$X$-secure and $T$-private information retrieval (XSTPIR) is a variant of private information retrieval where data security is guaranteed against collusion among up to $X$ servers and the user's retrieval privacy is guaranteed against collusion among up to $T$ servers. Recently, researchers have constructed XSTPIR schemes through the theory of algebraic geometry codes and algebraic curves, with the aim of obtaining XSTPIR schemes that have higher maximum PIR rates for fixed field size and $X,T$ (the number of servers $N$ is not restricted). The mainstream approach is to employ curves of higher genus that have more rational points, evolving from rational curves to elliptic curves to hyperelliptic curves and, most recently, to Hermitian curves. In this paper, we propose a different perspective: with the shared goal of constructing XSTPIR schemes with higher maximum PIR rates, we move beyond the mainstream approach of seeking curves with higher genus and more rational points. Instead, we aim to achieve this goal by enhancing the utilization efficiency of rational points on curves that have already been considered in previous work. By introducing a family of bases for the polynomial space $\text{span}_{\mathbb{F}_q}\{1,x,\dots,x^{k-1}\}$ as an alternative to the Lagrange interpolation basis, we develop two new families of XSTPIR schemes based on rational curves and Hermitian curves, respectively. Parameter comparisons demonstrate that our schemes achieve superior performance. Specifically, our Hermitian-curve-based XSTPIR scheme provides the largest known maximum PIR rates when the field size $q^2\geq 14^2$ and $X+T\geq 4q$. Moreover, for any field size $q^2\geq 28^2$ and $X+T\geq 4$, our two XSTPIR schemes collectively provide the largest known maximum PIR rates.

翻译：$X$安全$T$私有信息检索（XSTPIR）是私有信息检索的一种变体，其数据安全性可抵御最多$X$个服务器的合谋攻击，用户检索隐私性可抵御最多$T$个服务器的合谋攻击。近期，研究者通过代数几何码与代数曲线理论构建了XSTPIR方案，旨在针对固定的域规模及$X,T$参数（服务器数量$N$不受限制）获得具有更高最大PIR速率的方案。主流方法采用具有更多有理点的高亏格曲线，从有理曲线发展到椭圆曲线、超椭圆曲线，直至最近的Hermitian曲线。本文提出一种不同的视角：在构建更高最大PIR速率XSTPIR方案的共同目标下，我们突破寻求更高亏格与更多有理点曲线的主流路径，转而通过提升已有研究中曲线有理点的利用效率来实现该目标。通过引入多项式空间$\text{span}_{\mathbb{F}_q}\{1,x,\dots,x^{k-1}\}$的一组基函数族作为拉格朗日插值基的替代方案，我们分别基于有理曲线与Hermitian曲线构建了两个新的XSTPIR方案族。参数对比表明我们的方案具有更优性能：当域规模$q^2\geq 14^2$且$X+T\geq 4q$时，基于Hermitian曲线的XSTPIR方案提供了当前已知的最大PIR速率；此外，对于任意域规模$q^2\geq 28^2$且$X+T\geq 4$的情况，我们提出的两个XSTPIR方案共同提供了当前已知的最大PIR速率。