Two-Level Private Information Retrieval

In the conventional robust $T$ -colluding private information retrieval (PIR) system, the user needs to retrieve one of the possible messages while keeping the identity of the requested message private from any $T$ colluding servers. Motivated by the possible heterogeneous privacy requirements for different messages, we consider the $(N, T_{1}~:~K_{1}, T_{2}~:~K_{2})$ two-level PIR system with a total of $K_{2}$ messages in the system, where $T_{1}\geq T_{2}$ and $K_{1}\leq K_{2}$ . Any one of the $K_{1}$ messages needs to be retrieved privately against $T_{1}$ colluding servers, and any one of the full set of $K_{2}$ messages needs to be retrieved privately against $T_{2}$ colluding servers. We obtain a lower bound to the capacity by proposing two novel coding schemes, namely the non-uniform successive cancelation scheme and the non-uniform block cancelation scheme. A capacity upper bound is also derived. The gap between the upper bound and the lower bounds is analyzed, and shown to vanish when $T_{1}=T_{2}$ . Lastly, we show that the upper bound is in general not tight by providing a stronger bound for a special setting.