Practical security analysis of continuous-variable quantum key distribution with jitter in clock synchronization

Practical security analysis of continuous-variable quantum key distribution with jitter in clock synchronization

JID:PLA AID:24914 /SCO Doctopic: Quantum physics [m5G; v1.227; Prn:10/01/2018; 15:03] P.1 (1-7) Physics Letters A ••• (••••) •••–••• 1 67 Content...

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Practical security analysis of continuous-variable quantum key distribution with jitter in clock synchronization

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School of Information Science and Engineering, Central South University, Changsha 410083, China b State Key Laboratory of Advanced Optical Communication Systems and Networks, and Center of Quantum Information Sensing and Processing, Shanghai Jiao Tong University, Shanghai 200240, China

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Cailang Xie , Ying Guo , Qin Liao , Wei Zhao , Duan Huang , Ling Zhang Guihua Zeng b

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Article history: Received 30 October 2017 Received in revised form 22 December 2017 Accepted 4 January 2018 Available online xxxx Communicated by A. Eisfeld Keywords: CV-QKD Security analysis Clock synchronization Jitter

How to narrow the gap of security between theory and practice has been a notoriously urgent problem in quantum cryptography. Here, we analyze and provide experimental evidence of the clock jitter effect on the practical continuous-variable quantum key distribution (CV-QKD) system. The clock jitter is a random noise which exists permanently in the clock synchronization in the practical CV-QKD system, it may compromise the system security because of its impact on data sampling and parameters estimation. In particular, the practical security of CV-QKD with different clock jitter against collective attack is analyzed theoretically based on different repetition frequencies, the numerical simulations indicate that the clock jitter has more impact on a high-speed scenario. Furthermore, a simplified experiment is designed to investigate the influence of the clock jitter. © 2018 Published by Elsevier B.V.

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1. Introduction

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Quantum key distribution (QKD) allows two distant parties, the sender Alice and the receiver Bob, to establish a coincident secret key through an untrusted channel [1–3]. Unfortunately, there still exist a big gap between the theory and practice of QKD [4–6]. Indeed, many attacks which exploit security loopholes in practical realizations have been presented, such as the time-shift attack [7], faked states attacks [8,9], Trojan-horse attacks [10], phase-remapping attacks [11] for discrete-variable QKD systems, and the local oscillator attacks [12,13], saturation attacks [14], state-discrimination attack [15] for continuous-variable (CV) QKD systems. These afore-mentioned loopholes usually come from the imperfect devices or transmission channels, which can be exploited by Eve to break the unconditional security of quantum communication proved in theoretical security proofs. Specifically, there is a vulnerability exists permanently in practical CV-QKD system comes from the clock synchronization, namely clock jitter, has not been investigated throughly at present. The clock synchronization is of significant importance in a practical CVQKD system, as it can provide a common clock source for generating modulated signals and collecting transmitted signals [16,17].

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Corresponding author. E-mail address: [email protected] (L. Zhang).

https://doi.org/10.1016/j.physleta.2018.01.006 0375-9601/© 2018 Published by Elsevier B.V.

In a general practical CV-QKD system, the signal light and local oscillator (LO) are generated by the same laser and then modulated simultaneously to pulses using a modulator. Therefore, the LO and the signals have the identical frequency, which allows Bob to extract the synchronous clock from the LO [18–20]. However, the LO is notoriously vulnerable to attacks due to its classical features [21]. Indeed, by exploiting the loopholes of the LO, several attacks have been successfully launched against practical CV-QKD systems, such as the calibration attack [13], LO fluctuation attack [22] and wavelength attack [23,24]. Fortunately, all this attacks can be defeated by using an extra homodyne detector to monitor the real-time shot noise, monitoring the power of LO and adding wavelength filters before Bob’s detector, respectively. In contrast to the previous loopholes, the clock jitter is a random noise which exist permanently in clock signals due to the imperfect time base and phase locked logic [25–27]. Thus it can not be removed by a simple monitoring. In order to fix the vulnerability of the clock synchronization, we analyze the clock jitter effect on the practical CV-QKD system. In our scheme, the clock jitter exists in the clock synchronization signals which transmit from Alice to Bob through LO, thus affecting the preciseness of signals acquisition at Bob side. This clock jitter effect may leave security loopholes for Eve to attack the system. As we will show in this paper, the clock jitter effect on CV-QKD is characterized and the system security against collective attack is analyzed theoretically. In the aspect of experiment, a low com-

