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The traditional approach to formalizing ideal-model based definitions of security for multi-party protocols model adversaries (both real and ideal) as centralized entities that control all parties that deviate from the protocol. While this centralized-adversary modeling suffices for capturing basic security properties such as secrecy of local inputs and correctness of outputs against coordinated attacks, it turns out to be inadequate for capturing security properties that involve restricting the sharing of information between separate adversarial entities. Indeed, to capture collusion-freeness, Alwen et.al. propose a new ideal-model based definitional framework that involves a de-centralized adversary.
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