Completeness theorems for non-cryptographic fault-tolerant distributed computation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
Robustness for Free in Unconditional Multi-party Computation
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
New Public Key Cryptosystem Using Finite Non Abelian Groups
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
How to Solve any Protocol Problem - An Efficiency Improvement
CRYPTO '87 A Conference on the Theory and Applications of Cryptographic Techniques on Advances in Cryptology
Efficient Secure Multi-party Computation
ASIACRYPT '00 Proceedings of the 6th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Protocols for secure computations
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
General secure multi-party computation from any linear secret-sharing scheme
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
On secure multi-party computation in black-box groups
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
Robust multiparty computation with linear communication complexity
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
Scalable secure multiparty computation
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
Active security in multiparty computation over black-box groups
SCN'12 Proceedings of the 8th international conference on Security and Cryptography for Networks
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Recently, Desmedt et al. studied the problem of achieving securen -party computation over non-Abelian groups. Theyconsidered the passive adversary model and they assumed that theparties were only allowed to perform black-box operations over thefinite group G . They showed three results for then -product function f G (x 1 ,...,x n ) : =x 1 ·x 2 ·...·x n ,where the input of party P i isx i ∈ G for i ∈ {1,...,n }. First, if $t \geq \lceil \tfrac{n}{2}\rceil$ then it is impossible to have a t -private protocolcomputing f G . Second, theydemonstrated that one could t -privately compute f G for any $t \leq \lceil \tfrac{n}{2} \rceil -1$ in exponential communication cost. Third, they constructed arandomized algorithm with O (n t 2) communication complexity for anyt≤n/2.948 In this paper, we extend these results in two directions. First,we use percolation theory to show that for any fixedε 0, one can design a randomized algorithm forany $t\leq \frac{n}{2+\epsilon}$ using O (n 3) communication complexity, thus nearly matching theknown upper bound $\lceil \tfrac{n}{2} \rceil - 1$. This is thefirst time that percolation theory is used for multipartycomputation. Second, we exhibit a deterministic construction havingpolynomial communication cost for any t =O (n 1-ε ) (again forany fixed ε 0). Our results extend to the moregeneral 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 moreinput values.