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Converse Bounds for Sun-Jafar-type Weak Private Information Retrieval

Chandan Anand, Jayesh Seshadri, Prasad Krishnan, Gowtham R. Kurri

TL;DR

This work investigates the rate–privacy trade-offs of Sun–Jafar-type WPIR schemes under $MIL$ and $MaxL$ privacy metrics. It establishes MIL- and MaxL-based optimality results for the replicated, non-colluding setting, proving that the WSJ scheme is optimal under MIL in this regime and identifying explicit endpoint distributions for optimality. For MDS-coded and $T$-colluding WPIR, the paper derives threshold-based optimality results and provides counterexamples showing that these bounds can be surpassed when system parameters exceed the thresholds. The findings illuminate when intermediate choices of the number of undesired files $M'$ are necessary to maximize rate, and they underscore that tight information-theoretic converses for the entire WPIR class remain open. Together, the results tighten the understanding of WPIR performance limits and guide practical WPIR design under leakage constraints.$MIL$ and $MaxL$ metrics are central to assessing privacy leakage in these schemes.

Abstract

Building on the well-established capacity-achieving schemes of Sun-Jafar (for replicated storage) and the closely related scheme of Banawan-Ulukus (for MDS-coded setting), a recent work by Chandan et al. proposed new classes of weak private information retrieval (WPIR) schemes for the collusion-free (replication and MDS-coded) setting, as well as for the $T$-colluding scenario. In their work, Chandan et al. characterized the expressions for the rate-privacy trade-offs for these classes of WPIR schemes, under the mutual information leakage and maximal leakage metrics. Explicit achievable trade-offs for the same were also presented, which were shown to be competitive or better than prior WPIR schemes. However, the class-wise optimality of the reported trade-offs were unknown. In this work, we show that the explicit rate-privacy trade-offs reported for the Sun-Jafar-type schemes by Chandan et al. are optimal for the non-colluding and replicated setting. Furthermore, we prove the class-wise optimality for Banawan-Ulukus-type MDS-WPIR and Sun-Jafar-type $T$-colluding WPIR schemes, under threshold-constraints on the system parameters. When these threshold-constraints do not hold, we present counter-examples which show that even higher rates than those reported before can be achieved.

Converse Bounds for Sun-Jafar-type Weak Private Information Retrieval

TL;DR

This work investigates the rate–privacy trade-offs of Sun–Jafar-type WPIR schemes under and privacy metrics. It establishes MIL- and MaxL-based optimality results for the replicated, non-colluding setting, proving that the WSJ scheme is optimal under MIL in this regime and identifying explicit endpoint distributions for optimality. For MDS-coded and -colluding WPIR, the paper derives threshold-based optimality results and provides counterexamples showing that these bounds can be surpassed when system parameters exceed the thresholds. The findings illuminate when intermediate choices of the number of undesired files are necessary to maximize rate, and they underscore that tight information-theoretic converses for the entire WPIR class remain open. Together, the results tighten the understanding of WPIR performance limits and guide practical WPIR design under leakage constraints. and metrics are central to assessing privacy leakage in these schemes.

Abstract

Building on the well-established capacity-achieving schemes of Sun-Jafar (for replicated storage) and the closely related scheme of Banawan-Ulukus (for MDS-coded setting), a recent work by Chandan et al. proposed new classes of weak private information retrieval (WPIR) schemes for the collusion-free (replication and MDS-coded) setting, as well as for the -colluding scenario. In their work, Chandan et al. characterized the expressions for the rate-privacy trade-offs for these classes of WPIR schemes, under the mutual information leakage and maximal leakage metrics. Explicit achievable trade-offs for the same were also presented, which were shown to be competitive or better than prior WPIR schemes. However, the class-wise optimality of the reported trade-offs were unknown. In this work, we show that the explicit rate-privacy trade-offs reported for the Sun-Jafar-type schemes by Chandan et al. are optimal for the non-colluding and replicated setting. Furthermore, we prove the class-wise optimality for Banawan-Ulukus-type MDS-WPIR and Sun-Jafar-type -colluding WPIR schemes, under threshold-constraints on the system parameters. When these threshold-constraints do not hold, we present counter-examples which show that even higher rates than those reported before can be achieved.
Paper Structure (15 sections, 16 theorems, 72 equations, 1 table)

This paper contains 15 sections, 16 theorems, 72 equations, 1 table.

Table of Contents

  1. Introduction
  2. System model and Review of Weak Sun-Jafar-type Schemes WSJPIR2024
  3. Sun-Jafar-type WPIR scheme (${\mathcal{P}}_{\mathsf{WSJ}}$) for collusion-free, replicated storage scenario
  4. Banawan-Ulukus-type WPIR scheme (${\mathcal{P}}_{\mathsf{WBU}}$) for collusion-free, MDS-coded servers
  5. Sun-Jafar-type scheme (${\mathcal{P}}_{T-\mathsf{WSJ}}$) for replicated storage scenario and $T$-collusion
  6. Problem Formulation
  7. On the optimal rate-privacy tradeoffs of WPIR schemes WSJPIR2024 under $\mathsf{MIL}$ constraint
  8. Optimizing the rate of ${\mathcal{P}}_{\mathsf{WSJ}}$ scheme under a $\mathsf{MIL}$ constraint
  9. Optimizing the rate of the schemes ${\mathcal{P}}_{\mathsf{WBU}}$ and ${\mathcal{P}}_{T-\mathsf{WSJ}}$ under $\mathsf{MIL}$ constraint
  10. Improving the PIR rate beyond Theorem \ref{['thm:WBU_MIL']} and Theorem \ref{['thm:T-WSJ_MIL']}
  11. Optimal rate of weak Sun-Jafar type scheme under a fixed $\rho^{\mathsf{MaxL}}$ constraint
  12. Optimizing the rate of ${\mathcal{P}}_{\mathsf{WSJ}}$ scheme for the maximal leakage
  13. Optimal rate of ${\mathcal{P}}_{\mathsf{WBU}}$ scheme and ${\mathcal{P}}_{T-\mathsf{WSJ}}$ scheme under $\rho^{\mathsf{MaxL}}$
  14. Achieving higher PIR rates beyond Theorem \ref{['thm:WBU_MaxL']} and Theorem \ref{['thm:T-WSJ_MaxL']}
  15. Conclusion

Key Result

Theorem 1

WSJPIR2024 For any $\rho\geq 0$, the ${\mathcal{P}}_{\mathsf{WSJ}}$ scheme achieves the following rate-privacy trade-off. where $(x)_{+} \triangleq \max(x,0)$.

Theorems & Definitions (30)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • proof
  • Theorem 8
  • ...and 20 more