High performance frame synchronization for continuous variable quantum key distribution systems

Dakai Lin; Peng Huang; Duan Huang; Chao Wang; Jinye Peng; Guihua Zeng

doi:10.1364/OE.23.022190

1. Introduction

In the wake of the profound progress of Quantum Key Distribution (QKD), one of its branches, the Continuous Variable QKD (CVQKD) has also experienced a fast growing period [1,2]. QKD schemes allow two communicating parities, conventionally called Alice and Bob, to share secret key strings with unconditional security. In CVQKD schemes, information is encoded on continuous variables of quantum states. Different from Discrete Variable QKD (DVQKD) [3], CVQKD provides advantages of using standard telecommunication fiber only and has a prospect of achieving higher speed and longer distance. Many implementations and field tests of CVQKD systems are proposed with different distances and secure key rates [4,5].

Generally, a CVQKD system consists of two parts, the quantum channel communication part and the post-processing part, which have the following steps. Alice prepares quantum state |α〉 = |x + ip〉 and sends it to Bob, where x and p are Gaussian-distributed quadratures and Alice encodes information X_i on x or p randomly. After quantum channel transmission, Bob receives state |β〉 = |x̂+ ip̂〉, where x̂ = x + n_x, p̂ = p + n_p and n_x, n_p are independent Gaussian noise induced by transmission. During detection, Bob randomly choose a basis(x or p) to detect each state. For those states that Bob detects with the same basis chosen by Alice, the detection yields Y_i = X_i + N_i where N_i is the noise. After Alice and Bob pick out those variables encoded and decoded with the same basis, they are sharing two Gaussian strings satisfying Y = X + N. The mutual information between X and Y is expressed as I(X : Y), from which secret key are extracted.

During these steps, frame synchronization is of significant importance. Different from the clock synchronization, frame synchronization is used to determine the first bit in each string. Note that elements in string X are independent from each other, failed synchronization may result in Bob receiving a string Y′ that is independent from X. As a result, failed synchronization diminishes the mutual information to 0, i.e., I(X : Y′) = 0, and it is almost impossible for Alice and Bob to agree on secret key strings. On the other hand, a reliable synchronization scheme is hard to design due to the channel model of practical CVQKD systems. Theoretically, the involved channel is additive white Gaussian noise channel (AWGNC) thus one has Y = X + N where N is Gaussian noise. Unfortunately, a practical channel may induce a random phase shift denoted by a Hamiltonian operator Ĥ = 2θâ ⁺ â. Phase shift changes the phase space of quantum states, subsequently, alters the detection results of Y. What’s more, recent CVQKD systems have been explored for quite long-distance scenarios [6] where SNR is quite low. As a result, the noise variance V_N is much larger than the signal variance V_X, making it impossible to recognize the shape of signal from Y.

Previously, many synchronization schemes have been proposed for proper performance in AWGNC. In one method, training frame is used to identify the first bit of each frame [7], but unfortunately, it is hard to synchronize correctly under low SNR scenarios. In another method, different training frames are constructed for longer distance [8] and is further improved for higher speed [9]. However, it is weak against phase shift of channel. In conclusion, these methods may have several drawbacks. Firstly, they are not reliable in low SNR scenarios and are unable to cope with the random phase shift. Secondly, the formations of synchronization frame in these schemes are different from the Gaussian distributed signal, making them easily to be distinguished and are open to attackers. Thirdly, some privileges should be offered in these schemes for their proper operation, e.g., synchronization frame should have stronger power or the basis used in detection should be constant. In a word, these synchronization methods are not reliable in bad channels and are not secure enough.

To overcome the drawbacks of previous synchronization schemes, we propose a new method in this paper which is operable under low SNR scenarios and is irrelevant to the phase shift and random basis selection procedure. Besides, a synchronization frame is constructed by this method that has the same distribution with quantum signal which is easier to implement.

The rest part of this paper will be organized as follows. Section 2 proposes this new synchronization method and section 3 analyzes its performance, including its accuracy, complexity and security. The performance will be shown in different scenarios. And a conclusion is drawn in section 4.

2. Synchronization for CVQKD

Conventional synchronization methods are amplitude-based, i.e. either to use match filter [10] or to recognize signal shape involves no phase information of signal pulses. As a result, these methods are “phase-relevant” which is fragile in CVQKD systems as the phase of received signal may shift during transmission. For example, the two quadratures of state |α〉, x and p, becomes x^ps = |α|cos(θ + Δφ) and p^ps = |α|sin(θ + Δφ) where $| α | = \sqrt{x^{2} + p^{2}}$ , θ is the origin phase angle of |α〉 and Δφ is the phase shift value. Obviously, phase shift may greatly influence the states received by Bob, and as Bob haven’t synchronize the received string yet, it is hard to implement phase compensation and recover the signal. For this sake, phase shift is a great trouble during synchronization procedure.

