Necessary and Sufficient Conditions for Capacity-Achieving Private Information Retrieval with Non-Colluding and Colluding Servers

TL;DR

This paper establishes the necessary and sufficient matrix conditions for capacity-achieving private information retrieval schemes in both non-colluding and colluding server settings.

Contribution

It provides a complete characterization of PIR conditions in matrix form, clarifying the criteria for privacy, decodability, and optimality.

Findings

Derived matrix conditions for standard PIR

Derived matrix conditions for colluding PIR

Unified framework for PIR property verification

Abstract

Private Information Retrieval (PIR) is a mechanism for efficiently downloading messages while keeping the index secret. Here, PIRs in which servers do not communicate with each other are called standard PIRs, and PIRs in which some servers communicate with each other are called colluding PIRs. The information-theoretic upper bound on efficiency has been given in previous studies. However, the conditions for PIRs to keep privacy, to decode the desired message, and to achieve that upper bound have not been clarified in matrix form. In this paper, we prove the necessary and sufficient conditions for the properties of standard PIR and colluding PIR. Further, we represent the properties in matrix form.