MHz2k: MPC from HE over $\mathbb{Z}_{2^k}$ with New Packing, Simpler Reshare, and Better ZKP

We propose a multi-party computation (MPC) protocol over \(\mathbb {Z}_{2^k}\) secure against actively corrupted majority from somewhat homomorphic encryption. The main technical contributions are: (i) a new efficient packing method for \(\mathbb {Z}_{2^k}\)-messages in lattice-based somewhat homomorphic encryption schemes, (ii) a simpler reshare protocol for level-dependent packings, (iii) a more efficient zero-knowledge proof of plaintext knowledge on cyclotomic rings \({\mathbb Z}[X]/\varPhi _M(X)\) with M being a prime. Integrating them, our protocol shows from 2.2x upto 4.8x improvements in amortized communication costs compared to the previous best results. Our techniques not only improve the efficiency of MPC over \(\mathbb {Z}_{2^k}\) considerably, but also provide a toolkit that can be leveraged when designing other cryptographic primitives over \(\mathbb {Z}_{2^k}\).