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plexity setup is designed to demonstrate the influence of the clock jitter. Both the simulation and experimental results indicate that the system performance is reduced by manipulating the clock jitter. Moreover, we find that the jitter effect bring more negative influence in the CV-QKD system with a high repetition frequency. Therefore, with the rapid growth in the field of the high-speed CVQKD [28–31], the clock jitter will be increasingly important for the practical CV-QKD systems. This paper is organized as follows: In Sec. 2, we characterize the clock jitter effect on the detection in the practical CV-QKD system. Subsequently the system security with different clock jitter is analyzed in Sec. 3. Moreover, the low complexity experiment is given in Sec. 4. Finally, the conclusion is drawn in Sec. 5.

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2. The clock jitter effect on practical CV-QKD system

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Fig. 1. (Color online.) The practical homodyne detection scheme in the CV-QKD system.

In a general CV-QKD protocol, Alice usually encode information in the quadratures of the light field with Gaussian modulation. The modulated quantum states are subsequently transmitted to Bob over a lossy channel which is characterized by transmittance and excess noise. After receiving the states, Bob perform homodyne (or heterodyne) detection to measure randomly one of the quadrature (or both quadratures). A fraction of the measurements are applied to estimate the channel parameters, and the remaining measurements are used for key generation. Eventually, Alice and Bob extract a string of the secret key using information reconciliation and privacy amplification. According to the key establishment procedure of the CV-QKD system, both the parameter estimation and key establishment depend on the measurements of the receiver’s detector [32,33]. In order to collect the signal light accurately, Bob need to sample each pulse and integrate them together if the pulse period is longer than the photodiode response time [30]. This approach involves the data acquisition system with a high sampling rate, which results in more challenges for data processing. To avoid such a complex situation, an alternative method is proposed with an assumption that the optical pulse period is much short than the photodiode response time. In this case, the quadrature of the signal field is linearly proportional to the peak value of the balanced homodyne detector [34]. Therefore, it is deterministic whether or not Bob would sampling the accurate peak values. As shown in Fig. 1, the sampling clock of the data acquisition (DAQ) system is extracted from the LO using a beam splitter. It is one of the important factors to determine the accuracy of the pulse peak value. As known, there are two types of noises that reduce the accuracy of the DAQ system, i.e., the quantization noise and the clock jitter [35]. The former can be directly calculated as N q = (LSB)2 /12, where the LSB stands for the least significant byte of the DAQ system. With the factor that the value of N q is usually negligible for

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Fig. 2. (Color online.) The sampling noise of the signals acquisition due to the clock jitter.

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Table 1 The parameter definitions in Fig. 2.

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Parameter

Description

Ts t j Vp v T0 w0

Sampling period Clock jitter Peak value Jitter noise Pulse period Pulse width

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a high-precision system and it can not be controlled by Eve, we only confine our discussion to the clock jitter. The sampling process of the electric signal with clock jitter in the DAQ system is illustrated in Fig. 2 (see also Table 1). Here we assume the output signal of balanced homodyne detector to Gaussian shape [36]. Due to the presence of clock jitter t j , the kth sample will not be taken exactly at time kT s , but at kT s + t j . Thus the jitter noise  v reads

 v = s(kT s ) − s(kT s + t j ),

(1)

where s(t ) corresponds to the input signal that can be expressed by

s(t ) = V p e

− (t −μ2)

(2)

