Multi-Party Private Set Intersection (MP-PSI) with threshold enhances the flexibility of MP-PSI by disclosing elements present in at least $t$ participants' sets, rather than requiring elements to appear in all $n$ sets. In scenarios where each participant is responsible for its dataset, e.g., digital forensics, MP-PSI with threshold should disclose both intersection elements and corresponding holders such that elements are traceable and the reliability of intersection is guaranteed. We refer to MP-PSI with threshold supporting traceability as Traceable Over-Threshold MP-PSI (T-OT-MP-PSI). However, research on such protocols remains limited, and existing work tolerates at most $t-2$ semi-honest participants at considerable computational cost. We propose two novel Traceable OT-MP-PSI protocols. The first, Efficient Traceable OT-MP-PSI (ET-OT-MP-PSI), combines Shamir's secret sharing with an oblivious programmable pseudorandom function, achieving significantly improved efficiency with resistance to at most $t-2$ semi-honest participants. The second, Security-enhanced Traceable OT-MP-PSI (ST-OT-MP-PSI), achieves security against up to $n-1$ semi-honest participants by further leveraging the oblivious linear evaluation protocol. Compared to Mahdavi et al.'s protocol, ours eliminate the assumption that certain special parties do not collude. Experimental results demonstrate significant improvements: for $n=5$, $t=3$, and sets of size $2^{14}$, ET-OT-MP-PSI achieves $15056\times$ speedup and ST-OT-MP-PSI achieves $505\times$ speedup over Mahdavi et al.'s protocol.

翻译：带阈值的多方隐私集合求交（MP-PSI）通过披露存在于至少$t$个参与者集合中的元素，而非要求元素出现在所有$n$个集合中，从而增强了MP-PSI的灵活性。在每个参与者负责其自身数据集的场景中（例如数字取证），带阈值的MP-PSI应同时披露交集元素及相应的持有者，以确保元素可追溯且交集的可靠性得到保障。我们将支持可追溯性的带阈值MP-PSI称为可追溯超阈值多方隐私集合求交（T-OT-MP-PSI）。然而，此类协议的研究仍然有限，现有工作最多容忍$t-2$个半诚实参与者，且计算成本高昂。我们提出了两种新颖的可追溯OT-MP-PSI协议。第一种是高效可追溯OT-MP-PSI（ET-OT-MP-PSI），它将Shamir秘密共享与不经意可编程伪随机函数相结合，在最多抵抗$t-2$个半诚实参与者的同时，实现了显著提升的效率。第二种是安全性增强的可追溯OT-MP-PSI（ST-OT-MP-PSI），通过进一步利用不经意线性求值协议，实现了对最多$n-1$个半诚实参与者的安全性。与Mahdavi等人的协议相比，我们的协议消除了特定特殊方不共谋的假设。实验结果表明性能显著提升：在$n=5$、$t=3$、集合大小为$2^{14}$的设置下，ET-OT-MP-PSI相比Mahdavi等人的协议实现了$15056\times$的加速，ST-OT-MP-PSI实现了$505\times$的加速。