Multi-message private information retrieval with private side information

We consider the problem of private information retrieval (PIR) where a single user with private side information aims to retrieve multiple files from a library stored (uncoded) at a number of servers. We assume the side information at the user includes a subset of files stored privately (i.e., the server does not know the indices of these files). In addition, we require that the identity of requests and side information at the user are not revealed to any of the servers. The problem involves finding the minimum load to be transmitted from the servers to the user such that the requested files can be decoded with the help of received and side information. By providing matching lower and upper bounds, for certain regimes, we characterize the minimum load imposed to all the servers (i.e., the capacity of this PIR problem). Our result shows that the capacity is the same as the capacity of a multi-message PIR problem without private side information, but with a library of reduced size. The effective size of the library is equal to the original library size minus the size of side information.