## 3 Trojan Horsse Attacks
### Attacks at the Same Wavelength as the Original Pulses  
The security certification of the CV-QKD protocol provides a verifiable mechanism for the safe operation of the system. However, practical CV-QKD implementations may deviate from these assumptions, leading to security loopholes [1; 35; 36; 37]. Every optical element backscatters some amount of any incoming light. Consequently, every optical apparatus can be examined from the outside by shining into it a well-controlled light and analyzing the backscattered light. In a Trojan horse attack, an eavesdropper [PERSON] may probe the CV-QKD system by injecting bright light pulses and analyzing the back reflections (Figure 1). These reverse reflections stem from a change inrefractive index, such as a connection between two optical components or a density fluctuations within the optical device. A back-reflected signal containing an imprint of the optical modulator used to encode the quantum states may allow the eavesdropper to extract parts of the raw key. [PERSON]'s equipment is built around a standard optical time-domain reflectometer (OTDR). Send in short optical pulses and analyze the backscattered light intensity is a function of time. From the known speed of light, the time can be translated into distances. [PERSON] can indeed probe the setting of phase modulator by exploiting the change in birefringence in Titan-indiffus \(LiNbO_{3}\) integrated waveguides. In practical operations, when the wavelength of the Trojan horse attack pulses are the same as the original pulse of the CV-QKD system which is 1550 nm in this article, crosstalk may occur between the pulses (see in Figure 3). In order to judge the impact of our Trojan horse attacks strategy, we need to quantitatively describe the effect of crosstalk between the Trojan horse pulses and original signal pulses. Figure 2: Change of secret key rate with transmission distance under different excess noise levels. The excess noise without attacks is set to 0.01, the maximum excess noise when the transmission distance is 50 km is 0.0145. In this paper, we define a parameter \(\tau(0\leq\tau\leq 1)\) representing the degree of the crosstalk. The change of \(\tau\) will affect the excess noise of the system, which will be analyzed in the following part. The quantum channel involved in the CVQKD system is a normal linear model, and the relationship between Alice and Bob is as follows:  
\[y=tx+z, \tag{3}\]  
where \(t^{2}=\sqrt{\eta T}\), the parameter \(T\) is the transmission efficiency of the quantum channel and \(\eta\) denotes the efficiency of homodyne detector. Vector \(x=x_{1},x_{2},...x_{N},x\sim N(0,1)\), vector \(y=y_{1},y_{2},...y_{N}\), \(N\) is the total number of pulses received. Vector \(z\) is the noise term following a central normal distribution with variance \(\sigma^{2}=t^{2}+N_{0}+V_{el}\). \(N_{0}\) is the shot noise variance. The electronic noise \(V_{el}\) is not affected by the shift of pulses. The total channel-added noise expressed in shot noise units referred to the input is defined as:  
\[\chi_{line}=(1-T)/T+\epsilon=1/T-1+\epsilon, \tag{4}\]  
where \(\epsilon\) is the excess noise. In order to perform parameter estimation, i.e., to estimate the parameter \(T\) and \(\epsilon\) and then calculate the secret key rate, [PERSON] and [PERSON] reveal part of their data and use the following equation:  
\[ \begin{split}\langle x_{2}\rangle=V_{X},\langle xy\rangle=tV_{X },\\ \langle y_{2}\rangle=V_{Y}=t^{2}V_{X}+t^{2}\xi+N_{0}+V_{el}.\end{split}  \tag{5}\]  
In the actual model, [PERSON] has no access to the homodyne detector, so \(\eta\) and \(V_{el}\) would not be changed. For a chaotic source with an average noise photons number \(\langle N_{leak1}\rangle\), the excess noise contributed in matched mode is given by:  
\[\epsilon_{out1}=2\eta_{B}\langle N_{leak1}\rangle, \tag{6}\]  
where \(\eta_{B}\) is the transmittance of the [PERSON]'s system in the Gaussian modulation coherent state (GMCS).