Secret Sharing for Secure and Private Information Retrieval: A Construction Using Algebraic Geometry Codes

TL;DR

This paper introduces a new secure and private information retrieval scheme using algebraic geometry codes, enhancing privacy, security, and efficiency in distributed storage systems with colluding servers.

Contribution

It proposes a novel framework for homomorphic secret sharing in PIR, generalizing CSA codes and utilizing algebraic geometry codes for improved tradeoffs and broader parameter applicability.

Findings

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

Supports parameters where previous schemes do not apply.

Provides a flexible tradeoff between field size, file size, and server collusion.

Abstract

Private information retrieval (PIR) considers the problem of retrieving a data item from a database or distributed storage system without disclosing any information about which data item was retrieved. Secure PIR complements this problem by further requiring the contents of the data to be kept secure. Privacy and security can be achieved by adding suitable noise to the queries and data using methods from secret sharing. In this paper, a new framework for homomorphic secret sharing in secure and private information retrieval from colluding servers is proposed, generalizing the original cross-subspace alignment (CSA) codes proposed by Jia, Sun, and Jafar. We utilize this framework to give a secure PIR construction using algebraic geometry codes over hyperelliptic curves of arbitrary genus. 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 for some parameters where the original ones do not.