Practical continuous-variable quantum key distribution without finite sampling bandwidth effects

Huasheng Li; Chao Wang; Peng Huang; Duan Huang; Tao Wang; Guihua Zeng

doi:10.1364/OE.24.020481

1. Introduction

Quantum key distribution (QKD), an important application of the quantum information sciences, enables two correspondents to share a secret key, which in turn allows them to communicate with full security [1]. Its unconditional security is guaranteed by the Heisenberg’s uncertainty principle and the quantum no-cloning theorem. Compared with the discrete variable QKD, it has been shown theoretically that the continuous variable (CV) QKD has potential advantages, e.g., higher secret key rate and better compatibility with the current optical networks [2, 3]. Since it was proposed, the CVQKD has achieved great progress both in theory and experiment. The most valuable achievement is the Gaussian-modulated coherent state (GMCS) CVQKD protocol, which has been proven to be secure against general collective attacks [4–7] and coherent attacks [8–10]. Recently, long secure distance [11,12] and high speed [13,14] GMCS CVQKD have been experimentally demonstrated by using commercial components. Especially, field tests of the GMCS CVQKD protocol in telecommunication optical networks have also been successfully performed [15–17]. These pave the way for the practical application of CVQKD.

However, due to imperfections of the employed devices in the practical CVQKD system, there are deviations between the theoretical CVQKD protocol and its practical implementation. Recently, the practical security due to imperfections of the employed devices in the practical CVQKD system has attracted much attentions [18–23]. By far, several typical attack strategies, such as the local oscillator (LO) fluctuation attack, wavelength attack, LO calibration attack, saturation attack etc., have been found and their countermeasures have also been proposed.

Recently, it has been found that the finite sampling bandwidth (FSB) effects may influence the performance and the practical security of the practical CVQKD system [24]. In detail, the FSB of the employed analog-to-digital converter (ADC) will decrease the lower bound of secret key rate and restrain the linear relationship between secret key rate and system repetition rate. Moreover, the FSB effects open security loopholes for eavesdropper to implement attacks. To prevent the FSB effects, a countermeasure named dual sampling detection scheme was proposed in [24]. This scheme performs real-time parameter estimation by using two identical ADCs, which are triggered by the same electronic circuits. One ADC is used to sample the output of homodyne while the other ADC is used to sample the LO.

In this paper, we propose a more available scheme by making use of a dynamic delay adjusting module (DDM) and a statistical power feedback-control algorithm. In this way, the peak values of the transmitted pulses can be sampled accurately so that the FSB effects are removed. Subsequently, one can eliminate the deviations generated in the parameter estimation and the secret key rate calculation. In addition, the receiver, i.e., Bob, can detect easily any disturbances induced by environment or attackers through monitoring the statistical power of sampled data in real time. This helps to resist the practical attacks which do not change the distribution of receiver’s data. Clearly, the well-known LO calibration attack is invalid in the proposed scheme.

The remainder is organized as follows. In Section 2, we first recall the procedure of data acquisition in a practical GMCS CVQKD system, and then present the proposed high-precision data acquisition method. In Section 3, we investigate the effectiveness of the proposed scheme in removing the FSB effects. Finally, conclusions are drawn in Section 4.

2. High-precision data acquisition in practical CVQKD system

In the CVQKD protocol, the key information is generally encoded on continuous variables, i.e., the quadratures X and P, and usually transmitted using pulsed laser signal. To obtain the secret key, a homodyne or heterodyne detector at receiver’s side is employed to obtain the measurement results. As the output is an analog signal, a data acquisition card (commonly, the ADC) is necessary to obtain the transmitted quadratures, which are carried on the peaks of pulses. Theoretically, the sampling bandwidth of the employed device should be infinite so that all the selected quadratures are obtained accurately. However, it is impossible for Bob to sample all the peak values ideally, because the sampling bandwidth of commercial ADCs is finite. This results in the FSB effects, which will influence the performance and practical security of the involved CVQKD system. To remove the FSB effects, we propose a novel way of obtaining high-precision data in this section. Without loss of generality, we investigate how to remove the FSB effects in the GMCS CVQKD system.

