New $X$-Secure $T$-Private Information Retrieval Schemes via Rational Curves and Hermitian Curves
Yuan Gao, Weijun Fang, Jingke Xu, Jiejing Wen
TL;DR
The paper tackles XSTPIR by seeking higher maximum PIR rates without resorting to curves of higher genus, instead leveraging a novel basis for $\mathrm{span}_{\mathbb{F}_q}\{1,x,\dots,x^{k-1}\}$ to utilize rational points more efficiently. It develops two new XSTPIR schemes: one on rational curves and one on Hermitian curves, and proves improved rate expressions under concrete parameter regimes. Notably, the Hermitian-curve scheme achieves the largest known rates for $q^2\ge 14^2$ and $X+T\ge 4q$, and, for $q^2\ge 28^2$ with $X+T\ge 4$, the two schemes together attain the best-known rates. The results offer a new design principle for XSTPIR based on AG codes, with potential extensions to other curve families such as Norm-Trace, Ree, and Suzuki curves.
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.