Optimal configurations for peer-to-peer user-private information retrieval

User-private information retrieval systems should protect the userʹs anonymity when performing queries against a database, or they should limit the servers capacity of profiling users. Peer-to-peer user-private information retrieval (P2P UPIR) supplies a practical solution: the users in a group help each other in doing their queries, thereby preserving their privacy without any need of the database to cooperate. One way to implement the P2P UPIR uses combinatoric configurations to administrate the keys needed for the private communication between the peers. This article is devoted to the choice of the configuration in this system. First of all we characterize the optimal configurations for the P2P UPIR and see the relationship with the projective planes as described in finite geometry. Then we give a very efficient construction of such optimal configurations, i.e. finite projective planes. We finally check that the involved graphs are Ramanujan graphs, giving an additional justification of the optimality of the constructed configurations.