2.1. Previous data acquisition process

Before presenting the proposed high-precision data acquisition scheme, we introduce briefly the data acquisition process in previous GMCS CVQKD system. Generally, the sender Alice chooses two random numbers X_A and P_A from a set of Gaussian random numbers with variance V_A and zero mean, and modulates those random numbers on pulsed laser signal by using an amplitude and a phase modulator. After having been modulated, the information is carried on the peaks of pulsed signal. Making use of the polarization-multiplexing and time-multiplexing techniques, the quantum signal together with the LO signal are sent to the receiver Bob through a quantum channel. The structure of Bob’s apparatus in previous CVQKD system is described in Fig. 1. The inputs of PBS are quantum signal and LO which are multiplexed both in time domain and polarization. A 10:90 beam splitter is used to split out 10 percent of the LO to generate clock signal and monitor the intensity of LO in real time. A phase modulator is used to perform basis selection with random phase ψ (0 or π/2). Then the demultiplexed LO and signal interface in a homodyne detector. After detection, the output analog signal of the homodyne detector is send to the data acquisition sub-system to obtain the transmitted data. In general, the data acquisition sub-system of Bob’s system is consist of two modules, i.e., a ADC data acquisition card and a data pre-processing module. The former is driven by the clock generated using LO, and used to sample the output signal of homodyne detector. The latter is used to handle the sampled data (such as frame synchronization [25]) to obtain raw key strings. Subsequently, the raw key strings are sent to the data post-processing sub-system, which includes reconciliation, decoding, error correction and privacy amplification procedures to generate the final secret key.

Fig. 1 The structure of Bob’s apparatus in previous CVQKD system. PBS: polarization beam splitter; PD: phtodiode; LO: local oscillator; PM: phase modulator; ψ: random phase of 0 or π/2; 10:90 and 50:50 (reflectivity:transmittance), beam splitters. Data Pre-Processing: a module of the data acquisition sub-system, used to obtain raw key from the output data of ADC; Data Post-Processing Sub-system: used to obtain final secret key, including reconciliation, decoding, error correction and privacy amplification procedures.

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For the data acquisition sub-system in a practical CVQKD system, to ensure the peak values of all pulses are accurately obtained by using sampling techniques, two conditions should be satisfied. Firstly, the clock of DACs (used to generate modulation signals) at Alice’s side and the sampling clock of ADC at Bob’s side should have a very high relative stability. Secondly, the relative position between the rising edge of the sampling clock and the peak position of pulse signal in each clock cycle should be aligned precisely. The first condition can be satisfied by using homologous clocks as shown in Fig. 1, in which the sampling clock is generated by the LO that transmitted from Alice. However, it’s nearly impossible for previous schemes to meet the second condition, because the sampling bandwidth of commercial ADCs is finite, which is limited to GHz level currently [26]. Under this condition, the transmitted data can not be obtained accurately. Thus, if Bob still use the linear relationship between the shot noise and the intensity of LO to calibrate the shot noise, he will underestimate the excess noise introduced by inaccurate peak sampling, and derive an overestimated secret key rate bound.

Clearly, how to obtain accurately the peak values of the pulsed signal is the core task for the data acquisition sub-system of Bob’s system. Moreover, this is also very important for guaranteeing the practical security and high performance of the involved practical GMCS CVQKD system.

2.2. CVQKD with high-precision data acquisition

In this subsection, a new CVQKD system with high-precision data acquisition is proposed to improve the sampling accuracy and remove the FSB effects. As shown in Fig. 2, a DDM is adopted at Bob’s side in the proposed system. More specifically, the DDM is used to adjust dynamically the timing of the clock signal to a specified value under the control of certain feedback signals. In addition, the DDM is set on the clock path instead of signal path to minimize the introduced excess noise to the quantum signal. Moreover, compared with previous schemes, a statistical power feedback-control algorithm is employed in the data pre-processing module, which is used to generate feedback signals to control the DDM. What should be noted is that the LO is monitored in real time and the shot noise N₀ is still measured by using optical power detection in the proposed system.

Fig. 2 The structure of Bob’s apparatus in the proposed CVQKD system. PBS: polarization beam splitter; PD: photodiode; LO: local oscillator; PM: phase modulator; ψ: random phase of 0 or π/2; 10:90 and 50:50 (reflectivity:transmittance), beam splitters; DDM: Dynamic delay adjusting module; Data Pre-Processing: a module of data acquisition sub-system, used to obtain raw key from the output data of ADC; Data Post-Processing Sub-system: used to obtain final secret key, including reconciliation, decoding, error correction and privacy amplification procedures.

