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 f G (x 1,...,x n ) : = x 1 ·...·x n where each participant P i holds an input x i 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 f G . 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 f G 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 \(\lceil \frac{n}{2} \rceil - 1\).
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Qiao, Y., Tartary, C. (2008). Counting Method for Multi-party Computation over Non-abelian Groups. In: Franklin, M.K., Hui, L.C.K., Wong, D.S. (eds) Cryptology and Network Security. CANS 2008. Lecture Notes in Computer Science, vol 5339. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89641-8_12
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DOI: https://doi.org/10.1007/978-3-540-89641-8_12
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