Authentication and secrecy have been widely investigated in security protocols. They are closely related to each other and variants of definitions have been proposed, which focus on the concepts of corresponding assertion and key distribution. This paper proposes an onthe-fly model checking method based on the pushdown system to verify the authentication of recursive protocols with an unbounded number of principals. By experiments of the Maude implementation, we find the recursive authentication protocol, which was verified in the sense of (weak) key distribution, has a flaw in the sense of correspondence assertion.