, δs2

where V p is the pulse peak, μ and denote the mean and variance of the Gaussian pulse, respectively. In order to derive the mean value and the variance of the input signal, the repetition rate f rep and the duty cycle R duty of pulse have to be determined. These two parameters are related by R duty = w 0 / T 0 , where w 0 is the pulse width and T 0 is the pulse period. As T 0 = 1/ f rep , the relationship can be rewritten as w 0 = R duty / f rep . In a practical CV-QKD system, the parameters f rep and R duty are given, and hence we can assume the mean value and variance of the input signal to μ = w 0 /2 and δs = w 0 /8, respectively. For a general sampling process, the synchronous clock is usually multiplied at the DAQ circuit to restore the signal pulses as authentic as possible. Therefore, the sampling frequency can be expressed as f samp = M f rep , where M is the multiple. After substituting the above equations, we obtain

Ts =

1 M R duty

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w0.

(3)

For the sampling the peak value of pulse at μ = w 0 /2. For example, with the assumption that the parameters R duty = 50% and M = 16, the resulting jitter noise can be derived as

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m m x y 1  ˆt = i =m1 i i , σˆ 2 = ( y i − tˆxi )2 . 2

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Fig. 3. (Color online.) The ratio of the measured peak and true peak as a function of the clock jitter.



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 v = s(μ) − s(μ + t j ) = V p ⎝1 − e



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Thus the ratio of the measured peak V p and true peak V p can be given by

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where w 0 = 12 f rep . Fig. 3 shows the jitter effect on the high-speed DAQ system. It is obvious that the ratio k decreases with the increasing jitter, which means the raise of measurement error. Compared with the different frequencies, we find that the high-speed system is more sensitive to the clock jitter. 3. Parameter estimation and security analysis

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For the parameter estimation of the practical CV-QKD experiment, Alice and Bob choose the number m = N − n pairs (xi , y i )i =1,2···m to estimate the secret key rate, where N is the total received photons and n represents the fraction of photons using for key generation. Where N is the total number of signals and n signals are used for key generation. As a way to analyze the security, we consider the covariance matrix of the states shared by the two legal parties. Based on the assumption that the m variables are centered on 0 so that x =  y  = 0, the parameters of the covariance matrix depend on the variances of Alice and Bob, i.e., x2  and  y 2 , respectively, and the covariance of Alice and Bob, xy . Therefore x and y satisfy the following relations [37,6]:

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 y  = η T V X + η T ξ + N 0 + v el ,

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where V X is the modulation variance, T and ξ represent the transmittance and excess noise of the quantum channel, respectively, N 0 is the shot-noise, η and v el are the efficiency and electronic noise of the homodyne detector, respectively. Based on the given parameters η and v el , there are four estimators (V X , N 0 , T , ξ ) included in three equations. Therefore, a linearly normal model in practice √ is proposed which reads y = tx + z, where t = η T and z is a centered normal distribution with variance σ 2 = t 2 ξ + N 0 + v el . The maximum-likelihood estimators tˆ and σˆ 2 can be obtained directly by the normal linear model [37], that is

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where t and σ 2 are the true parameters. Combining the estimators with their confidence intervals together, the transmittance T and the excess noise ξ can be estimated by T = tˆ2 /η and ξ = (σˆ 2 − N 0 − v el )/tˆ2 , respectively. The homodyne detector measurements could be decomposed into two parts, the maximum measured values ym and the electronic noise zel . Thus we obtain the linear model y = ym + zel , where ym and zel follow the independent centered normal distribution with variance η T V X + η T ξ + N 0 and v el , respectively. Because of the clock jitter effect, the measurements of homodyne detector, ym , may not be equal to the input values. However, the electronic noise would not be changed. Thus we define the sam = ky , where k is achieved in Eq. (5). Then vector pled value ym m  + z and the estimators are given by of Bob becomes y  = ym el

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As the shot noise N 0 is usually calibrated using optical power detection [38], which means it would not be affected by the clock jitter, i.e., N 0 = N 0 . Therefore, the excess noise can be expressed in shot noise unit (SNU). The transmittance T and the excess noise ξ which are used to estimate the system security become