To overcome these difficulties, we consider to synchronize with a Gaussian-distributed signal. This signal, denoted as S, is public to Alice and Bob (also public to the eavesdropper, Eve) and is sent by Alice in the front of string X. At receiver’s side, Bob calculates the cross-correlation of S and the received signal Y for each pulse, i.e. cov(S, Y). According to the correlation theory, when the cross-correlation value reaches an obvious peak, which means S is aligned exactly with the noisy version of S in received signal, denoted as Ŝ, one can assert that the received signal is synchronized to S. Figure 1 shows the “correlation-peak” generated by calculating cov(S, Y). As can be seen, cov(S, Y) generates a peak just like cov(S, S), and channel noise may obscure the contrast between peak and other values. Fortunately, one may always get an obvious peak under different SNR values by adjusting the length of S. On the other hand, we note here that S is randomly constructed without any specific rules of formation, thus any Gaussian-distributed signal is available for synchronization. Furthermore, in order to generate a “phase-irrelevant” synchronization signal, we induce an extra step called phase disassembling, which transfers this arbitrary signal S into two parts, the amplitude part and the phase part.

Fig. 1 Demonstration of correlation peak. Figure 1 shows the self-correlation result of synchronization signal S, i.e. cov(S, S), and Fig. 2 shows the correlation result of cov(S, Y) with SNR = −6 dB.

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2.1. Phase disassembling

In CVQKD schemes, information are encoded on the quadrature position or the quadrature momentum of quantum pulses, and in a practical CVQKD system, the encoding process is achieved by using two modulators called amplitude modulator and phase modulator. Similarly, a synchronization signal S can be disassembled into the following two parts,

{\begin{matrix} \begin{array}{l} R_{i} = \max (S) \\ P_{i} = \cos^{- 1} (\frac{S_{i}}{\max (S)}) \end{array} & , i = 1, 2, \dots, L . \end{matrix}

where R and P are amplitude and phase part, respectively. One can easily tell that S_i = R_i · cos(P_i) and after transmitting S through a quantum channel with phase shift Δφ, Bob will receive Ŝ = R_i · cos(P_i + Δφ). As P is distributed in [0, π], Bob shall always receive Ŝ with an appropriate power.

We note here that the phase disassembling of Eq. (1) is not the only method. Setting amplitude part to a constant value is the simplest way with adequate performance. We will show more disassembling methods latter while Eq. (1) is clearest for exposition.

Phase disassemble overcomes phase shift by offering S with phase information, but one needs an extra step called phase matching to determine the correct synchronization position.

2.2. Phase matching

Phase disassemble insures that Bob will always receive Ŝ with appropriate power, but is unable to compensate the phase shift in Ŝ. Unfortunately, phase difference between S and Ŝ is destructive to their cross-correlation value as well as the synchronization. For example, given S = R · cos(P) and Ŝ = R · cos(P + Δφ), cov(S, Ŝ) follows,

\begin{array}{l} cov (S, \hat{S}) & = E {R \cdot \cos (P) \cdot R \cdot \cos (P + Δ φ)} \\ = E {R^{2}} \cdot E {\cos (P) \cdot \cos (P + Δ φ)} \\ \approx \cos (Δ φ) \cdot V_{A}, \end{array}

where Δφ denotes the phase difference between S and Ŝ. Clearly, Eq. (2) shows that cov(S, Ŝ) can be greatly affected by Δφ.

According to Eq. (2), consider Alice pre-adding a set of phase difference Δθ_k, k = 1, 2, ..., K in her frames, where Δθ_k is uniformly distributed in [0, π] and forms S′_k = R′_k · cos(P′_k − Δθ_k). By calculating the correlation value of each S′_k, Bob shall always get a correlation peak with S′_i when Δθ_i ≈ Δφ,

{\hat{S}}_{i}^{'} = R_{i}^{'} \cdot \cos (P_{i}^{'} - Δ θ_{i} + Δ φ) = R_{i}^{'} \cdot \cos (P_{i}^{'} + δ_{θ}) .

the maximum phase difference between S′_i and Ŝ′_i is

{\bar{δ}}_{θ} = \frac{π}{2 \cdot K}

and

cov (S^{'}, {\hat{S}}^{'}) = cov (\frac{π}{2 \cdot K}) \cdot V_{A}

. For example, minimum

cov (S^{'}, {\hat{S}}^{'}) = \sqrt{2} \cdot V_{A} / 2

when K = 2. As will be shown later, although larger K ensures larger cov(S′, Ŝ′), but K = 2 offers appropriate performance for most cases with low complexity.

