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Due to the limitations of symmetric-key techniques, authentication and key agreement (AKA) protocols based on public-key techniques have attracted much attention, providing secure access and communication mechanism for various application environments. Among these public-key techniques used for AKA protocols, chaotic-map is more effective than scalar multiplication and modular exponentiation, and it offers a list of desirable cryptographic properties such as un-predictability, un-repeatability, un-certainty, and higher efficiency than scalar multiplication and modular exponentiation. Furthermore, it is usually believed that three-factor AKA protocols can achieve a higher security level than single- and two-factor protocols. However, none of existing three-factor AKA protocols can meet all security requirements. One of the most prevalent problems is how to balance security and usability, and particularly how to …