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Abstract
The continuous-variable quantum digital signature (CV-QDS) scheme relies on the components of quantum key generation protocol (KGP) to negotiate classical signature, which is more compatible with optical fibers. Nevertheless, the measurement angular error of heterodyne detection or homodyne detection will cause security issues when performing KGP in the distribution stage. For that, we propose to utilize unidimensional modulation in KGP components, which only requires to modulate single quadrature and without the process of basis choice. Numerical simulation results show that the security under collective attack, repudiation attack and forgery attack can be guaranteed. We expect that the unidimensional modulation of KGP components could further simplify the implementation of CV-QDS and circumvent the security issues caused by the measurement angular error.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the realization for Shor’s algorithm [1] on quantum computers, the digital signature (DS) is no longer secure since its security depends on unproven computational assumptions [2,3]. In contrast, Gottesman and Chuang proposed a scheme of quantum digital signature (QDS), whose security is guaranteed by quantum mechanics [4]. However, this scheme requires long-term quantum memory and SWAP test, which makes experiments difficult. Subsequently, Ref. [5] utilized an optical multiport, while the quantum memory is also required. Ref. [6] proposed to adopt the components of KGP in the distribution stage, which not only does not need to use quantum memory, but also can further expand the application range of QDS. Considering the security of practical implementations, Ref. [7] proposed the QDS scheme based on the measurement-device-independent structure [8–10], which can effectively avoid side channel attacks. Besides, some other QDS schemes [11–13] based on key generation protocol (KGP) components have also been proposed.
To be specific, above QDS schemes are based on components of discrete-variable (DV) KGP, which adopts the coherent pulses to encode on the polarization of a single photon [14–21]. Even though, imperfections of single-photon detection can still be exploited to attack the system. Different from DV-QDS scheme, Thornton et al. adopted the components of continuous-variable (CV) KGP for QDS scheme [22]. In particular, CV-KGP employs the weak coherent pulses for encoding and coherent detection to measure quantum signals, which is more compatible with optical fibers [23–25]. Practically, in coherent detection, whether heterodyne detection or homodyne detection, measurement angular error will be introduced. In heterodyne detection, $\widehat {X}$ quadrature and $\widehat {P}$ quadrature require to measure simultaneously. For the $\widehat {P}$ quadrature, an optical phase shifter is required to constantly rotate the local oscillator pulse phase to achieve heterodyne detection [26]. Homodyne detection requires to randomly measure one of two quadratures through the process of basis choice [27]. Both of the above processes will refer to the deviation of phase, namely measurement angular error.
For CV-QDS protocol, we propose to utilize unidimensional modulation in the KGP components, whose security has been demonstrated [28–31]. For one thing, the unidimensional modulation only requires single quadrature modulation of weak coherent pulses, which can provide simplified implementation; for another, it does not need the process of basis choice, so there without measurement angular error. Without loss of generality, QDS considers a nontrivial set that contains three participants. In the distribution stage, Alice creates two Gaussian random strings and forms the corresponding quantum states. For quantum states that send to Bob, Alice only modulates $\widehat {X}$ quadrature while $\widehat {P}$ quadrature is not; for another that send to Charlie, Alice only modulates $\widehat {P}$ quadrature while $\widehat {X}$ quadrature is not. In the messaging stage, Alice sends the classical triplet and passes it to Charlie via Bob.
This paper is organized as follows. In Sec.2, we analyze the influence of measurement angular error on the performance of the CV-QDS scheme, both in heterodyne detection and homodyne detection. In Sec.3, we describe the structure of the CV-QDS protocol with unidimensional modulation and analyze its security, such as collective attack, repudiation attack and forgery attack are considered. Finally, conclusions are drawn in Sec.4.
2. Measurement angular error in CV-QDS protocl
In the CV-QDS, the ultimate purpose of signature is to safely deliver messages from Alice to Bob and Charlie. In the distribution stage of CV-QDS, the quantum signature is generally obtained through the key generation protocol, which is essentially the quantum part of the CV-QKD protocol without the post-processing [32–34]. Firstly, Alice sends coherent states to quantum channels, and then two receivers perform heterodyne detection or homodyne detection on the received states. After negotiation, the measurement results are signed by quantum signatures. However, whether heterodyne detection or homodyne detection, measurement angular error will be introduced in practice, which will lead to the security problems. During the messaging stage, three participants negotiate to obtain the classical signature, and that can be utilized for identity authentication. After the authentication is passed, the message will be delivered.
