**Multiparty Computation
from Somewhat Homomorphic Encryption**

Ivan Damgard (

Valerio Pastro (

Nigel
Smart (

Sarah Zakarias (

**Abstract:**

We propose a general multiparty computation protocol secure against an active
adversary corrupting up to $n-1$ of the $n$ players. The protocol may be used
to compute securely arithmetic circuits over any finite field $\F_{p^k}$. Our protocol consists
of a preprocessing phase that is both independent of the function to be
computed and of the inputs, and a much more efficient online phase where the
actual computation takes place. The online phase is unconditionally secure
and has total computational (and communication) complexity linear in $n$, the
number of players, where earlier work was quadratic in $n$. Hence, the work done
by each player in the online phase is independent of $n$ and moreover is only
a small constant factor larger than what one would need to compute the
circuit in the clear. It is the first protocol in the preprocessing model
with these properties. We show a lower bound implying that for computation in
large fields, our protocol is optimal. In practice, for 3 players, a secure
64-bit multiplication can be done in 0.05 ms. Our preprocessing is based on a
somewhat homomorphic cryptosystem. We extend a
scheme by Brakerski et al., so that we can perform
distributed decryption and handle many values in parallel in one ciphertext. The computational complexity of our
preprocessing phase is dominated by the public-key operations, we need $O(n^2/s)$ operations per secure multiplication where $s$
is a parameter that increases with the security parameter of the
cryptosystem. Earlier work in this model needed $\Omega(n^2)$
operations. In practice, the preprocessing prepares a secure 64-bit
multiplication for 3 players in about 13 ms, which is 2-3 order
of magnitude faster than the best previous results.