Everybody has its own constraint satisfaction problem, private concerns that owners prefer to keep as secret as possible. Resources may be shared and cause the need for cooperation. Here we consider the case where privacy is an overwhelming requirement and we assume that a majority of the participants are incorruptible. Namely, given n participants, at least an n/2 unknown subset of them are trustworthy and not corrupted or controlled by attackers. This is a common assumption in cryptographic multi-party computations where techniques exploiting such assumptions are known as threshold schemes. This work shows how a random solution of the described problem can be offered with a secure protocol that does not reveal anything except the existence of the solution and tells each participant the valuations corresponding to its subproblem. The technique is based on the properties of the recent Paillier cryptosystem and needs no external arbiter
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