# Security Analysis of Imperfect Gaussian Modulation Caused by Amplitude Modulator in Continuous–Variable Quantum Key Distribution

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## Abstract

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## 1. Introduction

## 2. Non–Ideal Gaussian Modulation in CV–QKD System

#### 2.1. The Realization Process of Gaussian Modulation

#### 2.2. Non–Ideal Gaussian Modulation Caused by Amplitude Modulator Curve

## 3. Practical Security Analysis of Non–ideal Gaussian Modulation

#### 3.1. Parameter Estimation under Non–ideal Gaussian Modulation

#### 3.2. Calculation of Secret Key Rate

## 4. Simulation and Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The Gaussian distribution under different fitting orders, including (

**a**) ideal Gaussian distribution, (

**b**) first–order fitting, (

**c**) second–order fitting, and (

**d**) third–order fitting.

**Figure 3.**The changes in excess noise and transmittance under different fitting orders. The red dotted line represents excess noise, and the blue dotted line represents transmittance. The ideal transmittance is 0.1 and the ideal excess noise is 0.01.

**Figure 4.**The relationship between excess noise and modulation variance under different fitting orders. The blue line represents first–order fitting, the red line represents second–order fitting, and the yellow line represents third–order fitting. The ideal excess noise is set to 0.01.

**Figure 5.**The relationship between transmittance and modulation variance under different fitting orders. The blue line represents first–order fitting, the red line represents second–order fitting, and the yellow line represents third–order fitting. The ideal transmittance is set to 0.1.

**Figure 6.**Under different fitting orders, the security key rate of CV–QKD. The blue solid line represents second–order fitting, the red dashed line represents third–order fitting, and the black solid line represents the ideal Gaussian distribution. ${V}_{\mathrm{A}}=4$, ${V}_{\mathrm{ele}}=0.01$, $\eta =0.613$, and $\beta =0.95$.

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**MDPI and ACS Style**

Li, Z.; Wang, X.; Chen, Z.; Xu, B.; Yu, S.
Security Analysis of Imperfect Gaussian Modulation Caused by Amplitude Modulator in Continuous–Variable Quantum Key Distribution. *Symmetry* **2023**, *15*, 1452.
https://doi.org/10.3390/sym15071452

**AMA Style**

Li Z, Wang X, Chen Z, Xu B, Yu S.
Security Analysis of Imperfect Gaussian Modulation Caused by Amplitude Modulator in Continuous–Variable Quantum Key Distribution. *Symmetry*. 2023; 15(7):1452.
https://doi.org/10.3390/sym15071452

**Chicago/Turabian Style**

Li, Zhenghua, Xiangyu Wang, Ziyang Chen, Bingjie Xu, and Song Yu.
2023. "Security Analysis of Imperfect Gaussian Modulation Caused by Amplitude Modulator in Continuous–Variable Quantum Key Distribution" *Symmetry* 15, no. 7: 1452.
https://doi.org/10.3390/sym15071452