Composable security analysis of continuous-variable measurement-device-independent quantum key distribution with squeezed states for coherent attacks

Measurement-device-independent quantum key distribution protocol, whose security analysis does not rely on any assumption on the detection system, can immune the attacking against detectors. We give a first composable security analysis for continuous-variable measurement-device-independent quantum key distribution using squeezed states against general coherent attacks. The security analysis is derived based on the entanglement-based scheme considering finite-size effect. A version of entropic uncertainty relation is exploited to give a lower bound on the conditional smooth min-entropy by trusting Alice's and Bob's devices. The simulation results indicate that, in the universal composable security framework, the protocol can tolerate 2.5 dB and 6.5 dB channel loss against coherent attacks with direct and reverse reconciliation, respectively.

Figure 1

EB scheme of the squeezed-state CV-MDI QKD protocol. EPR: two-mode squeezed state. Hom: homodyne detection. ${\mathrm{Hom}}_{1x}$: homodyne detection of measuring the $x$ quadrature. ${\mathrm{Hom}}_{2p}$: homodyne detection of measuring the $p$ quadrature. $X_C$ ($P_D$): measurement results of ${\mathrm{Hom}}_{1x}$ (${\mathrm{Hom}}_{2p}$). BS: 50:50 balanced beam splitter. ${\mathrm{Channel}}_{AC}$ (${\mathrm{Channel}}_{BC}$): totally untrusted quantum channel between Alice (Bob) and Charlie controlled by adversary. Public channel: authenticated channel used for classical communication.

Figure 2

Secret key rates of squeezed-state CV-MDI QKD protocol against coherent attack in the extremely asymmetric cases ($T_{BC}=0$) with direct reconciliation in the frame of composable security. Those lines are under the ideal conditions with ideal modulation variances $V_A=V_B={10}^5$ and perfect reconciliation efficiency $\beta =1$. The block lengths from left to right curves show $N={10}^{10}$ (green dot-dashed line), ${10}^{11}$ (blue dot line), ${10}^{12}$ (red dashed line), and $\infty$ (black solid line), respectively. Here the discretization parameter is set to $d=13$, the excess noise ${\varepsilon }_1={\varepsilon }_2=0.002$, and the overall security parameter is smaller than ${10}^{-20}$.

Figure 3

Secret key rates of squeezed-state CV-MDI QKD protocol against coherent attack in the extremely asymmetric cases ($T_{BC}=0$) with direct reconciliation. The protocol with practical modulation variances $V_A=V_B=5.04$ and imperfect reconciliation efficiency $\beta =96.9\%$ is considered. The block lengths from left to right curves are $N={10}^{10}$ (green dot-dashed line), ${10}^{11}$ (blue dot line), ${10}^{12}$ (red dashed line), and $\infty$ (black solid line), respectively. The discretization parameter, excess noises, and security parameters are chosen as in the case of ideal modulation.

Figure 4

Secret key rates vs block size for the extremely asymmetric case ($T_{BC}=0$) with direct reconciliation. The solid lines are under the ideal condition where modulation variances $V_A=V_B={10}^5$ and perfect reconciliation efficiency $\beta =1$. The dot-dashed lines are under the practical condition where practical modulation variances $V_A=V_B=5.04$ and imperfect reconciliation efficiency $\beta =96.9\%$. From left to right, the transmittance of the quantum channel corresponds to loss of 0.2 dB (black line), 0.4 dB (red line), and 0.5 dB (green line), respectively.

Figure 5

Secret key rates of squeezed-state CV-MDI QKD protocol against coherent attack in the extremely asymmetric case ($T_{BC}=0$) with reverse reconciliation. The protocol is under ideal modulation variances $V_A=V_B={10}^5$ and perfect reconciliation efficiency $\beta =1$. The block lengths from left to right curves correspond to $N={10}^{10}$ (green dot-dashed line), ${10}^{11}$ (blue dot line), ${10}^{12}$ (red dashed line), and $\infty$ (black solid line), respectively. Here the discretization parameter is set to $d=13$, the excess noise to ${\varepsilon }_1={\varepsilon }_2=0.002$, and the overall security parameter is smaller than ${10}^{-20}$.

Figure 6

Secret key rates of squeezed-state CV-MDI QKD protocol against coherent attack in the extremely asymmetric case ($T_{BC}=0$) with reverse reconciliation. The protocol is under practical modulation variances $V_A=V_B=5.04$ and imperfect reconciliation efficiency $\beta =96.9\%$. The block lengths from left to right curves are $N={10}^{10}$ (green dot-dashed line), ${10}^{11}$ (blue dot line), ${10}^{12}$ (red dashed line), and $\infty$ (black solid line), respectively. The discretization parameter, excess noises, and security parameters are chosen as in the case of ideal modulation.

Figure 7

Secret key rates vs block size for the extremely asymmetric case ($T_{BC}=0$) with reverse reconciliation. The solid lines are under the ideal condition that modulation variances $V_A=V_B={10}^5$ and perfect reconciliation efficiency $\beta =1$. The dot-dashed lines are under the practical condition that practical modulation variances $V_A=V_B=5.04$ and imperfect reconciliation efficiency $\beta =96.9\%$. From left to right, the transmittance of the quantum channel corresponds to loss of 1 dB (black line), 2 dB (red line), and 3 dB (green line), respectively.

Figure 8

EB scheme of the squeezed-state CV-MDI QKD protocol with Eve's attacks. After two channels, mode $A_2$ becomes $A^{\prime }$ and mode $B_2$ becomes $B^{\prime }$. QM is the quantum memory.

Figure 9

Comparison of the secret key between the independent entangling cloner attacks model and correlated mode attacks model in the symmetric case. The ideal case is under ideal modulation variances $V_A=V_B={10}^5$ and perfect reconciliation efficiency $\beta =1$. The practical case is with practical modulation variances $V_A=V_B=5.04$ and imperfect reconciliation efficiency $\beta =96.9\%$. The solid lines are the secret key rates under two independent entangling cloner attacks models. The dashed lines are the secret key rates under two correlated mode attacks models.

Figure 10

Comparison of the secret key between the extremely asymmetric results and the PLOB bound. Our protocol is under ideal modulation variances $V_A=V_B={10}^5$ and perfect reconciliation efficiency $\beta =1$. The red solid line is the bound of secret key capacity of asymmetric CV-MDI-QKD. The green dot-dashed line and the blue dashed line are the secret key rates of CV-MDI-QKD with squeezed states in direct and reverse reconciliation cases, respectively.