3. Performance analysis

There are two key factors to valuate the performance of a synchronization method, one is “synchronous accuracy” and the other one is its complexity. Accuracy reflects the stability of synchronization, and complexity is quite important when one needs to implement the method to a resource-limited system. More importantly, low complexity ensures the instantaneity of synchronization which is an important issue. In this section, we will analyse the performances of this proposed method such as its accuracy, complexity, security and practical performance.

3.1. Synchronous accuracy

To analysis the accuracy (stability) of this synchronization algorithm, it’s essential to understand the cross-correlation process of signals. For this proposed algorithm, the synchronization is based on the fact that the cross-correlation value of synchronization signal S and received signal generates a peak when S is aligned precisely with its noisy version Ŝ in the signal. As a result, the cross-correlation value of cov(S, Ŝ) and cov(S, Y) is crucial to the synchronization, that’s because correct synchronization relies on the correct distinguish of cov(S, Ŝ) from cov(S, Y). However, the cross-correlation value of specific S may vary due to the randomly distributed noise and signal, so the accuracy analysis should be based on probabilities.

First, one need to consider the distribution of the product of Gaussian signals. That’s because the correlation value of cov(S, N) and cov(S, Y) are both related to this distribution. Mathematically, the product of two independent Gaussian variables $X ~ N (0, σ_{X}^{2})$ and $Y ~ N (0, σ_{Y}^{2})$ (here N(μ, σ ²) denotes a Gaussian distributed signal with mean value μ and variance σ ²) has the following distribution,

P_{Z} (u) = P_{X Y} (u) = \frac{K_{0} (\frac{| u |}{σ_{X} \cdot σ_{Y}})}{π \cdot σ_{X} \cdot σ_{Y}},

where K_n(z) is a modified Bessel function of the second kind. To calculate the expectation of Z, we have

E (Z) = \frac{1}{π \cdot σ_{Z} \cdot σ_{Y}} \int_{- \infty}^{\infty} u \cdot K_{0} (\frac{| u |}{σ_{X} \cdot σ_{Y}}) \cdot d u,

Theoretically, E(Z) = 0. But Z is practically of finite length thus

E^{'} (Z_{L}) = \sum_{i = 1}^{L} z_{i}

follows a distribution centered on 0, which is a Gaussian distribution follows

E^{'} (Z_{L}) ~ N (0, σ_{X}^{2} σ_{Y}^{2} L)

. Here E′(Z_L) reflects the correlation value of two independent Gaussian signals with finite length L, eg. given S with length L and the variance of signal and noise are V_A and V_N, respectively, one has

cov (S, N) ~ N (0, V_{A} V_{N} L) .

On the other hand, the self-correlation value of S is,

cov (S, S) = L \cdot E {S \cdot S} = L \cdot E {R^{2}} \cdot E {\cos^{2} (P)} = V_{A} L,

thus one has,

cov (S, \hat{S}) = cov (S, S) + cov (S, N) ~ N (V_{A} L, V_{A} V_{N} L) .

Similarly,

cov (S, Y) = cov (S, X) + cov (S, N) ~ N (0, (V_{A}^{2} + V_{A} V_{N}) L) .

A correct synchronization relies on that cov(S, Ŝ) > max(cov(S, Y)). Here the max(·) means to get the maximum value because synchronization procedure gets one value of cov(S, Ŝ) with numerous cov(S, Y). Although the value of a Gaussian distribution varies from (−∞, ∞), it can be practically bounded on the area ( $- \sqrt{3} σ$ , $\sqrt{3} σ$ ) with probability more than 99.9%. Thus, one can estimate a theoretical synchronization successful rate as

P_{suc} = P (cov (S, \hat{S}) > \max (cov (S, Y)))

= P (x > 0 | x ~ N (μ_{1}, σ_{1}^{2})),

μ_{1} = V_{A} L - \sqrt{3 V_{A} (V_{A} + V_{N}) L},

σ_{1} = \sqrt{V_{A} V_{N} L},

Furthermore, as will be analysed in section. 3.3, it is prefered to synchroinze with randomly chosen basis. As a result, when considering the random basis selection, nearly half of the pulses in S can be regarded as noise becaues variables received with different basis is independent from its sent value (the two quadratures of quantum states are independent from each other). So one has,

cov (S_{L}, {\hat{S}}_{L}) = cov (S_{L / 2}, {\hat{S}}_{L / 2}) + cov (S_{L / 2}, Y_{L / 2})

