Blake-Wilson, Johnson & Menezes protocol revisited

In this paper, we investigate the famous Blake-Wilson, Johnson & Menezes (BJM) authenticated key exchange protocols. We observe that the BJM model fails to model the adversary's capability in the public setting well. We modify the BJM model by providing it with a new Register query and a modified Corrupt query. This way, we bring the BJM model further to practice. Moreover, our modification has a significant impact on the security proof of the BJM protocols. Specifically, the security proofs using CDH assumption will no longer work in the modified BJM model. With some modifications, we show that the BJM protocols are secure in the modified BJM model under the gap Diffie-Hellman assumption (GDH).