# Efficient High-Dimensional Quantum Key Distribution with Hybrid Encoding
## 4 Security Analysis  
Before we analyze security of \(d\)-QKD with hybrid encoding, we need to assume constraints to construct secure \(d\)-QKD with hybrid encoding as it is studied in [28]: (i) [PERSON]'s and [PERSON]'s random number generators and their classical post-processing should be trusted. (ii) [PERSON]'s and [PERSON]'s encoding systems should be fully characterized and not be influenced by [PERSON]. (iii) [PERSON] cannot physically access to the output ports of the interferometer, in our protocol, the tritter. (iv) The detectors may have some imperfections, but the defects is not from [PERSON]. The first assumption is essential for all QKD schemes to ensure security. The first and second assumptions are necessary for MDI-QKD as well. The third and final assumptions are different from the scenario of MDI-QKD. They are necessary to prevent particular classes of side channel attacks [28; 29]. The third assumption can be considered not impractical, since \(d\)-QKD with hybrid encoding has the similar experimental situation to prepare-and-measure QKD protocols like original BB84. In the situation, [PERSON] can have full measurement setup in his room and he can block access from the outside. Let us consider several side channel attacks against \(d\)-QKD with hybrid encoding. Since its similarity of principles to the original DDI-QKD, the security of \(d\)-QKD with hybrid encoding against side channel attacks is comparable to that of the original DDI-QKD studied in [28; 29]. Faked-state attack [10], detector efficiency mismatch attack [11], and time-shift attack [14] are not compatible with assumption (iv) since the attacks require a prior knowledge of imperfections of the single photon detectors. Trojan-horse attack based on back reflection [71; 72; 73] is considerable. In Trojan-horse attack based on back reflection, [PERSON] sends multi-photon states into [PERSON]'s([PERSON]'s) encoding system. The photons are reflected at the elements in the encoding system. Then [PERSON] can obtain information about a generated single photon state by analyzing the reflected beam. The attack can be prevented by using frequency filters and isolators like in MDI-QKD case. Trojan-horse attack proposed in [74] is forbidden by assumption (iv), since the detectors in the measurement setup should be manufactured by [PERSON] to accomplish this attack. Detector blinding attack can threaten QKD systems as well. An essential procedure of the detector blinding attack is that [PERSON] shines strong classical light onto detectors, avalanche photodiodes, to change their mode from Geiger mode to linear mode [12]. In the linear mode, a detection signal can be generated by the strong light pulse that exceeds a threshold. This means that [PERSON] can control a detector  
Figure 3: A schematic diagram of experimental setup of three-fold cyclic transformation of OAM modes. There are OAM beam splitters (OAM BSs) which consist of a Mach-Zehnder interferometer with Dove prisms. \(\alpha/2\) means relative angle between the two Dove prisms. The first OAM BS (\(\alpha=\pi\)) and the final OAM BS (\(\alpha=-\pi\)) change a direction of propagation of a photon whose OAM value is odd and even, respectively. The second OAM BS (\(\alpha=\pi/2\)) separates a photon whose OAM value is 0 and 2. With OAM holograms, the three-fold cyclic transformation of OAM modes \(\{-1,0,1\}\) is accomplished. click by regulating amplitude of the light pulse. If a threshold of the all detectors is identical, the basic detector blinding attack can be detected by [PERSON]. Let us define the threshold \(\mu\). Then the amplitude of [PERSON]'s light pulse should be larger than \(\mu\) when it arrives at detectors. [PERSON] intercepts [PERSON]'s signal and resends a strong light corresponding to the measured quantum state. When [PERSON]'s and [PERSON]'s bases are matched, for example OAM modes and path modes, the amplitude of [PERSON]'s light pulse should be larger than \(d\mu\) to make a detector click since the tritter splits the light pulse into four output ports identically.