## 5 Security Analysis of LDAKM-EIoT
In the same way, it can also be proven that LDAKM-EIoT provides protection against edge node and cloud server impersonation attacks as the creation of other messages \(Msg_{2}\), \(Msg_{3}\), and \(Msg_{4}\) employs both \"long term\" and \"short term\" secrets. Therefore, LDAKM-EIoT is secure against impersonation attacks. **Proposition 4**: _LDAKM-EIoT protects Ephemeral secret leakage (ESL) attack._  
**Proof. ** In LDAKM-EIoT, the session key is computed by a genuine cloud server \(CS_{j}\) and the accessed \(D_{i}\) during the the authentication and key management phase as \(SK_{CS_{j}-D_{i}}=h(TID_{D_{i}}||\)\(T_{3}||\)\(h(r_{en_{1}}\)\(||T_{1}\)\(||AD_{i}\)\(||ID_{D_{i}}\)\(||r_{d_{i}}\)\(||TC_{D_{i}})\)\(||h(r_{CS_{j}}\)\(||ACS_{j}\)\(||ID_{CS_{j}}\)\(||T_{3}))\). Here, \(AD_{i}=h(K_{EN-D_{i}}||\)\(ID_{D_{i}}||ID_{EN}||\)\(RTS_{D_{i}})\) where \(K_{EN-D_{i}}\) is the shared secret key of the IoT device and edge node generated by the \(TA\) and \(K_{EN-CS_{j}}\) is the shared secret key of the \(EN\) and cloud server generated by the \(TA\). Further, the identities of the IoT device, \(ID_{D_{i}}\), the identity of the edge node \(ID_{EN}\), and the identity of the cloud server \(ID_{CS_{j}}\) are also utilized. It is understandable that the \"session key\" is the combination of both the session temporary (ephemeral) information, also called \"short term secrets\" (different random nonces and timestamps), as well as the \"long term secrets\" (different secret keys and identities). Thus, the session key can only be disclosed if \(\mathcal{A}\) compromises both the session temporary and long term secrets. Moreover, as different random nonces and timestamps are used in the computation of the session keys between \(D_{i}\) and \(CS_{j}\) in distinct sessions, even if a session key is disclosed for a specific session, it will not result in computing the session keys of other sessions because of the combination of short and long term secrets. Thus, LDAKM-EIoT provides protection against session temporary information attack, and it also preserves the \"perfect forward secrecy\" goal. Therefore, LDAKM-EIoT is resilient against \"ESL attack\". Consequently, LDAKM-EIoT preserves the session key security under the \"CK adversary model\" [40]. **Proposition 5**: _LDAKM-EIoT is resilient against privileged-insider attack._  
**Proof. ** The privileged-insider user of a cloud server, say \(\mathcal{A}\), knows the registration information \(\{RID_{CS_{j}},ACS_{j},ID_{CS_{j}},h(\cdot)\}\). However, he/she cannot calculate the \"session key\" on behalf of the cloud server as the session key utilizes the credentials that are only known to the IoT devices or the edge node. The \"session key\" calculated by the legitimate cloud server is \(SK_{CS_{j}-D_{i}}=h(TID_{D_{i}}||\)\(T_{3}||\)\(h(r_{en_{1}}\)\(||T_{1}\)\(||AD_{i}\)\(||ID_{D_{i}}\)\(||r_{d_{i}}\)\(||TC_{D_{j}})\)\(||h(r_{CS_{j}}\)\(||ACS_{j}\)\(||ID_{CS_{j}}\)\(||T_{3}))\). Moreover, \(r_{en_{1}}\) is the random nonce of the \(EN\). The identities of the IoT device, \(ID_{D_{i}}\), the identity of the edge node \(ID_{EN}\), and identity of the cloud server \(ID_{CS_{j}}\) are also utilized in \(SK_{CS_{j}-D_{i}}\). The privileged-insider user of the cloud server does not have any information about \(ID_{D_{i}}\), \(ID_{EN}\), and \(K_{EN-D_{i}}\). Therefore, \(\mathcal{A}\) cannot calculate the \"session key\" on behalf of the cloud server, as well as he/she cannot impersonate the cloud server, as explained in Proposition 3. Hence, LDAKM-EIoT is resilient against \"privileged-insider attack\". **Proposition 6**: _LDAKM-EIoT preserves the anonymity and untraceability properties._  
**Proof.