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For a practical CVQKD system, after power up, the initial relative position difference (represented in time scale) between the rising edge of the sampling clock and the peak of pulse in each clock cycle is uncertain. Denote the initial relative position difference by t_init, the pulse period of the system by T_r, and the step size of DDM by t_s, which is the minimum time delay unit, then one has the following relationship,

T_{r} = M * t_{s} + Δ t,

where Δt ∈ [0, t_s], T_r = 1/f_r with f_r being the system repetition rate, and M is an integer. It means that if the timing of sampling clock can be adjust by the unit of t_s, Bob could sample the pulsed signal at M different positions within one pulse period through setting different clock delays of DDM. Under this condition, the statistical power value of the sampled data at a given position i (i ∈ [0, M − 1]) within a period of time T_c can be calculated by using the following ways, i.e.,

P_{i} = \sum_{j = 1}^{L} A_{i (j)}^{2},

and

t_{i} = i * t_{s}, i \in [0, M - 1],

where A_i(j) is the amplitude of the jth sampled value at the relative position i, L = T_c/T_r is the total number of sampling points within T_c, and t_i is the adjusted timing delay at the relative position i. Therefore, Bob can obtain M different statistical power values by setting the delay of the sampling clock to all M different timing delays.

Because the statistical power value of the sampled data at the peak position in each pulse period is the maximum one of all, i.e.,

\begin{array}{l} P_{\max} & = & \max {P_{i}, i = 0, 1, 2, \dots, M - 2, M - 1} \\ = & P_{m_{0}}, m_{0} \in [0, 1, 2, \dots, M - 2, M - 1] . \end{array}

Then

t_{final} = | t_{init} - m_{0} * t_{s} | = Δ t_{0},

where t_final is the final relative position difference between the rising edge of sampling clock and peak position in each pulse period, and Δt₀ ∈ [0, t_s/2]. It means that after m₀ steps’ timing adjustment, one may minimize the relative position difference between the rising edge of sampling clock and the peak position of pulse in each pulse period to Δt₀, and finally the peak values of all pulses can be sampled accurately.

Here, three aspects have attracted our attentions. Firstly, when t_s → 0 one has Δt₀ → 0, which means the sampling precision may be controlled by choosing appropriate t_s. Secondly, the longer Tc, the higher accuracy of statistical power value. However, there is a trade-off between accuracy and efficiency in a practical CVQKD system. Thirdly, the whole process of finding the optimal sampling position in each pulse period can be viewed as an initial calibration procedure of the system. Once this calibration procedure is completed, Bob can determine the optimal sampling position and monitor the corresponding statistical power in the whole process of quantum communication.

3. Effectiveness of high sampling precision

In this section, we investigate the effectiveness of the proposed high-precision data acquisition scheme in removing the FSB effects. Three aspects of the FSB effects are addressed, including the estimation of the excess noise, the calculation of the secret key rate bound, and the security loopholes for practical security.

3.1. Influence of high sampling precision on excess noise

In a GMCS CVQKD system, to obtain a better detection response, the bandwidth of the detection device should be significantly higher than the repetition rate of the pulsed laser signal [28]. Currently, bandwidth of the available homodyne detectors for quantum communication may reach 1 GHz [13, 29], which is much higher than the system repetition rates that reported yet [11, 13, 14], and guarantees us preferable detection response. Therefore, it’s reasonable to assume that the homodyne detector works in its linear region, and the response of the homodyne detector is linear with respect to the input optical filed quadrature. Under the circumstances, the pulse signals before and after detection are generally Gaussian with the following shape function [24,30],

r (t) = A_{p} e^{- {(t - μ)}^{2} / (2 σ^{2})},

where A_p is the peak value of the Gaussian pulse, μ and σ² are the mean and variance, respectively. Choosing μ = t_p/2 and σ² = (t_p/8)², where t_p = 1/f_p is the duration of each pulse, the amplitude of sampled data can be expressed as,

A_{s} = A_{p} e^{- 8 δ^{2}},

with δ ∈ [−t_s/t_p, t_s/t_p]. From Eq. (7), we can see that the precision of sampled data is only determined by the step size of the DDM, and the maximum normalized sampling error is

E_{norml} = 1 - e^{- 8 δ^{2}} .

