## Security Analysis  
### Mutual Authentication  
In this research, BAN logic is used to prove mutual authentication in the authentication algorithm [31], mainly to ensure that data is not modified during the phases. The notation of BAN logic is described as below: \begin{tabular}{c c} \hline \hline \(P|\equiv X\) & \(P\) believes \(X\). \\ \(P\lhd X\) & \(P\) sees \(X\). \\ \(P|\sim X\) & \(P\) once said \(X\). \\ \(P|\Rightarrow X\) & \(P\) has jurisdiction over \(X\). \\ \(\#(X)\) & Formula \(X\) is new. \\ \(P|\stackrel{{ K}}{{\rightarrow}}X\) & \(P\) has \(X\) as a public key. \\ \(P\) and \(X\) may use the session key \(K\) to \\ \(P\stackrel{{ K}}{{\rightarrow}}Q\) & communicate with each other. \\ \(\{X\}_{K}\) & The formula \(X\) is encrypted by \(K\). \\ \hline \hline \end{tabular}   
In the initializing phase, the scheme makes sure that the legality is authenticated by the accessing party (\(AP\)) and the \(SCC\). The main goals of the scheme are: \begin{tabular}{l c} \(G1\): \(AP|\equiv AP\stackrel{{ SK_{AP}-SCC}}{{\leftrightarrow}}SCC\) \\ \(G2\): \(AP|\equiv SCC|\equiv AP\stackrel{{ SK_{AP}-SCC}}{{\leftrightarrow}}SCC\) \\ \(G3\): \(SCC|\equiv AP\stackrel{{ SK_{AP}-SCC}}{{\leftrightarrow}}SCC\) \\ \(G4\): \(SCC|\equiv AP|\equiv AP\stackrel{{ SK_{AP}-SCC}}{{\leftrightarrow}}SCC\) \\ \(G5\): \(AP|\equiv ID_{SCC}\) \\ \(G6\): \(AP|\equiv SCC|\equiv ID_{SCC}\) \\ \(G7\): \(SCC|\equiv ID_{AP}\) \\ \(G8\): \(SCC|\equiv AP|\equiv ID_{AP}\) \\ \end{tabular}   
According to the authenticate algorithm, BAN logic is used to produce an idealized form as follows: \begin{tabular}{l} \(M1\): \(AP\)\(\rightarrow\)\(SCC(\{ID_{AP},Pwd_{AP},R_{AP},h_{AP_{1}},M_{Access},T_{1}\}_{Puk_{SCC}})\) \\ \(M2\): \(SCC\)\(\rightarrow\)\(AP(\{ID_{SCC},R_{SCC},h_{SCC_{1}},Token_{1},T_{3}\}_{Puk_{AP}})\) \\ \end{tabular}   
To analyze the proposed scheme, the following assumptions are made: \begin{tabular}{l} \(A1\): \(AP|\equiv\#(R_{AP})\) \\ \(A2\): \(SCC|\equiv\#(R_{AP})\) \\ \(A3\): \(AP|\equiv\#(R_{SCC})\) \\ \(A4\): \(SCC|\equiv\#(R_{SCC})\) \\ \(A5\): \(AP|\equiv SCC|\Rightarrow SCC\stackrel{{ SK_{AP}-SCC}}{{ \leftrightarrow}}AP\) \\ \end{tabular}   
_A6_: \(SCC\equiv AP|\Rightarrow AP^{\text{ }SKAP\text{-}SCC}\) _SCC_  
_A7_: \(AP|\equiv SCC|\Rightarrow ID_{SCC}\)  
_A8_: \(SCC|\equiv AP|\Rightarrow ID_{AP}\)  
Security control center authenticates access party. By _M1_ and the seeing rule, derive:  
**Statement 1. **\(SCC\lhd(\{ID_{AP},Pwd_{AP},R_{AP},h_{AP_{1}},M_{Access},T_{1}\}_{Puk_{SCC}})\)  
By _A2_ and the freshness rule, derive:  
**Statement 2. **\(SCC|\equiv\#(\{ID_{AP},Pwd_{AP},R_{AP},h_{AP_{1}},M_{Access},T_{1}\}_{Puk_{SCC}})\)  
By (Statement 1) and the message meaning rule derive:  
**Statement 3. **\(SCC|\equiv AP|\sim(ID_{AP},Pwd_{AP},R_{AP},h_{AP_{1}},M_{Access},T_{1})\)  
By (Statement 2), (Statement 3) and the nonce verification rule, derive:  
**Statement 4. **\(SCC|\equiv AP|\equiv(ID_{AP},Pwd_{AP},R_{AP},h_{AP_{1}},M_{Access},T_{1})\)  
By (Statement 4) and the belief rule, derive (_G4_):  
**Statement 5. **\(SCC|\equiv AP|\equiv AP^{\text{ }SKAP\text{-}SCC}\) _SCC_  
By (Statement 5), _A6_, and the jurisdiction rule, derive (_G3_):  
**Statement 6. **\(SCC|\equiv AP^{\text{ }SKAP\text{-}SCC}\) _SCC_  
By (Statement 4) and the belief rule, derive (_G8_):  
**Statement 7. **\(SCC|\equiv AP|\Rightarrow ID_{AP}\)  
By (Statement 7), _A8_, and the belief rule, derive (_G7_):  
**Statement 8. **\(SCC|\equiv ID_{AP}\)  
Seccess party authenticates security control center. By _M1_ and the seeing rule, derive:  
**Statement 9. **\(AP\lhd(\{ID_{SCC},R_{SCC},h_{SCC_{1}},Token_{1},T_{3}\}_{Puk_{AP}})\)  
By _A3_ and the freshness rule, derive:  
**Statement 10. **\(AP|\equiv\#(\{ID_{SCC},R_{SCC},h_{SCC_{1}},Token_{1},T_{3}\}_{Puk_{AP}})\)  
By (Statement 9) and the message meaning rule derive:  
**Statement 11. **\(AP|\equiv SCC|\sim(ID_{SCC},R_{SCC},h_{SCC_{1}},Token_{1},T_{3})\)  
By (Statement 10), (Statement 11) and the nonce verification rule, derive:  
**Statement 12.