International Association
for Cryptologic Research

Traditional private set intersection (PSI) involves a receiver and a sender holding sets $X$ and $Y$, respectively, with the receiver learning only the intersection $X\cap Y$. We turn our attention to its fuzzy variant, where the receiver holds $|X|$ hyperballs of radius $\delta$ in a metric space and the sender has $|Y|$ points. Representing the hyperballs by their center, the receiver learns the points $x\in X$ for which there exists $y\in Y$ such that $\dist(x,y)\leq \delta$ with respect to some distance metric. Previous approaches either require general-purpose multi-party computation (MPC) techniques like garbled circuits or fully homomorphic encryption (FHE), leak details about the sender’s precise inputs, support limited distance metrics, or scale poorly with the hyperballs' volume. This work presents the first blackbox construction for fuzzy PSI (including other variants such as PSI cardinality, labeled PSI, and circuit PSI), which can handle polynomially large radius and dimension (i.e., a potentially exponentially large volume) in two interaction messages, supporting general $L_{p\in[1,\infty]}$ distance, without relying on garbled circuits or FHE. The protocol excels in both asymptotic and concrete efficiency compared to existing works. For security, we solely rely on the assumption that the Decisional Diffie-Hellman (DDH) holds in the random oracle model.

In this paper we study cryptographic finite abelian groups of unknown order and hardness assumptions in these groups. Abelian groups necessitate multiple group generators, which may be chosen at random. We formalize this setting and hardness assumptions therein. Furthermore, we generalize the algebraic group model and strong algebraic group model from cyclic groups to arbitrary finite abelian groups of unknown order. Building on these formalizations, we present techniques to deal with this new setting, and prove new reductions. These results are relevant for class groups of imaginary quadratic number fields and time-lock cryptography build upon them.