The Complexity of Multiparty PSM Protocols and Related Models.

We study the efficiency of computing arbitrary k-argument functions in the Private Simultaneous Messages (PSM) model of [10,14]. This question was recently studied by Beimel et al. [6], in the two-party case (k = 2). We tackle this question in the general case of PSM protocols for k >= 2 parties. Our motivation is two-fold: On one hand, there are various applications (old and new) of PSM protocols for constructing other cryptographic primitives, where obtaining more efficient PSM protocols imply more efficient primitives. On the other hand, improved PSM protocols are an interesting goal on its own. In particular, we pay a careful attention to the case of small number of parties (e.g., k = 3, 4, 5), which may be especially interesting in practice, and optimize our protocols for those cases. Our new upper bounds include a k-party PSM protocol, for any k > 2 and any function f : [N](k) -> {0, 1}, of complexity O(poly(k) . N-k/2) (compared to the previous upper bound of O(poly(k) . Nk-1)), and even better bounds for small values of k; e.g., an O(N) PSM protocol for the case k = 3. We also handle the more involved case where different parties have inputs of different sizes, which is useful both in practice and for applications. As applications, we obtain more efficient Non-Interactive secure Multi-Party (NIMPC) protocols (a variant of PSM, where some of the parties may collude with the referee [5]), improved ad-hoc PSM protocols (another variant of PSM, where the subset of participating parties is not known in advance [4,7]), secret-sharing schemes for uniform access structures with smaller share size than previously known, and better homogeneous distribution designs [4] (a primitive with many cryptographic applications on its own).