## 2 Review of [PERSON] et al.'s Scheme
### Authentication and Key Agreement Phase between UE and SN  
This phase is mutual authentication and key agreement process between UE and SN, which is performed through a public channel. A detailed description of the process is presented as follows.  
1. \(UE_{i}\) pre-computes \(T_{U_{i}}(K_{sn}||ID_{SN_{j}})\) mod \(p\) and \(T_{u_{i}}(K_{ue}||ID_{UE_{i}})\) mod \(p\). Then, \(UE_{i}\) generates \(x_{i}\) and computes \(T_{x_{i}}(K_{sn}||ID_{SN_{j}})\) mod \(p\), \(K_{1}=T_{x_{i}}(T_{S_{i}}(K_{sn}||ID_{SN_{j}})\) mod \(p)\) mod \(p\), \(K_{2}=T_{u_{i}}(T_{S_{j}}(K_{sn}||ID_{SN_{j}})\) mod \(p)\) mod \(p\), \(MAC_{1}=H_{1}(K_{2},ID_{UE_{i}},ID_{SN_{j}},T_{u_{i}}\)\((K_{ue_{i}}||ID_{UE_{i}})\) mod \(p||T_{u_{i}}(K_{ue}||ID_{UE_{i}})\) mod \(p||T_{u_{i}}(K_{su}||ID_{SN_{j}})\) mod \(p)\). \(UE_{i}\) encrypts \(E_{1}=ENK_{K_{1}}(ID_{UE_{i}}||T_{U_{i}}(K_{sn}||ID_{SN_{j}})\) mod \(p||T_{u_{i}}(K_{ue}||ID_{UE_{i}})\) mod \(p||T_{u_{i}}(K_{ue}||ID_{UE_{i}})\) mod \(p)\) by the secret parameter \(K_{1}\) and sends the access request including \(\{ID_{SN_{j}},T_{x_{i}}\)\((K_{sn}||ID_{SN_{j}})\) mod \(p,E_{1},MAC_{1}\}\) to \(SN_{j}\).
2. After receiving the access request, \(SN_{j}\) computes \(K_{1}^{\prime}=T_{s_{j}}(T_{x_{i}}(K_{sn}||ID_{SN_{j}})\) mod \(p)\) mod \(p\) and decrypt \(E_{1}=ID_{UE_{i}}\), \(T_{u_{i}}(K_{ue}||ID_{UE_{i}})\) mod \(p\), \(T_{u_{i}}(K_{ue}||ID_{UE_{i}})\) mod \(p\). \(SN_{j}\) checks whether \(ID_{UE_{i}}\) exist in a database, If it exist, \(SN_{j}\) verifies that \(H_{1}(T_{u_{i}}(K_{ue_{i}}||ID_{UE_{i}})\) mod \(p||ID_{UE_{i}})\) is correct.
3. \(SN_{j}\) computes \(K_{2}^{\prime}=T_{s_{j}}(T_{u_{i}}(K_{sn}||ID_{SN_{j}})\) mod \(p)\) mod \(p\) and verifies that \(MAC_{1}\) is correct. If \(MAC_{1}\) is correct, \(SN_{j}\) generates \(y_{i}\) and computes \(T_{y_{i}}(K_{sn}||ID_{SN_{j}})\) mod \(p\), \(T_{s_{j}}(K_{ue}||ID_{UE_{i}})\) mod \(p\), \(K_{3}=T_{s_{j}}(T_{u_{i}}(K_{ue}||ID_{UE_{i}})\) mod \(p)\) mod \(p\)) mod \(p\)), \(MAC_{2}=H_{1}(K_{3},ID\)\(u_{E_{i}}\), \(ID_{SN_{j}}\),\(T_{y_{i}}(K_{sn}||ID_{SN_{j}})\) mod \(p,T_{x_{i}}(K_{sn}||ID_{SN_{j}})\) mod \(p\)).
4. \(SN_{j}\) computes the session key \(SK_{ij}=H_{2}(T_{y_{i}}(T_{x_{i}}(K_{sn}||ID_{SN_{j}})\) mod \(p)) mod \(p||K_{1}^{\prime}||K_{2}^{\prime}||K_{3}\)\(||ID_{UE_{i}}||ID_{SN_{j}})\) and sends authentication request encrypted with \(K_{1}^{\prime}\) including \(\{ENC_{K_{1}^{\prime}}\)\((T_{y_{i}}(K_{sn}||ID_{SN_{j}})\) mod \(p,T_{s_{j}}(K_{ue}||ID_{UE_{i}})\) mod \(p,MAC_{2})\}\).
5. On receiving the authentication request, \(UE_{i}\) decrypt \(ENC_{K_{1}^{\prime}}\) and get the \(\{(T_{y_{i}}(K_{sn}||ID_{SN_{j}})\) mod \(p,T_{s_{j}}(K_{ue}||ID_{UE_{i}})\) mod \(p,MAC_{2})\}\). Then, \(UE_{i}\) computes \(K_{2}^{\prime}=T_{u_{i}}(T_{s_{j}}(K_{ue}||ID_{UE_{i}})\) mod \(p)\) mod \(p\) and verify that \(MAC_{2}\) is correct. If it is correct, \(UE_{i}\) computes the session key \(SK_{ij}^{\prime}=H_{2}(T_{y_{i}}(T_{x_{i}}(K_{sn}||ID_{SN_{j}})\) mod \(p)) mod \(p||K_{1}||K_{2}||K_{3}^{\prime}||ID_{UE_{i}}||ID_{SN_{j}})\), \(MAC_{3}=H_{1}(SK_{ij}^{\prime}||ID_{UE_{i}}||ID_{SN_{j}}||T_{x_{i}}(K_{sn}||ID_{ SN_{j}})\) mod \(p\), \(T_{y_{i}}(K_{sn}||ID_{SN_{j}})\) mod \(p\)) and sends \(MAC_{3}\) to \(SN_{j}\)
6. Finally, \(SN_{j}\) verifies that \(MAC_{3}\) is correct. If it is correct, \(UE_{i}\) and \(SN_{j}\) authenticate and correctly establish the session key each other.