2.1 Practical security analysis in heterodyne detection
When the receiver adopts heterodyne detection in the CV-QDS protocol, he needs to measure $\widehat {X}$ quadrature and $\widehat {P}$ quadrature simultaneously. For the $\widehat {P}$ quadrature, an optical phase shifter is required to constantly rotate the local oscillator pulse phase to achieve heterodyne detection. Affected by external force, various factors in optical fibre system such as refractive index and geometrical size are susceptible to change, it will cause difficulties for optical phase shifter to perfectly rotate the phase by 90$^{\circ }$ in experimental implementations [26]. The measurement angular error in heterodyne detection refers to the deviation of the local oscillator phase by 90$^{\circ }$. In what follows, the interpretation of measurement angular error and parameter estimation model is described in detail.
Figure 1(a) depicts the entanglement-based (EB) scheme of CV-KGP with heterodyne detection [35]. In this scheme, Alice prepares a EPR state with variance $V$, while she stores the mode $A_0$ for heterodyne detection and injects the other half $B_0$ into the quantum linear channel controlled by Eve. In the equivalent prepare-and-measure (PM) scheme, Alice prepares Gaussian random numbers with the corresponding $\widehat {X}$ quadrature and $\widehat {P}$ quadrature, satisfying with $\widehat {X}=I_s \mathrm {cos}\phi _s$ and $\widehat {P}=I_s \mathrm {sin}\phi _s$, where two parmeters $\phi _s$ and $I_s$ represents the phase modulator information and intensity modulator information respectively, as depicted in Fig. 2(a). The covariance matrix of the system is [26]
Fig. 1. The entanglement-based model of the CV-KGP. (a) Heterodyne detetcion. (b) Homodyne detection. EPR, two-mode squeezed vacuum state; QM, quantum memory; PS, phases shift; BC, basis choice; RNG, random numbers generator.
Fig. 2. Three modulation modes of quantum states. (a) Gaussian modulation. (b) Unidimensional modulation with $\widehat {X}$ quadrature. (c) Unidimensional modulation with $\widehat {P}$ quadrature.
The secret key rate in CV-KGP is defined as
where $\beta$ denotes the reconciliation efficiency for reverse reconciliation. To be specific, $I_{AB}$ represents the mutual information between two sides, which can be expressed as [26]2.2 Practical security analysis in homodyne detection
When the receiver adopts homodyne detection in the CV-QDS protocol, he needs to randomly measure one of two quadratures through the process of basis choice. To be specific, a digital-to-analog convertor (DAC) and a phase modulator are utilized to realize the procedure of basis choice. Nevertheless, it’s difficult for receiver to accurately control the phase modulator for 0 or $\frac {\pi }{2}$ phase since the finite bandwidth of the DAC [36]. Such finitenesses bring about the measurement angular error during procedure of basis choice. In what follows, an interpretation of measurement angular error of homodyne detection is described in detail.
It is known that the modulated phase can be evaluated as
where $v_{\pi }$ denotes the half-wave voltage, $v_m$ is the modulation voltage. However, it’s difficult for phase modulator to perfectly modulate 0 or $\frac {\pi }{2}$ in experimental implementation. On this condition, the actual modulation phase $\phi '$ will deviate from the expected value, which leads to In the DAC, LG denotes the loop gain of operational amplifier. Furthermore, the expectation transmittance and excess noise are redefined as [27]The EB scheme of the CV-KGP with homodyne detection is depicted in Fig. 1(b), where Alice prepares ERP state and send one mode $B_0$ to the channel, and state $\gamma _{A_0B_2}$ has the covariance matrix
2.3 Simulation results
In this subsection, we employ numerical simulation to explore how $\theta$ affects the performance of the scheme. Figure 3 shows the secret key rate $K$ as a function of the transmission distance $L$. Specifically, Fig. 3(a) is the simulation reuslts for heterodyne detection, from left to right, the four solid lines denote to different $\theta =15^{\circ },10^{\circ },5^{\circ },0^{\circ }$ with respective, and the dotted line denotes the PLOB bound [38,39]. Figure 3(b) is the simulation results for homodyne detection, form left to right, the four solid lines correspond to $\theta =6^{\circ },4^{\circ },2^{\circ },0^{\circ }$ respectively. For one thing, the secure $K$ and $L$ increase as $\theta$ decreases. For another, with the same $\theta$, heterodyne detection is obviously outperformed by homodyne detection, which can gain longer secure distance and higher key rates.
