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

2025-08-26

AI Summary

This work addresses the fundamental trade-off between privacy and communication efficiency in distributed private information retrieval (PIR). We propose a novel algebraic geometric PIR scheme based on Hermitian curves, marking the first application of $mathbb{F}_ell$-maximal Hermitian curves to construct cross-subspace alignment (CSA) codes. Leveraging the fact that such curves attain the Hasse–Weil upper bound on the number of rational points, our construction achieves significantly higher coding dimension and message retrieval rate compared to low-genus or hyperelliptic curve-based alternatives under identical parameters. Theoretical analysis and experimental evaluation demonstrate that the maximal-curve-driven CSA-PIR protocol guarantees information-theoretic privacy while substantially reducing total communication overhead. Our results establish the unique advantage and practical viability of maximal curves for designing efficient, scalable PIR protocols.

Technology Category

• Search and Optimization: Distributed Search
• Machine Learning: Privacy
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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.

Problem

• Improving private information retrieval rates using Hermitian curve codes
• Constructing cross-subspace alignment codes from algebraic geometry
• Enhancing PIR performance through maximal curves with rational points

Innovation

• Hermitian curve AG codes for PIR
• Cross-subspace alignment with maximal curves
• Higher retrieval rates via rational points