Counting Method for Multi-party Computation over Non-abelian Groups

  • Youming Qiao
  • Christophe Tartary
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5339)

Abstract

In the Crypto’07 paper [5], Desmedt et al. studied the problem of achieving secure n-party computation over non-Abelian groups. The function to be computed is fG(x1,...,xn) : = x1 ·...·xn where each participant Pi holds an input xi from the non-commutative group G. The settings of their study are the passive adversary model, information-theoretic security and black-box group operations over G.

They presented three results. The first one is that honest majority is needed to ensure security when computing fG. Second, when the number of adversary \(t\leq\lceil\frac{n}{2}\rceil-1\), they reduced building such a secure protocol to a graph coloring problem and they showed that there exists a deterministic secure protocol computing fG using exponential communication complexity. Finally, Desmedt et al. turned to analyze random coloring of a graph to show the existence of a probabilistic protocol with polynomial complexity when t < n/μ, in which μ is a constant less than 2.948.

We call their analysis method of random coloring the counting method as it is based on the counting of the number of a specific type of random walks. This method is inspiring because, as far as we know, it is the first instance in which the theory of self-avoiding walk appears in multiparty computation.

In this paper, we first give an altered exposition of their proof. This modification will allow us to adapt this method to a different lattice and reduce the communication complexity by 1/3, which is an important saving for practical implementations of the protocols. We also show the limitation of the counting method by presenting a lower bound for this technique. In particular, we will deduce that this approach would not achieve the optimal collusion resistance \(\lceil \frac{n}{2} \rceil - 1\).

Keywords

Multiparty Computation Passive Adversary Non-Abelian Groups Graph Coloring Neighbor-Avoiding Walk Random Walk 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Youming Qiao
    • 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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