Succinct Oblivious Tensor Evaluation and Applications: Adaptively-Secure Laconic Function Evaluation and Trapdoor Hashing for All Circuits

Abstract: We propose the notion of succinct oblivious tensor evaluation (OTE), where two parties compute an additive secret sharing of a tensor product of two vectors $\mathbf{x} \otimes \mathbf{y}$, exchanging two simultaneous messages. Crucially, the size of both messages and of the CRS is independent of the dimension of $\mathbf{x}$. We present a construction of OTE with optimal complexity from the standard learning with errors (LWE) problem. Then we show how this new technical tool enables a host of cryptographic primitives, all with security reducible to LWE, such as: * Adaptively secure laconic function evaluation for depth-$D$ functions $f:{0, 1}^{m\rightarrow{0,} 1}<sup>\ell$ with communication $m+\ell+D\cdot \mathrm{poly}(\lambda)$. * A trapdoor hash function for all functions. * An (optimally) succinct homomorphic secret sharing for all functions. * A rate-$1/2$ laconic oblivious transfer for batch messages, which is best possible. In particular, we obtain the first laconic function evaluation scheme that is adaptively secure from the standard LWE assumption, improving upon Quach, Wee, and Wichs (FOCS 2018). As a key technical ingredient, we introduce a new notion of \emph{adaptive lattice encodings}, which may be of independent interest.