The Capacity of Private Information Retrieval from Coded Databases

TL;DR

This paper derives the maximum efficiency of private information retrieval from coded distributed databases, revealing a fundamental tradeoff between retrieval cost and storage cost, generalizing previous results for replicated databases.

Contribution

It establishes the information-theoretic capacity of PIR from coded databases, linking capacity to code rate and number of messages, extending classical PIR results.

Findings

Capacity formula: C=(1+R_c+R_c^2+...+R_c^{M-1})^{-1}

Capacity depends only on code rate and number of messages

Generalizes PIR capacity results from replication to coding

Abstract

We consider the problem of private information retrieval (PIR) over a distributed storage system. The storage system consists of $N$ non-colluding databases, each storing a coded version of $M$ messages. In the PIR problem, the user wishes to retrieve one of the available messages without revealing the message identity to any individual database. We derive the information-theoretic capacity of this problem, which is defined as the maximum number of bits of the desired message that can be privately retrieved per one bit of downloaded information. We show that the PIR capacity in this case is $C=\left(1+\frac{K}{N}+\frac{K^2}{N^2}+\cdots+\frac{K^{M-1}}{N^{M-1}}\right)^{-1}=(1+R_c+R_c^2+\cdots+R_c^{M-1})^{-1}=\frac{1-R_c}{1-R_c^M}$, where $R_c$ is the rate of the $(N,K)$ code used. The capacity is a function of the code rate and the number of messages only regardless of the explicit structure of the storage code. The result implies a fundamental tradeoff between the optimal retrieval cost and the storage cost. The result generalizes the achievability and converse results for the classical PIR with replicating databases to the case of coded databases.