Secure Arithmetic Computation with No Honest Majority (0811.0475v3)

Abstract: We study the complexity of securely evaluating arithmetic circuits over finite rings. This question is motivated by natural secure computation tasks. Focusing mainly on the case of two-party protocols with security against malicious parties, our main goals are to: (1) only make black-box calls to the ring operations and standard cryptographic primitives, and (2) minimize the number of such black-box calls as well as the communication overhead. We present several solutions which differ in their efficiency, generality, and underlying intractability assumptions. These include: 1. An unconditionally secure protocol in the OT-hybrid model which makes a black-box use of an arbitrary ring $R$, but where the number of ring operations grows linearly with (an upper bound on) $\log|R|$. 2. Computationally secure protocols in the OT-hybrid model which make a black-box use of an underlying ring, and in which the number of ring operations does not grow with the ring size. These results extend a previous approach of Naor and Pinkas for secure polynomial evaluation (SIAM J. Comput., 35(5), 2006). 3. A protocol for the rings $\mathbb{Z}_m=\mathbb{Z}/m\mathbb{Z}$ which only makes a black-box use of a homomorphic encryption scheme. When $m$ is prime, the (amortized) number of calls to the encryption scheme for each gate of the circuit is constant. All of our protocols are in fact UC-secure in the OT-hybrid model and can be generalized to multiparty computation with an arbitrary number of malicious parties.