## 5 Performance Evaluation
### Handover Authentication
#### 5.3.1 Computational Cost
In the quantum-resistant handover authentication protocol, presented in Figure 17, the process unfolds as follows: During lines 1 and 2, the foreign agent ([PERSON]) computes the authentication message and sends it to the home agent ([PERSON]). Subsequently, in line 3, [PERSON] generates its authentication message and transmits it back to FA. In line 5, both [PERSON] and [PERSON] mutually authenticate each other, leading to the generation of a temporary session key shared between them. Moving to line 6, the mobile device (MD) calculates its authentication message and transmits it to FA for verification. Upon successful authentication of [PERSON] by FA in line 9, FA proceeds to generate a new session key. It then securely forwards the encrypted session key and relevant parameters to MD. In line 13, MD performs verification of [PERSON]'s identity and calculates the new session key, effectively completing the quantum-resistant handover authentication protocol. The mobile device performs two NTRU (number theory research unit) encryptions and one NTRU decryption in lines 6 and 13, respectively. Additionally, it involves seven hash functions. Considering the overall computational cost, there are five NTRU encryptions, five NTRU encryptions, and 23 hash functions in total. In [PERSON] et al. 's scheme, shown in Figure 18, the user initiates the process by generating a timestamp, elliptic curve point parameters, and encrypted ID. These are then transmitted to low-Earth-orbit satellites (LEOs) in line 3. Subsequently, LEOs forward the message and its ID to the network control center (NCC) in the following line. Upon receiving the message, NCC first verifies the identity of LEOs in line 5. In line 6, NCC proceeds to verify the user's identity. Upon successful verification, NCC calculates the session key and the response message in line 7. Once the user receives the response message, they performs a verification of NCC's identity in line 11. Subsequently, in line 11, the user generates their session key. To ensure the session keys held by both the user and NCC are identical, the message \(Z\) is transmitted to NCC in lines 12 and 13. NCC then checks the session key in line 14, ensuring consistency and completing the authentication and session key establishment process. Lines 1 and 2 involve the user inserting their smart card into a card reader, where they perform two elliptic curve point multiplication operations, four hash function operations, and one symmetric encryption operation. In line 11, the user executes one elliptic curve point multiplication operation and two hash function operations. As for the network control center (NCC), lines 6 and 7 require one symmetric decryption operation, five hash function operations, and three elliptic curve point multiplication operations. In line 14, the NCC performs one hash function operation. Table 13 shows the abbreviations used in the FogHA protocol. Table 14 shows the abbreviations used in the quantum-resistant handover authentication protocol. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline FAPk & kth fog access point \\ \hline MDi & Mobile devices of user i \\ \hline PID & Pseudo identities \\ \hline APK & Pre-negotiation temporary key \\ \hline \(\Delta T\) & Maximum transmission delay \\ \hline TC & Credentials of user \\ \hline TID & Temporary identity \\ \hline SK & Session key \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Abbreviations in FogHA. Figure 16: FogHA handover authentication procedure. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(FA\) & Foreign agent \\ \hline \(HA\) & Home agent \\ \hline \(MD\) & Mobile device \\ \hline \(L_{m},L_{m,A},L_{m,A},p,r\) & Polynomial \\ \hline \(h_{r}F\) & Public/private key of entity \\ \hline \(h_{SC}\) & Trusted third system center public key \\ \hline TS & Timestamp \\ \hline SK & Session key \\ \hline \(MID_{s}\), \(MPW_{i}\) & Hashed ID and password \\ \hline \end{tabular}
\end{table}