A Numerical Study on the Wiretap Network with a Simple Network Topology

In this paper, we study a security problem on a simple wiretap network, consisting of a source node S, a destination node D, and an intermediate node R. The intermediate node connects the source and the destination nodes via a set of noiseless parallel channels, with sizes $n_1$ and $n_2$, respectively. A message $M$ is to be sent from S to D. The information in the network may be eavesdropped by a set of wiretappers. The wiretappers cannot communicate with one another. Each wiretapper can access a subset of channels, called a wiretap set. All the chosen wiretap sets form a wiretap pattern. A random key $K$ is generated at S and a coding scheme on $(M, K)$ is employed to protect $M$. We define two decoding classes at D: In Class-I, only $M$ is required to be recovered and in Class-II, both $M$ and $K$ are required to be recovered. The objective is to minimize $H(K)/H(M)$ {for a given wiretap pattern} under the perfect secrecy constraint. The first question we address is whether routing is optimal on this simple network. By enumerating all the wiretap patterns on the Class-I/II $(3,3)$ networks and harnessing the power of Shannon-type inequalities, we find that gaps exist between the bounds implied by routing and the bounds implied by Shannon-type inequalities for a small fraction~($<2\%$) of all the wiretap patterns. The second question we investigate is the following: What is $\min H(K)/H(M)$ for the remaining wiretap patterns where gaps exist? We study some simple wiretap patterns and find that their Shannon bounds (i.e., the lower bound induced by Shannon-type inequalities) can be achieved by linear codes, which means routing is not sufficient even for the ($3$, $3$) network. For some complicated wiretap patterns, we study the structures of linear coding schemes under the assumption that they can achieve the corresponding Shannon bounds....