Considering that the excess noise plays an important role, we firstly investigate the influence of sampling precision on the excess noise in a practical CVQKD system. After the quantum communication, Alice and Bob share N couples of correlated variables {(x_i, y_i)|i = 1, 2, . . . , N}. Here, x is the quadrature modulated by Alice, y is the quadrature measured by Bob, N is the total number of transmitted data, and N = m + n. Then, n couples of variables are randomly selected to estimate the quantities that required to compute the secret key rate, and the rest m pairs of variables are used for the key establishment. Consider the involved quantum channel as a normal linear model, the correlated variables shared by Alice and Bob can be expressed by

y = t x + z,

where

t = \sqrt{η T}

and z is the noise term which follows a centered normal distribution with variance σ_z² = N₀ + ηTξ + V_el. The involved parameter η is the detection efficiency, T is the channel transmittance, N₀ is the variance of shot noise, ξ is the excess noise, and V_el is the electric noise (all expressed in their respective units).

If the transmitted signals are obtained ideally, one has

〈 x^{2} 〉 = V_{A}, 〈 x y 〉 = \sqrt{η T} V_{A},

〈 y^{2} 〉 = η T (V_{A} + ξ) + N_{0} + V_{el},

where V_A is the modulation variance. The maximum-likelihood estimators t̂, σ̂_z² and V̂_A are known as [6,27]:

\hat{t} = \frac{\sum_{i = 1}^{m} x_{i} y_{i}}{\sum_{i = 1}^{m} {x_{i}}^{2}},

{\hat{σ}}_{z}^{2} = \frac{1}{m} \sum_{i = 1}^{m} {(y_{i} - \hat{t} x_{i})}^{2},

{\hat{V}}_{A} = \frac{1}{m} \sum_{i = 1}^{m} {x_{i}}^{2} .

These are independent estimators with the following distributions

\hat{t} ~ 𝒩 (t, \frac{{σ_{z}}^{2}}{\sum_{i = 1}^{m} {x_{i}}^{2}}),

\frac{m {\hat{σ}}_{z}^{2}}{{σ_{z}}^{2}}, \frac{m {\hat{V}}_{A}}{V_{A}} ~ χ^{2} (m - 1),

where t, σ_z² and V_A are the true values of the parameters. In the limit of large N, the confidence intervals for these parameters can be computed as

t \in [\hat{t} - Δ T, \hat{t} + Δ T],

{σ_{z}}^{2} \in [{\hat{σ}}_{z}^{2} - Δ {σ_{z}}^{2}, {\hat{σ}}_{z}^{2} + Δ {σ_{z}}^{2}],

V_{A} \in [{\hat{V}}_{A} - Δ V_{A}, {\hat{V}}_{A} + Δ V_{A}],

where

Δ T = z_{∊_{PE} / 2} \sqrt{\frac{{\hat{σ}}_{z}^{2}}{m V_{A}}}

,

Δ {σ_{z}}^{2} = z_{∊_{PE} / 2} \frac{{\hat{σ}}_{z}^{2}}{\sqrt{m}}

,

Δ V_{A} = z_{∊_{PE} / 2} \frac{{\hat{V}}_{A} \sqrt{2}}{\sqrt{m}}

and z_{∊_PE/2} satisfies

1 - \erf (z_{∊_{PE} / 2} / \sqrt{2}) = ∊_{PE} / 2

. ∊_PE is the probability that the estimated parameters do not belong to the confidence region computed from the parameter estimation procedure, and the error function erf(x) is defined as

\erf (x) = \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- t^{2}} d t .

Using the previous estimators, we obtain T and ξ with the following formulas

T = {\hat{t}}^{2} / η,

ξ = ({\hat{σ}}_{z}^{2} - N_{0} - V_{el}) / {\hat{t}}^{2} .

Actually, as implied by Eq. (8), there are deviations between the actual sampling values and the ideal sampling values. In the realistic model [28, 30], Eve has no ability to manipulate Bob’s detection devices. Thus, the detection efficiency η and the electronic noise V_el does not change during the whole QKD process. Moreover, when the shot noise N₀ is measured using optical power, if the intensity of LO does not change, N₀ is considered to remain unchanged whether sampling errors or attacks are introduced or not. Therefore, Eqs. (10) and (11) should be modified as follows when sampling errors are considered,

〈 x^{2} 〉 = V_{A}, 〈 x y_{real} 〉 = e^{- 8 δ^{2}} \sqrt{η T_{real}} V_{A},

〈 y_{real}^{2} 〉 = e^{- 16 δ^{2}} [η T_{real} (V_{A} + ξ_{real}) + N_{0}] + V_{el} .

