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

TL;DR

This paper introduces new XSTPIR schemes using rational and Hermitian curves, leveraging an innovative basis for polynomial spaces to improve PIR rates over existing methods.

Contribution

It proposes a novel approach that enhances the utilization of rational points on algebraic curves, leading to superior XSTPIR schemes based on rational and Hermitian curves.

Findings

Hermitian-curve-based scheme achieves highest known PIR rates for large fields.

New schemes outperform previous constructions in specific parameter regimes.

Parameter comparisons confirm the efficiency of the proposed methods.

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.