ASIACRYPT 2008: Advances in Cryptology - ASIACRYPT 2008 pp 37-53

# Graph Design for Secure Multiparty Computation over Non-Abelian Groups

• Xiaoming Sun
• Andrew Chi-Chih Yao
• Christophe Tartary
Conference paper

DOI: 10.1007/978-3-540-89255-7_3

Part of the Lecture Notes in Computer Science book series (LNCS, volume 5350)
Cite this paper as:
Sun X., Yao A.CC., Tartary C. (2008) Graph Design for Secure Multiparty Computation over Non-Abelian Groups. In: Pieprzyk J. (eds) Advances in Cryptology - ASIACRYPT 2008. ASIACRYPT 2008. Lecture Notes in Computer Science, vol 5350. Springer, Berlin, Heidelberg

## Abstract

Recently, Desmedt et al. studied the problem of achieving secure n-party computation over non-Abelian groups. They considered the passive adversary model and they assumed that the parties were only allowed to perform black-box operations over the finite group G. They showed three results for the n-product function fG(x1,...,xn) : = x1 ·x2 ·...·xn, where the input of party Pi is xi ∈ G for i ∈ {1,...,n}. First, if $$t \geq \lceil \tfrac{n}{2} \rceil$$ then it is impossible to have a t-private protocol computing fG. Second, they demonstrated that one could t-privately compute fG for any $$t \leq \lceil \tfrac{n}{2} \rceil - 1$$ in exponential communication cost. Third, they constructed a randomized algorithm with O(nt2) communication complexity for any $$t < \tfrac{n}{2.948}$$.

In this paper, we extend these results in two directions. First, we use percolation theory to show that for any fixed ε> 0, one can design a randomized algorithm for any $$t\leq \frac{n}{2+\epsilon}$$ using O(n3) communication complexity, thus nearly matching the known upper bound $$\lceil \tfrac{n}{2} \rceil - 1$$. This is the first time that percolation theory is used for multiparty computation. Second, we exhibit a deterministic construction having polynomial communication cost for any t = O(n1 − ε) (again for any fixed ε> 0). Our results extend to the more general function $$\widetilde{f}_{G}(x_{1},\ldots,x_{m}) := x_{1} \cdot x_{2} \cdot \ldots \cdot x_{m}$$ where m ≥ n and each of the n parties holds one or more input values.

### Keywords

Multiparty Computation Passive Adversary Non-Abelian Groups Graph Coloring Percolation Theory

## Authors and Affiliations

• Xiaoming Sun
• 1
• Andrew Chi-Chih Yao
• 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