The Capacity of Uncoded Storage Constrained PIR

Private information retrieval (PIR) allows a user to retrieve a desired message out of $K$ possible messages from $N$ databases (DBs) without revealing the identity of the desired message. In this work, we consider the problem of PIR from uncoded storage constrained DBs. Each DB has a storage capacity of $\mu KL$ bits, where $L$ is the size of each message in bits, and $\mu\in[1/N,\ 1]$ is the normalized storage. In the storage constrained PIR problem, there are two key challenges: a) construction of communication efficient schemes through storage content design at each DB that allow download efficient PIR; and b characterizing the optimal download cost via information-theoretic lower bounds. The novel aspect of this work is to characterize the optimum download cost of PIR with storage constrained DBs for any value of storage. In particular, for any $(N,\ K)$, we show that the optimal tradeoff between storage $(\mu)$ and the download cost $(D(\mu))$ is given by the lower convex hull of the pairs $(\frac{t}{N}(1+\frac{1}{t}+\frac{1}{t^{2}}+\cdots+\frac{1}{t^{K-1}}))$ for $t$ = 1,2, …, N. The main contribution of this paper is the converse proof, i.e., obtaining lower bounds on the download cost for PIR as a function of the available storage.