Previous FPSI works have demonstrated a linear scaling with the distance threshold $δ$, while some recent works have achieved a poly-logarithmic dependence on $δ$. However, these protocols either support only the $L_\infty$ distance, or they support general $L_{p\in[1,\infty]}$ distances but rely on expensive additive homomorphic encryption (AHE). Achieving exact logarithmic dependence on $δ$ for general $L_{p\in[1,\infty]}$ distances without relying on costly AHE would constitute a theoretical breakthrough in optimal threshold scaling and a practical advance toward scalable FPSI applications. In this work, we present new FPSI protocols for $L_{p\in[1,\infty]}$ distances that are entirely built from oblivious transfer (OT) and symmetric-key primitives. We propose FPSI protocols based on both the apart and the separate assumptions, which are applicable to low- and high-dimensional settings, respectively. Our constructions achieve strictly logarithmic complexity in $δ$, which is optimal in the sense that distinguishing all values in an interval of length $O(δ)$ necessarily requires $Ω(\log δ)$ bits of information. Our core idea is to perform fuzzy matching via prefix representation and interactively determine the correct prefix using equality conditions. To this end, we propose a suite of new components that can be implemented efficiently using only OT and symmetric-key operations. We implement our FPSI protocols and compare them with the state-of-the-art FPSI protocols for $L_{p\in[1,\infty]}$ distance. Experiments show that our protocols outperform the prior state-of-the-art by up to $43.7\times$ in runtime and $31.3\times$ in communication.

翻译：以往的FPSI工作在距离阈值$δ$上呈现线性缩放，而近期部分工作实现了对$δ$的多对数依赖。然而，这些协议要么仅支持$L_\infty$距离，要么支持一般$L_{p\in[1,\infty]}$距离但依赖昂贵的加法同态加密（AHE）。在不依赖高成本AHE的前提下，针对一般$L_{p\in[1,\infty]}$距离实现$δ$的精确对数依赖，将在最优阈值缩放理论上取得突破，并在实践中推动可扩展FPSI应用的发展。本文提出了针对$L_{p\in[1,\infty]}$距离的全新FPSI协议，该协议完全基于不经意传输（OT）和对称密钥原语构建。我们分别提出了基于apart假设和separate假设的FPSI协议，前者适用于低维场景，后者适用于高维场景。我们的构造在$δ$上实现了严格的对数复杂度——这是理论最优的，因为要区分长度为$O(δ)$区间内的所有值，至少需要$Ω(\log δ)$比特信息。核心思想是通过前缀表示实现模糊匹配，并利用等式条件交互确定正确的前缀。为此，我们提出了一系列仅需OT和对称密钥操作即可高效实现的新组件。我们对提出的FPSI协议进行实现，并与现有最先进的$L_{p\in[1,\infty]}$距离FPSI协议进行对比。实验表明，我们的协议在运行时间上最高提速$43.7$倍，通信量最高降低$31.3$倍。