It means that, if Alice and Bob do not aware the deviations introduced by the limited capability of data acquisition part in a practical CVQKD system, they will still use Eq. (10) and Eq. (11) to calculate the related parameters and the secret key rate. Accordingly, one has,

〈 x y 〉 = 〈 x y_{real} 〉,

〈 y^{2} 〉 = 〈 {y_{real}}^{2} 〉 .

Namely

\sqrt{η T_{est}} V_{A} = e^{- 8 δ^{2}} \sqrt{η T_{real}} V_{A},

η T_{est} (V_{A} + ξ_{est}) + N_{0} + V_{el} = e^{- 16 δ^{2}} [η T_{real} (V_{A} + ξ_{real}) + N_{0}] + V_{el},

and yield

T_{real} = e^{16 δ^{2}} T_{est},

ξ_{real} = ξ_{est} + \frac{1}{η T_{est}} (1 - e^{- 16 δ^{2}}) N_{0},

where the subscript ‘real’ indicates the true parameters that should be used in the secret key rate calculation procedure, while the subscript ‘est’ means the estimated parameters without considering the FSB effects.

Now we consider a common CVQKD system setup with a repetition rate of f_r. Generally, the duty cycle of the generated light pulses is less than 10% [11, 13, 15, 30] in a practical CVQKD system. Moreover, there is pulse broadening when the light pulses are transmitted in fiber channel, and it has been well studied in classical optical communications. In our case, the system repetition rate and the light power of pulses are relatively low, so we do not consider the nonlinear effect. Therefore, the pulse broadening factor can be expressed [31]

B_{f} (z) = {[1 + \frac{1}{4 σ^{4}} {(\int_{0}^{z} β_{2} (z) d z)}^{2}]}^{1 / 2},

and the group velocity dispersion β₂(z) = 0, then we have B_f (z) = 1, which means pulse broadening can be ignored in our condition. Therefore, we choose a typical value t_p = 1/(10 f_r) in the following analyses and simulations for simplicity, which does not loss of generality. Then Eqs. (29) and (30) may be rewritten as

T_{real} = e^{1600 {f_{r}}^{2} {t_{s}}^{2}} T_{est},

ξ_{real} = ξ_{est} + \frac{1}{η T_{est}} (1 - e^{- 1600 {f_{r}}^{2} {t_{s}}^{2}}) N_{0} .

Expressed in shot noise units, Eq. (33) can be rewritten as

ε_{real} = ε_{e s t} + \frac{1}{η T_{est}} (1 - e^{- 1600 {f_{r}}^{2} {t_{s}}^{2}}) .

Subsequently, the excess noise ε_fsb due to the FSB effects is given by

ε_{f s b} = ε_{real} - ε_{est} = \frac{1}{η T_{est}} (1 - e^{- 1600 {f_{r}}^{2} {t_{s}}^{2}}) .

The estimated excess noise as a function of the system repetition rate is shown in Fig. 3 with transmission distance L = 25 km and ε_real = 0.04. From top to bottom, the three dashed curves correspond to t_s = 5 ps, 10 ps and 20 ps, and the three solid curves correspond to f_samp = 5 GHz, 3 GHz and 1 Ghz, respectively. We find that the estimated excess noise ε_est decreases with the increase of system repetition rate. In previous schemes, ε_est decreases quickly even at the sampling rate of 5 GHz. While in the proposed scheme, the decrease rate of ε_est is much slow with the increase of system repetition rate. When t_s is chosen to 5 ps, the estimated excess noise ε_est decreases slightly even at a high system repetition rate of 45 MHz. The result indicates that the proposed high-precision data acquisition scheme can remove the deviation between the estimated excess noise and the real excess noise, which is induced by the FSB effects in the practical GMCS CVQKD system.

Fig. 3 The estimated excess noise as a function of system repetition rate. The transmission distance L = 25 km and ε_real = 0.04. From top to bottom, the three dashed curves are obtained with t_s = 5 ps, 10 ps and 20 ps, while the three solid curves are obtained with f_samp = 5 GHz, 3 GHz and 1 GHz, respectively.

