Lower Bound on the Redundancy of PIR Codes

TL;DR

This paper establishes a lower bound on the redundancy of multi-server PIR codes, matching known upper bounds, and precisely determines the minimum redundancy for 3 and 4 servers, advancing understanding of PIR code efficiency.

Contribution

It proves a tight lower bound on PIR code redundancy for all server counts and exactly characterizes the minimum redundancy for 3 and 4 servers, using novel and independent methods.

Findings

Redundancy of $k$-server PIR codes is at least proportional to $\

ext{Exact minimum redundancy for 3 and 4 servers is determined.}

Lower and upper bounds on PIR code redundancy are shown to coincide.

Abstract

We prove that the redundancy of a $k$-server PIR code of dimension $s$ is $\Omega(\sqrt{s})$ for all $k \ge 3$. This coincides with a known upper bound of $O(\sqrt{s})$ on the redundancy of PIR codes. Moreover, for $k=3$ and $k = 4$, we determine the lowest possible redundancy of $k$-server PIR codes exactly. Similar results were proved independently by Mary Wootters using a different method.