  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 (retriever) of cache-size $M$ 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 where the user acquires side information
and the retrieval phase where the user downloads desired information. In the
prefetching phase, the user receives $m_n$ full messages from the $n$th
database, under the cache memory size constraint $\sum_{n=1}^N m_n \leq M$. 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 \textit{partially known
private side information}. We characterize the capacity of the PIR with
partially known private side information to be
$C=\left(1+\frac{1}{N}+\cdots+\frac{1}{N^{K-M-1}}\right)^{-1}=\frac{1-\frac{1}{N}}{1-(\frac{1}{N})^{K-M}}$.
Interestingly, this result is the same if none of the databases knows any of
the prefetched side information, i.e., when the side information is obtained
externally, a problem posed by Kadhe et al. and settled by Chen-Wang-Jafar
recently. Thus, our result implies that there is no loss in using the same
databases for both prefetching and retrieval phases.