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3.2. Influence of sampling precision on secret key rate bound

Next we address the influence of sampling precision on the secret key rate bound. After the parameters and their respective confidence intervals been determined, the secret key rate for collective attacks including the finite size effects can be computed as [6],

K_{finite} = \frac{n}{N} [β I (X : Y) - S_{∊_{PE}} (Y : E) - Δ (n)],

R_{finite} = f_{r} * K_{finite},

where β ∈ [0, 1] is the reconciliation efficiency, I(X:Y) is the amount of mutual information shared by Alice and Bob, S_{∊_PE} (Y:E) is the maximal value of the Holevo information compatible with the statistics except with probability ∊_PE and Δ(n) is related to the security of the privacy amplification [6].

Figure 4 shows the relationship between sampling precision and the secret key rate. The solid and dashed curves represent the secret key rates obtained in previous schemes and the proposed scheme, respectively. The secret key rate bound as a function of the transmission distance is shown in Fig. 4(a). Compared with the ideal data acquisition condition, the secret key rate bound has a very slight deviation in the proposed scheme (the four curves are overlapped and can not be distinguished clearly). While the secret key rate bound deteriorates a lot in previous schemes because of the FSB effects [24], even a very high sampling bandwidth (such as f_samp = 5 GHz) of ADC is used. Furthermore, as shown in Fig. 4(b), the linear relationship between the secret key rate and the system repetition rate is restrained in previous schemes, and the secret key rate reaches to a maximal value at a relatively low system frequency (about 5 MHz when f_samp is as high as 5 GHz). In comparison, the secret key rate increases proportionately with the system repetition rate in the proposed scheme. Clearly, the FSB effects on the secret key rate can be removed completely with the high-precision data acquisition approach.

Fig. 4 Secret key rates under different conditions. Parameters are set as: V_A = 5N₀, β = 0.95, ξ = 0.01N₀, V_el = 0.01N₀, ∊_PE = 10⁻¹⁰, N = 10⁹, η = 0.6. (a) The secret key rate bound of CVQKD system as a function of distance. The system repetition rate f_r =10 MHz, the three solid curves from left to right are obtained with f_samp = 1 GHz, 3 GHz and 5 GHz, corresponding to previous schemes. The three dashed curves are obtained with t_s =5 ps, 10 ps and 20 ps, corresponding to the new scheme. (b) The secret key rate of CVQKD system as a function of the system repetition rate. The transmission distance L = 50 km.

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3.3. Influence of sampling precision on security loopholes

Now we analyze the influence of sampling precision on the security loopholes which may compromise the practical security of the involved CVQKD system. For the practical security of CVQKD system, several practical attack scenarios have been proposed so far as addressed in the introduction. Some of these attacks can be resisted by simply inserting optical devices, for example, the wavelength attacks [19–21] can be prevented with a fiber Bragg grating at Bob’s side. Since the Gaussian distribution of the received data is necessary in GMCS CVQKD, here we mainly discuss the influence of sampling precision on the security loopholes, which do not change the distribution of Bob’s data. A practical attack which meets this requirement is the LO calibration attack.

As mentioned in Sec. 3, Bob’s detection devices are not accessible to Eve in the realistic model, so the transmitted X and P quadratures reserved by Bob are always carried on the peaks of the pulses. Suppose a t_dly delay is introduced on the pulse signal under the attack of Eve, then we have

A_{sdly} = A_{p} exp [- 8 {(2 t_{dly} + t_{s})}^{2} / {t_{p}}^{2}],

r = A_{sdly} / A_{s} = exp [- 32 t_{dly} (t_{dly} + t_{s}) / {t_{p}}^{2}],

P_{sdly} = \sum_{i = 1}^{L} A_{sdly (i)}^{2}, P_{s} = \sum_{i = 1}^{L} A_{s (i)}^{2},

R = P_{sdly} / P_{s} = r^{2} P_{s} / P_{s} = exp [- 64 t_{dly} (t_{dly} + t_{s}) / {t_{p}}^{2}],

where A_sdly is the amplitude value, P_sdly is the statistical power value of the sampled data, and they are obtained under the condition of pulse signal been shifted t_dly. While P_s is the statistical power value of the sampled data without this practical attack. We note that P_s is a calibrated value, which is obtained after the optimal peak sampling position in each pulse period has been determined. It is computed by using the statistical power method proposed in Sec. 2.2. Therefore, under the assumption that t_p = 1/(10 f_r), Eq. (41) can be expressed as

R = exp [- 64 t_{dly} (t_{dly} + t_{s}) / {(t_{r} / 10)}^{2}] = exp [- 6400 {f_{r}}^{2} t_{dly} (t_{dly} + t_{s})],

where R represents the relative change of statistical power with t_dly.

