Optimality of routing on the wiretap network with simple network topology

In this paper, we study the performance of routing in the Level-I/II (n₁, n₂) wiretap networks, consisting of a source node, a destination node, and an intermediate node. The intermediate node connects the source and the destination nodes via a set of noiseless parallel channels, with sizes n₁ and n₂, respectively. The information in the network may be eavesdropped by a wiretapper, who can access at most one set of channels, called a wiretap set. All the possible wiretap sets which may be accessed by the wiretapper form a wiretap pattern. A random key K is used to protect the message M. We define two decoding levels: in Level-I, only M is decoded and in Level-II, both M and K are decoded. The objective is to minimize H(K)/H(M) under perfect secrecy constraint. Our concern is whether routing is optimal in this simple network. By harnessing the power of Shannon-type inequalities, we enumerate all the wiretap patterns in the Level-I/II (3, 3) networks, and find out that gaps exist between the bounds by routing and the bounds by Shannon-type inequalities for a small fraction of all the wiretap patterns. Furthermore, we show that for some wiretap patterns, the Shannon bounds can be achieved by a linear code; i.e, routing is not optimal even in the (3, 3) case. Some subtle issues on the network models are discussed and interesting open problems are introduced.