# On secure multi-party computation in black-box groups

Desmedt, Yvo, Pieprzyk, Josef, Steinfeld, Ron, & Wang, Huaxiong
(2007)
On secure multi-party computation in black-box groups.
*Lecture Notes in Computer Science : Advances in Cryptology*, *4622*, pp. 591-612.

## Abstract

We study the natural problem of secure n-party computation (in the passive, computationally unbounded attack model) of the n-product function f G (x 1,...,x n ) = x 1 ·x 2 ⋯ x n in an arbitrary finite group (G,·), where the input of party P i is x i ∈ G for i = 1,...,n. For flexibility, we are interested in protocols for f G 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 f G exists. Second, on the positive side, we initiate an approach for construction of black-box protocols for f G 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*2t+1^2/t) 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(n*t^2) group elements. Furthermore, we believe that our results can be improved by further study of the associated combinatorial problems.

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ID Code: | 74285 |
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Item Type: | Journal Article |

Refereed: | Yes |

Additional Information: | Proceedings of the 27th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 19-23, 2007. |

DOI: | 10.1007/978-3-540-74143-5_33 |

ISSN: | 0302-9743 |

Divisions: | Current > QUT Faculties and Divisions > Science & Engineering Faculty |

Deposited On: | 23 Jul 2014 01:24 |

Last Modified: | 24 Jul 2014 23:47 |

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