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log2

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(12)

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It is notable that both the practical transmittance T and excess noise ξ are underestimated. In numerical simulations, the estimated excess noise, ξ  , vs jitter, t j , for different repetition frequencies is shown in Fig. 4. For a fixed repetition frequency, the excess noise decreases as the clock jitter increases. Moreover, this mis-estimated error of the excess noise is more obvious for the cases with higher repetition frequency. For example, when jitter is lower than 5 ps, the value of estimated excess noise with the repetition rate of 1 GHz drops more quickly than the case of 50 MHz. The practical security of the CV-QKD system against the collective attack is investigated in [37], the secret key rate in the finite-size analysis reads

n

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χtot + 1/ V X



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(14)

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Table 2 The parameters for the security analysis of the CV-QKD system. All the variances and noises are in shot noise unit.

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Fig. 4. (Color online.) The estimated excess noise as a function of jitter. The channel transmittance is set to 0.1 and excess noise ξ = 0.05.

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where χtot = χh + χ B /tˆ is the total added noise related to the input, χh = 1/tˆ − 1 + ξˆ is the channel added noise and χ B = (1 − η) v el /η is the detector added noise. The modulation variance V X can be obtained by V X = V A + 1 in a practical Gaussian modulated CV-QKD system, thus the covariance matrix between Alice and Bob can be written as



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γAB = ⎣ 

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( V A + 1)I

T min ( V 2A + 2V A )σz

T min ( V 2A + 2V A )σz [ T min ( V A + ξmax ) + 1]I

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⎤ ⎦,

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(15)

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where I = diag(1, 1) and σz = diag(1, −1), T min and ξmax are the minimum of T and the maximum of ξ , respectively. The values T min and ξmax can be obtained as [37]:

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T min = ⎝tˆ − z P E /2



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σˆ 2 ⎠ /η ,

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mV A

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ξmax = σˆ + z P E /2 √

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error function defined as

ˆ2

− N 0 − v el /(t N 0 ),

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1 erf( z P E /2 / 2) = 12 2 x 2 erf(x) = √2 0 e −t dt. π ˆ ˆ2

and erf(x) is the For a given imple-

mentation of CV-QKD system, t and σ can be computed directly. From the theoretical perspective, we can also assign them the expected values



ηT ,

(17)

E[σˆ ] = η T ξ + N 0 + v el . 2



Based on the covariance matrix in Eq. (15), the value of S BPEE can be computed by

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(16)



where z P E /2 follows 1 −

E[tˆ] =

P E

SBE =

2  λi − 1 G

i =1

2

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⎞2

σˆ

2

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i =3

2

,

(18)

where G (x) = (x + 1) log2 (x + 1) − x log2 x, λ1,2 are symplectic eigenvalues of covariance matrix which represents the states including Alice and quantum channel, λ3,4,5 are the symplectic eigenvalues of covariance matrix which characterizes the sates after Bob’s projective measurement. These symplectic eigenvalues can be expressed by

Fig. 5. (Color online.) Secret key rate as a function of distance for different repetition frequencies and clock jitters. In (a), the main figure and inset show the secret bounds for f rep = 50 MHz and f rep = 100 MHz, respectively. Curves from top to bottom represent different jitter 100 ps, 150 ps, 200 ps, respectively. In (b), the main figure and inset show the secret bounds for f rep = 500 MHz and f rep = 1 GHz, respectively. And the corresponding clock jitters from top to bottom are 15 ps, 20 ps, 25 ps, respectively.

λ21,2

1



2 1



= (A ±

λ23,4 = (C ± 2 λ5 = 1,

(19)

C 2 − 4D ),

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B ( V A + 1) + T min ( V A + ξmax + 1)

T min ( V A + ξmax ) + 1 + χh VA +1+



B χh

T min ( V A + ξmax ) + 1 + χh



log2 (2/ ¯ ) n

+

2 n

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(20)

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.

