The Exponential Challenge Response (XRC) and Dual Exponential Challenge Response (DCR) signature schemes are the building blocks of the HMQV protocol. We pro-pose a complementary analysis of these schemes; on the basis of this analysis we show how impersonation and man in the middle attacks can be mounted against the HMQV protocol when some session specific information leakages happen. We define the Full Exponential Challenge Response (FXRC) and Full Dual Exponential Challenge Response (FDCR) sig-nature schemes; using these schemes we propose the Fully Hashed MQV protocol (with security arguments), which preserves the remarkable performance of the (H)MQV protocols and resists the attacks we present.