Efficient multi‐key fully homomorphic encryption over prime cyclotomic rings with fewer relinearisations

Abstract

Multi-key fully homomorphic encryption (MKFHE) allows computations on ciphertexts encrypted by different users, which can be applied to implement secure multi-party computing (MPC). The current NTRU-based MKFHE has the following two drawbacks: One is that the relinearisation process during homomorphic evaluation is so complicated that the corresponding computation time is costly. The other is that a class of subfield attacks are proposed and affects the security of NTRU schemes over power-of-2 cyclotomic rings for large moduli q, especially for the NTRU-based fully homomorphic encryption (FHE) schemes. In this work, an efficient MKFHE scheme is proposed over prime cyclotomic rings with fewer relinearisations, which seems a good choice because of its potential to resist a subfield attack. More specifically, the time of the relinearisation process is reduced by half in homomorphic evaluations by separating the homomorphic multiplication and the relinearisation process (implementing two homomorphic multiplication operations together before relinearisation), while in current NTRU-type MKFHE schemes, these two processes are usually performed together. The error bound of the basic function components is re-analysed over prime cyclotomic rings in the average case, which can be used in the error analysis of our scheme. We construct an efficient NTRU-based single-key FHE scheme and an efficient MKFHE scheme over prime cyclotomic rings through relinearisation and modulus-switching techniques. The MKFHE scheme proposed has the on-the-fly property and has a tight ciphertext size compared with the GSW-type and BGV-type MKFHE schemes. An experiment shows that the homomorphic evaluation of the optimised single-key FHE scheme proposed is 1.9 times faster than an efficient NTRU-type MKFHE DHS16 proposed at DCC 2016.