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

TL;DR

This paper investigates how shared randomness among servers in graph-replicated systems enhances symmetric private information retrieval capacity, providing bounds and exact capacities for specific graph structures.

Contribution

It introduces a method to convert PIR schemes into SPIR schemes in graph-replicated systems and derives capacity bounds for path and cyclic graphs.

Findings

Capacity bounds for path and cyclic graphs.

Exact SPIR capacity for path graph with three vertices.

Shared randomness improves SPIR capacity.

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 $\frac{1}{2}$.