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A Low-Complexity Scheme for Multi-Message Private Information Retrieval

Ningze Wang, Anoosheh Heidarzadeh, Alex Sprintson

TL;DR

The paper tackles MPIR with N non-colluding servers and K messages, where a user aims to privately retrieve D messages. It proposes a vector-linear MPIR scheme with subpacketization degree L for all parameters satisfying N=DL+1, achieving capacity when D divides K and close-to-capacity performance otherwise, while keeping subpacketization linear in N. The scheme relies on a randomized, three-step process that constructs linear combinations of message subpackets to satisfy privacy and recoverability with a provable rate, and proves optimality of the employed probability distribution over the randomization indices. The work advances practical MPIR by reducing subpacketization and enabling capacity-achieving performance in a broad parameter regime, with concrete exemplifications and comparisons to prior high-subpacketization approaches.

Abstract

Private Information Retrieval (PIR) is a fundamental problem in the broader fields of security and privacy. In recent years, the problem has garnered significant attention from the research community, leading to achievability schemes and converse results for many important PIR settings. This paper focuses on the Multi-message Private Information Retrieval (MPIR) setting, where a user aims to retrieve \(D\) messages from a database of \(K\) messages, with identical copies of the database available on \(N\) remote servers. The user's goal is to maximize the download rate while keeping the identities of the retrieved messages private. Existing approaches to the MPIR problem primarily focus on either scalar-linear solutions or vector-linear solutions, the latter requiring a high degree of subpacketization. Furthermore, prior scalar-linear solutions are restricted to the special case of \(N = D+1\). This limitation hinders the practical adoption of these schemes, as real-world applications demand simple, easily implementable solutions that support a broad range of scenarios. In this work, we present a solution for the MPIR problem, which applies to a broader range of system parameters and requires a limited degree of subpacketization. In particular, the proposed scheme applies to all values of \(N=DL+1\) for any integer \(L\geq 1\), and requires a degree of subpacketization \(L\). Our scheme achieves capacity when \(D\) divides \(K\), and in all other cases, its performance matches or comes within a small additive margin of the best-known scheme that requires a high degree of subpacketization.

A Low-Complexity Scheme for Multi-Message Private Information Retrieval

TL;DR

The paper tackles MPIR with N non-colluding servers and K messages, where a user aims to privately retrieve D messages. It proposes a vector-linear MPIR scheme with subpacketization degree L for all parameters satisfying N=DL+1, achieving capacity when D divides K and close-to-capacity performance otherwise, while keeping subpacketization linear in N. The scheme relies on a randomized, three-step process that constructs linear combinations of message subpackets to satisfy privacy and recoverability with a provable rate, and proves optimality of the employed probability distribution over the randomization indices. The work advances practical MPIR by reducing subpacketization and enabling capacity-achieving performance in a broad parameter regime, with concrete exemplifications and comparisons to prior high-subpacketization approaches.

Abstract

Private Information Retrieval (PIR) is a fundamental problem in the broader fields of security and privacy. In recent years, the problem has garnered significant attention from the research community, leading to achievability schemes and converse results for many important PIR settings. This paper focuses on the Multi-message Private Information Retrieval (MPIR) setting, where a user aims to retrieve messages from a database of messages, with identical copies of the database available on remote servers. The user's goal is to maximize the download rate while keeping the identities of the retrieved messages private. Existing approaches to the MPIR problem primarily focus on either scalar-linear solutions or vector-linear solutions, the latter requiring a high degree of subpacketization. Furthermore, prior scalar-linear solutions are restricted to the special case of . This limitation hinders the practical adoption of these schemes, as real-world applications demand simple, easily implementable solutions that support a broad range of scenarios. In this work, we present a solution for the MPIR problem, which applies to a broader range of system parameters and requires a limited degree of subpacketization. In particular, the proposed scheme applies to all values of for any integer , and requires a degree of subpacketization . Our scheme achieves capacity when divides , and in all other cases, its performance matches or comes within a small additive margin of the best-known scheme that requires a high degree of subpacketization.

Paper Structure

This paper contains 11 sections, 2 theorems, 79 equations, 1 table.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Main Results
  4. Proof of Theorem \ref{['thm:1']}
  5. Proposed Scheme
  6. Proof of Validity of $\{P_{i,j}\}$
  7. Proofs of Recoverability and Privacy
  8. Proof of Achievable Rate
  9. Proof of Optimality of $\{P_{i,j}\}$
  10. Proof of Theorem \ref{['thm:2']}
  11. An Illustrative Example

Key Result

Theorem 1

For linear MPIR with a subpacketization degree of $L$, a total of $K$ messages, $D$ demand messages, and ${N=DL+1}$ servers, the capacity is lower bounded by and upper bounded by

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2