We present a new automatic technique for proving strong secrecy for security protocols. Strong secrecy means that an adversary cannot see any difference when the value of the secret changes. Our technique relies on an automatic translation of the protocol into Horn clauses, and a resolution algorithm on the clauses. It requires important extensions with respect to previous work for the proof of (standard) secrecy and authenticity. This technique can handle a wide range of cryptographic primitives, and yields proofs valid for an unbounded number of sessions and an unbounded message space; it is also flexible and efficient. We have proved its correctness, implemented it, and tested it on several examples of protocols including JFK [ACM Conference on Computer and Communications Security (CCS'02)] (a proposed replacement for IPsec).