## 1 Introduction
### Related Works  
Many authentication and key-agreement schemes have been proposed for e-health systems. In 2010, [PERSON] et al. [3] proposed an authentication scheme with contract-oriented hybrid public-key infrastructure based on the HIPAA specification design electronic medical method in [4]. In 2012, [PERSON] and [PERSON] [5] pointed out shortcomings of previous schemes, including that of [PERSON] et al., including the fact that without patient authorization, medical service providers can access patient information without restriction. They also revealed that the previous schemes in [4,6] raise the problem that a round-oriented health smartcard cannot verify the need for health information in multiple places at once, and so proposed a contract-oriented CA-based electronic health service system. In 2014, [PERSON] and [PERSON] [7] developed a CA-based authentication scheme for e-healthcare systems. Their developed scheme uses the existing PKI and public-key certificate to set up a contract-based system with a medical-center server located at hospitals, and is compliant with HIPAA privacy/security regulations.  
In 2019, [PERSON] et al. [8] proposed a lightweight authentication and ownership transfer protocol for e-health systems in the context of IoT. Their protocol not only provides authentication and key agreement but also satisfies access control and preserves the privacy of doctors and patients. In 2020, [PERSON] et al. [9] proposed a new biometric-based key-management scheme to facilitate remote-access authorization anytime and anywhere. In their scheme, patients and doctors realize mutual authentication by using their biometric information through real-time video-communication technology. Additionally, their scheme also provided a safety channel in delivering their access authorization and secret data between patient and doctor. In the same year, [PERSON] et al. [10] presented a robust authentication and access-control protocol for securing wireless healthcare sensor networks. Their proposed scheme employed three factors, including smart-card authentication, biometric authentication and password authentication, to overcome the pitfalls in previous schemes [11,12]. Additionally, [PERSON] et al. [13] proposed a hash-chain-based authentication scheme for wireless body-area networks in healthcare IoT. Their scheme provides perfect forward secrecy and resists potential attacks, including key-compromise impersonation attacks and known session-specific temporary information attacks. In 2021, [PERSON] et al. [14] considered the entire process of data from data generation through transmission by wearable devices to mobile devices and then to a medical center server, and developed an efficient authentication scheme based on extended chaotic maps. Their scheme reduces the amount of computation on wearable devices, while also taking advantage of the immutability of the blockchain to ensure that data cannot be tampered with, enhancing security requirements. In 2022, [PERSON] et al. [15] performed cryptanalysis of the scheme of [PERSON] et al. [8] and stated that it is insecure against some possible attacks. They also proposed a lightweight authentication scheme for smart healthcare applications in IoMT as an alternative. At the same year, [PERSON] and [PERSON] [16] proposed an effective multiserver biometric-authentication scheme based on extended chaotic maps for TMIS to overcome the weaknesses of [PERSON] et al.'s scheme [17] in terms of authentication and revocation. In 2022, [PERSON] et al. [18] introduced a new method of high-speed symmetric encryption using the Chebyshev chaotic map and developed a multiserver/multiclient authentication scheme using this symmetric map to overcome the weaknesses of [PERSON] et al.'s scheme [19] in terms of revocation and user anonymity.