New $X$-Secure $T$-Private Information Retrieval Schemes via Rational Curves and Hermitian Curves

2026-01-12

AI Summary

This study addresses the problem of maximizing the private information retrieval (PIR) rate in X-secure T-private information retrieval (XSTPIR) under fixed finite field size and given parameters X and T, while ensuring data security and user privacy. To this end, the work proposes a novel XSTPIR scheme that constructs a family of polynomial-space bases as alternatives to Lagrange interpolation bases, significantly improving the utilization efficiency of rational points on algebraic curves—such as rational and Hermitian curves—without requiring curves of higher genus or more rational points. Leveraging algebraic geometry code theory, the scheme designs an optimized encoding and retrieval structure to enhance both information distribution and privacy guarantees. The Hermitian-based construction achieves the highest known PIR rate when $q^2 \geq 14^2$ and $X+T \geq 4q$, while both constructions attain optimal performance for $q^2 \geq 28^2$ and $X+T \geq 4$.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Privacy
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

$X$-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.

Problem

• XSTPIR
• PIR rate
• algebraic curves
• information retrieval
• data security

Innovation

• XSTPIR
• algebraic geometry codes
• Hermitian curves
• rational points utilization
• polynomial basis