## 2 Related Work
### Secure Data Energy Evaluation Modeling Method  
Secure data energy efficiency of the network can be evaluated by our proposed SDAA protocol and is used to determine energy efficiency in a comparable way in which the energy efficiency of secure MAC protocols, including SCFSC, SDA, and EEST, was determined. This is possible to assess the overall performance metrics of the network, based on the reliability and reduced delay of the entire network, for the measurement of the energy efficiency of the network. To comply with the performance metrics measurement in the entire network, our proposed SDAA protocol, therefore, optimizes the energy efficiency method used in [28] to evaluate the performance metrics of the network. Based upon this, the energy consumption of the proposed SDAA protocol is divided into two parts: the first is secure data energy consumption \(E_{SecureCollect}\), and the second is secure data transmission energy consumption \(E_{SecureSend}\). Consequently, we define the secure total energy consumption as:  
\[E_{TotalSecure}=E_{SecureCollect}\ E_{SecureSend} \tag{6}\]  
Here, \(E_{SecureCollect}=xP_{SecureCollect}\cdot T_{SecureCollect}\) is the energy of secure energy consumption of authenticated and aggregated data collected in the CBND, \(P_{SecureCollect}\) is the secure energy consumption of authenticated and aggregated data collected, and \(T_{SecureCollect}\) denotes the secure time collection of \(x\) data bits of the authenticated and aggregated data in the network. In addition, the energy consumption of secure data transmission is influenced by bandwidth, transmission loss, and transmission delay. Secure data transmission energy consumption comprises of two parts: secure transmission energy consumption due to vulnerability or malicious node attacker (MNA) detection. The energy consumption of the secure data transmission is defined based on [4; 23; 28]:  
\[E_{SecureSend}=P_{SecureSend}\cdot T_{SecureSend}\cdot L(d) \tag{7}\]  
Here, \(P_{SecureSend}\) is the energy of the secure data transmission by the legitimate USN and vulnerable nodes in the UVWSN, \(T_{SecureSend}\) is the time of secure data transmission, \(L(d)\) is the distance between CH and legitimate USN and the vulnerable sender and receiver underwater sensor nodes (USNs) energy loss, and \(d\) is the distance between the sender and receiver of the USN utilized in UVWSN. In this research, the UVWSN network comprises the USN of the target legitimate USN and vulnerable nodes that securely collect the data through secure data aggregation and authentication methods deployed in the proposed SDAA protocol. This securely transmits data in the network using multiple mobile sinks (MMS) nodes, utilizing radio links in the CBND. Since the speed is very rapid in underwater monitoring conditions, the secure data transmission time that utilizes the UVWSN monitoring application, which deploys MMS, CH, and BS, should be given priority. This will assist in measuring accurate energy efficiency and reduce delays in the network while utilizing USN links. This process is securely protected through the authentication of the secure aggregation methods in the network. Consequently, different delays experienced in the network lead to obtaining accurate secure data collection delays, which include secure a computation delay \(T_{SecureCompute}\) and time \(T_{SecureIVS}\) used in the UVWSN network that involves target legitimate USN and vulnerable sensor nodes during secure data collection and waiting time \(T_{w}\), determined as:  
\[T_{SecureTotal}=T_{SecureCompute}\ T_{SecureIVS}+T_{w} \tag{8}\]  
\[T_{SecureIVS}=\sum_{i\in TNi}\frac{D_{UV\to i}}{S_{UV\to i}} \tag{9}\]  
Here, \(D_{UV\to i}\) denotes the distance between USNs and other entities, such as MMS, CH, and BS, deployed in the UVWSN. Thus, target legitimate USN and vulnerable sensor nodes are denoted as \(TNi\), which include cluster head (CH) target legitimate USN, and vulnerable nodes used to collect secure data.