Client-to-client password-authenticated key agreement (C2C-PAKA) protocol deals with the authenticated key agreement process between two clients of different realms, who only share their passwords with their own servers. Recently, Byun et al.  proposed an efficient C2C-PAKA protocol and carried a claimed proof of security in a formal model of communication and adversarial capabilities. In this paper, we show that the protocol is insecure against
attack and the claim of provable security is seriously incorrect. To draw lessons from these results, we revealed fatal flaws in Byun et al.’s security model and their proof of security. Then, we modify formal security model and corresponding security definitions. In addition, a new
C2C-PAKA protocol is presented with security proof.