AG codes from the Hermitian curve for Cross-Subspace Alignment in Private Information Retrieval

TL;DR

This paper introduces a new private information retrieval scheme utilizing algebraic geometry codes from the Hermitian curve, achieving higher rates by leveraging the curve's many rational points.

Contribution

It presents a novel PIR scheme based on Hermitian curve AG codes, demonstrating improved retrieval rates over previous schemes using lower-genus curves.

Findings

Higher PIR rates achieved with Hermitian curve codes

Utilizes maximal curves for efficient PIR construction

Longer code lengths due to abundant rational points

Abstract

Private information retrieval (PIR) addresses the problem of retrieving a desired message from distributed databases without revealing which message is being requested. Recent works have shown that cross-subspace alignment (CSA) codes constructed from algebraic geometry (AG) codes on high-genus curves can improve PIR rates over classical constructions. In this paper, we propose a new PIR scheme based on AG codes from the Hermitian curve, a well-known example of an $F_\ell$-maximal curve, that is, a curve defined over the finite field with $\ell$ elements which attains the Hasse-Weil upper bound on the number of its $F_\ell$-rational points. The large number of rational points enables longer code constructions, leading to higher retrieval rates than schemes based on genus 0, genus 1, and hyperelliptic curves of arbitrary genus. Our results highlight the potential of maximal curves as a natural source of efficient PIR constructions.