Distribution Design

Motivated by applications in cryptography, we introduce and study the problem of distribution design. The goal of distribution design is to find a joint distribution on \$n\$ random variables that satisfies a given set of constraints on the marginal distributions. Each constraint can either require that two sequences of variables be identically distributed or, alternatively, that the two sequences have disjoint supports. We present several positive and negative results on the existence and efficiency of solutions for a given set of constraints. Distribution design can be seen as a strict generalization of several well-studied problems in cryptography. These include secret sharing, garbling schemes, and non-interactive protocols for secure multiparty computation. We further motivate the problem and our results by demonstrating their usefulness towards realizing non-interactive protocols for ad-hoc secure multiparty computation, in which any subset of the parties may choose to participate and the identity of the participants should remain hidden to the extent possible.