## 3 System Model and Design Objectives
### Authentication Formation
#### 3.4.3 Energy Efficiency  
Communication overhead (CO) leads to additional energy consumption. Therefore, we attempted to reduce communication overhead that occurs in the contending WSN approaches, such as SD, EEHA, HAS, IIF, and RHC protocols as compared to the proposed SDAAA approach, which resolves CO as follows:  
In the proposed SDAAA protocol, let us assume the sensor network '\(N_{s}\)' consists of the sensor nodes '\(k_{n}\)', which exchange the messages '\(M_{1}\)' and '\(M_{2}\)'. Determining the message exchange process between intermediate nodes is of paramount significance for determining the consumed energy of the nodes. Thus, we determine the communication overhead '\(C_{0}\)' of the source node as:  
\[C_{0}=(1-M_{1})*k_{n}*M_{2} \tag{21}\]  
When forwarding data to either the next hop or the destination hops, data dependency '\(D_{d}\)' may occur at each hop, increasing communication overhead. Thus, the communication overhead for the next hop nodes or destination node, except the source node, is given by:  
\[C_{0}=(1-M_{1})*k_{n}\times\sum k*(D_{d}-1) \tag{22}\]  
Equations (21) and (22) show the communication overhead of the messages exchanged between source and destination nodes. Therefore, the total communication overhead '\(C_{to}\)' of the message exchange can be determined as follows:  
\[C_{to}=(1-M_{1})*k_{n}+(1-M_{1})*k_{n}\times\sum k*(D_{d}-1) \tag{23}\]  
Equation (23) shows that when the overhead ratio '\(R_{o}\)' of our proposed SDAAA approach is compared with other known approaches, our proposed SDAAA approach communication overhead is much lower, as determined by Equation (24):  
\[R_{o}=\frac{((1-M_{1})*k_{n}+(1-M_{1})*k_{n})+\sum k*(D_{d}-1)}{M_{1}*M_{2}} \tag{24}\]  
Thus, the communication overhead of our proposed SDAAA approach is less than that of the other contending approaches, which helps preserve energy consumption.  
\[R_{o}=\frac{((1-M_{1})*k_{n}+(1-M_{1})*k_{n})+\sum k*(D_{d}-1)}{M_{1}*M_{2}}<1 \tag{25}\]To determine the energy consumption, the sensing energy '\(E_{s}\)' of the sensor node '\(K\)' can be computed as:  
\[E_{s}=\hat{\eta}\times R_{s} \tag{26}\]  
where \(\hat{\eta}\): a constant representing the energy to sense a bit of data and \(R_{s}\): sensing rate in bit/s.  
For communication purposes, we calculate the energy dissipated '\(E_{d}\)' by sensor node '\(K\)' to transmit the data packet of size '\(P_{s}\)' to destination sensor node '\(k_{d}\)'s as:  
\[E_{d}=(k,k_{a},r,P_{s})=((\tau_{1}+\tau_{2}\times r(k,k_{d})^{nh})\times(R_{s} \times P_{s})+R_{o}) \tag{27}\]  
After the energy dissipation for transmitting the packet is obtained, the energy for receiving the packets '\(E\),' can be obtained as:  
\[E_{r}=(k,k_{a},P_{s})=((\tau_{4}*R_{s}\times P_{s})+R_{o}) \tag{28}\]  
Finally, we determine the total energy consumption '\(E_{t}\)' for the entire communication process as follows:  
\[E_{t}=\left((\hat{\eta}\times R_{s})+((\tau_{4}*R_{s}\times P_{s})+R_{o})+(( \tau_{1}+\tau_{2}\times r(k,k_{d})^{nh})\times(R_{s}\times P_{s})+R_{o})\right) \tag{29}\]  
By substituting the values, we obtain:  
\[E_{t}=(E_{s}+E_{d}+E_{r}) \tag{30}\]  
where \(r\): 1 hop distance between two sensor nodes set to 1, \(\tau_{2}\), \(\tau_{3}\), \(\tau_{4}\): constant parameters, \(nh\): Number of hops between source and destination nodes.