Algebraic Geometry Codes for Cross-Subspace Alignment in Private Information Retrieval

TL;DR

This paper introduces a novel algebraic geometry code-based framework for private information retrieval that improves upon existing schemes by offering better tradeoffs and higher retrieval rates in certain scenarios.

Contribution

It generalizes cross-subspace alignment codes using algebraic geometry codes and provides explicit constructions over low-genus curves, expanding the applicability of PIR schemes.

Findings

Offers better tradeoffs between field size, file size, and server collusion.

Achieves higher retrieval rates in some cases with fixed field size.

Extends the existence of schemes beyond the original limitations.

Abstract

A new framework for interference alignment in secure and private information retrieval (PIR) from colluding servers is proposed, generalizing the original cross-subspace alignment (CSA) codes proposed by Jia, Sun, and Jafar. The general scheme is built on algebraic geometry codes and explicit constructions with replicated storage are given over curves of genus zero and one. It is shown that the proposed scheme offers interesting tradeoffs between the field size, file size, number of colluding servers, and the total number of servers. When the field size is fixed, this translates in some cases to higher retrieval rates than those of the original scheme. In addition, the new schemes exist also in cases where the original ones do not.