Private Set Multi-Party Computations are protocols that allow parties to jointly and securely compute functions: apart from what is deducible from the output of the function, the input sets are kept private. Then, a Private Set Union (PSU), resp. Intersection (PSI), is a protocol that allows parties to jointly compute the union, resp. the intersection, between their private sets. Now a structured PSI, is a PSI where some structure of the sets can allow for more efficient protocols. For instance in Fuzzy PSI, elements only need to be close enough, instead of equal, to be part of the intersection. We present in this paper, Fuzzy PSU protocols (FPSU), able to efficiently take into account approximations in the union. For this, we introduce a new efficient sub-protocol, called Oblivious Key Homomorphic Encryption Retrieval (OKHER), improving on Oblivious Key-Value Retrieval (OKVR) techniques in our setting. In the fuzzy context, the receiver set $X=\{x_i\}_{1..n}$ is replaced by ${\mathcal B}_δ(X)$, the union of $n$ balls of dimension $d$ with radius $δ$, centered at the $x_i$. The sender set is just its $m$ points of dimension $d$. Then the FPSU functionality corresponds to $X \sqcup \{y \in Y, y \notin {\mathcal B}_δ(X)\}$. Thus, we formally define the FPSU functionality and security properties, and propose several protocols tuned to the patterns of the balls using the $l_\infty$ distance. Using our OKHER routine and homomorphic encryption, we are for instance able to obtain a FPSU protocols with an asymptotic communication volume bound ranging from $O(dm\log(δ{n}))$ to $O(d^2m\log(δ^2n))$, depending on the receiver data set structure.

翻译：私有集合多方计算协议允许参与方联合且安全地计算函数：除可从函数输出推导的信息外，输入集合始终保持私有。私有集合并集协议与私有集合交集协议分别使参与方能够联合计算其私有集合的并集与交集。结构化私有集合交集指集合的某些结构特性可支持更高效协议的场景，例如在模糊私有集合交集中，元素仅需足够接近（而非完全相等）即可被视为交集成员。本文提出模糊私有集合并集协议，能够高效处理并集计算中的近似问题。为此，我们引入名为"不经意密钥同态加密检索"的新型高效子协议，该协议在我们设定的场景中改进了现有不经意键值检索技术。在模糊计算场景中，接收方集合$X=\{x_i\}_{1..n}$被替换为${\mathcal B}_δ(X)$——即$n$个$d$维空间中以$x_i$为中心、$δ$为半径的超球体的并集。发送方集合则为其$m$个$d$维点集。此时模糊私有集合并集功能对应$X \sqcup \{y \in Y, y \notin {\mathcal B}_δ(X)\}$的计算。我们正式定义了模糊私有集合并集的功能需求与安全特性，并基于$l_\infty$距离针对超球体分布模式提出了多种协议方案。通过运用不经意密钥同态加密检索例程与同态加密技术，我们获得的模糊私有集合并集协议具有渐进通信量边界，其范围从$O(dm\log(δ{n}))$到$O(d^2m\log(δ^2n))$，具体取决于接收方数据集的结构特征。