## 5 Security Analysis of LDAKM-EIoT
** Let us assume an adversary \(\mathcal{A}\) intercepts the messages \(Msg_{1}=\{TID_{D_{i}}\), \(M_{1}\), \(M_{2}\), \(T_{1}\}\), \(Msg_{2}\)\(=\{TID_{D_{i}}^{\prime}\), \(M_{3}\), \(M_{4}\), \(T_{2}\}\), \(Msg_{3}=\{M_{5}\), \(M_{6}\), \(M_{7}\), \(T_{3}\}\), and \(Msg_{4}=\{M_{8}\), \(\alpha\), \(\beta\), \(M_{9}\), \(M_{10}\), \(T_{3}\), \(T_{4}\}\) during the \"authentication and key management phase\" between \(D_{i}\) and \(CS_{j}\) via the \(EN\). We used random nonces and current timestamps in the construction of messages, which helped to generate dynamic and unique messages for different sessions. Moreover, the identities \(ID_{D_{i}}\), \(ID_{EN}\), and \(ID_{CS_{j}}\) are not transmitted in the plaintext format during transmission. This helps to attain both the anonymity and untraceability objectives of LDAKM-EIoT. **Proposition 7**: _LDAKM-EIoT provides protection against IoT device physical capture attack._  
**Proof. ** Each IoT device stores the credentials \(\{RID_{D_{i}}\), \(TID_{D_{i}}\), \(TC_{D_{i}}\), \(AD_{i}\), \(h(\cdot)\}\), which are used for the \"authentication and key management\" purposes with other network communicating entities. The protection against IoT device physical capture attack is a very essential security requirement [50; 51]. Suppose \(n_{c}\) IoT devices are physically captured by an adversary \(\mathcal{A}\). We assess the IoT device physical capture attack as the fraction of total secure communications that are compromised from the capturing of \(n_{c}\) IoT devices not including the communication in which the \"compromised IoT devices\" are straightforwardly extended. For example, one can estimate the probability of \(\mathcal{A}\)'s ability to decrypt the secure communication between a cloud server \(CS_{j}\) and a non-compromised IoT device \(D_{i}\) when \(n_{c}\) IoT devices are already physically stolen and compromised. Suppose this probability is represented by \(P_{e}(n_{c})\). If \(P_{e}(n_{c})=0\), an \"authentication and key management\" protocol is termed as \"unconditionally secure against IoT device physical capture attack\". From a captured IoT device \(D_{i}\), \(\mathcal{A}\) will have the extracted credentials \(\{RID_{D_{i}}\), \(TID_{D_{i}}\), \(TC_{D_{i}}\), \(AD_{i}\), \(h(\cdot)\}\) along with the secret pairwise session key \(SK_{CS_{j}-D_{i}}\) shared between \(D_{i}\) and \(CS_{j}\) from its memory. However, it is very important to notice that all \(RID_{D_{i}}\), \(TID_{D_{i}}\), \(TC_{D_{i}}\), \(AD_{i}\), \(RTS_{D_{i}}\), and \(K_{EN-D_{i}}\) are distinct for all installed IoT devices. Thus, the stolen \(D_{i}\) by \(\mathcal{A}\) can only help to find the secret keys between that \(D_{i}\) and \(CS_{j}\). Furthermore, this results in all other secret keys between that \(CS_{j}\) and other non-compromised IoT devices \(D_{i}\) to be not still revealed. Therefore, compromising an IoT device will not cause the compromise of the entire communication, and we get the secure communications among the cloud server and other non-compromised IoT devices. This means that LDAKM-EIoT is unconditionally secure against IoT device physical capture attack.