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Necessary and Sufficient Conditions for Capacity-Achieving Private Information Retrieval with Adversarial Servers

Atsushi Miki, Toshiyasu Matsushima

Abstract

Private information retrieval (PIR) is a mechanism for efficiently downloading messages while keeping the index of the desired message secret from the servers. PIR schemes have been extended to various scenarios with adversarial servers: PIR schemes where some servers are unresponsive or return noisy responses are called robust PIR and Byzantine PIR, respectively; PIR schemes where some servers collude to reveal the index are called colluding PIR. The information-theoretic upper bound on the download efficiency of these PIR schemes has been proved in previous studies. However, systematic ways to construct PIR schemes that achieve the upper bound are not known. In order to construct a capacity-achieving PIR schemes systematically, it is necessary to clarify the conditions that the queries should satisfy. This paper proves the necessary and sufficient conditions for capacity-achieving PIR schemes.

Necessary and Sufficient Conditions for Capacity-Achieving Private Information Retrieval with Adversarial Servers

Abstract

Private information retrieval (PIR) is a mechanism for efficiently downloading messages while keeping the index of the desired message secret from the servers. PIR schemes have been extended to various scenarios with adversarial servers: PIR schemes where some servers are unresponsive or return noisy responses are called robust PIR and Byzantine PIR, respectively; PIR schemes where some servers collude to reveal the index are called colluding PIR. The information-theoretic upper bound on the download efficiency of these PIR schemes has been proved in previous studies. However, systematic ways to construct PIR schemes that achieve the upper bound are not known. In order to construct a capacity-achieving PIR schemes systematically, it is necessary to clarify the conditions that the queries should satisfy. This paper proves the necessary and sufficient conditions for capacity-achieving PIR schemes.

Paper Structure

This paper contains 20 sections, 11 theorems, 72 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Statement
  4. Related Work
  5. Sun's Colluding PIR 8119895
  6. Sun's Robust PIR 8119895
  7. Banawan's Byzantine PIR 8457293
  8. Wang's Colluding PIR 8262858
  9. Wang's Robust and Byzantine PIR 8262858
  10. Conditions for PIR properties
  11. Conditions for Correctness
  12. Conditions for Robust and Byzantine Privacy
  13. Conditions for Robust and Byzantine Capacity
  14. Conditions for Colluding Privacy
  15. Conditions for Colluding Capacity
  16. ...and 5 more sections

Key Result

Theorem 1

there exists a matrix $D_m$ and the following equation should be satisfied. The $\mathbf{O}_{l_1 \times l_2}$ denotes zero matrix of size $l_1 \times l_2$ and the $\mathbf{E}$ denotes identity matrix of size $L_w \times L_w$.

Figures (1)

  • Figure 1: PIR flow

Theorems & Definitions (21)

  • Theorem 1: Condition for Correctness
  • Lemma 1: Sufficient Condition for Robust and Byzantine Privacy
  • proof
  • Lemma 2: Uniqueness of the Desired Index
  • proof
  • Lemma 3: Entropy of the Query for the Unresponsive or Byzantine Servers
  • proof
  • Theorem 2: Conditions for Robust and Byzantine Privacy
  • proof
  • Lemma 4: Conditions for Robust and Byzantine Capacity Achievability
  • ...and 11 more