On the Capacity of Single-Server Multi-Message Private Information Retrieval with Side Information

We study Private Information Retrieval with Side Information (PIR-SI) in the single-server multi-message setting. In this setting, a user wants to download D messages from a database of $K \geq D$ messages, stored on a single server, without revealing any information about the identities of the demanded messages to the server. The goal of the user is to achieve information-theoretic privacy by leveraging the side information about the database. The side information consists of a random subset of M messages. The identities of the messages forming the side information are initially unknown to the server. Our goal is to characterize the capacity of this setting, i.e., the maximum achievable download rate. In our previous work, we have established the PIR-SI capacity for the special case in which the user wants a single message, i.e., $D = 1$ and showed that the capacity can be achieved through the Partition and Code scheme. In this paper, we focus on the case when the user wants multiple messages, i.e., $D\lt /p\gt \gt 1$. Our first result is that if the user wants more messages than what they have as side information, i.e., $D \gt M$, then the capacity is $\frac{D}{K-M}$, and it can be achieved using a scheme based on the Generalized Reed-Solomon codes. Our second result shows that when $D\leq M$ the capacity can be higher. We present a lower bound on the capacity based on an achievability scheme which we call Generalized Partition and Code.