Graph Design for Secure Multiparty Computation over Non-Abelian Groups

Recently, Desmedt et al. studied the problem of achieving secure nparty computation over non-Abelian groups. They considered the passive adversary model and they assumed that the parties were only allowed to perform black-box operations over the finite group G. They showed three results for the n-product function fG(x1, . . . , xn) := x1 · x2 · . . . · xn, where the input of party Pi is xi ∈ G for i ∈ {1, . . . , n}. First, if t ≥ ⌈n2 ⌉ then it is impossible to have a t-private protocol computing fG. Second, they demonstrated that one could t-privately compute fG for any t ≤ ⌈n2 ⌉−1 in exponential communication cost. Third, they constructed a randomized algorithm with O(n t) communication complexity for any t < n 2.948 . In this paper, we extend these results in two directions. First, we use percolation theory to show that for any fixed ǫ > 0, one can design a randomized algorithm for any t ≤ n 2+ǫ using O(n) communication complexity, thus nearly matching the known upper bound ⌈ 2 ⌉ − 1. This is the first time that percolation theory is used for multiparty computation. Second, we exhibit a deterministic construction having polynomial communication cost for any t = O(n) (again for any fixed ǫ > 0). Our results extend to the more general function e fG(x1, . . . , xm) := x1 · x2 · . . . · xm where m ≥ n and each of the n parties holds one or more input values.