# A Secure and Lightweight Group Mobility Authentication Scheme for 6 LoWPAN Networks
## 4 System Models
### Analysis of SVO Logic
#### 5.2.4 Security Proof  
* **Establishing Trust Between Entities**  
S1: AAA received \(|\{\textit{TID}_{GL}\mid\textit{TCI}_{G}\mid\textit{|MAC}(6\textit{GLM})\}|\)\(\textit{KGL}\leftrightarrow_{A}\). The AAA server receives a message containing the necessary identifiers, timestamp, and a MAC value securely encrypted using the shared key, \(K_{GL}\leftrightarrow_{A}\). This indicates that the initial communication between the group leader and the AAA server has been successfully initiated. From P5, SA1, and P3, we can obtain the following:  
S2: AAA believes 6 LMNGL says \(\{\textit{TID}_{GL}\mid\textit{1}\mid\textit{TCI}_{G}\mid\textit{|MAC}_{ KGL} \leftrightarrow_{A}(6\textit{GLM})\}\). From S2, P4, and F1, we can obtain the following:  
S3: AAA believes fresh \(\{\textit{TID}_{GL}\mid\textit{1}\mid\textit{TCI}_{G}\mid\textit{|MAC}_{ KGL} \leftrightarrow_{A}(6\textit{GLM})\}\). From S2, S3, and J1, we can obtain the following:  
S4: AAA believes 6 LMNGL received \(|\{\textit{TID}_{GL}\mid\textit{1}\mid\textit{TCI}_{G}\mid\textit{|MAC}_{ KGL} \leftrightarrow_{A}(6\textit{GLM})\}\). From S4, P6, and BA2, we can obtain the following:  
S5: AAA believes 6 LMNGL received \(\textit{MAC}(6\textit{GLM})\); the AAA believes each corresponding 6 MNi truly sends the handover. **Step S2-S5:** Using principles of authentication theory, the AAA server deduces that the GL genuinely sent the received message. This assurance is built through the following:  
* Trust in the cryptographic properties of the MAC. * Validation of the timestamp to ensure message freshness. * Assurance of integrity, as the MAC guarantees, is that the message has not been altered. The AAA server thus believes the handover request originates from the legitimate group leader. * **Authenticating the Local Mobility Anchor (LMA)**  
S6: LMA believes \(|\textit{ID}_{LMA}\mid\textit{1}\mid\textit{TID}_{GL}\mid\textit{1}\textit{ TID}_{GC}\mid\textit{|CT}_{A}\}|\textit{kLMA}\leftrightarrow_{A}\). S7: LMA believes AAA believes \(\{\textit{ID}_{LMA}\mid\textit{1}\mid\textit{TLI}_{MA}\mid\textit{TID}_{GL} \mid\textit{|TID}_{G}\mid\textit{|CT}_{A}\}\). From S7, P17, P16, and F1, we can obtain the following:  
S8: LMA believes fresh \(|\textit{ID}_{LMA}\mid\textit{1}\textit{TLI}_{GL}\mid\textit{TID}_{GL} \mid\textit{TID}_{GG}\mid\textit{|CT}_{A}\). From S7, S8, and NV1, we can obtain the following:  
S9: LMA believes AAA believes \(|\textit{ID}_{LMA}\mid\textit{1}\mid\textit{TLI}_{MA}\mid\textit{TID}_{GL} \mid\textit{|TID}_{GG}\mid\textit{|CT}_{A}\). **Step S6-S9:** Similar logic is applied to the communication between the LMA and the AAA. By analyzing the timestamps, MAC values, and cryptographic keys used in their interaction, the LMA becomes convinced that the AAA is authentic and that the information exchanged is fresh and untampered. This step is critical because the LMAserves as a central anchor in managing mobility and must trust the AAA for further secure handover processes. * **Ensuring Handover Security Between MAGs** S10: 6 MAG\({}^{\rm new}\) believes \(\{T\!G\!ID\!\mid\!\mid T_{GL},M_{LMA}\}\). From S10, P11, BA2, we can obtain the following: S11: 6 MAG\({}^{\rm new}\) believes LMA says \(\{T\!I\!D_{6G}\}\). From S1, P16, and F1, we can obtain the following: S12: 6 MAG\({}^{\rm new}\) believes fresh \(\{T\!I\!D_{6G}\}\). From S11, S12, and NV1, we can obtain the following: S13: 6 MAG\({}^{\rm new}\) believes LMA believes \(\{T\!I\!D_{6G}\}\). From S13, J1, we can obtain the following: S14: 6 MAG\({}^{\rm new}\) believes \(\{T\!I\!D_{6G}\}\). S15: 6 MAG\({}^{\rm new}\) says \(\{T\!I\!D_{LMA}\!\mid\!\mid\!\mid\!T\!I\!D_{GL}\!\mid\!\mid\!T\!I\!D_{6G}\! \mid\!\mid\!N_{4}\}_{KG}+_{A}\). From S15, P9, and BP2, we can obtain the following: S16: 6 MAG\({}^{\rm new}\) believes AAA says \(\{T\!I\!D_{LMA}\!\mid\!\mid\!T\!I\!D_{GL}\!\mid\!\mid\!T\!I\!D_{6G}\!\mid\! \mid\!\mid\!N_{4}\}\). From S16, P10, and F1, we can obtain the following: S17: 6 MAG\({}^{\rm new}\) believes fresh \(\{T\!I\!D_{LMA}\!\mid\!\mid\!T\!I\!D_{GL}\!\mid\!\mid\!T\!I\!D_{6G}\!\mid\!