HMQV is one of the most efficient (provably secure) authenticated key-exchange protocols based on public-key cryptography, and is widely standardized. In spite of its seemingly conceptual simplicity, the HMQV protocol was actually very delicately designed. The provable security of HMQV is conducted in the Canetti-Krawczyk framework (CK-framework, in short), which is quite complicated and lengthy with many subtleties actually buried there. However, lacking a full recognition of the precise yet subtle interplay between HMQV protocol structure and provable security can cause misunderstanding of the HMQV design, and can cause potential flawed design and analysis of HMQV protocol variants. In this work, we explicitly make clear the interplay between HMQV protocol structure and provable security, showing the delicate design of HMQV. We then re-examine the security model and analysis of a recently proposed HMQV protocol variant, specifically, the FHMQV protocol proposed by Sarr et al. in . We clarify the relationship between the traditional CK-framework and the CK-FHMQV security model proposed for FHMQV, and show that CK-HMQV and CK-FHMQV are incomparable. Finally, we make a careful investigation of the CDH-based analysis of FHMQV in the CK-FHMQV model, which was considered to be one of the salient advantages of FHMQV. We identify that the CDH-based security analysis of FHMQV is actually flawed. The flaws identified in the security proof of FHMQV just stem from lacking a full realization of the precise yet subtle interplay, as clarified in this work, between HMQV protocol structure and provable security.