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IEEE International Symposium on Information Theory (ISIT)

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

Shaofeng Zou, Yingbin Liang, Lifeng Lai, Shlomo Shitz Shamai

Published Jun 1, 2015• 8 citations
Publisher

Abstract

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.

DOI
10.1109/isit.2015.7282677

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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.
A K-receiver degraded broadcast channel with secrecy outside a bounded range is studied, in which a transmitter sends K messages respectively to K receivers, and the channel quality gradually degrades from receiver K to receiver 1. Each receiver k is required to decode messages W1, …, Wk, for 1 ≤ k ≤ K. Furthermore, each message Wkshould be kept secure from receivers with two-level worse channel quality, i.e., receivers 1, …, k − 2. The secrecy capacity region is fully characterized. The achievable scheme designates one superposition layer to each message with random binning employed for each layer for protecting all upper-layer messages from lower-layer receivers. Furthermore, the scheme allows adjacent layers to share rates so that part of the rate of each message can potentially be shared with its immediate upper-layer message to enlarge the rate region. More importantly, an induction approach is developed to perform Fourier-Motzkin elimination over 2K variables among Θ(K2) bounds to obtain a close-form achievable rate region. A converse proof is developed that matches the achievable rate region, which involves recursive construction of the rate bounds.
A three-receiver degraded broadcast channel with secrecy outside of a bounded range is studied, in which the channel quality gradually degrades from receiver 3 to receiver 1. The transmitter has three messages intended for the receivers with receiver 3 decoding all messages, receiver 2 decoding the first two messages, and receiver 1 decoding only the first message. Furthermore, the third message should be kept secure from receiver 1. The discrete memoryless channel is studied and the secrecy capacity region is characterized. The achievable scheme is based on superposition coding and random binning, in which one superposition layer and random binning together provide secrecy. The converse proof is derived based on the insight obtained from the achievable scheme so that manipulations of terms yield tight rate bounds.
A K-receiver degraded broadcast channel with layered decoding and secrecy constraints is investigated, in which receivers are ordered by their channel quality. Each receiver is required to decode one more message compared to the receiver with one level worse channel quality, and this message should be kept secure from all receivers with worse channel quality. For both the discrete memoryless channel and the Gaussian channel, the secrecy capacity region is characterized. The achievability scheme is based on stochastic encoding and superposition coding schemes. Novel generalization of the analysis of leakage rates and of the proof of the converse is developed for the K-receiver scenario.
In this paper, we consider a scenario where a source node wishes to broadcast two confidential messages for two respective receivers, while a wire-tapper also receives the transmitted signal. This model is motivated by wireless communications, where individual secure messages are broadcast over open media and can be received by any illegitimate receiver. The secrecy level is measured by equivocation rate at the eavesdropper. We first study the general (non-degraded) broadcast channel with confidential messages. We present an inner bound on the secrecy capacity region for this model. The inner bound coding scheme is based on a combination of random binning and the Gelfand-Pinsker bining. This scheme matches the Marton's inner bound on the broadcast channel without confidentiality constraint. We further study the situation where the channels are degraded. For the degraded broadcast channel with confidential messages, we present the secrecy capacity region. Our achievable coding scheme is based on Cover's superposition scheme and random binning. We refer to this scheme as secret superposition scheme. In this scheme, we show that randomization in the first layer increases the secrecy rate of the second layer. This capacity region matches the capacity region of the degraded broadcast channel without security constraint. It also matches the secrecy capacity for the conventional wire-tap channel. Our converse proof is based on a combination of the converse proof of the conventional degraded broadcast channel and Csiszar lemma.
The Gaussian fading channel is studied, in which the channel from the transmitter to the receiver is corrupted by a multiplicative fading coefficient H and an additive Gaussian random noise. It is assumed that the channel is experiencing block fading, and the transmitter does not know the channel state information (CSI). The receiver is assumed to have full knowledge of the CSI. If the channel state is better, then more information is required to be decoded by the receiver, and if the channel state is worse, more information is required to be secure from the receiver. Furthermore, the information intended to be decoded by the receiver with a better state (e.g., |H| ≥ |h0|) is required to be secure from the receiver if it has a state worse than |h0| by Δ (i.e., |H| ≤ |h0| — Δ), which is referred to as secrecy outside a bounded range. A (layered) broadcast approach is studied for this problem, which views the fading channel as a degraded broadcast channel with a number of receivers each experiencing a different fading coefficient. The achievable scheme designates one superposition layer to each message with binning employed to protect all upper-layer messages from lower-layer receivers. Furthermore, the scheme allows adjacent layers to share rates so that part of the rate of each message can be shared with its upper-layer messages to enlarge the rate region. The achievable secrecy rate region via the broadcast approach is characterized. The developed scheme can adapt the transmission rate to the actual unknown channel state without exploiting the CSI at the transmitter.
In this paper, we consider a scenario where a source node wishes to broadcast two confidential messages for two respective receivers, while a wire-tapper also receives the transmitted signal. This model is motivated by wireless communications, where individual secure messages are broadcast over open media and can be received by any illegitimate receiver. The secrecy level is measured by equivocation rate at the eavesdropper. We first study the general (non-degraded) broadcast channel with confidential messages. We present an inner bound on the secrecy capacity region for this model. The inner bound coding scheme is based on a combination of random binning and the Gelfand-Pinsker bining. This scheme matches the Marton's inner bound on the broadcast channel without confidentiality constraint. We further study the situation where the channels are degraded. For the degraded broadcast channel with confidential messages, we present the secrecy capacity region. Our achievable coding scheme is based on Cover's superposition scheme and random binning. We refer to this scheme as Secret Superposition Scheme. In this scheme, we show that randomization in the first layer increases the secrecy rate of the second layer. This capacity region matches the capacity region of the degraded broadcast channel without security constraint. It also matches the secrecy capacity for the conventional wire-tap channel. Our converse proof is based on a combination of the converse proof of the conventional degraded broadcast channel and Csiszar lemma. Finally, we assume that the channels are Additive White Gaussian Noise (AWGN) and show that secret superposition scheme with Gaussian codebook is optimal. The converse proof is based on the generalized entropy power inequality.