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 party 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 function (in the information theoretic sense). The question we address in this paper is what functions can be privately computed in a given incomplete network. Every function can be privately computed in two-connected networks with at least three parties. Thus the question is interesting only for non two-connected networks. Generalizing results of (Bläser et al. in J. Cryptol 19(3): 341-357 2006) we characterize the functions that can be computed privately in simple networks-networks with one separating vertex and no leaves. We then deal with private computations in arbitrary non two-connected networks: we reduce this question to private computations of related functions on trees and give some sufficient conditions and necessary conditions on the functions that can be privately computed on trees.