标题： 从赫尔米蒂安曲线生成的AG码用于私有信息检索中的交叉子空间对齐 显示英文标题

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

摘要： 私有信息检索（PIR）解决的是从分布式数据库中检索所需消息而不泄露正在请求的消息的问题。 最近的研究表明，通过代数几何（AG）码在高亏格曲线上的构造得到的交叉子空间对齐（CSA）码可以在经典构造之上提高PIR速率。 在本文中，我们提出了一种基于赫尔米特曲线（Hermitian curve）的AG码的新PIR方案，赫尔米特曲线是一个$F_\ell$-极大曲线的著名例子，即在一个包含$\ell$个元素的有限域上定义的曲线，其$F_\ell$-有理点数量达到了Hasse-Weil上界。 大量的有理点使得可以构造更长的码，从而比基于0亏格、1亏格和任意亏格双曲曲线的方案具有更高的检索速率。 我们的结果突显了极大曲线作为高效PIR构造自然来源的潜力。 显示英文摘要

摘要： 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.