LaMacchia, Lauter and Mityagin  proposed a novel security definition for authenticate key exchange (AKE) that gives an adversary the power to obtain ephemeral information regarding a target test session. To demonstrate feasibility of secure protocols in the new definition, henceforth called eCK, the authors described a protocol called NAXOS. NAXOS combines an ephemeral private key x with a static private key a to generate an ephemeral public key X (more precisely in what we call the NAXOS' approach X=gH(x,a)). Thus no one is able to query the discrete logarithm of X without knowing both the ephemeral and static private keys. This idea is crucial in the security argument to guard against leaked ephemeral secrets belonging to the test session. Another important assumption is the gap assumption that allows the protocol to remain secure even in the presence of malicious insiders. Both ideas have been successfully used in creating various protocols secure in the eCK model. In this paper, we construct two eCK-secure protocols without the above mentioned ideas. KFU1 is secure under the GDH assumption without using the NAXOS' approach. KFU2 builds upon KFU1 and drops the gap requirement, thus it is secure under the CDH assumption. Efficiency and security of the proposed protocols are comparable to the well-known HMQV  protocol. Furthermore, unlike HMQV and NAXOS the use of the random oracle in KFU1 and KFU2 is restricted to the key derivation function making them more suitable for practical applications.
2012 The Institute of Electronics, Information and Communication Engineers