On Structure-Preserving Cryptography and Lattices

Abstract

The Groth-Sahai proof system is a highly efficient pairing-based proof system for a specific class of group-based languages. Cryptographic primitives that are compatible with these languages (such that we can express, e.g., that a ciphertext contains a valid signature for a given message) are called structure-preserving . The combination of structure-preserving primitives with Groth-Sahai proofs allows to prove complex statements that involve encryptions and signatures, and has proved useful in a variety of applications. However, so far, the concept of structure-preserving cryptography has been confined to the pairing setting. In this work, we propose the first framework for structure-preserving cryptography in the lattice setting. Concretely, we - define structure-preserving sets as an abstraction of (typically noisy) lattice-based languages, - formalize a notion of generalized structure-preserving encryption and signature schemes capturing a number of existing lattice-based encryption and signature schemes), - construct a compatible zero-knowledge argument system that allows to argue about lattice-based structure-preserving primitives, - offer a lattice-based construction of verifiably encrypted signatures in our framework. Along the way, we also discover a new and efficient strongly secure lattice-based signature scheme. This scheme combines Rückert\u27s lattice-based signature scheme with the lattice delegation strategy of Agrawal et al., which yields more compact and efficient signatures. We hope that our framework provides a first step towards a modular and versatile treatment of cryptographic primitives in the lattice setting