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Single Server Private Information Retrieval Protocols With Codes Over Rings

Şeyma Bodur, Edgar Martínez-Moro, Diego Ruano

TL;DR

This work addresses private information retrieval (PIR) from a single server by introducing a ring-based coding framework that achieves computational privacy while curbing rank-based attacks. It uses a two-code construction: an inner non-free $\mathbb{Z}_m$-code $C_{IN}$ inside $\mathcal{R}=\mathbb{Z}_m[x]/\langle x^n-1\rangle$ and an outer code $C_{OUT}$ formed as a matrix-product code over $\mathcal{R}$, enabling modular arithmetic at the server and recovery stages. The scheme resists Bordage’s rank-difference attack through non-free, non-Hensel projection codes and CRT-based lifting, while delivering a calculable PIR rate and transparent computational costs. The approach generalizes PIR to rings, offering practical privacy benefits for large-scale data access scenarios and potential applications to privacy-preserving use of AI tools, with clear trade-offs between rate and security relative to field-based schemes.

Abstract

A Private Information Retrieval (PIR) protocol based on coding theory for a single server is proposed. It provides computational security against linear algebra attacks, addressing the main drawback of previous PIR proposals based on coding theory. The approach involves two types of codes each one over a different ring, an inner non-free linear code that will be used as a distinguisher of some elements added to the query matrix, and an outer code that will be used for generating the query matrix. Moreover, it only uses modular arithmetic at the server level and the recovering stage if the base ring chosen for the inner code is $\mathbb Z_m$.

Single Server Private Information Retrieval Protocols With Codes Over Rings

TL;DR

This work addresses private information retrieval (PIR) from a single server by introducing a ring-based coding framework that achieves computational privacy while curbing rank-based attacks. It uses a two-code construction: an inner non-free -code inside and an outer code formed as a matrix-product code over , enabling modular arithmetic at the server and recovery stages. The scheme resists Bordage’s rank-difference attack through non-free, non-Hensel projection codes and CRT-based lifting, while delivering a calculable PIR rate and transparent computational costs. The approach generalizes PIR to rings, offering practical privacy benefits for large-scale data access scenarios and potential applications to privacy-preserving use of AI tools, with clear trade-offs between rate and security relative to field-based schemes.

Abstract

A Private Information Retrieval (PIR) protocol based on coding theory for a single server is proposed. It provides computational security against linear algebra attacks, addressing the main drawback of previous PIR proposals based on coding theory. The approach involves two types of codes each one over a different ring, an inner non-free linear code that will be used as a distinguisher of some elements added to the query matrix, and an outer code that will be used for generating the query matrix. Moreover, it only uses modular arithmetic at the server level and the recovering stage if the base ring chosen for the inner code is .
Paper Structure (12 sections, 2 theorems, 20 equations, 1 figure, 1 table)

This paper contains 12 sections, 2 theorems, 20 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Single Server PIR Protocols
  3. HHWZ PIR protocol
  4. Single Server PIR protocol Over Rings
  5. Data Storage Process
  6. Query Generation
  7. Response
  8. Recovering Stage
  9. Analysis
  10. PIR Rate
  11. Computational cost
  12. Security Analysis

Key Result

Theorem 5

Let $C_{\mathtt{OUT}}$ be a linear code in $\mathcal{R}^s$, and $C_{\mathtt{IN}} \subseteq \mathcal{R}$ an $R$-linear code. Let $\boldsymbol{\Delta}$ be chosen as in Section sec: 3 and $\boldsymbol{\Delta}={\mathbf W} + {\mathbf E} + {\mathbf U}$ the decomposition in Equation (delta). Then we have t where $\mathrm{rowspan}$ is taken in $R$ and $\Gamma_s(C_{\mathtt{IN}})=[C_{\mathtt{IN}},\ldots,C_{

Figures (1)

  • Figure 1: Query matrix in the original HHWZ PIR protocol vs. the modified one. White places means entries equal to $0$

Theorems & Definitions (10)

  • Definition 1: blackmore2001matrixasch
  • Remark 2
  • Remark 3: This protocol over a field
  • Example 4: Toy example
  • Theorem 5
  • proof
  • Remark 6: Choosing the base ring and the projection codes
  • Corollary 7
  • proof
  • Remark 8