International Association
for Cryptologic Research

Laconic cryptography focuses on designing two-message protocols that allow secure computation over large datasets while minimizing communication costs. While laconic cryptography protocols achieve asymptotically optimal communication complexity for many tasks, their concrete efficiency is prohibitively expensive, due to heavy use of public-key techniques or non-black-box cryptographic primitives. In this work, we initiate the study of {\it laconic cryptography with preprocessing}, introducing a model that includes an offline phase to generate database-dependent correlations, which are then used in a lightweight online phase. These correlations are conceptually simple, relying on linear-algebraic techniques. This enables us to develop a protocol for private laconic vector oblivious linear evaluation (plvOLE). In such a protocol, the receiver holds a large database $\mathsf{DB}$, and the sender has two messages $v$ and $w$ along with an index $i$. The receiver learns the value $v \cdot \mathsf{DB}_i + w$ without revealing other information. Our protocol, which draws from ideas developed in the context of private information retrieval with preprocessing, serves as the backbone for two applications of interest: laconic private set intersection (lPSI) for large universes and laconic function evaluation for RAM programs (RAM-LFE). Based on our plvOLE protocol, we provide efficient instantiations of these two primitives in the preprocessing model.

Laconic cryptography is an emerging paradigm that enables cryptographic primitives with sublinear communication complexity in just two messages. In particular, a two-message protocol between Alice and Bob is called \emph{laconic} if its communication and computation complexity are essentially independent of the size of Alice's input. This can be thought of as a dual notion of fully-homomorphic encryption, as it enables ``Bob-optimized'' protocols. This paradigm has led to tremendous progress in recent years. However, all existing constructions of laconic primitives are considered only of \emph{theoretical interest}: They all rely on non-black-box cryptographic techniques, which are highly impractical. This work shows that non-black-box techniques are not necessary for basic laconic cryptography primitives. We propose a \emph{completely algebraic} construction of laconic encryption, a notion that we introduce in this work, which serves as the cornerstone of our framework. We prove that the scheme is secure under the standard Learning With Errors assumption (with polynomial modulus-to-noise ratio). We provide proof-of-concept implementations for the first time for laconic primitives, demonstrating the construction is indeed practical: For a database size of $2^{50}$, encryption and decryption are in the order of single digit \emph{milliseconds}. Laconic encryption can be used as a black box to construct other laconic primitives. Specifically, we show how to construct: \begin{itemize} \item Laconic oblivious transfer \item Registration-based encryption scheme \item Laconic private-set intersection protocol \end{itemize} All of the above have essentially optimal parameters and similar practical efficiency. Furthermore, our laconic encryption can be preprocessed such that the online encryption step is entirely combinatorial and therefore much more efficient. Using similar techniques, we also obtain identity-based encryption with an unbounded identity space and tight security proof (in the standard model).