= cov (S_{L / 2}, S_{L / 2}) + cov (S_{L / 2}, N_{L / 2}) + cov (S_{L / 2}, Y_{L / 2})

~ N (\frac{1}{2} V_{A} L, V_{A} (3 V_{A} + 4 V_{N}) \frac{L}{2}),

where the subscript L denotes the length of the string, and the synchronization successful rate becomes,

{\ddot{P}}_{suc} = P (cov (S_{L}, {\hat{S}}_{L}) > \max (cov (S_{L}, Y_{L})))

= P (x > 0 | x ~ N (μ_{2}, σ_{2}^{2})),

μ_{2} = \frac{1}{2} V_{A} L - \sqrt{3 V_{A} (V_{A} + V_{N}) L},

σ^{2} = \sqrt{V_{A} (3 V_{A} + 4 V_{N}) \frac{L}{2})} .

Then, one can calculate the theoretical synchronization successful rate. The theoretical performance curves are presented later along with the experimental results.

In a practical system, instantaneity is an essential property for Bob. That’s because Bob can’t judge the synchronization position by searching the maximum match value after receiving the entire transmission string. What’s more, if Bob don’t even know whether Alice is sending messages, synchronization position found by Bob is of nonsense. A proper solution is that Bob sets a threshold 𝒯 and verdict synchronization when match value outnumbers the threshold. Consequently, the performance of synchronization is sensitive to this threshold. An appropriate threshold should meet two rules: First, match value should be larger than 𝒯 when S is aligned to Ŝ. Second, match value should be lesser than 𝒯 at any unsynchronized situation. The latter is more important than the former as miss-synchronization is more acceptable than error-synchronization. Obviously, a proper choice of the threshold is $𝒯 = \max (cov (S, Y)) = \sqrt{3 V_{A} (V_{A} + V_{N}) L}$ with the error-synchronization probability less than 0.01%. On the other hand, when picking a proper L for a synchronization scheme, one should refer to the theoretical precision results above.

3.2. Complexity reduction

An important factor for evaluating a synchronization scheme is its calculation complexity. Different from any other post-processing steps such as phase compensation and reconciliation, instantaneity is essential during synchronizing otherwise one need to storage all received signals. In that case, memory burden will increase with time and the system will finally crash. In this paper, the proposed method has higher complexity than original methods [7–9 ] and we present a method to reduce its complexity for practical instantaneity.

The primary calculation of the proposed synchronization method is the cross-correlation operation. It needs L multiplications and L − 1 additions for each pulse when the length of S is L. An obvious realization of synchronization is shown below, where L + T memory units(shift register) are employed with L multipliers and L − 1 adders. Here we use extra T memory units to match the time delay induced by multipliers and adders.

In Fig. 2, the L multipliers is quite costly for realization. In order to simplify the multiplications, we can quantize S to S̄ which is consisted of finite values, e.g., given a S ranging from (−4, 4), we can quantize it into 9 levels as {−4, −3, −2, −1, 0, 1, 2, 3, 4}. Besides, one have cov(C·S, Ŝ) = C·cov(S, Ŝ) and it is much easier to multiply with multiplier x = 2^N, N = 0, 1, 2, ... since these multiplications can be realized by N-step left-shift operation in binary field. Consequently, we can always quantize S into N + 1 levels for faster synchronization. e.g., given a Gaussian synchronization signal with arbitrary amplitude, one can pick a proper constant C such that C · S ∈ (−2^N, 2^N) and quantize C · S into 2N + 1 levels. Obviously, such quantization of S should induce quantization noise, which will influence the match value. However, such sacrificing of precision is worthy for reducing the complexity.

Fig. 2 Realization of synchronization scheme

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As a result, during a hardware implementation of this scheme, the complexity of multiplier can be reduced to the level same to the adder. The time delay of synchronization is T = 1 + ⌈log₂ L⌉ (clocks), and the instantaneity is insured by the flow structure of circuit (for each new pulse, signal flows through the circuit from up to down). For example, when L = 512, the time delay induced by synchronization is T = 1 + log₂ (512) = 10. We need 512 left-shift operators, 511 adders and a shift register with L + T = 522 memory units. We note that such scale of memory and logical resource cost is acceptable in most platforms.

3.3. Security analysis

Although CVQKD systems are theoretically secure, security loopholes have been found in many systems [11,12]. Similar to clock synchronization, most known frame synchronization schemes may experience attacks through security loopholes induced by inappropriate calibration routines. That’s because the synchronization frames of known methods are specially constructed or transmitted with strong power, making them easy to be distinguished and can be modified by Eve without being spotted.

