In simulation based proof of cryptographic scheme, if k parties are invovled in the scheme, then k simulators should be constructed for all the parties？
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If in the threat model, these parties are controlled by the same adversary, i.e. these participants are colluding with each other, then we only need a simulator. If each party is corrupted by a different adversary, then we must construct a different simulator for each party. The analysis of the latter case will be much more complicated and will be rarely seen because the adversary is more powerful in the previous model and we usually prefer to analyze the former case.