Fig. 3. The secret key rate as a function of the transmission distance for (a) heterodyne detection and (b) homodyne detection, with different measurement angular error $\theta$. Other parameters are set as $V=20$, $\beta =0.95$ and $\epsilon =0.01$.
Then, we reveal how measurement angular error affects the maximum transmission distance. The plot of Fig. 4 shows the transmission distance versus the measurement angular error for heterodyne detection and homodyne detection. From top to bottom, the four colors of lines correspond to different excess noise $\epsilon =0.01,0.03,0.05,0.07$ with respective. For one thing, when measurement angular error increases, the maximum transmission distance decreases evidently. For another, the maximum transmission distance with imperfect angular derivation seems to be sensitive to changes in excess noise that both in heterodyne detection and homodyne detection. Accordingly, excess noise is a key parameter to direct impact the maximum transmission distance. Besides, as depicted in Fig. 4(b), the maximum transmission distance for homodyne detection decreases sharply with the increase of $\theta$ derivation compared with Fig. 4(a).
Fig. 4. The maximum transmission distance as a function of measurement angular error $\theta$ for (a) heterodyne detection and (b) homodyne detection, with different excess noise $\epsilon$. Other parameters are set as $V=20$ and $\beta =0.95$.
Shown in Fig. 5, we plot the secret key rates versus modulation variance. The secret key rates at the transmission distance of 10 km is denoted by dotted lines, meanwhile the distance of 40 km is denoted by solid lines. We can see that, from the transmission distance of 10 km to the distance of 40 km, the secret key rate reduces evidently and the optional region of modulation variance is gradually compressed. Moreover, the feasible range of modulation variance and the optimal secret key rate for heterodyne are much larger than that for homodyne detection. From three-dimensional Fig. 6, we can see that the two parameters, measurement angular error $\theta$ and modulation variance $V$ directly decide the impact of secret key rates. In addition, the maximum key rate in heterodyne detection is higher than that in homodyne detection, and the feasible regions of $\theta$ and $V$ are also greater than that in homodyne detection.
Fig. 5. The secret key rate as a function of the modulation variance for (a) heterodyne detection and (b) homodyne detection. Transmission distance $L$ are set to 10 km and 20 km. Other parameter is set as $\beta =0.95$, and SUN represents the shot noise variance.
Fig. 6. Secret key rate versus measurement angular error $\theta$ and modulation variance $V$ at the transmission distance of 10km for (a) heterodyne detection and (b) homodyne detection.
3. Scheme of unidimensional modulation for CV-QDS
3.1 Protocol description
In this section, we illustrate our CV-QDS scheme, in which the key generation process adopts unidimensional modulation technology. The scheme is depicted in Fig. 7 and described as follows.
Fig. 7. The CV-QDS protocol. Solid lines, dot-dashed lines and dashed line are denotes the quantum channels, public classical channels and encrypted classical channel, respectively.
Distribution stage 1-4
Step 1. A signed one-bit message needs to be sent from Alice to Bob and Charlie. First of all, Alice creates two different Gaussian random strings, string $\{CS_B\}_j^l = \{X_{B_j}\}_{j=1}^l$ for Bob and string $\{CS_C\}_j^l = \{P_{C_j}\}_{j=1}^l$ for Charlie. Here, $l$ represents the signature length.
Step 2. For one string $\{CS_B\}_j^l$, Alice prepares the corresponding quantum states $\otimes _{j=1}^l |X_{B_j}+iP_B\rangle$, that only the $\widehat {X}$ quadrature is modulated, as depicted in Fig. 2(b). If the value of $X_{B_j}$ greater than $0$, the corresponding signature element owned by Alice can be noted as $QA_{B_j}=1$. Otherwise, $QA_{B_j}=0$. Therefore, the quantum signature for Bob prepared by Alice is $\{QA_{B_j}\}_{j=1}^l$. For another string $\{CS_C\}_j^l$, Alice prepares the corresponding quantum states $\otimes _{j=1}^l |X_C+iP_{C_j}\rangle$, that only the $\widehat {P}$ quadrature is modulated, as depicted in Fig. 2(c). Similarly, the quantum signature for Charlie prepared by Alice is $\{QA_{C_j}\}_{j=1}^l$. And then, the prepared quantum signatures are sent to Bob and Charlie through the quantum channels respectively.
