On the Optimal Message Size in PIR Under Arbitrary Collusion Patterns

TL;DR

This paper characterizes the minimum message size needed for capacity-achieving private information retrieval schemes under arbitrary server collusion patterns, extending previous results to more general settings and providing matching schemes for certain patterns.

Contribution

It provides a complete characterization of capacity-achieving PIR schemes under arbitrary collusion patterns and derives a general lower bound on message size based on hitting numbers.

Findings

Derived a lower bound on message size for arbitrary collusion patterns.

Established matching schemes for cyclically contiguous collusion patterns.

Extended the understanding of PIR capacity to general collusion scenarios.

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.