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Performance Bounds on a Wiretap Network With Arbitrary Wiretap Sets

机译：具有任意窃听集的窃听网络上的性能界限

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Consider a communication network represented by a directed graph ${cal G}=({cal V},{cal E})$ , where ${cal V}$ is the set of nodes and ${cal E}$ is the set of point-to-point channels in the network. On the network, a secure message $M$ is transmitted, and there may exist wiretappers who want to obtain information about the message. In secure network coding, we aim to find a network code, which can protect the message against the wiretapper whose power is constrained. Cai and Yeung studied the model in which the wiretapper can access any one but not more than one set of channels, called a wiretap set, out of a collection ${cal A}$ of all possible wiretap sets. In order to protect the message, the message needs to be mixed with a random key $K$ . They proved tight fundamental performance bounds when ${cal A}$ consists of all subsets of ${cal E}$ of a fixed size $r$ . However, beyond this special case, obtaining such bounds is much more difficult. In this paper, we investigate the problem when ${cal A}$ consists of arbitrary subsets of ${cal E}$ and obtain the following results: 1) an upper bound on $H(M)$ and 2) a lower boun- on $H(K)$ in terms of $H(M)$ . The upper bound on $H(M)$ is explicit, while the lower bound on $H(K)$ can be computed in polynomial time when $vert{cal A}vert$ is fixed. The tightness of the lower bound for the point-to-point communication system is also proved.
机译：考虑由有向图$ {cal G} =（{cal V}，{cal E}）$表示的通信网络，其中$ {cal V} $是节点集合，而$ {cal E} $是集合网络中点对点通道的数量。在网络上，安全消息$ M $被传输，并且可能存在想要获取有关消息信息的窃听者。在安全的网络编码中，我们的目标是找到一种网络代码，该网络代码可以保护消息免受功率受限的窃听者的攻击。蔡和杨研究了一种模型，在该模型中，窃听者可以访问所有可能的窃听集的集合中的任何一个，但最多只能访问一套称为窃听集的通道。为了保护消息，需要将消息与随机密钥$ K $混合。当$ {cal A} $由固定大小$ r $的$ {cal E} $的所有子集组成时，它们证明了严格的基本性能界限。但是，除了这种特殊情况之外，获得这样的界限要困难得多。在本文中，我们研究了当$ {cal A} $由$ {cal E} $的任意子集组成时的问题，并获得以下结果：1）$ H（M）$的上限，以及2）下限-关于$ H（M）$。 $ H（M）$的上限是明确的，而$ H（K）$的下限可以在多项式时间内计算，只要固定$ vert {cal A} vert $即可。还证明了点对点通信系统下限的紧密性。

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