# A Hybrid Trust Model against Insider Packet Drop Attacks in Wireless Sensor Networks
## 3 Motivation of the Proposed Work
### Design of the New Trust Model
#### 4.2.1 New Trust Model
However, since \(\alpha_{n=1}\) in Equation (11) is not inversely proportional to a node's initial trust value \(T(0)\), and thus it does not satisfy _Property 2_, we use a different base penalty by dividing \(\alpha\) by the initial trust value \(T(0)\) as:  
\[\alpha_{n\geq 2}=\frac{\alpha_{n=1}}{T(0)}=\frac{1}{s+f+3}. \tag{12}\]  
By Equation (12), the base penalty of a node is adjusted according to the node's initial trust value such that the base penalty of a node with lower trust is larger than that of a node with higher trust. **Theorem 2**. : _The trust function defined above satisfies Property 1 and Property 2._  
**Proof. ** We prove _Property 1_ by showing that \(PT(n)\) is an increasing function and for \(n\geq 2\), \(PT(n)-PT(n-1)>PT(n-1)-PT(n-2)\). According to Equation (7):  
\[PT(n)-PT(n-1)=\alpha\sum_{i=1}^{n}\frac{(1-P[f]^{i})}{(1-P[f])}. \tag{13}\]  
\[PT(n-1)-PT(n-2)=\alpha\sum_{i=1}^{n-1}\frac{(1-P[f]^{i})}{(1-P[f])}. \tag{14}\]  
By Equations (13) and (14), \((PT(n)-PT(n-1))-(PT(n-1)-PT(n-2))=\alpha(1-P[f]^{n})/(1-P[f])>0\), since \(\alpha>0\), \(n>0\) and \(0\leq P[f]<1\). We assume \(P[f]\
eq 1\). Thus, \(PT(n)\) satisfies _Property 1_. Next, to prove _Property 2_, let nodes 1 and 2 have the same number of past behaviors (\(s_{1},f_{1}\)) and (\(s_{2},f_{2}\)), respectively, where \(s_{1}+f_{1}=s_{2}+f_{2}\). Assume that node 1's trust value is higher than node 2's trust value, such that \(s_{1}\geq s_{2}\) and \(f_{1}\leq f_{2}\). They are generating \(n\) consecutive failures equally. Let \(\alpha_{1}\) and \(\alpha_{2}\) be base penalties of node 1 and node 2, respectively. Then, when \(n\geq 2\), by (12), \(\alpha_{1}=(s_{1}+f_{1}+3)^{-1}\) and \(\alpha_{2}=(s_{2}+f_{2}+3)^{-1}\), and thus \(\alpha_{1}=\alpha_{2}\) since \(s_{1}+f_{1}=s_{2}+f_{2}\). Given the same base penalty, since \(P[f]\) of node 1 is less than \(P[f]\) of node 2, according to Equations (7) and (10), the total penalty of node 2 is larger than that of node 1. Therefore, the trust function satisfies _Property 2_. We demonstrate how our model works by using the same example as in Section 3 and compare our trust model with the beta trust model and the entropy trust model. Again, a node's trust value is approximately 1 after it successfully forwarded 1000 packets (that is, \(s=1000\)), then the node starts dropping all received packets. As the number of consecutive drops \(n\) goes from 1 to 30, Figure 2 shows how their trust values \(T\) drop; the trust value in our new trust model is calculated by Equations (9) and (10). For the first failure (\(n=1\)), our new trust model penalizes the node the same way as the beta trust model does. However, as \(n\) grows, our trust model gradually increases the penalty to the node (_Property 1_), and the trust value goes below 0.6 when \(n=30\). Assuming that the same trust threshold is used, we can see that our trust model detects inside packet drop attackers much faster than the beta trust model and the entropy trust model. In addition, if the node is an intelligent attacker that knows how our trust model works against its packet drops, to avoid being detected by our trust model, the attacker will not launch a long number of consecutive drops, which may lead to a failure in dropping consecutive critical packets completely, as discussed in Section 3. In the rest of our paper, we conduct extensive simulations based on the OPNET Modeler to show the performance of our trust model in practical network environments. Figure 2: Trust value decreases as the number of consecutive packet drops increases according to beta, entropy, and the new trust models, respectively (from top to bottom).