On private computation in incomplete networks

Suppose that some parties are connected by an incomplete network of reliable and private channels. The parties cooperate to execute some protocol. However, the parties are curious - after the protocol terminates each processor tries to learn information from the communication it heard. We say that a function can be computed privately in a network if there is a protocol in which each processor learns only the information implied by its input and the output of the protocol. The question we address in this paper is what functions can be computed privately in a given incomplete network. It is known that if a network is 2-connected then every pair of parties can communicate privately. Thus, the question is interesting only for non-2-connected networks. We first characterize the functions that can be computed privately in simple networks - networks with one separating vertex and two 2-connected components. We then deal with private computations in arbitrary networks: we reduce this question to private computations of related functions on trees, and give sufficient and necessary conditions on the functions that can be computed privately on trees.