## 4 Results
### APT Detection  
In the next phase of the assessment, we showcase the behaviour of the developed detection system via synthetically generated APT behaviour. As documented by [PERSON] in [40], in the particular case of the VAM process, an attacker may periodically manipulate certain variables for various reasons. In one of the described scenarios, [PERSON] shows that periodically injected pulses (e.g., increase followed by decrease) on specially chosen control variables may have a significant economical impact. For example, according to [40], the repeated pulse on the vaporizer steam duty valve (control variable #4) may yield an economic loss of at least $10,000.  
To showcase the effectiveness of the developed detection system, and to analyze the advantage of the APT, we leverage the intervention on the vaporizer steam duty valve, as described in [40]. The behaviour exhibited by this input is shown in Figure 14(a). However, the impact of the repeated opening/closing of the valve may be visible on observable variables (Figure 14(b)). Therefore, the attacker may need to compromise a large number of observable variables to ensure that the input of each detector is according to the expected value.  
Next, we presume that, in parallel to the APT, periodically, attestation is performed on an attack-free control variable. Namely, attestation was performed on the C\({}_{2}\)H\({}_{4}\) feed valve every 600 min, while the attacker performed an intervention every 300 min (see Figure 15(a),b). The impact of these interventions on a selected output, namely the production of VAc, is illustrated in Figure 15(c). In case the attacker does not hide his/her presence, all detectors will report an anomaly. This behaviour is depicted in Figure 15(d).  
Figure 16: _Cont._  
Figure 15: APT example: repeated pulses on the vaporizer steam duty valve (control variable #4).  
However, as mentioned earlier, in the case of an APT, the attacker may be aware of the implemented monitoring and detection solutions. Therefore, he/she may try to compromise as many elements as possible. The extreme case when the attacker is able to compromise all output, and manipulate the value reported for all observable variables is shown in Figure 17. Here, two scenarios are explored: a first scenario in which the attacker uses recently reported legitimate measurements to estimate the value of observable variables (Figure 17a); and, a second scenario where past values are used in the attacker's estimations (Figure 17b). In both cases, we presume that all variables have been compromised. In the first scenario, the system is unable to detect the APT. However, in the second scenario, the ANOMALY state is reported by most of the detectors. This is owed to the fact that more recent measurements better reflect the current process state than past ones.  
Next, we empirically assessed the advantage of the attacker, as detailed in Equation (17). As mentioned earlier, the advantage of APT is affected by the number of compromised detectors, and by the precision of the manipulated variables. Figure 18 details the result of this experiment in the same contextual scenario (attacker controls input #4, while attester uses input #2). As shown here, while the attacker leverages immediate prior values to estimate the next output, there is still a level of uncertainty, that, accumulated, will require a larger number of compromised elements. According to this experiment, the attacker needs to compromise at least 14 components (out of 18) in order for the APT not to be detected. This is a significant result that confirms our previous observation according to which it is not sufficient for the attacker to compromise the majority of components. It must do so by leveraging high-quality logic that reproduces the behaviour of the underlying physical process.  
Figure 16: APT and attestation example.  
Figure 17: Detectors input compromised by attacker’s estimation via linear regression.