It is well-known that n players connected only by pairwise secure channels can achieve multi-party computation secure against an active adversary if and only if – t < n/2 of the players are corrupted with respect to computational security, or – t < n/3 of the players are corrupted with respect to unconditional security. In this paper we examine to what extent it is possible to achieve conditional (such as computational) security based on a given intractability assumption with respect to some number T of corrupted players while simultaneously achieving unconditional security with respect to a smaller threshold t ≤ T. In such a model, given that the intractability assumption cannot be broken by the adversary, the protocol is secure against T corrupted players. But even if it is able to break it, the adversary is still required to corrupt more than t players in order to make the protocol fail. For an even more general model involving three different thresholds tp, tσ, and T,...