Secure multiparty computation with general interaction patterns

We present a unified framework for studying secure multi-party computation (MPC) with arbitrarily restricted inter-Action patterns such as a chain, a star, a directed tree, or a directed graph. Our study generalizes both standard MPC and recent models for MPC with specific restricted interac-Tion patterns, such as those studied by Halevi et al. (Crypto 2011), Goldwasser et al. (Eurocrypt 2014), and Beimel et al. (Crypto 2014). Since restricted interaction patterns cannot always yield full security for MPC, we start by formalizing the notion of "best possible security" for any interaction pattern. We then obtain the following main results: Completeness theorem. We prove that the star interaction pattern is complete for the problem of MPC with general interaction patterns. Positive results. We present both information-Theoretic and computationally secure protocols for computing arbitrary functions with general interaction patterns. We also present more efficient protocols for computing symmetric functions, both in the computational and in the information-Theoretic setting. Our computationally secure protocols for general func-Tions necessarily rely on indistinguishability obfusca-Tion while the ones for computing symmetric functions make simple use of multilinear maps. Negative results. We show that, in many cases, the complexity of our information-Theoretic protocols is es-sentially the best that can be achieved. All of our protocols rely on a correlated randomness setup, which is necessary in our setting (for computing general functions). In the computational case, we also present a generic procedure to make any correlated randomness setup reusable, in the common random string model. Although most of our information-Theoretic protocols have exponential complexity, they may be practical for function on small domains (e.g., f0; 1g20), where they are concretely faster than their computational counterparts.