Multi-key Fully Homomorphic Encryption without CRS from RLWE

Fully Homomorphic Encryption (FHE) is a powerful encryption system in cloud computing that allows homomorphic computations on encrypted data without decrypting them. Multi-key fully homomorphic encryption (MFHE), as an extension to FHE, allows homomorphic computations on ciphertexts encrypted under different keys.   However, most MFHE schemes require a Common Random\slash Reference String (CRS), while the few that do not are based on the Learning With Errors (LWE) problem, which means that they can only deal with single bit plaintext.  Consequently, MFHE schemes based on the Ring Learning With Errors (RLWE) problem are more desirable, as they can handle polynomial plaintext.  Requiring the CRS seems to weaken the semantic definition of MFHE, where all users generate their own keys independently.

In this paper, we study the RLWE-based MFHE in the CRS model and propose the first RLWE-based MFHE without CRS.   To this end, we remove the CRS by designing a new relinearization algorithm.  Like previous MFHE schemes, our RLWE-based MFHE without CRS has a simple 1-round threshold decryption, which implies a $3$-round secure MPC protocol in the plain model from the RLWE assumption.