Round Efficiency of Multi-party Computation with a Dishonest Majority

We consider the round complexity of multi-party computation in the presence of a static adversary who controls a majority of the parties. Here, n players wish to securely compute some functionality and up to n − 1 of these players may be arbitrarily malicious. Previous protocols for this setting (when a broadcast channel is available) require O(n) rounds. We present two protocols with improved round complexity: The first assumes only the existence of trapdoor permutations and dense cryptosystems, and achieves round complexity O(log n) based on a proof scheduling technique of Chor and Rabin [[13]]; the second requires a stronger hardness assumption (along with the non-black-box techniques of Barak [[2]]) and achieves O(1) round complexity.