On secure multi-party computation in black-box groups

Yvo Desmedt*, Josef Pieprzyk, Ron Steinfeld, Huaxiong Wang

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference proceeding contributionpeer-review

9 Citations (Scopus)


We study the natural problem of secure n-party computation (in the passive, computationally unbounded attack model) of the n-product function f G(x1,...,xn) = x1 · x 2 ⋯ 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(t 2t+1)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 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.

Original languageEnglish
Title of host publicationAdvances in Cryptology - CRYPTO 2007 - 27th Annual International Cryptology Conference, Proceedings
EditorsAlfred Menezes
Place of PublicationBerlin
PublisherSpringer, Springer Nature
Number of pages22
Volume4622 LNCS
ISBN (Print)9783540741428
Publication statusPublished - 2007
Event27th Annual International Cryptology Conference, CRYPTO 2007 - Santa Barbara, CA, United States
Duration: 19 Aug 200723 Aug 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4622 LNCS
ISSN (Print)03029743
ISSN (Electronic)16113349


Other27th Annual International Cryptology Conference, CRYPTO 2007
Country/TerritoryUnited States
CitySanta Barbara, CA


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