# Graph Layer Security: Encrypting Information via Common Networked Physics
## 3 GLS Encryption Using Physical Networked Dynamic
### Active and Passive Attackers
#### 3.3.1 Passive Eavesdropper  
In the context of GLS, where the physical networked dynamic is used for encryption, a passive eavesdropper is defined to intercept the transmitted information without degrading the physical networked dynamics. In this view, the eavesdropper can (1) intercept only the transmitted information, or (2) in a more sophisticated way, equip eavesdropping sensors on parts of the network nodes and try to recover the whole physical networked dynamic matrix \(\mathbf{X}\). The former has been discussed in Equation (9), i.e., the eavesdropper can intercept the transmitted information at Tx or Rx, and the further optimal relay selection and weight computation in Equations (22) and (23) can ensure the maximal secrecy rate. We will show the GLS performance with the eavesdropper that only intercepts the transmitted information in Section 4.2.  
For the latter, the GLS encryption performance will be degraded in terms of how accurately the eavesdropper recovers the unknown physical networked dynamic \(\mathbf{X}\), and this depends on how many network nodes have been hacked by the eavesdropping sensors. As such, the goal for this passive eavesdropper is to recover the networked dynamics and then use the recovered dynamics to decrypt the transmitted information between legitimate nodes. According to the graph sampling theory [30; 31; 32; 33; 34; 35], the prerequisites of this eavesdropper are to know the data-driven GFT surrogate, \(\mathbf{\Gamma}\), and the weight vector \(\mathbf{\alpha}^{\left(Tx,Rx\right)}\) for each (Tx,Rx) pair. Then, the eavesdropping process can be divided into the following three steps.  
Step 1: [PERSON] identifies the critical subset of network nodes \(\mathcal{S}\subset\{1,\cdots,N\}\) using the graph sampling theory, i.e.,  
\[rank(\mathbf{\Gamma}_{\mathcal{S},z})=r. \tag{24}\]  
Here, Equation (24) is implemented via a _greedy_ algorithm that minimises the condition number of \(\mathbf{\Gamma}_{\mathcal{S},z}\) by finding and adding row index to \(\mathcal{S}\), i.e., \(\mathcal{S}\leftarrow\mathcal{S}\cup\{i\}\), such that \(i=\operatorname*{argmin}_{j\in\mathcal{N}\setminus\mathcal{S}}cond(\mathbf{\Gamma}_ {\mathcal{S}+\{j\},z})\).  
Step 2: [PERSON] deploys sensors on the selected nodes, i.e., \(\mathcal{S}\), and eavesdrops the physical networked dynamics under these nodes, denoted as \(\mathbf{X}_{S,:}\). Then, [PERSON] can reconstruct the complete networked dynamics as [30; 31; 32; 33; 34; 35]:  
\[\mathbf{\hat{X}}=\mathbf{\Gamma}_{;1,1^{r}}\cdot\mathbf{\Gamma}_{\mathcal{S},1 ^{r}}^{\dagger}\cdot\mathbf{X}_{\mathcal{S},:}, \tag{25}\]  
which will then be used for decrypting the intercepted information.  
Step 3: [PERSON] intercepts the communication data from Tx node, and uses its reconstructed Tx underlying signals to decrypt the transmitted information.  
It is noteworthy that hacking different network nodes for physical networked dynamic recovery is a strong assumption for eavesdroppers, as they have to know the data-driven GFT surrogate \(\mathbf{\Gamma}\) and the weight vector \(\boldsymbol{\alpha}^{(Tx,Rx)}\) for each (Tx,Rx) pair, not to mention how difficult it is to embed eavesdropping sensors on a number of network nodes without being detected. We will show the number of nodes being hacked versus the GLS performance in Section 4.2