Completeness theorems for non-cryptographic fault-tolerant distributed computation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
A threshold of ln n for approximating set cover (preliminary version)
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
PODC '97 Proceedings of the sixteenth annual ACM symposium on Principles of distributed computing
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Robustness for Free in Unconditional Multi-party Computation
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
CRYPTO '91 Proceedings of the 11th Annual International Cryptology Conference on Advances in Cryptology
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A secure threshold protocol for n players tolerating an adversary structure A is feasible iff maxa驴A |a| n/c, where c = 2 or c = 3 depending on the adversary being eavesdropping (passive) or Byzantine (active) respectively [1]. However, there are situations where the threshold protocol 驴 for n players tolerating an adversary structure A may not be feasible but by letting each player Pi to act for a number of similar players, say wi, a new secure threshold protocol 驴驴 tolerating A may be devised. Note that the new protocol 驴驴 has N = 驴i=1n wi players and works with the same adversary structure A used in 驴. The integer quantities wi's are called weights and we are interested in computing wi's so that 1. 驴驴 tolerates A even if 驴 does not tolerate A. 2. N = 驴i=1n wi is minimum. Since the best known secure threshold protocol over N players has a communication complexity of O(mN2 lg |F|) bits [9], where m is the number of multiplication gates in the arithmetic circuit, over the finite field F, that describes the functionality of the protocol, it is evident that the weights assigned to the players have a direct influence on the complexity of the resulting secure weighted threshold protocol. In this work, we focus on computing the optimum N. We show that computing the optimum N is NP-Hard. Furthermore, we prove that the above problem of computing the optimum N is inapproximable within (1 - 驴) ln (|A|/c) + ln((|A|/c)(1-驴))-1/N* (c - 1), for any 驴 0 (and hence inapproximable within 驴 (lg |A|)), unless NP 驴 DTIME(nlog log n), where N* is the optimum solution.