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Cryptanalysis of a PIR Scheme based on Linear Codes over Rings

Luana Kurmann, Svenja Lage, Violetta Weger

Abstract

In this paper we present an attack on a recently proposed code-based Private Information Retrieval (PIR) scheme. Indeed, the server can retrieve the index of the desired file with high probability in polynomial time. The attack relies on the fact that random codes over finite rings are free with high probability and that the dimension of the rowspan of the query matrix decreases when the rows corresponding to the desired index are removed.

Cryptanalysis of a PIR Scheme based on Linear Codes over Rings

Abstract

In this paper we present an attack on a recently proposed code-based Private Information Retrieval (PIR) scheme. Indeed, the server can retrieve the index of the desired file with high probability in polynomial time. The attack relies on the fact that random codes over finite rings are free with high probability and that the dimension of the rowspan of the query matrix decreases when the rows corresponding to the desired index are removed.

Paper Structure

This paper contains 18 sections, 8 theorems, 49 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Codes over $\mathbb{Z}_{p^{e}}$
  4. Codes over $\mathbb{Z}_m$
  5. Codes over $\mathbb{Z}_{p_i^{e_i}}[x] /\langle x^n-1\rangle$
  6. Description of the PIR Scheme
  7. Query Generation
  8. Server Response
  9. Recovering Stage
  10. Example where Unique Recovery fails and Additional Condition
  11. Example
  12. Set up
  13. Query Generation
  14. Recovering Stage
  15. Repair
  16. ...and 3 more sections

Key Result

Theorem 2.3

Norton2000Hamming Any linear code $C$ over $\mathbb{Z}_{p^{e}}$ has a generator matrix in standard form. Moreover, we call $k \coloneqq \sum_{i=0}^{e-1} k_i (e-i)$ the $\mathbb{Z}_{p}$-dimension. The number of codewords of $C$ is

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 18 more