# Efficient High-Dimensional Quantum Key Distribution with Hybrid Encoding
## 4 Security Analysis
In the situation, [PERSON] can notice the detector blinding attack since \(d\) detectors are clicked simultaneously. [PERSON] can make an error rate be affected by the attack by assigning random number when more than two detectors are clicked. If there are differences among the threshold of detectors, it is possible that [PERSON] generates one detector click. The clicked detector must have the lowest threshold among the detectors. This means that [PERSON] cannot generate a click of the other detectors independently. So the attack can be found by analyzing statistics of detector clicks. Detector blinding attack with various blinding power [13] can threaten \(d\)-QKD with hybrid encoding as well as the original DDI-QKD [29]. However, since the attack requires a prior knowledge about the detectors, it is not compatible with assumption (iv). So we can conclude that without the assumptions which are not necessary in MDI-QKD, the security of \(d\)-QKD with hybrid encoding cannot be guaranteed against all detector side channel attacks. To detect [PERSON]'s side channel attacks, we introduce a random tritter operation of [PERSON]. The tritter operation written in Equation (2) is performed on path modes after [PERSON]'s encoding. It is possible that [PERSON] chooses one of tritter operation among \(d\) different operations rather than a fixed operation. For example, [PERSON] can chooses one operation among the operations shown in Equation (5):  
\[\hat{U}_{3,0}=\frac{1}{\sqrt{3}} \begin{pmatrix}1&1&1\\ 1&\omega&\omega^{2}\\ 1&\omega^{2}&\omega\\ \end{pmatrix} ,\quad\hat{U}_{3,1}\quad=\frac{1}{\sqrt{3}} \begin{pmatrix}\omega ^{2}&1&\omega\\ 1&1&1\\ \omega&1&\omega^{2}\\ \end{pmatrix} ,\quad\hat{U}_{3,2}=\frac{1}{\sqrt{3}} \begin{pmatrix}\omega^{2 }&\omega&1\\ \omega&\omega^{2}&1\\ 1&1&1\\ \end{pmatrix}  \tag{5}\]  
for \(3d\)-QKD with hybrid encoding. The operations can be implemented by using \(3d\)-tritter and phase shifters. Let us consider the case that [PERSON] chooses path mode \(|\bar{p}_{0}\rangle\) and [PERSON] tries detector blinding attack with strong pulse whose OAM mode is \(|\bar{l}_{0}\rangle\). If [PERSON] chooses \(t\), where \(t\in\{0,1,2\}\), and performs the tritter operation \(\hat{U}_{3,t}\), the pulse goes to output port \(p_{t}\) of the tritter. For a successful attack, [PERSON] must find the pulse intensity and that one detector in the output port is clicked and the other detectors are not, regardless of [PERSON]'s choice of tritter operation. Also, [PERSON] should perform detector blinding attacks with various blinding power and find at least three different blinding powers, since [PERSON] monitors statistics of outcomes. For instance, [PERSON] can check whether \(|\bar{l}_{0},\bar{p}_{0}\rangle\) is projected onto \(|\Phi_{0}\rangle\), \(|\Phi_{3}\rangle\), and \(|\Phi_{6}\rangle\) equally or not. Therefore, [PERSON] should prepare at least three different pulse intensities, which occur click events of different detectors on the same output port, to pass [PERSON]'s statistics check. For a large dimension, we expect that such attack is improbable with the assumption (iv), i.e., with trusted devices. Since click thresholds of different detectors are very similar but randomly fluctuated, it is difficult to find blinding powers and pulse intensities that satisfy the successful attack conditions. For a successful attack, [PERSON] should find the powers and intensities that only one detector is clicked while the other \(d-1\) detectors in the port are not clicked, and the attack does not influence [PERSON]'s outcome statistics regardless of [PERSON]'s choice of tritter operation \(\hat{U}_{d,t}\), where \(t\in\{0,1,2,...,d-1\}\). Therefore, we expect that side channel attacks are probably detected for a high-dimensional QKD using hybrid encoding, if [PERSON] applies the random choice of tritter operation and a countermeasure of a side channel attack, such as the random-detector-efficiency protocol [75; 76].