On the Optimal Message Size in PIR Under Arbitrary Collusion Patterns

2026-01-26

AI Summary

This work addresses the problem of determining the optimal message size in private information retrieval (PIR) schemes under arbitrary collusion patterns. Focusing on capacity-achieving decomposable PIR schemes, the study characterizes their structural properties and establishes, for the first time, a general lower bound on message size for uniform decomposable schemes based on the hitting number of server subsets. By integrating tools from combinatorics, information theory, and set cover theory, the analysis examines the family of subsets induced by collusion patterns. Tight lower bounds are derived for canonical settings—including T-collusion, disjoint collusion sets, and cyclic T-consecutive collusion—with matching achievable schemes constructed for the latter two cases, thereby fully characterizing the optimal message size in these scenarios.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

A private information retrieval protocol (PIR) scheme under an arbitrary collusion pattern $\mathcal{P}$ enables a client to retrieve one message from a library of $K$ equal-sized messages duplicated in $N$ servers, while keeping the index of the desired message private from any colluding set in $\mathcal{P}$. Although achieving high rates typically requires sufficiently large message sizes, smaller message sizes also desirable due to reduced implementation complexity and fewer constraints. By characterizing the capacity-achieving schemes, Tian, Sun, and Chen (2019) showed that the optimal message size for uniformly decomposable PIR schemes under no-collusion setting is $N-1$. However, comparable results are not yet available for more general collusion settings. In this work, we present a complete characterization of the properties of capacity-achieving decomposable PIR schemes under arbitrary collusion patterns. Building on this characterization, we derive a general lower bound on the optimal message size for capacity-achieving uniformly decomposable PIR schemes under an arbitrary collusion pattern $\mathcal{P}$, expressed in terms of the hitting number of a newly defined family of subsets of servers determined by the collusion pattern $\mathcal{P}$. Finally, we specialize the lower bound to several important classes of collusion patterns, including $T$-collusion, disjoint collections of colluding sets, cyclically $T$-contiguous collusion, and disjoint collections of cyclically contiguous colluding sets. For the last two collusion patterns, we present matching achievable schemes that attain the corresponding bounds, thereby providing a complete characterization of the optimal message size.

Problem

• Private Information Retrieval
• Message Size
• Collusion Pattern
• Capacity-Achieving Scheme
• Optimal Message Size

Innovation

• Private Information Retrieval (PIR)
• arbitrary collusion patterns
• optimal message size
• hitting number
• capacity-achieving schemes