Automated verification has become an essential part in the security evaluation of cryptographic protocols. Recently, there has been a considerable effort to lift the theory and tool support that existed for reachability properties to the more complex case of equivalence properties. In this work we contribute both to the theory and practice of this verification problem. We establish new complexity results for static equivalence, trace equivalence and labelled bisimilarity and provide a decision procedure for these equivalences in the case of a bounded number of sessions. We implemented the procedure in a new tool, DEEPSEC. We showed through extensive experiments that it is significantly more efficient than other similar tools, while at the same time raises the scope of the protocols that can be analysed.