Privacy-Preserving Distributed Averaging via Homomorphically Encrypted Ratio Consensus

In this article, we develop distributed iterative algorithms that enable the components of a multicomponent system, each with some integer initial value, to asymptotically compute the average of their initial values, without having to reveal to other components the specific value they contribute to the average calculation. We assume a communication topology captured by an arbitrary strongly connected digraph, in which certain nodes (components) might be curious but not malicious (i.e., they execute the distributed protocol correctly, but try to identify the initial values of other nodes). We first develop a variation of the so-called ratio consensus algorithm that operates exclusively on integer values and can be used by the nodes to asymptotically obtain the average of their initial (integer) values, by taking the ratio of two integer values they maintain and iteratively update. Assuming the presence of a trusted node (i.e., a node that is not curious and can be trusted to set up a cryptosystem and not reveal any decrypted values of messages it receives), we describe how this algorithm can be adjusted using homomorphic encryption to allow the nodes to obtain the average of their initial values while ensuring their privacy (i.e., without having to reveal their initial value). We also extend the algorithm to handle situations where multiple nodes set up cryptosystems and privacy is preserved as long as one of these nodes can be trusted (i.e., the ratio of trusted nodes over the nodes that set up cryptosystems decreases).