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.

We show that it is possible to perform $n$ independent copies of $1$-out-of-$2$ oblivious transfer in two messages, where the communication complexity of the receiver and sender (each) is $n(1+o(1))$ for sufficiently large $n$. Note that this matches the information-theoretic lower bound. Prior to this work, this was only achievable by using the heavy machinery of rate-$1$ fully homomorphic encryption (Rate-$1$ FHE, Brakerski et al., TCC 2019). To achieve rate-$1$ both on the receiver's and sender's end, we use the LPN assumption, with slightly sub-constant noise rate $1/m^{\epsilon}$ for any $\epsilon>0$ together with either the DDH, QR or LWE assumptions. In terms of efficiency, our protocols only rely on linear homomorphism, as opposed to the FHE-based solution which inherently requires an expensive ``bootstrapping'' operation. We believe that in terms of efficiency we compare favorably to existing batch-OT protocols, while achieving superior communication complexity. We show similar results for Oblivious Linear Evaluation (OLE). For our DDH-based solution we develop a new technique that may be of independent interest. We show that it is possible to ``emulate'' the binary group $\bbZ_2$ (or any other small-order group) inside a prime-order group $\bbZ_p$ \emph{in a function-private manner}. That is, $\bbZ_2$ operations are mapped to $\bbZ_p$ operations such that the outcome of the latter do not reveal additional information beyond the $\bbZ_2$ outcome. Our encoding technique uses the discrete Gaussian distribution, which to our knowledge was not done before in the context of DDH.

Threshold Private Set Intersection (PSI) allows multiple parties to compute the intersection of their input sets if and only if the intersection is larger than $n-t$, where $n$ is the size of the sets and $t$ is some threshold. The main appeal of this primitive is that, in contrast to standard PSI, known upper-bounds on the communication complexity only depend on the threshold $t$ and not on the sizes of the input sets. Current Threshold PSI protocols split themselves into two components: A Cardinality Testing phase, where parties decide if the intersection is larger than some threshold; and a PSI phase, where the intersection is computed. The main source of inefficiency of Threshold PSI is the former part. In this work, we present a new Cardinality Testing protocol that allows $N$ parties to check if the intersection of their input sets is larger than $n-t$. The protocol incurs in $\tilde{ \mathcal{O}} (Nt^2)$ communication complexity. We thus obtain a Threshold PSI scheme for $N$ parties with communication complexity $\tilde{ \mathcal{O}}(Nt^2)$.

Consider a server with a \emph{large} set $S$ of strings $\{x_1,x_2\ldots,x_N\}$ that would like to publish a \emph{small} hash $h$ of its set $S$ such that any client with a string $y$ can send the server a \emph{short} message allowing it to learn $y$ if $y \in S$ and nothing otherwise. In this work, we study this problem of two-round private set intersection (PSI) with low (asymptotically optimal) communication cost, or what we call \emph{laconic} private set intersection ($\ell$PSI) and its extensions. This problem is inspired by the recent general frameworks for laconic cryptography [Cho et al. CRYPTO 2017, Quach et al. FOCS'18]. We start by showing the first feasibility result for realizing $\ell$PSI~ based on the CDH assumption, or LWE with polynomial noise-to-modulus ratio. However, these feasibility results use expensive non-black-box cryptographic techniques leading to significant inefficiency. Next, with the goal of avoiding these inefficient techniques, we give a construction of $\ell$PSI~schemes making only black-box use of cryptographic functions. Our construction is secure against semi-honest receivers, malicious senders and reusable in the sense that the receiver's message can be reused across any number of executions of the protocol. The scheme is secure under the $\phi$-hiding, decisional composite residuosity and subgroup decision assumptions. Finally, we show natural applications of $\ell$PSI~to realizing a semantically-secure encryption scheme that supports detection of encrypted messages belonging to a set of ``illegal'' messages (e.g., an illegal video) circulating online. Over the past few years, significant effort has gone into realizing laconic cryptographic protocols. Nonetheless, our work provides the first black-box constructions of such protocols for a natural application setting.

Quantum pseudorandom functions (QPRFs) extend the classical security of a PRF by allowing the adversary to issue queries on input superpositions. Zhandry [Zhandry, FOCS 2012] showed a separation between the two notions and proved that common construction paradigms are also quantum secure, albeit with a new ad-hoc analysis. In this work, we revisit the question of constructing QPRFs and propose a new method starting from small-domain (classical) PRFs: At the heart of our approach is a new domain-extension technique based on bipartite expanders. Interestingly, our analysis is almost entirely classical. As a corollary of our main theorem, we obtain the first (approximate) key-homomorphic quantum PRF based on the quantum intractability of the learning with errors problem.