The Capacity of Private Information Retrieval from Uncoded Storage Constrained Databases

Private information retrieval (PIR) allows a user to retrieve a desired message from a set of databases without revealing the identity of the desired message. The replicated databases scenario was considered by Sun and Jafar, 2016, where $N$ databases can store the same $K$ messages completely. A PIR scheme was developed to achieve the optimal download cost given by $\left(1+ \frac{1}{N}+ \frac{1}{N^{2}}+ \cdots + \frac{1}{N^{K-1}}\right)$. In this work, we consider the problem of PIR from storage constrained databases. Each database 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. On one extreme, $\mu=1$ is the replicated databases case. On the other hand, when $\mu= 1/N$, then in order to retrieve a message privately, the user has to download all the messages from the databases achieving a download cost of $1/K$. We aim to characterize the optimal download cost versus storage trade-off for any storage capacity in the range $\mu \in [1/N, 1]$. For any $(N,K)$, we show that the optimal trade-off between storage, $\mu$, and the download cost, $D(\mu)$, is given by the lower convex hull of the $N$ pairs $\left(\mu= \frac{t}{N},D(\mu) = \left(1+ \frac{1}{t}+ \frac{1}{t^{2}}+ \cdots + \frac{1}{t^{K-1}}\right)\right)$ for $t=1,2,\ldots, N$. To prove this result, we first present the storage constrained PIR scheme for any $(N,K)$. We next obtain a general lower bound on the download cost for PIR, which is valid for the following storage scenarios: replicated or storage constrained, coded or uncoded, and fixed or optimized. We then specialize this bound using the uncoded storage assumption to obtain lower bounds matching the achievable download cost of the storage constrained PIR scheme for any value of the available storage.