Graph Design for Secure Multiparty Computation over Non-Abelian Groups

  • Xiaoming Sun
  • Andrew Chi-Chih Yao
  • Christophe Tartary
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5350)

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 fG(x1,...,xn) : = x1 ·x2 ·...·xn, where the input of party Pi is xi ∈ 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 fG. Second, they demonstrated that one could t-privately compute fG for any \(t \leq \lceil \tfrac{n}{2} \rceil - 1\) in exponential communication cost. Third, they constructed a randomized algorithm with O(nt2) 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(n3) 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(n1 − 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 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Xiaoming Sun
    • 1
  • Andrew Chi-Chih Yao
    • 1
  • Christophe Tartary
    • 1
    • 2
  1. 1.Institute for Theoretical Computer ScienceTsinghua UniversityBeijingPeople’s Republic of China
  2. 2.Division of Mathematical Sciences School of Physical and Mathematical SciencesNanyang Technological UniversitySingapore

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