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, where
$K_{1}$
messages need to be retrieved privately against
$T_{1}$
colluding servers, and all the messages need to be retrieved privately against
$T_{2}$
colluding servers where
$T_{2}\leq T_{1}$
. We obtain a lower bound to the capacity by proposing a novel coding scheme, namely the non-uniform successive cancellation scheme. A capacity upper bound is also derived. The gap between the upper bound and the lower bound is analyzed, and shown to vanish when
$T_{1}=T_{2}$
.