How to play any mental game, or a completeness theorem for protocols with honest majority

Permission to copy without fee all or part of this material is granted provided that the copies are not made or Idistributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machimery. To copy otherwise, or to republish, requires a fee and/or specfic permission. correctly run a given Turing machine hi on these 2;‘s while keeping the maximum possible pniracy about them. That is, they want to compute Y~(~l,..., 2,) without revealing more about the Zi’s than it is already contained in the value y itself. For instance, if M computes the sum of the q’s, every single player should not be able to learn more than the sum of the inputs of the other parties. Here A4 ma.y very well be a probabilistic Turing machine. In this case, all playen want to agree on a single string y, selected with the right probability distribution, as M’s output.