Optimal-round preprocessing-MPC of polynomials over non-zero inputs via distributed random matrix

Secure multiparty computation (MPC) is an extensively studied research field in cryptography. It considers two main models—the plain model and the preprocessing model. The latter enables achieving objectives known to be unachievable for the plain model. One prominent example of such an objective is the perfectly secure evaluation of functions over private inputs, with the majority of the parties being dishonest. Recent results have shown that this objective can even be achieved in an optimal number of rounds of communication—two rounds. However, when the function to be evaluated is a polynomial with a possibly high degree over inputs taken from a large domain, existing solutions require an exponential amount of memory in the size of the domain. This paper presents preprocessing-MPC protocols for high-degree polynomials over non-zero inputs. These protocols are the first to have optimal round complexity, perfect security against coalitions of up to N- 1 out of N parties, and communication and space complexities that grow linearly with the number of monomials in the polynomial (independent of its degree). Furthermore, the results are extended to the client-server model. Namely, this paper presents a scheme that enables a user to outsource the storage of non-zero secrets to N distrusted servers and have the servers obliviously evaluate polynomials over the secrets in a single round of communication, perfectly secure against coalitions of up to N- 1 servers. These schemes are based on a novel secret sharing scheme, Distributed Random Matrix (DRM), first presented here. The DRM secret sharing scheme supports homomorphic multiplications and, after a single round of communication, supports homomorphic additions.