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A Linear Programming Approach to Private Information Retrieval

Anoosheh Heidarzadeh, Ningze Wang, Alex Sprintson

TL;DR

This work tackles private information retrieval with multiple servers and multiple demanded messages by focusing on AB-PIR, where downloads are restricted to $0$/$1$-coefficient linear combinations. It introduces an LP-based framework to optimize AB-PIR schemes over all $(N,K,D)$, yielding a capacity lower bound $\underline{R}=\max_t g_t/f_t$ and an upper bound $\overline{R}$, with optimality when $D\mid K$ and potential improvements when $D\nmid K$. The authors show how to realize the lower bound via a concrete AB-PIR scheme that leverages subpacketization and a set of linear constraints, and they connect their approach to the BU2018 schemes, explaining when the new method matches or surpasses prior results. A detailed illustrative example demonstrates practical gains: for $K=5$, $D=2$, $N=2$ and $N=3$, the LP-based scheme achieves $82/135$ and $57/80$ respectively, exceeding the prior $17/28$ and $42/59$. The framework is extensible to non-binary coefficients and related PIR variants, offering a systematic route to improved privacy-preserving data retrieval across broad parameter regimes.

Abstract

This work presents an algorithmic framework that uses linear programming to construct \emph{addition-based Private Information Retrieval (AB-PIR)} schemes, where retrieval is performed by downloading only linear combinations of message symbols with coefficients set to 0 or 1. The AB-PIR schemes generalize several existing capacity-achieving PIR schemes and are of practical interest because they use only addition operations -- avoiding multiplication and other complex operations -- and are compatible with any finite field, including binary. Our framework broadens the search space to include all feasible solutions and can be used to construct optimal AB-PIR schemes for the entire range of problem parameters, including the number of servers, the total number of messages, and the number of messages that need to be retrieved. The framework enables us to identify schemes that outperform the previously proposed PIR schemes in certain cases and, in other cases, achieve performance on par with the best-known AB-PIR solutions. Additionally, the schemes generated by our framework can be integrated into existing solutions for several related PIR scenarios, improving their overall performance.

A Linear Programming Approach to Private Information Retrieval

TL;DR

This work tackles private information retrieval with multiple servers and multiple demanded messages by focusing on AB-PIR, where downloads are restricted to /-coefficient linear combinations. It introduces an LP-based framework to optimize AB-PIR schemes over all , yielding a capacity lower bound and an upper bound , with optimality when and potential improvements when . The authors show how to realize the lower bound via a concrete AB-PIR scheme that leverages subpacketization and a set of linear constraints, and they connect their approach to the BU2018 schemes, explaining when the new method matches or surpasses prior results. A detailed illustrative example demonstrates practical gains: for , , and , the LP-based scheme achieves and respectively, exceeding the prior and . The framework is extensible to non-binary coefficients and related PIR variants, offering a systematic route to improved privacy-preserving data retrieval across broad parameter regimes.

Abstract

This work presents an algorithmic framework that uses linear programming to construct \emph{addition-based Private Information Retrieval (AB-PIR)} schemes, where retrieval is performed by downloading only linear combinations of message symbols with coefficients set to 0 or 1. The AB-PIR schemes generalize several existing capacity-achieving PIR schemes and are of practical interest because they use only addition operations -- avoiding multiplication and other complex operations -- and are compatible with any finite field, including binary. Our framework broadens the search space to include all feasible solutions and can be used to construct optimal AB-PIR schemes for the entire range of problem parameters, including the number of servers, the total number of messages, and the number of messages that need to be retrieved. The framework enables us to identify schemes that outperform the previously proposed PIR schemes in certain cases and, in other cases, achieve performance on par with the best-known AB-PIR solutions. Additionally, the schemes generated by our framework can be integrated into existing solutions for several related PIR scenarios, improving their overall performance.
Paper Structure (7 sections, 2 theorems, 67 equations, 1 table)

This paper contains 7 sections, 2 theorems, 67 equations, 1 table.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Main Results
  4. Proofs of Theorems
  5. Proof of Theorem \ref{['thm:1']}
  6. Proof of Theorem \ref{['thm:2']}
  7. An Illustrative Example

Key Result

Theorem 1

The capacity of AB-PIR with $N$ servers, $K$ messages, and $D$ demand messages is lower bounded by where $\mathrm{f} = [f_1,\dots,f_D]$ and $\mathrm{g} = [g_1,\dots,g_D]$ are defined in eq:f and eq:g, respectively, and is upper bounded by

Theorems & Definitions (4)

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