Overdrive2k: Efficient Secure MPC over $\mathbb {Z}_{2^k}$ from Somewhat Homomorphic Encryption

Abstract

Recently, Cramer et al. (CRYPTO 2018) presented a protocol, $\mathsf {SPDZ2k}$, for actively secure multiparty computation for dishonest majority in the pre-processing model over the ring $\mathbb {Z}_{2^k}$, instead of over a prime field $\mathbb {F}_p$. Their technique used oblivious transfer for the pre-processing phase, more specifically the MASCOT protocol (Keller et al. CCS 2016). In this paper we describe a more efficient technique for secure multiparty computation over $\mathbb {Z}_{2^k}$ based on somewhat homomorphic encryption. In particular we adapt the Overdrive approach (Keller et al. EUROCRYPT 2018) to obtain a protocol which is more like the original SPDZ protocol (Damgård et al. CRYPTO 2012). To accomplish this we introduce a special packing technique for the BGV encryption scheme operating on the plaintext space defined by the $\mathsf {SPDZ2k}$ protocol, extending the ciphertext packing method used in SPDZ to the case of $\mathbb {Z}_{2^k}$. We also present a more complete pre-processing phase for secure computation modulo $2^k$ by adding a new technique to produce shared random bits.