Private Information Retrieval is Graph Based Replication Systems

Replication is prevalent in both theory and practice as a means for obtaining robustness in distributed storage systems. A system in which every data entry is stored on two separate servers gives rise to a graph structure in a natural way, and the combinatorial properties of this graph shed light on the possible features of the system. One possible feature of interest, that has recently gained renewed attention, is private information retrieval (PIR). A PIR protocol enables a user to obtain a data entry from a storage system without revealing the identity of the requested entry to sets of colluding servers. In this paper we suggest a simple PIR protocol for graph based replication systems, which guarantees perfect secrecy against any set of colluding servers that does not induce a cycle. Furthermore, it is shown that the secrecy deteriorates gracefully with the number of cycles in the colluding set, and that the upload complexity can be reduced for graphs of certain specialized structure.