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The last part of Eq. (13), privacy amplification (n), can be expressed as [37]:

(n) = 7

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A χh +

B

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B = ( V A + 1)( T min ξmax + 1) − T min V A ,



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− 4B ),

A = ( V A + 1)2 + [ T min ( V A + ξmax ) + 1]2 − 2T min ( V 2A + 2V A ),

D=

106

111

A2

with the notations

C=

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log2 (1/ P A ),

(21)

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Fig. 6. (Color online.) Secret bit rate as a function of jitter for different repetition frequencies. The corresponding distance of (a), (b), (c) and (d) are 20 km, 25 km, 30 km and 35 km, respectively.

where ¯ and P A are the smoothing parameter and failure probability of the privacy amplification, respectively. As known, (n) mainly related to total photon number n, thus it can be equal to P E . Based on different repetition frequencies and the parameters in Table 2 which are standard in CV-QKD experiments [18], we investigate the impact of the clock jitter on the CV-QKD system, the results are illustrated in Fig. 5. Firstly, by observing the two figures separately, we can find that the system performance decreases with the rise of jitter noise. Moreover, with the comparison of the system performance of different repetition frequencies, the phenomenon is more obvious in the higher repetition rate situations. In particular, the clock jitter effects on the secret key rate may differ by orders of magnitude, for example, to obtain similar secret key rate, the clock jitter changes from 102 ps to 101 ps when system repetition rate increases from 50 MHz to 1 GHz. The bit rate of secret key in practical CV-QKD system, R, can also be calculated by

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R = f rep K .

(22)

In the certain distances of 20 km, 25 km, 30 km and 35 km, the simulations of R are illustrated in Fig. 6. It is obvious that the secret bit rate is decreasing with the jitter increasing. Particularly, the higher repetition even achieves a lower secret key bit rate when the clock jitter increases to large enough, for example, with the distance 20 km and the fixed clock jitter t j = 500 ps, the secret bit rate of f rep = 100 MHz is lower than the case of f rep = 50 MHz. In a result, from the theoretical perspective, the clock jitter is an inescapable loophole in the practical CV-QKD system, especially for high-speed situations.

4. Experimental realization

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In order to investigate the influence of the clock jitter in the practical CV-QKD system, we design a simplified experiment setup as shown in Fig. 7. In this implementation, the coherent light at wavelength 1550 nm and power 15 dBm is generated by a tunable laser source and modulated with two electro-optical amplitude modulators whose electro-optical bandwidth are both 12 GHz, the first modulator is used for modulation of pulse and the second one is used for modulation of Gaussian data. Subsequently, the modulated light is attenuated to quantum level and sent to Bob through 25 km fiber. At Bob side, the signals are received by a photodiode whose bandwidth is 300 MHz, and sampled using a 10-bit data acquisition system with bandwidth of 5 GSa/s and range of 1 V (it means that the quantization noise is around 7.95 × 10−8 V2 which can be neglected). In the electrical part, a 10-bit AWG whose maximum sample rate can reach 10 GSa/s is used to generate simultaneously analog signals of pulse and Gaussian data. Meanwhile, a clock signal which is produced by the AWG is used for the time base of the clock source. The outputs of the clock source is monitored using an oscilloscope with the sample rate of 50 GSa/s and sent to the DAQ system at Bob side. Based on this setup we perform two experiments to investigate the effect of the clock jitter. Firstly, the ratio of the measured peak and true peak in Eq. (5) is one of the key factors. In order to find out the relationship between the ratio and clock jitter, a pulse sequence with constant amplitude is measured based on three kinds of time bases. For detail, we remove the second amplitude modulator, only the first one is used to modulate the light to a pulse sequence with the repetition frequency from 10 MHz to 200 MHz and the duty cycle of 50%. Then three kinds of time bases is applied to create various

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Fig. 7. (Color online.) The simplified experiment setup to analyze the jitter effect. Alice sends the modulated coherent light and the clock synchronization signals to Bob. Eve applies a clock source to intercept the clock signals, and then regenerates them based on different time bases, i.e., Rb clock, internal clock and OCXO clock. In order to calculate the clock jitter of the fake clock signals, an oscilloscope is placed at the entrance of Bob’s side. Meanwhile, the fake clock signals are used for sampling the quantum signals in the DAQ system.

