Anonymity, as an instance of information hiding, is one of the security properties intensively studied nowadays due to its applications to various fields such as e-voting, e-commerce, e-mail, e -cash, and so on. In this paper we study the decidability and complexity status of the anonymity property in security protocols. We show that anonymity is undecidable for unrestricted security protocols, is NEXPTIME-complete for bounded security protocols, and it is NP-complete for 1-session bounded security protocols. In order to reach these objectives, an epistemic language and logic to reason about anonymity properties for security protocols under an active intruder, are provided. Agent states are endowed with facts derived from actions performed by agents in protocol executions, and an inference system is provided. To define anonymity, an observational equivalence is used, which is shown to be decidable in deterministic polynomial time.

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