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The integration of IoT with the cloud infrastructure is essential for designing smart applications. However, such integration may lead to security issues. Authentication and session key establishment is an essential security requirement for secure communication between IoT devices and cloud servers. For evaluating authentication key agreement schemes, the extended Canetti–Krawczyk (eCK) adversary model is regarded to be a more strict and relevant adversary model. Many schemes for authenticated key exchange between IoT devices and cloud servers have been proposed in the literature but have been assessed under Dolev and Yoa (DY) adversary model. Recently, Rostampour et al. introduced an ECC-based approach for enabling authentication between IoT devices and cloud servers that is secure and robust to various attacks under the Dolev and Yoa adversary model. In this paper, a detailed review and the automated security verification of the Rostampour et al. scheme are carried out under the eCK adversary model using Scyther-Compromise. The validation indicates that the scheme is not secure and is susceptible to various attacks under the eCK adversary model. To overcome the limitation of the Rostampour et al. scheme, a design of an ECC-based scheme for authentication between IoT devices and cloud servers under the eCK adversary model is proposed. The Scyther verification indicates that the scheme is safe under the eCK adversary model. The soundness of the correctness of the proposed scheme has been analyzed using BAN logic. Comparative analysis indicates that the scheme is resilient under the eCK adversary model with an energy overhead of 278.16 mJ for a resource constraint IoT device and a communication overhead of 1,408 bits.