Towards the Capacity of Private Information Retrieval from Coded and Colluding Servers

TL;DR

This paper introduces practical PIR concepts and derives capacity results for coded, colluding, and adversarial server scenarios, confirming several conjectures and establishing the optimality of existing schemes.

Contribution

It defines full support-rank and strongly linear PIR, derives capacity formulas for these, and proves the optimality of the star product scheme under certain conditions.

Findings

Capacity of MDS-coded, linear, full support-rank PIR with colluding servers is derived.

Capacity of symmetric, linear PIR with colluding, adversarial, and nonresponsive servers is established.

Star product scheme is shown to be essentially optimal under specific restrictions.

Abstract

In this work, two practical concepts related to private information retrieval (PIR) are introduced and coined full support-rank PIR and strongly linear PIR. Being of full support-rank is a technical, yet natural condition required to prove a converse result for a capacity expression and satisfied by almost all currently known capacity-achieving schemes, while strong linearity is a practical requirement enabling implementation over small finite fields with low subpacketization degree. Then, the capacity of MDS-coded, linear, full support-rank PIR in the presence of colluding servers is derived, as well as the capacity of symmetric, linear PIR with colluding, adversarial, and nonresponsive servers for the recently introduced concept of matched randomness. This positively settles the capacity conjectures stated by Freij-Hollanti et al. and Tajeddine et al. in the presented cases. It is also shown that, further restricting to strongly-linear PIR schemes with deterministic linear interference cancellation, the so-called star product scheme proposed by Freij-Hollanti et al. is essentially optimal and induces no capacity loss.