On Secure Multi-party Computation in Black-Box Groups

  • Yvo Desmedt
  • Josef Pieprzyk
  • Ron Steinfeld
  • Huaxiong Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4622)

Abstract

We study the natural problem of secure n-party computation (in the passive, computationally unbounded attack model) of the n-product function fG(x1,...,xn) = x1 ·x2 ⋯ xn in an arbitrary finite group (G,·), where the input of party Pi is xi ∈ G for i = 1,...,n. For flexibility, we are interested in protocols for fG which require only black-box access to the group G (i.e. the only computations performed by players in the protocol are a group operation, a group inverse, or sampling a uniformly random group element).

Our results are as follows. First, on the negative side, we show that if (G,·) is non-abelian and n ≥ 4, then no ⌈n/2⌉-private protocol for computing fG exists. Second, on the positive side, we initiate an approach for construction of black-box protocols for fG based on k-of-k threshold secret sharing schemes, which are efficiently implementable over any black-box group G. We reduce the problem of constructing such protocols to a combinatorial colouring problem in planar graphs. We then give two constructions for such graph colourings. Our first colouring construction gives a protocol with optimal collusion resistance t < n/2, but has exponential communication complexity \(O(n\frac{2t+1}{t}^2)\) group elements (this construction easily extends to general adversary structures). Our second probabilistic colouring construction gives a protocol with (close to optimal) collusion resistance t < n/μ for a graph-related constant μ ≤ 2.948, and has efficient communication complexity O(nt2) group elements. Furthermore, we believe that our results can be improved by further study of the associated combinatorial problems.

Keywords

Multi-Party Computation Non-Abelian Group Black-Box Planar Graph Graph Colouring 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Yvo Desmedt
    • 1
  • Josef Pieprzyk
    • 2
  • Ron Steinfeld
    • 2
  • Huaxiong Wang
    • 2
    • 3
  1. 1.Dept. of Computer Science, University College LondonUK
  2. 2.Centre for Advanced Computing – Algorithms and Cryptography (ACAC), Dept. of Computing, Macquarie University, North RydeAustralia
  3. 3.Division of Math. Sci., Nanyang Technological UniversitySingapore

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