Fortunately, the synchronization method proposed in this paper is secure because the synchronization frame has the same distribution with information string X and both of them are quantum signals. As a result, it’s impossible for Eve to distinguish or modify S without detecting the quantum signals. However, transmitting S in quantum level causes other problems. One is the random basis selection process during Bob’s detection. In such cases, Alice chooses two sets of measuring bases which produces two sets of orthogonal states, and transmits signal with random basises. Bob also chooses basises randomly for detection, and only when Bob’s choice is the same with Alice can Bob receive the signal correctly. Signals received by wrong basis can be regarded as independent from the sent value and it’s obvious that, independent value of sent and received signal cannot be used for synchronization. Most classical synchronization schemes solve this problem by setting a constant modulation and measurement basis. However, Eve is able to know this basis and modify synchronization frames with intercept-resend attack. In order to keep S invisible, the modulation and measurement of S should also be done with randomly chosen basis. As a result, there are averagely half of the basises being correctly chosen by Bob, and the cross-correlation of S and Ŝ is decreased by half. This phenomenon, which may reduce the performance of synchronization, can be overcame by increasing the length of S.

Besides, improper phase disassemble undermines security as well. As shown in Eq. (1), phase part P is not uniformly distributed. As a result, different phase shift Δφ may lead to different power of S and make it conspicuous. To solve this problem, one can pick a phase disassemble method which yields uniformly distributed phase, and luckily, a Gaussian variable is always able to be disassembled into a Rayleigh distributed amplitude and uniformly distributed phase,

{\begin{matrix} \begin{array}{l} R_{i} = \sqrt{S_{i}^{2} + G_{i}^{2}} \\ P_{i} = \cos^{- 1} (\frac{S_{i}}{\sqrt{S_{i}^{2} + G_{i}^{2}}}) \end{array} & , i = 1, 2, \dots, L \end{matrix} .

where G is a random Gaussian variable which is independent from S but has the same distribution with S. Eq. (17) shows a method offering uniformly distributed phase and Rayleigh distribution amplitude. Consequently, Eve is unable to distinguish S from X anymore, because S and X have same distribution. Meanwhile, it is easy to note the existence of Eve if Eve perform the same synchronization process as Bob, which is because the synchronization of Eve will increase the noise in S.

3.4. Performance test

The performance of this proposed synchronization method can be characterized by the synchronization success rate against different SNR values and phase shift values. Figure 3 shows the successful rate to SNR, which is highly relevant to the communication distance, and different color represent different length of S.

Fig. 3 Synchronization success rate for SNR from −10dB to −20dB. From the bottom to the top we used the following length of S: 1024, 2048, 4096, 8192. The solid curves are the theoretical performance line according to section 3.1, the dash lines with triangle markers are practical performance without quantization and the dash lines with square markers are practical performance with S quantized to 9 levels.

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In Fig. 3, we can notice that the synchronization success rate is highly related to L and SNR while larger L provides more reliable synchronization. In a practical scheme, an estimation of communication channel is applicable in order to choose an appropriate value of L according to the theoretical performance.

In section. 2, we mentioned that phase matching can be realized by preset 2 synchronization frames representing Δφ = 0, π/2 (π and 3π/2 are the corresponding negative version). As a result, the maximum phase difference between S and Y is π/2/2 = π/4 which results in the success rate drop shown in Fig. 4. Definitely, preset more synchronization frames may improve the performance against phase shift, but it will increase the complexity and it’s better to extend the length of S than to preset more phase biases.

Fig. 4 Synchronization success rate for Δφ from 0° to 180°. The corresponding length of S from the bottom to the top is: 512,1024,2048.

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4. Conclusion

In this paper, we proposed a high performance synchronization scheme for CVQKD systems which provides reliable synchronization under low SNR situations. A theoretical analysis and a model for implementation are presented, along with a method to reduce the complexity of this scheme. The result of accuracy analysis shows this method outperforms any known synchronization technologies. As a result, this synchronization method is able to offer more reliable synchronization for long-distance CVQKD systems.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants No. 61170228. No 61332019 61471239) and the Hi-Tech Research and Development Program of China (Grant No: 2013AA122901).

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High performance frame synchronization for continuous variable quantum key distribution systems

Abstract

1. Introduction

2. Synchronization for CVQKD

2.1. Phase disassembling

2.2. Phase matching

3. Performance analysis

3.1. Synchronous accuracy

3.2. Complexity reduction

3.3. Security analysis

3.4. Performance test

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (21)

Optics Express