Toward the Capacity of Private Information Retrieval From Coded and Colluding Servers

Abstract

In this work, two practical concepts related to private information retrieval (PIR) are introduced and coined <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">full support-rank</i> PIR and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">strongly linear</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> and Tajeddine <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> is essentially optimal and induces no capacity loss.