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Fig. 8. (Color online.) The ratio between measured and true peak as a function of the repetition frequency, and the comparison between the theory and experiment based on different clock jitters. The root mean square of 68 ps, 126 ps, and 200 ps correspond to Rb clock, internal clock and OCXO clock, respectively.

Fig. 9. (Color online.) The experimental variance as a function of the repetition frequency based on different clock jitters. The root mean square of 68 ps, 126 ps, and 200 ps correspond to Rb clock, internal clock and OCXO clock, respectively.

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jitter clock signals, namely internal clock of the AWG, OCXO time base and Rb time base. On account of the randomness of the clock jitter, it is represented by its root mean square which can be calculated based on the monitoring outcomes of the oscilloscope. As drawn in Fig. 8, the root mean square of the three time bases Rb clock, internal clock and OCXO clock are 68 ps, 126 ps and 200 ps, respectively. Based on all three time bases, the ratio between the measured peak and true peak decreases with the increasing repetition frequency. Moreover, the larger the clock jitter, the greater will be the influence of the ratio. However, the decreasing trend of the measured ratio tend to be slower than the theoretical values, the probable causes of the phenomenon come from the internal and external amplifier of the electronic parts. That is, as the pulse width lessens, the pulse amplitude significantly lessens too, which has been full discussed in [39]. This attenuation is a system error and can be overcome by post-compensation, thus it would not be covered here. Secondly, the affects of the clock jitter on the measured variance are further explored. We add the second amplitude modulator back to the setup to modulate the Gaussian data which is generated by the AWG. Corresponding to the first experiment, three time bases are used to be the reference clock of the clock source. In particular, the coherent light is modulated first to a constant pulse sequence with the duty cycle of 10% and the frequency from 15 MHz to 50 MHz, and immediately modulated to be the pulses

whose peaks follows Gaussian distribution. The results based on the measurements of the DAQ system in this experiment are illustrated in Fig. 9, it shows that the variance decreases with the increasing repetition frequency if the jitter exists. Another noteworthy phenomenon is the relationship of the variance and the clock jitter at the same frequency, that is, the higher the jitter noise, the less value of the variance is achieved. In addition, according to Eq. (9), one of the key factors is that Bob employ the variance of the shot noise, N 0 , to normalize the modulated variance, excess noise and electrical noise. However, the precise shot noise is k2 N 0 . Therefore, in order to fix the clock jitter vulnerability, the clock synchronization signals should be collected in time to obtain the true shot noise. Another method is canceling the clock transfer during key establishment and replacing by two independent clock sources for modulation and detection respectively. Nevertheless, both the approaches would sharply increase the system cost and complexity, or bring extra sampling errors.

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We have proposed the theoretical and experimental analysis of the clock jitter effect on the practical CV-QKD system. Compared to the reported loopholes of the classical LO and imperfect devices, the clock jitter is a random noise which exist permanently in the clock synchronization in a realistic CV-QKD system. In this paper, the clock jitter effect on the CV-QKD system has been char-

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acterized and analyzed in theory. Moreover, a simplified setup is designed to investigate the relationship between the clock jitter and data sampling. The results indicate that the clock jitter has a greater impact on the CV-QKD system with a higher repetition frequency. Therefore, with the rapid development in the field of the high-speed CV-QKD, we believe the clock jitter is a key security issue that has to be considered in the implementation of CV-QKD.

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Acknowledgements

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This work was supported by the Fundamental Research Funds for the Central Universities of Central South University and the National Natural Science Foundation of China (Grant Nos. 61379153, 61572529).

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