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.
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Sun, X., Yao, A.CC., Tartary, C. (2008). Graph Design for Secure Multiparty Computation over Non-Abelian Groups. In: Pieprzyk, J. (eds) Advances in Cryptology - ASIACRYPT 2008. ASIACRYPT 2008. Lecture Notes in Computer Science, vol 5350. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89255-7_3
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