Deciding key cycles for security protocols

Many recent results are concerned with interpreting proofs of security done in symbolic models in the more detailed models of computational cryptography. In the case of symmetric encryption, these results stringently demand that no key cycle (e.g. {k}(k)) can be produced during the execution of protocols. While security properties like secrecy or authentication have been proved decidable for many interesting classes of protocols, the automatic detection of key cycles has not been studied so far. In this paper, we prove that deciding the existence of key-cycles is NP-complete for a bounded number of sessions. Next, we observe that the techniques that we use are of more general interest and apply them to reprove the decidability of a significant existing fragment of protocols with timestamps.