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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.

Wyner introduced wiretap channel in his seminal paper [

Körner and Marton considered a general two-receiver DM-BC with degraded message sets in [

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 [

Different from [

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

We denote discrete random variables and their realizations with uppercase and lowercase letters, respectively; for example,

We consider a class of three-receiver discrete memoryless broadcast channels (DM-BC) with input alphabet

The states

A broadcast channel

A

and a tuple of decoding maps

The rate tuples of the code are defined as follows:

A rate tuple (

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

Rate tuples

Setting

Setting

The most important part of the proof of the achievability is the code generation. We use superposition technique and Wyner’s wiretap coding [

Consider a DM-BC with

At receiver

At receive

At receive

Subsequently, we use Gel’fand-Pinsker coding scheme and determine that the encoder can choose the proper

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

First, we consider the rate

where (a) follows from Fano’s inequality.

Next, we consider the rate

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

(f) follows the fact

which is due to the fact that

Finally, we consider the

where (a) and (c) follow Fano’s inequality, (b) follows from the secrecy condition, (d) follows the fact that

which is due to the degradation condition, and (e) is due to the following Markov Chain:

Finally, let

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,

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.

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