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

TL;DR

This paper proves the optimality of certain weak private information retrieval schemes in specific settings and provides counterexamples showing potential for higher rates when constraints are relaxed.

Contribution

It establishes the class-wise optimality of Sun-Jafar-type and Banawan-Ulukus-type WPIR schemes under certain conditions, and identifies scenarios with improved achievable rates.

Findings

Optimality of Sun-Jafar-type schemes in non-colluding, replicated setting.

Optimality of Banawan-Ulukus-type MDS-WPIR schemes under threshold constraints.

Existence of counterexamples with higher rates when constraints are not met.

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.