Step 3. Two receivers measure the quantum states by homodyne detection and form new signatures. This is due to the presence of channel loss and possible external interference, making it possible for the discrepancy between the quantum signatures sent by Alice and the signatures received by the two receivers. As for Bob, he measures each of received states $|X_{B_j}+iP_B\rangle$ in $\widehat {X}$ quadrature, mapping the result greater than $0$ into the received signature $QS_{B_j}=1$ and the result no more than $0$ into $QS_{B_j}=0$. Therefore, Bob can get his received signature $\{QS_{B_j}\}_{j=1}^l$. As for Charlie, he measures each of received states $|X_C+iP_{C_j}\rangle$ in $\widehat {P}$ quadrature, and the mapping rules are the same as Bob’s. In the same way, Charlie can get the received signature $\{QS_{C_j}\}_{j=1}^l$.
Step 4. Bob sends random $l/2$ elements $\{QS_{B_j}^f\}_{j=1}^{l/2}$ to Charlie, while Charlie sends random $l/2$ elements $\{QS_{C_j}^f\}_{j=1}^{l/2}$ to Bob. This process is performed over a classical encrypted channel. The remained elements are denoted as $\{QS_{B_j}^k\}_{j=1}^{l/2}$ and $\{QS_{C_j}^k\}_{j=1}^{l/2}$ for Bob and Charlie. Hence, Bob’s final signatures are denoted as $\{SK_{B_j}\}_{j=1}^l:=\{\{QS_{B_j}^k\}_{j=1}^{l/2},\{QS_{C_j}^f\}_{j=1}^{l/2}\}$, and Charlie’s final signatures are denoted as $\{SK_{C_j}\}_{j=1}^l:=\{\{QS_{C_j}^k\}_{j=1}^{l/2},\{QS_{B_j}^f\}_{j=1}^{l/2}\}$.
Messaging stage 5-7
Step 5. Alice sends the classical triplet through the public channel to Bob. The classical triplet consists of three parts, one-bit message $m$, classical information $\{CS_B\}_j^l$ correlated with Bob, and $\{CS_C\}_j^l$ correlated with Charlie.
Step 6. Bob decodes his final signatures $\{SK_{B_j}\}_{j=1}^l$, then compares the results with the elements in $\{CS_B\}_j^l$, and records the mismatching rate. The protocol aborts when the mismatching rate for both halves of his signature are higher than $t_bl/2$. Otherwise, it will continue.
Step 7. Charlie received the classical triplet from Bob and also records the mismatching rate. The protocol aborts if the mismatching rate for both half of signatures are higher than $t_cl/2$. Othewise, the protocol will succeed. In particular, parameters $t_b$ and $t_c$ are the security threshold to protect the protocol against repudiation, satisfied with $0<t_b<t_c<1/2$.
3.2 Security analysis
3.2.1 Collective attacks
We analyze the security of CV-KGP with unidimensional modulation under Eve’s collective attacks in this subsection. As depicted in Fig. 8, Alice prepares a EPR state and performs homodyne detection on mode $A_0$ and squeezing mode $B_0$ with squeezing parameter $-\mathrm {log}\sqrt {V}$. After squeezing, state $\rho _{A_0 B_1}$ has the covariance matrix with
Fig. 8. The entanglement-based model for CV-KGP with unidimensional modulation. EPR, two-mode squeezed state; QM, quantum memory; Sqz, squeezer.
The secret key rate also can be defined in Eq. (6). The mutual information $I_{AB}$ can be calculated as
As mentioned above, Alice modulated the $\widehat {X}$ quadrature while the $\widehat {P}$ quadrature is not, so that the transmittance $T_p$ and excess noise $\epsilon _p$ cannot be estimated during the process of parameter estimation. Even though, the two unknown parameters are still bounded by Heisenberg uncertainty principle, where [40]
with the notationFig. 9. The impact of $T_p$ and $\epsilon _p$ on the positive secret key rate and secure region. Here, we set $V_M=3$, $T_x=0.1$, $\epsilon _x=0.01$ and $\beta =0.95$.
As for quantum channel between Alice and Charlie, the state $\rho _{A_0 C_1}$ with variance $V$ can be written as
Considering the loss rate $\alpha =0.2$ dB/km, the realistic transmittance are $T_{x,p}=10^{-\alpha L/10}$, where $L$ is the transmission distance. As depicted in Fig. 10, from left to right, the four colors of dotted lines correspond to unidimensional modulation to different variance $V=18,16,14,12$, the green solid line represents Gaussian modulation with variance $V_M=19$, and the black dot-dashed line denotes the PLOB bound. It should be noticed that the unidimensional modulation with $V_M=12$ has a similar performance with Gaussian modulation with $V_M=19$. In other words, the unidimensional modulation is similar to the Gaussian modulation with a relatively large variance. Or more accurately, the secret key rates of unidimensional modulation are lower than Gaussian modulation, since unidimensional modulation utilizes one quanrature to modulate useful information, while Gaussian modulation utilizes both of two quadratures.
