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A Unified Framework for Constructing Information-Theoretic Private Information Retrieval

Liang Feng Zhang

TL;DR

This survey formalizes a unified framework, FOASC, built on families of orthogonal arrays with span capability to construct information-theoretic private information retrieval protocols. It shows how the most influential IT-PIR constructions—ranging from covering codes and polynomial interpolation to matching-vector schemes over finite fields and rings—fit naturally within FOASC, enabling systematic analysis of communication complexity and privacy guarantees. The paper also outlines open problems aimed at further reducing communication costs and extending the framework to broader parameter regimes, highlighting directions for developing next-generation IT-PIR protocols. Overall, FOASC provides a cohesive lens to analyze, compare, and extend IT-PIR constructions, with practical implications for private data access across multiple servers and settings.

Abstract

Retrieving up-to-date information from a publicly accessible database poses significant threats to the user's privacy. {\em Private information retrieval} (PIR) protocols allow a user to retrieve any entry from a database, without revealing the identity of the entry being retrieved to the server(s). Such protocols have found numerous applications in both theoretical studies and real-life scenarios. The existing PIR constructions mainly give multi-server {\em information-theoretic} PIR (IT-PIR) protocols or single-server computational PIR (CPIR) protocols. Compared with CPIR, IT-PIR protocols are computationally more efficient and secure in the presence of unbounded servers. The most classical and challenging problem in the realm of IT-PIR is constructing protocols with lower {\em communication complexity}. In this review, we introduce a new discrete structure called {\em families of orthogonal arrays with span capability} (FOASC) and propose a unified framework for constructing IT-PIR protocols. We show how the most influential IT-PIR protocols in the literature can be captured by the framework. We also put forward several interesting open problems concerning FOASC, whose solutions may result in innovative IT-PIR protocols.

A Unified Framework for Constructing Information-Theoretic Private Information Retrieval

TL;DR

This survey formalizes a unified framework, FOASC, built on families of orthogonal arrays with span capability to construct information-theoretic private information retrieval protocols. It shows how the most influential IT-PIR constructions—ranging from covering codes and polynomial interpolation to matching-vector schemes over finite fields and rings—fit naturally within FOASC, enabling systematic analysis of communication complexity and privacy guarantees. The paper also outlines open problems aimed at further reducing communication costs and extending the framework to broader parameter regimes, highlighting directions for developing next-generation IT-PIR protocols. Overall, FOASC provides a cohesive lens to analyze, compare, and extend IT-PIR constructions, with practical implications for private data access across multiple servers and settings.

Abstract

Retrieving up-to-date information from a publicly accessible database poses significant threats to the user's privacy. {\em Private information retrieval} (PIR) protocols allow a user to retrieve any entry from a database, without revealing the identity of the entry being retrieved to the server(s). Such protocols have found numerous applications in both theoretical studies and real-life scenarios. The existing PIR constructions mainly give multi-server {\em information-theoretic} PIR (IT-PIR) protocols or single-server computational PIR (CPIR) protocols. Compared with CPIR, IT-PIR protocols are computationally more efficient and secure in the presence of unbounded servers. The most classical and challenging problem in the realm of IT-PIR is constructing protocols with lower {\em communication complexity}. In this review, we introduce a new discrete structure called {\em families of orthogonal arrays with span capability} (FOASC) and propose a unified framework for constructing IT-PIR protocols. We show how the most influential IT-PIR protocols in the literature can be captured by the framework. We also put forward several interesting open problems concerning FOASC, whose solutions may result in innovative IT-PIR protocols.

Paper Structure

This paper contains 22 sections, 1 theorem, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Private Information Retrieval
  4. Orthogonal Arrays
  5. A Framework based on Families of Orthogonal Arrays
  6. Families of Orthogonal Arrays with Span Capability
  7. The Framework
  8. PIR Constructions within the Proposed Framework
  9. Protocols based on Covering Codes
  10. Protocols based on Polynomial Interpolations
  11. Lagrange Interpolations
  12. Hermite Interpolations
  13. Protocols based on Matching Vectors over Finite Fields
  14. Yekhanin's Construction
  15. Raghavendra's Interpretation
  16. ...and 7 more sections

Key Result

Theorem 1

If there is an ${\sf FOASC}(N,k,s,t;\bm \alpha)$, where $\bm \alpha$ are $n$ functions from $\mathbb{S}$ to $\mathbb{R}$, then there is a $(t,k)$-PIR protocol $\cal P$ with communication complexity ${\bf C}_{\cal P}(n,k)=k(\log|\mathbb{S}|+\log|\mathbb{R}|)$.

Figures (2)

  • Figure 1: $k$-Server information-theoretic PIR system
  • Figure 2: A unified framework for constructing $(t,k)$-PIR from ${\sf FOASC}(N,k,s,t;\bm \alpha)$

Theorems & Definitions (6)

  • Definition 1: Private Information Retrieval
  • Definition 2: Orthogonal Array
  • Example 1
  • Definition 3: Families of Orthogonal Arrays with Span Capability
  • Example 2
  • Theorem 1