The Capacity of Private Information Retrieval with Private Side Information

The problem of private information retrieval with private side information (PIR-PSI) is recently introduced by Kadhe et al. In this problem, $N$ replicated databases each store $K$ messages, of which $M$ are known to the user as side-information. The identity of these $M$ messages is unknown to the databases. The user wishes to retrieve, as efficiently as possible, a new desired message without revealing any information about the joint indices of the desired and side-information messages. We show that the capacity of PIR-PSI is $\left(1+\frac{1}{N}+\frac{1}{N^2}+\cdots+\frac{1}{N^{K-M-1}}\right)^{-1}$. Thus, the capacity of PIR-PSI with $K$ messages, $N$ databases and side-information of $M$ messages is identical to the capacity of PIR with $K-M$ messages, $N$ databases and no side-information, which was found previously by Sun and Jafar. The achievability and converse proofs are essentially inherited from the original work of Sun and Jafar.