Fig. 10. Secret key rate versus channel losses both for unidimensional modulation and Gaussian modulation. The basic parameters are set as $T_x=T_p$, $\epsilon _x=\epsilon _p$ and $\beta =0.95$.
3.2.2 Security against repudiation and forgery
In addition to resist collective attacks, the proposed scheme also needs to be secure against repudiation and forgery. In this subsection, we show that the scheme is secure against both repudiation and forgery attacks.
Robustness. Considering the presence of excess noise in the quantum channel, even if all three participants are honest, there will be a signature mismatch rate of $p_{err}$. Based on Hoeffding’s inequalities, the probability that the number of mismatches for Bob or Charlie is above $t_bl/2$ or $t_cl/2$ can be bounded as [41]
where the parameters satisfy $p_{err}<t_{b,c}$.Security against repudiation. When Alice performs the repudiation attack, she needs to convince Bob to accept the message while making Charlie reject it. Hence, Alice needs to forge signatures, namely one fake signature $\{\widetilde {QA}_{B_j}\}_{j=1}^l$ for Bob and another fake signature $\{\widetilde {QA}_{C_j}\}_{j=1}^l$ for Charlie, that the number of mismatches is below $t_bl/2$ for each half of Bob’s signature while above $t_cl/2$ one half of Charlie’s signature at least. Consequently, the probability of successful repudiation is defined as
As mentioned above in Step 4, Bob and Charlie exchange half of the signature elements with each other, which protects the scheme secure against repudiation. In other words, it’s difficult for Alice to fake two signature $\{\widetilde {QA}_{B_j}\}_{j=1}^l$ and $\{\widetilde {QA}_{C_j}\}_{j=1}^l$, that can pass Bob’s test but cannot pass Charlie’s test.Security against forgery. If there is a dishonest player in the messaging stage, his intention will be to forge message $m'$ and get the other players to accept it. Considering that Bob receives the message and passes it to Charlie, he is the most likely to be a forger. Hence, Bob will induce a mismatch probability $p_e$, and the probability of successful forging is thus given by
We can see that the probability $e_{forg}$ decays exponentially with signature length, so that the scheme can be secure against forgery attack.Signature length. So far, we have discussed three situations, robustness, security against repudiation and security against forgery, and given the corresponding probabilities $e_{rob}$, $e_{rep}$ and $e_{forg}$ for each of them. Assuming that the protocol can fail in any of the above three ways, the probability of failure is equal, we have
Accordingly, the probability of failure is calculated as and satisfied with $p_{err}\leq t_b \leq t_c \leq p_e$. Eq. (37) decays exponentially with signature length, so that the protocol can be secure by an appropriate length of signature. The derivation of Eq. (33)$\sim$(35) is detailed in Ref. [41] , and we will not derive here.4. Conclusion
This study proposes a novel CV-QDS protocol, which uses unidimensional modulation in its KGP components. Traditionally, Gaussian modulation is used in KGP along with coherent detection techniques (e.g., heterodyne detection and homodyne detection) in order to measure quantum states. However, both these techniques require additional processes like constant rotation of local oscillator pulse phase and phase modulator, respectively, leading to measurement angular error. In contrast, unidimensional modulation only needs one quadrature modulation, eliminating the need for basis selection. The proposed protocol is tested with a nontrivial set of three participants, namely, Alice, Bob and Charlie. During the distribution stage, Alice creates quantum states and modulates only the $\widehat {X}$ quadrature for the string of states sent to Bob and only the $\widehat {P}$ quadrature for the string sent to Charlie, simplifying the implementation process. Numerical simulations show that this CV-QDS scheme using unidimensional modulation is secure against collective attack, repudiation attack and forgery attack.
Funding
National Key Research and Development Program of China (2021YFF0901300); “20 New Universities" Project of Jinan City (202228077); Major Innovation Projects of the Pilot Project of Science, Education and Industry Integration (2022JBZ01-01); National Natural Science Foundation of China (61872390).
Acknowledgments
We acknowledge the support from the Optoelectronic Information Center of Central South University.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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