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On the Optimal Message Size in PIR Under Arbitrary Collusion Patterns

Guru S. Dornadula, Manikya Pant, Gowtham R. Kurri, Prasad Krishnan

TL;DR

This work addresses the problem of identifying the minimum message size required for capacity-achieving uniformly decomposable PIR schemes under arbitrary collusion patterns. It introduces a complete characterization of capacity-achieving decomposable PIR schemes through four properties (P1–P4) and derives a general lower bound on the optimal message size expressed via the hitting number $\alpha(\mathcal{F}(\mathcal{P}))$ of a newly defined family $\mathcal{F}(\mathcal{P})$, leveraging a fractional Shearer's lemma argument. When specialized to cyclically $T$-contiguous collusion, the bound yields $\alpha(\mathcal{F}(\mathcal{P}))=\lceil N/T \rceil - 1$, and a capacity-achieving scheme with $L = N/T - 1$ exists whenever $T$ divides $N$, achieving the bound and matching the known capacity. The results extend to other patterns such as disjoint collections and disjoint cyclic contiguity with accompanying (where available) matching schemes, providing a near-complete picture of optimal message sizes under these collusion structures and offering practical guidance for implementing PIR with smaller, capacity-achieving message sizes.

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.

On the Optimal Message Size in PIR Under Arbitrary Collusion Patterns

TL;DR

This work addresses the problem of identifying the minimum message size required for capacity-achieving uniformly decomposable PIR schemes under arbitrary collusion patterns. It introduces a complete characterization of capacity-achieving decomposable PIR schemes through four properties (P1–P4) and derives a general lower bound on the optimal message size expressed via the hitting number of a newly defined family , leveraging a fractional Shearer's lemma argument. When specialized to cyclically -contiguous collusion, the bound yields , and a capacity-achieving scheme with exists whenever divides , achieving the bound and matching the known capacity. The results extend to other patterns such as disjoint collections and disjoint cyclic contiguity with accompanying (where available) matching schemes, providing a near-complete picture of optimal message sizes under these collusion structures and offering practical guidance for implementing PIR with smaller, capacity-achieving message sizes.

Abstract

A private information retrieval protocol (PIR) scheme under an arbitrary collusion pattern enables a client to retrieve one message from a library of equal-sized messages duplicated in servers, while keeping the index of the desired message private from any colluding set in . 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 . 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 , expressed in terms of the hitting number of a newly defined family of subsets of servers determined by the collusion pattern . Finally, we specialize the lower bound to several important classes of collusion patterns, including -collusion, disjoint collections of colluding sets, cyclically -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.
Paper Structure (17 sections, 9 theorems, 54 equations)

This paper contains 17 sections, 9 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. System Model and Preliminaries
  3. The Characterizing Properties of Capacity-Achieving PIR Schemes Under Arbitrary Collusion
  4. A Lower Bound on the Message Size and Optimality for Cyclically Contiguous Collusion
  5. A General Lower Bound on the Optimal Message Size
  6. The Optimal Message Size Under Cyclically Contiguous Collusion
  7. An Illustrative Example
  8. Proof of Theorem \ref{['thm:characterizing-properties']}
  9. Proof of Theorem \ref{['thm:minimum-message size']}
  10. Proof of Corollary \ref{['corollory_1']}
  11. Proof of Corollary \ref{['corollary2']}
  12. Proof of Theorem \ref{['thm:T-contiguous']}
  13. Proof of Theorem \ref{['them:disjointcontiguous']}
  14. Missing Details in Remark \ref{['remark2']}
  15. Missing Details in Remarks \ref{['remark5']} and \ref{['remark7']}
  16. ...and 2 more sections

Key Result

Proposition 1

Let $\alpha: \mathcal{P}\to \mathbb{Q}_{+}$ be any fractional covering with respect to a family $\mathcal{P}$ of subsets of $[1:n] , i.e,\sum \limits_{P \in \mathcal{P} : i\in \mathcal{P}} \alpha(P) \geq 1 , \forall i$. For jointly distributed random variables $X_{1}, \dots , X_{n}$, with equality if and only if $X_{i}$'s for $i$ such that $\sum\limits_{P \in \mathcal{P}: i\in \mathcal{P}} \alpha

Theorems & Definitions (22)

  • Definition 1: TianSC19
  • Proposition 1: MadimanT10 JakharKCP25
  • Theorem 1
  • Remark 1
  • Remark 2
  • proof : Proof Sketch of Theorem \ref{['thm:characterizing-properties']}
  • Theorem 2
  • proof : Proof Sketch
  • Corollary 1
  • Remark 3
  • ...and 12 more