The Canetti-Krawczyk (CK) and extended Canetti-Krawczyk (eCK) security models, are widely used to provide security arguments for key agreement protocols. We discuss security shades in the (e)CK models, and some practical attacks unconsidered in (e)CK-security arguments. We propose a strong security model which encompasses the eCK one. We also propose a new protocol, called Strengthened MQV (SMQV), which in addition to provide the same efficiency as the (H)MQV protocols, is particularly suited for distributed implementations wherein a tamper-proof device is used to store long-lived keys, while session keys are used on an untrusted host machine. The SMQV protocol meets our security definition under the Gap DiffieHellman assumption and the Random Oracle model.

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