The Capacity of Private Information Retrieval

TL;DR

This paper determines the maximum efficiency of private information retrieval from multiple databases, providing a precise capacity formula and revealing a scheme that maintains optimality even when messages are eliminated.

Contribution

The paper derives the exact information-theoretic capacity of PIR for any number of messages and databases, introducing a capacity-achieving scheme with a unique robustness property.

Findings

PIR capacity formula: (1 + 1/N + ... + 1/N^{K-1})^{-1}

Capacity-achieving scheme is robust to message elimination

The scheme achieves optimal privacy and efficiency for all message subsets

Abstract

In the private information retrieval (PIR) problem a user wishes to retrieve, as efficiently as possible, one out of $K$ messages from $N$ non-communicating databases (each holds all $K$ messages) while revealing nothing about the identity of the desired message index to any individual database. The information theoretic capacity of PIR is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. For $K$ messages and $N$ databases, we show that the PIR capacity is $(1+1/N+1/N^2+\cdots+1/N^{K-1})^{-1}$. A remarkable feature of the capacity achieving scheme is that if we eliminate any subset of messages (by setting the message symbols to zero), the resulting scheme also achieves the PIR capacity for the remaining subset of messages.