## 4 The Proposed 2 FAKA-C/S Protocol  
To enhance security for both mobile users and aggregated servers, our proposed protocol integrates key methodologies, which are shown as follows:  
1. Utilization of Elliptic Curve Cryptography (ECC): We harness ECC's capability to generate robust cryptographic keys of reduced length, significantly bolstering encryption without increasing computational demands. This is achieved by combining the strengths of ECC with the Diffie-Hellman key exchange protocol. The Diffie-Hellman algorithm allows two users, without a secure communication channel, to generate a shared secret key using random numbers. The Diffie-Hellman of ECC ensures forward secrecy.  
\begin{table}
\begin{tabular}{c c c l} \hline \hline
**Notation** & **Description** & **Notation** & **Description** \\ \hline \(U_{i}\) & User/Client & \(x\) & Long \(-\) term key of \(S_{j}\) \\ \(S_{j}\) & Aggregated Server & \(\mathcal{A}\) & Malicious adversary \\ \(ID_{i}\) & Unique identity of \(U_{i}\) & \(\|\) & String concatenation operation \\ \(PID_{i}\) & Pseudo \(-\) identity of \(U_{i}\) & T & Current timestamp \\ \(PW_{i}\) & Password chosen by \(U_{i}\) & \(\oplus\) & Bitwise XOR operation \\ \(b\) & Random numbers of \(S_{j}\) & \(H(\cdot)\) & One-way hash function \\ \(a\) & Random numbers of \(U_{i}\) & \(SK\) & Session key shared between \(U_{i}\) and \(S_{j}\) \\ \(\mathcal{D}_{ID}\) & Space of user identities. & \(r_{i}\) & Random number chosen by \(S_{j}\) during registration \\ \(\mathcal{D}_{PW}\) & Space of user passwords & \(R_{i}\) & Pre \(-\) sec ret value chosen by \(U_{i}\) \\ \(G\) & Abelian group on the elliptic curve & \(SC_{i}\) & Smart card issued to \(U_{i}\) \\ \(p\) & Generator of group \(G\) & \(Sum\) & Counter for allowed login failures \\ \(n_{0}\) & Random number within the range \(2^{4}\) to \(2^{8}\) & \(t_{u}\) & Timestamp generated by \(SC_{i}\) \\ \(PW_{i}^{new}\) & New password set by \(U_{i}\) & \(l\) & Any instance if no distinction is needed \\ \(U^{l}\) & \(i_{th}\) instance of user \(U\). & \(S^{j}\) & \(j_{th}\) instance of server \(S\) \\ \(q_{h}\) & Number of hash queries made by adversary \(\mathcal{A}\) & \(q_{e}\) & Number of execute queries made by adversary \(\mathcal{A}\) \\ \(q_{s}\) & Number of send queries made by adversary \(\mathcal{A}\) & \(l\) & Bit length of the hash value \\ \hline \hline \end{tabular}
\end{table}
Table 1: Notations with related descriptions.  
2 Implementation of Hash-chain: The proposed protocol incorporates Hash-chain by applying a one-way hash function, H(-), which plays a pivotal role in safeguarding data integrity and strengthening authentication processes. Specifically, Hash-chains generate a unique hash value for each session, ensuring that even if data is intercepted, it cannot be reused in subsequent sessions. This approach is crucial in mitigating risks associated with replay attacks.
3 Mutual Authentication and Session Key Agreement: A pivotal aspect of the proposed protocol is the mutual authentication process, where both the user and the aggregated server authenticate each other's identity before establishing a shared session key for encrypted communications, ensuring both confidentiality and integrity of data.
4 Enhanced Anonymity and DoS Attacks Mitigation: The proposed protocol introduces a pseudo-identity \(PID_{i}\) to preserve user anonymity and prevent traceability. Furthermore, the integration of timestamps T plays a crucial role in defending against DoS attacks by verifying the timeliness of each session.  
In the forthcoming sections, this paper gives the protocol's system setup phase, registration phase, subsequent login and authentication phase, and finally, the password update phase for user.