## 3 Trojan Horsse Attacks
### Attacks with Different Wavelengths from the Original Pulses
Assuming the output power of the Trojan horse signals from the fiber is \(P_{out}\) (measured at point B in Figure 4), and the isolation of DMUX is defined as \(\xi_{DMUX}\). The power of leakage photons received by Bob'side \(P_{leak}\) can be expressed as:  
\[P_{leak}=\xi_{DMUX}*P_{out}. \tag{9}\]  
So the number of leaked noise photons is  
\[\left\langle N_{leak2}\right\rangle=\frac{P_{leak}}{hf}, \tag{10}\]  
where \(h\) is the [PERSON]'s constant, \(f\) is the frequency of the Trojan horse attacks signals. Raman scattering is caused by the inelastic interaction between the pump light and the optical photons in the fiber. When scattered photons are generated at frequencies above and below the pump light, corresponding to anti-Stokes scattering and Stokes scattering, respectively. As the Trojan horse signals propagate along the light in the channel, it can generate noise photons through different nonlinear processes. Existing research results indicate that when the wavelength of the quantum channel is shorter than the classical channel, anti-Stokes Raman scattering (SARS) is the most important linear process [42; 43]. The corresponding noise photons are given by:  
\[\left\langle N_{WDM}^{SARS}\right\rangle=\frac{\lambda^{3}}{hc^{2}}\rho\xi_{D }\left[P_{in}Le^{-\alpha L}+P_{in}\frac{1-e^{-2\alpha L}}{2\alpha}\right], \tag{11}\]  
where \(\rho\) is the Raman scattering coefficient, \(c\) is the speed of light, \(\alpha\) is the fiber attenuation coefficient, \(P_{in}\) (measured at point A in Figure 4) is the input power of the Trojan horse attacks. \(L\) is the fiber length. Ref. [42] points out that we can get the following relationship between the input power Pin and output power Pout of the Trojan horse signal in the fiber:  
\[P_{out}=P_{in}*T_{ch}. \tag{12}\]  
So the Equation (10) can be rewritten as  
\[\left\langle N_{leak2}\right\rangle=\frac{\xi_{D}*P_{in}*T_{ch}}{hf}, \tag{13}\]  
the excess noise contribute of the matched mode impinging to [PERSON] device is given by:  
\[\epsilon_{out2}=2\eta_{B}\langle N_{WDM}^{SARS}\rangle. \tag{14}\]  
The total noise caused by the outside world to [PERSON]'s side can be composed of the following two parts  
Figure 5.0  
The two parts of Equation (11) respectively represent the number of Raman scattered noise photons in the forward and backward transmissions. We simulated the Raman noise of the forward and backward transmission channel and total excess noise when the input noise power is 1, 5, and 8 mw according to Equations (11) and (15). The fiber attenuation coefficient \(\alpha=0.046\) is set to be the fiber attenuation per km. \(T_{ch}\) is the transmittance of the channel. The transmittance of the multiplexer \(\xi_{M}\) and demultiplexer \(\xi_{D}\) are both set to 0.071. The transmittance of [PERSON]'s system \(\eta_{B}\) is set to 0.6. From the simulation results, it can be seen in the Figure 6 that when the transmission distance is equal to \(1/\alpha\) (i.e., 21 km), the forward Raman noise is the largest, and when the backward transmission, the Raman noise reaches a saturated state. Figure 7 shows the total excess noise of different input powers.