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

2025-10-29

AI Summary

This work investigates the impact of fully shared randomness on the capacity of symmetric private information retrieval (SPIR) in graph-structured database replication systems. Addressing the conventional setting of local randomness induced by graph-based replication, we propose a generic construction that transforms any PIR scheme into an SPIR scheme. Leveraging rigorous information-theoretic and graph-theoretic modeling, we derive tight upper and lower bounds on the SPIR capacity. For the first time, we establish exact capacity results for path and cycle graphs—e.g., the SPIR capacity of a three-node path graph is precisely 1/2, strictly exceeding the capacity achievable under graph-replication-induced randomness alone. Our results demonstrate that full randomness sharing circumvents inherent topological constraints on the privacy-efficiency trade-off in graph-replicated systems, thereby establishing a new theoretical benchmark and constructive paradigm for distributed private information retrieval.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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 $frac{1}{2}$.

Problem

• Modeling symmetric private information retrieval on graph-based database replication systems
• Quantifying SPIR capacity improvement using common randomness replication
• Establishing capacity bounds for path and cyclic graph topologies

Innovation

• Modeled database replication using simple graphs
• Developed algorithm converting PIR to SPIR schemes
• Established tighter capacity bounds for path graphs