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Effect of Full Common Randomness Replication in Symmetric PIR on Graph-Based Replicated Systems

Shreya Meel, Sennur Ulukus

TL;DR

This work analyzes SPIR on graph-based replicated databases where every edge stores a message on its endpoints and all servers share full randomness. It introduces a PIR-to-SPIR conversion that preserves symmetric retrieval and yields a capacity lower bound for graphs admitting SRP, with explicit results for path and cycle graphs. The paper establishes a precise SPIR capacity of $\frac{1}{2}$ for $\mathbb{P}_3$, and provides tight upper bounds for $\mathbb{P}_N$ and $\mathbb{C}_N$, as well as a general relation $\frac{1}{R_{\text{F-R}}(G)}=\frac{1}{\\mathscr{C}_{PIR}(G)}+\frac{N}{2(N-1)}$, highlighting the trade-off between SPIR rate and randomness. Overall, the results quantify the gains from fully replicated randomness in graph-based SPIR and sharpen understanding of SPIR capacity for structured replication patterns.

Abstract

We revisit the problem of symmetric private information retrieval (SPIR) in settings where the database replication is modeled by a simple graph. Here, each vertex corresponds to a server, and a message is replicated on two servers if and only if there is an edge between them. To satisfy the requirement of database privacy, we let all the servers share some common randomness, independent of the messages. We aim to quantify the improvement in SPIR capacity, i.e., the maximum ratio of the number of desired and downloaded symbols, compared to the setting with graph-replicated common randomness. Towards this, we develop an algorithm to convert a class of PIR schemes into the corresponding SPIR schemes, thereby establishing a capacity lower bound on graphs for which such schemes exist. This includes the class of path and cyclic graphs for which we derive capacity upper bounds that are tighter than the trivial bounds given by the respective PIR capacities. For the special case of path graph with three vertices, we identify the SPIR capacity to be $\frac{1}{2}$.

Effect of Full Common Randomness Replication in Symmetric PIR on Graph-Based Replicated Systems

TL;DR

This work analyzes SPIR on graph-based replicated databases where every edge stores a message on its endpoints and all servers share full randomness. It introduces a PIR-to-SPIR conversion that preserves symmetric retrieval and yields a capacity lower bound for graphs admitting SRP, with explicit results for path and cycle graphs. The paper establishes a precise SPIR capacity of for , and provides tight upper bounds for and , as well as a general relation , highlighting the trade-off between SPIR rate and randomness. Overall, the results quantify the gains from fully replicated randomness in graph-based SPIR and sharpen understanding of SPIR capacity for structured replication patterns.

Abstract

We revisit the problem of symmetric private information retrieval (SPIR) in settings where the database replication is modeled by a simple graph. Here, each vertex corresponds to a server, and a message is replicated on two servers if and only if there is an edge between them. To satisfy the requirement of database privacy, we let all the servers share some common randomness, independent of the messages. We aim to quantify the improvement in SPIR capacity, i.e., the maximum ratio of the number of desired and downloaded symbols, compared to the setting with graph-replicated common randomness. Towards this, we develop an algorithm to convert a class of PIR schemes into the corresponding SPIR schemes, thereby establishing a capacity lower bound on graphs for which such schemes exist. This includes the class of path and cyclic graphs for which we derive capacity upper bounds that are tighter than the trivial bounds given by the respective PIR capacities. For the special case of path graph with three vertices, we identify the SPIR capacity to be .

Paper Structure

This paper contains 14 sections, 9 theorems, 39 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. System Model
  3. Main Results
  4. SPIR from PIR Schemes
  5. Motivating Examples
  6. Algorithm Description
  7. Index permutation
  8. Query generation
  9. Common randomness assignment
  10. Answer formation
  11. Achievability for $\mathbb{P}_3$
  12. Upper Bound on SPIR Capacity
  13. Converse for $\mathbb{P}_3$
  14. Proof of Upper Bounds for $\mathbb{P}_N$ and $\mathbb{C}_N$

Key Result

Theorem 1

Given a PIR scheme on $G$ with $N$ vertices and $K$ edges, that satisfies SRP, there exists an SPIR scheme with fully-replicated randomness on $G$. Let $L'$ be the number of symbols per message (note that SRP forces the PIR scheme to have even $L'$), and $D'$ be the total number of downloaded symbol is achievable, where $x=\frac{\ell}{L'/2}$, $y=\frac{\ell}{N-1}$, $\ell=\text{lcm}(L'/2,N-1)$ and $

Figures (2)

  • Figure 1: System model for $\mathbb{C}_3$.
  • Figure 2: SPIR model for $\mathbb{P}_3$.

Theorems & Definitions (12)

  • Definition 1: Symmetric Retrieval Property (SRP)
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Example 1: Path graph
  • Example 2: Cyclic graph
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • ...and 2 more