . . , W K,1 , W K,2 to be secured from receiver Y k−2 , for 3 ≤ k ≤ K. Furthermore, due to the degradedness condition, W k−1,2 can be decoded by receiver Y k . Thus, W k−1,2 satisfies both the decoding and secrecy requirements as W k . Hence, the rate of W k−1,2 can be counted towards either R k−1 or R k . Based on such an understanding, we design the following rate sharing scheme. We define R 2 = R 2,1 , R k = R k−1,2 + R k,1 for 3 ≤ k ≤ K − 1, and R K = R K−1,2 + R K,1 + R K,2 , and include these equations to the above achievable region. We then perform Fourier-Motzkin elimination to eliminate R k,1 , R k,2 for 2 ≤ k ≤ K and obtain a closed-form achievable rate region. Such a process involves eliminating 2K − 2 variables from the order of K 2 bounds, which is intractable for arbitrary K. We propose an inductive Fourier Motzkin elimination approach as shown in Appendix C, and obtain the achievable region given in Theorem 1.

C Induction Algorithm for Fourier-Motzkin Elimination
As we have shown in Appendix A, we need to eliminate R k,1 , R k,2 for 2 ≤ k ≤ K in the following region: where the bounds (47a), (47b) and (47c) correspond to the achievable region after rate splitting, which are expressed in terms of component rates, and the bounds (47d), (47e) and (47f) are corresponding to the rate sharing strategy.
It can be seen that the total number of bounds in the above region is on the order of K 2 over which 2K − 2 variables need to be eliminated. Directly applying Fourier-Motzkin elimination is not analytically tractable. We design an inductive algorithm, in which we eliminate the rate pairs (R k−1,2 , R k,1 ) for 3 ≤ k ≤ K − 1 one at each step, and finally eliminate (R K−1,2 , R K,1 , R K,2 ). We first replace R 2,1 with R 2 , R k−1,2 + R k,1 with R k for 3 ≤ k ≤ K − 1, and R K−1,2 + R K,1 + R K,2 with R K , and we obtain the following region: To start the elimination process, we first eliminate (R 2,2 , R 3,1 ) from the inequalities given below, corresponding to the decoding and secrecy requirements of receiver 1 to receiver 3: We then obtain the following inequalities after elimination: which we denote as R 3 .
We then eliminate (R 3,2 , R 4,1 ) from the inequalities in R 3 and the inequalities given below, which together are corresponding to the decoding and secrecy requirements of receiver 1 to receiver 4: We then obtain the following bounds after elimination: which we denote as R 4 .
As we observe, the region R 3 and R 4 conform to the following structure for k = 3 and k = 4: We next show that the region R k takes the structure (53) for any 3 ≤ k ≤ K − 1 using induction. We have verified such a claim for k = 3, 4. If such a claim holds for R k , we eliminate R k,2 and R k+1,1 from the inequalities in R k and the inequalities given below, which together are corresponding to the decoding and secrecy requirements of receiver 1 to receiver k + 1: Then the resulting region, following standard steps of Fourier-Motzkin elimination to eliminate R k,2 and R k+1,1 , equals (53) for k + 1.

D Converse Proof of Theorem 1
By Fano's inequality and the secrecy requirements, we have the following inequalities: both of which implies that We denote Y i−1 k := (Y k,1 , . . . , Y k,i−1 ), and Y n k,i+1 := (Y k,i+1 , . . . , Y k,n ). We set U 1,i := Due to the degradedness condition, it can be verified that (U 1,i , U 2,i , . . . , U K−1,i , U K,i , X i ) satisfy the following Markov chain condition: We first bound the rate R 1 . Since W 1 is only required to be decoded by receiver Y 1 , we obtain the following bound: where (a) is due to Fano's inequality.
We further bound the rate R 2 as follows: where (a) is due to Fano's inequality, and (b) is due to the Markov chain condition We then bound the sum rate bounds on k i=2 R i , for 3 ≤ k ≤ K: where (a) is due to the independence between the messages (W 1 , . . . , W k ), (b) is due to Fano's inequality, and (c) is due to the facts that −I( ) ≤ 0 and the following inequalities: where (a) is due to Csiszár's sum identity property [9], and (b) and (c) are due to the degradedness condition (1).
We next bound the sum rate bounds on k j=l R j , for 3 ≤ l ≤ k ≤ K, which correspond to the secrecy constraints: where (a) is due to the secrecy requirement (57) and the independence of the messages, and (b) is due to the following fact: where (a) is due to the independence of the messages.
We next bound each term in (63) one by one. We first bound H(W j ) for l ≤ j ≤ k as follows: where (a) is due to the independence of the messages and the Fano's inequality (57), (b) is due to Csiszár sum identity property, (c) is due to the degradedness condition (1) and the fact that the inequality (d) is due to the degradedness condition (1) and the fact that and (e) is due to Csiszár's sum identity property.
Following the intermediate step in (65), H(W j ) is also upper bounded as follows: Hence, substituting (65) for l ≤ j ≤ k, and (68) for j = l − 1 into the first term in (63), we obtain, We then bound the third term in (63) for 3 ≤ l ≤ k ≤ K as follows: where (a) is due to the following fact: where (a) is due to the following two facts. The first fact is shown as follows: Furthermore, based on (61), we bound K j=2 R j as follows: And based on (72), we bound K j=l R j as follows: where (a) is due to the Markov chain condition (58).
The proof of the converse is then completed by defining a uniformly distributed random variable Q ∈ {1, . . . , n}, and setting U k (Q, U k,Q ), Y k Y k,Q , for k ∈ [1 : K], and X (Q, X Q ).