## 5 Security Analysis of LDAKM-EIoT  
In this section, we provide the security analysis of LDAKM-EIoT in Propositions 1-7, which prove its robustness against the following possible attacks. Proposition 1: \(LDAKM\)-EIoT is resilient against replay attack. Proof: In LDAKM-EIoT, we utilized various current timestamp values for the communicating entities (\(D_{i}\), \(EN\), and \(CS_{j}\)). In the each exchanged message of LDAKM-EIoT, we used the maximum transmission delay \(\Delta\ T\) factor, which is a small value. Consequently, an adversary \(\mathcal{A}\) can not gain any advantage in replaying the old transmitted messages, which are used in the authentication and key management procedure among \(D_{i}\), \(EN\), and \(CS_{j}\) within \(\Delta\ T\). Thus, LDAKM-EIoT is secure against replay attack. Proposition 2: \(LDAKM\)-EIoT is resilient against man-in-the-middle attack (MITM). Proof: Suppose an adversary \(\mathcal{A}\) seizes an authentication request message \(\{TID_{D_{i}},M_{1},M_{2},T_{1}\}\) exchanged between \(D_{i}\) and \(EN\) and further tries to update this message in such a way that it looks similar to a genuine authentication message, as \(\{TID_{D_{i}}^{a},M_{1}^{a},M_{2}^{a},T_{1}^{a}\}\) by using the parameters such as \(M_{1}^{a}=AD_{i}\oplus r_{d_{i}}^{a}\) and \(M_{2}^{a}=h(AD_{i}||T_{1}^{a}||\ TID_{D_{i}}^{a}||\ RID_{D_{i}}||\ TCD_{i}||\ TCD_{i}||\)\(r_{d_{i}})\). The values of the used variables are \(RID_{D_{i}}=h(n||ID_{D_{i}})\), \(TC_{D_{i}}=h(RID_{D_{i}}||ID_{EN}||ID_{TA}||\)\(n||RTS_{D_{i}})\), \(AD_{i}=h(K_{EN-D_{i}}||\)\(ID_{D_{i}}||\)\(RTS_{D_{i}})\) where \(RTS_{D_{i}}\) is the registration timestamp for \(D_{i}\) and \(K_{EN-D_{i}}\) is the shared secret key of IoT device and edge node generated by the \(TA\). To launch this attack, \(\mathcal{A}\) can initiate the generation of random once \(r_{d_{i}}^{a}\) and current timestamp \(T_{1}^{a}\). However, without obtaining the information about \"long term secret\" \(RID_{D_{i}}\), \(ID_{EN}\), \(ID_{TA}\), \(n\), and \(K_{EN-D_{i}}\), \(\mathcal{A}\) cannot re-create another valid authentication request message \(\{TID_{D_{i}}^{a},M_{1}^{a},M_{2}^{a},T_{1}^{a}\}\). By using a similar approach, we can also explain that other messages cannot be recomputed by \(\mathcal{A}\), which are used in the \"authentication and key management\" phase without the \"long term secrets\" used by \(D_{i}\), \(EN\), and \(CS_{j}\). This proves the resilience of LDAKM-EIoT against MITM. Proposition 3: \(LDAKM\)-EIoT provides protection against different impersonation attacks. **Proof. ** Suppose an adversary \(\mathcal{A}\) attempts to create a valid authentication request message on behalf of a communicating party (for example, \(D_{i}\)) after obtaining \(D_{i}\)'s authentication request \(M_{S}g_{1}=\{TID_{D_{i}},M_{1},M_{2},T_{1}\}\) sent towards the \(EN\) by \(D_{i}\), where \(M_{1}=AD_{i}\oplus r_{d_{i}}\) and \(M_{2}=h(AD_{i}||T_{1}||\)\(TID_{D_{i}}||RD_{D_{i}}||\)\(TCD_{D_{i}}||r_{d_{i}})\). The values of the used variables are \(RID_{D_{i}}=h(n||ID_{D_{i}})\), \(TC_{D_{i}}=h(RID_{D_{i}}||ID_{EN}||ID_{TA}||n||RTS_{D_{i}})\), and \(AD_{i}=h(K_{EN-D_{i}}||\)\(ID_{D_{i}}||ID_{EN}||\)\(RTS_{D_{i}})\) where \(RTS_{D_{i}}\) is the registration timestamp for \(D_{i}\) and \(K_{EN-D_{i}}\) is the shared secret key of IoT device and edge node generated by the \(TA\). Note that \(Msg_{1}\) uses \(M_{1}\) and \(M_{2}\), which are computed using the \"long term secrets\"(\(RID_{D_{i}}\), \(ID_{EN}\), \(ID_{TA}\), \(n\), \(K_{EN-D_{i}}\)) and also the \"short term secrets\" (random nonce \(r_{d_{i}}\)). Without knowing these secret values, \(\mathcal{A}\) is not able to create a valid authentication request message on behalf of the genuine IoT device \(D_{i}\). Hence, we can say that LDAKM-EIoT is resilient against IoT device impersonation attack.