Performance Bounds on a Wiretap Network with Arbitrary Wiretap Sets

Consider a communication network represented by a directed graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ is the set of nodes and $\mathcal{E}$ 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 \cite{cai2002secure} 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 $\mathcal{A}$ 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 $\mathcal{A}$ consists of all subsets of $\mathcal{E}$ 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 $\mathcal{A}$ consists of arbitrary subsets of $\mathcal{E}$ and obtain the following results: 1) an upper bound on $H(M)$; 2) a lower bound 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 $|\mathcal{A}|$ is fixed. The tightness of the lower bound for the point-to-point communication system is also proved.