A framework consisting of the Protocol Derivation System (PDS) and the Protocol Composition Logic (PCL) has been recently proposed by Datta et al. for the design and analysis of a secure composition of cryptographic protocols. However, the PDS in this proposed framework can only be used for the protocols of the Station-to-Station family, which are signature-based authenticated Diffie–Hellman key exchange protocols. In this paper, the PDS is extended to support key exchange protocols using a trusted third party and an encryption-based authentication such as those in the Needham–Schroeder family. This is achieved by means of adding new components, refinements, and transformations to the PDS. In addition, the PCL is applied to prove the correctness of the derived protocols. Then, the derivation graph of the Needham−Schroeder family is developed by using the extended PDS. Finally, the derivations and proofs of the protocols in the Needham–Schroeder family are shown in this paper. Copyright © 2012 John Wiley & Sons, Ltd.