Chapter

Advances in Cryptology - ASIACRYPT 2008

Volume 5350 of the series Lecture Notes in Computer Science pp 37-53

Graph Design for Secure Multiparty Computation over Non-Abelian Groups

  • Xiaoming SunAffiliated withInstitute for Theoretical Computer Science, Tsinghua University
  • , Andrew Chi-Chih YaoAffiliated withInstitute for Theoretical Computer Science, Tsinghua University
  • , Christophe TartaryAffiliated withInstitute for Theoretical Computer Science, Tsinghua UniversityDivision of Mathematical Sciences School of Physical and Mathematical Sciences, Nanyang Technological University

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.

Keywords

Multiparty Computation Passive Adversary Non-Abelian Groups Graph Coloring Percolation Theory