# The Capacity of Cache Aided Private Information Retrieval

## TL;DR

This paper precisely characterizes the maximum efficiency of cache-aided private information retrieval, showing how storage size influences download cost and proving that simple memory-sharing schemes are optimal.

## Contribution

It provides the exact capacity formula for cache-aided PIR as a function of storage, including a tight converse proof establishing the optimality of memory sharing.

## Key findings

- Capacity formula: D*(S)/L = (1 - S/K)(1 + 1/N + ... + 1/N^{K-1})
- Optimality of memory sharing schemes for intermediate storage levels
- Special cases recover known results for S=0 and S=K

## Abstract

The problem of cache enabled private information retrieval (PIR) is considered in which a user wishes to privately retrieve one out of $K$ messages, each of size $L$ bits from $N$ distributed databases. The user has a local cache of storage $SL$ bits which can be used to store any function of the $K$ messages. The main contribution of this work is the exact characterization of the capacity of cache aided PIR as a function of the storage parameter $S$. In particular, for a given cache storage parameter $S$, the information-theoretically optimal download cost $D^{*}(S)/L$ (or the inverse of capacity) is shown to be equal to $(1- \frac{S}{K})\left(1+ \frac{1}{N}+ \ldots + \frac{1}{N^{K-1}}\right)$. Special cases of this result correspond to the settings when $S=0$, for which the optimal download cost was shown by Sun and Jafar to be $\left(1+ \frac{1}{N}+ \ldots + \frac{1}{N^{K-1}}\right)$, and the case when $S=K$, i.e., cache size is large enough to store all messages locally, for which the optimal download cost is $0$. The intermediate points $S\in (0, K)$ can be readily achieved through a simple memory-sharing based PIR scheme. The key technical contribution of this work is the converse, i.e., a lower bound on the download cost as a function of storage $S$ which shows that memory sharing is information-theoretically optimal.