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Private Information Retrieval on Multigraph-Based Replicated Storage

Shreya Meel, Xiangliang Kong, Thomas Jacob Maranzatto, Itzhak Tamo, Sennur Ulukus

TL;DR

This work studies private information retrieval under multigraph-based replicated storage, modeling storage as an $r$-multigraph and deriving fundamental capacity bounds. It presents a constructive lower bound: any graph-based PIR scheme with symmetric retrieval property (SRP) on the base graph lifts to an $r$-multigraph with rate scaled by $\frac{2^{r-1}}{2^r-1}$, achieving $\mathscr{C}(G^{(r)}) \ge R(G) \cdot \frac{1}{2-\frac{1}{2^{r-1}}}$. An LP-based general upper bound is established, $\mathscr{C}(G^{(r)}) \le \min\left( \frac{\Delta(G)}{|E(G)|}, \frac{1}{\nu(G)} \right) \cdot \frac{1}{2-2^{1-r}}$, which is tight for certain graphs such as even-vertex multipaths. Together, these results illuminate the trade-offs and limits of PIR in multigraph-replicated storage and guide design of efficient retrieval schemes for such systems.

Abstract

We consider the private information retrieval (PIR) problem for a multigraph-based replication system, where each set of $r$ files is stored on two of the servers according to an underlying $r$-multigraph. Our goal is to establish upper and lower bounds on the PIR capacity of the $r$-multigraph. Specifically, we first propose a construction for multigraph-based PIR systems that leverages the symmetry of the underlying graph-based PIR scheme, deriving a capacity lower bound for such multigraphs. Then, we establish a general upper bound using linear programming, expressed as a function of the underlying graph parameters. Our bounds are demonstrated to be tight for PIR systems on multipaths for even number of vertices.

Private Information Retrieval on Multigraph-Based Replicated Storage

TL;DR

This work studies private information retrieval under multigraph-based replicated storage, modeling storage as an -multigraph and deriving fundamental capacity bounds. It presents a constructive lower bound: any graph-based PIR scheme with symmetric retrieval property (SRP) on the base graph lifts to an -multigraph with rate scaled by , achieving . An LP-based general upper bound is established, , which is tight for certain graphs such as even-vertex multipaths. Together, these results illuminate the trade-offs and limits of PIR in multigraph-replicated storage and guide design of efficient retrieval schemes for such systems.

Abstract

We consider the private information retrieval (PIR) problem for a multigraph-based replication system, where each set of files is stored on two of the servers according to an underlying -multigraph. Our goal is to establish upper and lower bounds on the PIR capacity of the -multigraph. Specifically, we first propose a construction for multigraph-based PIR systems that leverages the symmetry of the underlying graph-based PIR scheme, deriving a capacity lower bound for such multigraphs. Then, we establish a general upper bound using linear programming, expressed as a function of the underlying graph parameters. Our bounds are demonstrated to be tight for PIR systems on multipaths for even number of vertices.

Paper Structure

This paper contains 6 sections, 4 theorems, 29 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. A Lower Bound on the PIR Capacity
  4. Motivating Example
  5. General Scheme for $r$-Multigraph
  6. An Upper Bound on the PIR Capacity

Key Result

Theorem 1

Let $T$ denote a scheme for the simple graph $G$, that satisfies SRP and yields the rate $R(G)$. Then, there exists a scheme $T^{(r)}$ for the corresponding $r$-multigraph $G^{(r)}$ which gives the capacity lower bound,

Figures (3)

  • Figure 1: The PIR system corresponding to $\mathbb{P}_3^{(2)}$.
  • Figure 2: The PIR system corresponding to $\mathbb{P}_3$
  • Figure 3: PIR systems for cycle $\mathbb{C}_3$ (left) and $2$-multicycle $\mathbb{C}_3^{(2)}$ (right).

Theorems & Definitions (8)

  • Definition 1: Symmetric Retrieval Property
  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Lemma 1
  • Lemma 2