Simulations in Fig. 5 show the relationship between the statistical power and the delay introduced by Eve under different system repetition rates. From top to bottom, the curves are obtained with f_r = 10MHz, 25MHz, 50MHz, 80MHz and 100MHz, respectively. It’s clear that the sensitivity of statistical power increases quickly with the system repetition rate, and the delay t_dly has a significant effect on the statistical power of the sampled data. We note that the statistical power of the sampled data will change slightly even the practical attack, e.g., a calibration attack, is absent, because statistical errors can not be eliminated completely in the proposed scheme. Accordingly, the proposed method may not sensitive enough in a practical CVQKD system when the pulse shift is very small, e.g., several picoseconds. However, as implied by Eqs. (5)–(9) in [23], a successful LO calibration attack with a full intercept-resend implementation should satisfied the following inequality

\frac{{\hat{ξ}}_{calib}^{IR}}{N_{0}^{'}} = \frac{N_{0}}{N_{0}^{'}} [2.1 + \frac{1}{η T} (1 - \frac{N_{0}^{'}}{N_{0}})] \leq 0.1,

\frac{N_{0}}{N_{0}^{'}} \leq \frac{0.1 + \frac{1}{η T}}{2.1 + \frac{1}{η T}},

where

{\hat{ξ}}_{calib}^{IR}

is the excess noise introduced by the LO calibration attack and the full intercept-resend attack, N′₀ is the variance of the calibrated shot noise, N₀ is the variance of the true shot noise under the LO calibration attack in [23], and 0.1 is a typical value of excess noise represented in shot noise units. Then, for a channel transmittance T = 0.5 and a homodyne detection efficiency η = 0.5, we have

\frac{N_{0}}{N_{0}^{'}} \leq \frac{41}{61} \approx 0.6721 .

As analyzed in previous sections and in [24], the variance of true shot noise N₀ is affected by the shift of pulse signal in time domain, the relationship between N₀ and N′₀ satisfies the following equation

N_{0} = r^{2} N_{0}^{'} .

Combining the Eqs. (39)–(41) and Eq. (45), we have

\frac{P_{sdly}}{P_{s}} = \frac{N_{0}}{N_{0}^{'}} \leq \frac{41}{61} \approx 0.6721,

which means the statistical power of the received data will decrease more than 32.79% after the LO calibration attack is implemented.

Fig. 5 The relative change of statistical power with the change of the t_dly. Curves from top to bottom are obtained with f_r = 10 MHz, 25 MHz, 50 MHz, 80 MHz, 100 MHz and t_s is set to 10 ps.

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The result indicates that the LO calibration attack will introduce a large disturbance on Bob’s data. Therefore, Bob can perceive this disturbance from the perspective of statistical power of the received data, and re-evaluate the accuracy of peak sampling to determine the optimal sampling position in each pulse period. Thus, in this way, the LO calibration attack with full intercept-resend implementation can always be resisted by Bob.

4. Conclusion

In conclusion, we propose a novel scheme for practical CVQKD systems with a high-precision data acquisition method which consists of two parts, i.e., a dynamic delay adjusting module and a statistical power feedback-control algorithm. Making use of the proposed scheme, Bob can accurately obtain the peak values of the transmitted pulses, and completely remove the FSB effects in the practical CVQKD system. Moreover, with the help of the statistical power feedback-control algorithm, the proposed scheme may help to resist the practical attacks which do not change the distribution of Bob’s data. For instance, the typical LO calibration attack is invalid in the proposed scheme.

Funding

National Natural Science Foundation of China (NSFC) (61170228, 61332019, 61471239 and 61501290); National Program On Key Research Project of China (2016YFA0302600).

References and links

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Practical continuous-variable quantum key distribution without finite sampling bandwidth effects

Abstract

1. Introduction

2. High-precision data acquisition in practical CVQKD system

2.1. Previous data acquisition process

2.2. CVQKD with high-precision data acquisition

3. Effectiveness of high sampling precision

3.1. Influence of high sampling precision on excess noise

3.2. Influence of sampling precision on secret key rate bound

3.3. Influence of sampling precision on security loopholes

4. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (47)

Optics Express