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The Role of Common Randomness Replication in Symmetric PIR on Graph-Based Replicated Systems

Shreya Meel, Sennur Ulukus

TL;DR

A general lower bound on the SPIR capacity is derived, and it is established that the minimum size of common randomness required for SPIR is equal to the message size, and the SPIR capacity improves over the first, more restrictive setting.

Abstract

In symmetric private information retrieval (SPIR), a user communicates with multiple servers to retrieve from them a message in a database, while not revealing the message index to any individual server (user privacy), and learning no additional information about the database (database privacy). We study the problem of SPIR on graph-replicated database systems, where each node of the graph represents a server and each link represents a message. Each message is replicated at exactly two servers; those at which the link representing the message is incident. To ensure database privacy, the servers share a set of common randomness, independent of the database and the user's desired message index. We study two cases of common randomness distribution to the servers: i) graph-replicated common randomness, and ii) fully-replicated common randomness. Given a graph-replicated database system, in i), we assign one randomness variable independently to every pair of servers sharing a message, while in ii), we assign an identical set of randomness variable to all servers, irrespective of the underlying graph. In both settings, our goal is to characterize the SPIR capacity, i.e., the maximum number of desired message symbols retrieved per downloaded symbol, and quantify the minimum amount of common randomness required to achieve the capacity. To this goal, in setting i), we derive a general lower bound on the SPIR capacity, and show it to be tight for path and regular graphs through a matching converse. Moreover, we establish that the minimum size of common randomness required for SPIR is equal to the message size. In setting ii), the SPIR capacity improves over the first, more restrictive setting. We show this through capacity lower bounds for a class of graphs, by constructing SPIR schemes from PIR schemes.

The Role of Common Randomness Replication in Symmetric PIR on Graph-Based Replicated Systems

TL;DR

A general lower bound on the SPIR capacity is derived, and it is established that the minimum size of common randomness required for SPIR is equal to the message size, and the SPIR capacity improves over the first, more restrictive setting.

Abstract

In symmetric private information retrieval (SPIR), a user communicates with multiple servers to retrieve from them a message in a database, while not revealing the message index to any individual server (user privacy), and learning no additional information about the database (database privacy). We study the problem of SPIR on graph-replicated database systems, where each node of the graph represents a server and each link represents a message. Each message is replicated at exactly two servers; those at which the link representing the message is incident. To ensure database privacy, the servers share a set of common randomness, independent of the database and the user's desired message index. We study two cases of common randomness distribution to the servers: i) graph-replicated common randomness, and ii) fully-replicated common randomness. Given a graph-replicated database system, in i), we assign one randomness variable independently to every pair of servers sharing a message, while in ii), we assign an identical set of randomness variable to all servers, irrespective of the underlying graph. In both settings, our goal is to characterize the SPIR capacity, i.e., the maximum number of desired message symbols retrieved per downloaded symbol, and quantify the minimum amount of common randomness required to achieve the capacity. To this goal, in setting i), we derive a general lower bound on the SPIR capacity, and show it to be tight for path and regular graphs through a matching converse. Moreover, we establish that the minimum size of common randomness required for SPIR is equal to the message size. In setting ii), the SPIR capacity improves over the first, more restrictive setting. We show this through capacity lower bounds for a class of graphs, by constructing SPIR schemes from PIR schemes.
Paper Structure (26 sections, 28 theorems, 126 equations, 3 figures, 16 tables)

This paper contains 26 sections, 28 theorems, 126 equations, 3 figures, 16 tables.

Table of Contents

  1. Introduction
  2. System Model
  3. Main Results
  4. SPIR with Graph-Replicated Randomness
  5. Achievability Proof
  6. General Scheme for Graphs
  7. Illustrative Examples
  8. Converse Proofs
  9. Proof of Theorem \ref{['thm:bound_rho']}
  10. Proof of Theorem \ref{['thm:upperbnd_capacity']}
  11. SPIR Schemes from PIR Schemes
  12. SPIR with Fully-Replicated Randomness
  13. SPIR from PIR Schemes
  14. Algorithm Description
  15. SPIR Schemes on Star Graphs
  16. ...and 11 more sections

Key Result

Theorem 1

For any graph $G$ with $N$ vertices, its SPIR capacity $\mathscr{C}(G)$ is bounded as provided that the randomness ratio $\rho=1$, i.e., $\rho^*\leq\rho = 1$.

Figures (3)

  • Figure 1: System model with graph-replicated randomness for $N=3$ servers.
  • Figure 2: System model with fully-replicated randomness for $N=3$ servers.
  • Figure 3: SPIR systems with $N=4$ servers.

Theorems & Definitions (47)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2
  • Remark 3
  • Definition 1: Symmetric Retrieval Property (SRP)our_journal2025
  • Theorem 4
  • Remark 4
  • Theorem 5
  • ...and 37 more