A Numerical Study on the Capacity Region of a Three-Layer Wiretap Network

Abstract

In this paper, we study a three-layer wiretap network including the source node in the top layer, N nodes in the middle layer and L sink nodes in the bottom layer. Each sink node recovers the message generated from the source node correctly via the middle layer nodes that it has access to. Furthermore, it is required that an eavesdropper eavesdropping a subset of the channels between the top layer and the middle layer learns absolutely nothing about the message. For each pair of decoding and eavesdropping patterns, we are interested in finding the capacity region consisting of (N+1)-tuples, with the first element being the size of the message successfully transmitted and the remaining elements being the capacity of the N channels from the source node to the middle layer nodes. This problem can be seen as a generalization of the secret sharing problem. We show that when the number of middle layer nodes is no larger than four, the capacity region is fully characterized as a polyhedral cone. When such a number is 5, we find the capacity regions for 74,222 decoding and eavesdropping patterns. For the remaining 274 cases, linear capacity regions are found. The proving steps are: (1) Characterizing the Shannon region, an outer bound of the capacity region; (2) Characterizing the common information region, an outer bound of the linear capacity region; (3) Finding linear schemes that achieve the Shannon region or the common information region.