The Capacity of Symmetric Private Information Retrieval

TL;DR

This paper determines the maximum efficiency of symmetric private information retrieval (SPIR) with multiple databases, showing it depends on the number of databases and shared randomness, regardless of message count.

Contribution

It establishes the exact capacity of SPIR with common randomness, extending understanding of privacy constraints in distributed data retrieval.

Findings

Capacity of SPIR is 1-1/N with common randomness

Shared randomness must be at least 1/(N-1) bits per message bit

Capacity is independent of the number of messages K

Abstract

Private information retrieval (PIR) is the problem of retrieving as efficiently as possible, one out of $K$ messages from $N$ non-communicating replicated databases (each holds all $K$ messages) while keeping the identity of the desired message index a secret from each individual database. Symmetric PIR (SPIR) is a generalization of PIR to include the requirement that beyond the desired message, the user learns nothing about the other $K-1$ messages. The information theoretic capacity of SPIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. We show that the capacity of SPIR is $1-1/N$ regardless of the number of messages $K$, if the databases have access to common randomness (not available to the user) that is independent of the messages, in the amount that is at least $1/(N-1)$ bits per desired message bit, and zero otherwise. Extensions to the capacity region of SPIR and the capacity of finite length SPIR are provided.