Graph Design for Secure Multiparty Computation over Non-Abelian Groups

Xiaoming Sun, Andrew Chi-Chih Yao, Christophe Tartary

Abstract: Recently, Desmedt et al. studied the problem of achieving secure n-party 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 f G (x 1,...,x n ) : = x 1 ·x 2 ·...·x n , where the input of party P i is x i  ∈ G for i ∈ {1,...,n}. First, if $t \geq \lceil \tfrac{n}{2} \rceil$ then it is impossible to have a t-private protocol computing f G . Second, they demonstrated that one could t-privately compute f G for any $t \leq \lceil \tfrac{n}{2} \rceil - 1$ in exponential communication cost. Third, they constructed a randomized algorithm with O(n t 2) communication complexity for any $t < \tfrac{n}{2.948}$ . In this paper, we extend these results in two directions. First, we use percolation theory to show that for any fixed ε> 0, one can design a randomized algorithm for any $t\leq \frac{n}{2+\epsilon}$ using O(n 3) communication complexity, thus nearly matching the known upper bound $\lceil \tfrac{n}{2} \rceil - 1$ . 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 1 − ε ) (again for any fixed ε> 0). Our results extend to the more general function $\widetilde{f}_{G}(x_{1},\ldots,x_{m}) := x_{1} \cdot x_{2} \cdot \ldots \cdot x_{m}$ where m ≥ n and each of the n parties holds one or more input values.