Private information retrieval with partially known private side information

We consider the problem of private information retrieval (PIR) of a single message out of K messages from N replicated and non-colluding databases where a cache-enabled user of cache-size M messages possesses side information in the form of full messages that are partially known to the databases. In this model, the user and the databases engage in a two-phase scheme, namely, the prefetching phase and the retrieval phase. In the prefetching phase, the user receives m _n full messages from the nth database, under the cache memory size constraint Σ ^N _n=1 m _n ≤ M. In the retrieval phase, the user wishes to retrieve a message such that no individual database learns anything about the identity of the desired message. In addition, the identities of the side information messages that the user did not prefetch from a database must remain private against that database. Since the side information provided by each database in the prefetching phase is known by the providing database and the side information must be kept private against the remaining databases, we coin this model as partially known private side information. We characterize the capacity of the PIR with partially known private side information to be C = (1 + 1/N +...+1/N ^K-M-1 ) ^-1 = 1-1/N/1-(1/N) ^K-M , which is the same if none of the databases knows any of the prefetched side information. Thus, our result implies that there is no loss in using the same databases for both prefetching and retrieval phases.