Graph Design for Secure Multiparty Computation over Non-Abelian Groups

Recently, Desmedt et al. studied the problem of achieving securen -party computation over non-Abelian groups. Theyconsidered the passive adversary model and they assumed that theparties were only allowed to perform black-box operations over thefinite group G . They showed three results for then -product function f G (x 1 ,...,x n ) : =x 1 ·x 2 ·...·x n ,where the input of party P i isx i ∈ G for i ∈ {1,...,n }. First, if $t \geq \lceil \tfrac{n}{2}\rceil$ then it is impossible to have a t -private protocolcomputing f G . Second, theydemonstrated 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 arandomized algorithm with O (n t 2) communication complexity for anyt≤n/2.948 In this paper, we extend these results in two directions. First,we use percolation theory to show that for any fixede > 0, one can design a randomized algorithm forany $t\leq \frac{n}{2+\epsilon}$ using O (n 3) communication complexity, thus nearly matching theknown upper bound $\lceil \tfrac{n}{2} \rceil - 1$. This is thefirst time that percolation theory is used for multipartycomputation. Second, we exhibit a deterministic construction havingpolynomial communication cost for any t =O (n 1-e ) (again forany fixed e > 0). Our results extend to the moregeneral 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 moreinput values.