Quantum hacking of free-space continuous-variable quantum key distribution by using a machine-learning technique

The transmission coefficient of the free-space channel is fluctuating due to the occurrence of random fluctuations of the refractive index and noise. The Gaussian-modulated quantum states of light may degrade into a non-Gaussian mixture at a certain probability after being transmitted through the free-space channel. In such a situation, Eve can apply an entanglement-distillation attack to steal secret information without it being revealed if Alice and Bob regard the practical fluctuation losses as constant. However, the entanglement-distillation attack is available only when the transmitting states are a non-Gaussian mixture; otherwise, Eve will expose herself. Exploiting this loophole and considering the limitations of the attack, we enhance Eve's eavesdropping ability by integrating a machine-learning technique with the entanglement-distillation attack. With the help of convolutional neural network (CNN, one of the most popular machine leaning tools), Eve can choose the best opportunity to launch the entanglement-distillation attack. Therefore, Eve has a chance to bias the estimation of the parameters by improving the entanglement resource of the mixed states, which allows Eve to render the final keys shared between the legitimate parties insecure. Our proposed CNN-based entanglement-distillation attack does not require additional resources and can be implemented using simple means based on the present technologies.

Figure 1

Convolutional neural network structure with input matrix of size $100\times 100$. The single small square denotes one element in each matrix. The matrix number and the dimension of each matrix for all input, convolutional, and max-pooling layers are given at the top of each layer. The small gray squares denote the convolutional kernel sizes and the pooling region sizes of convolutional and max-pooling layers, respectively.

Figure 2

Training and testing procedures of the CNN model. The collected values are combined into ordered matrices (instances) for training sets of the CNN model. After multiple iterations of training, the CNN model is able to distinguish the distribution of the input data set.

Figure 3

Performance of the CNN model with different training data sets. The number of values in each instance of the data sets ranges from 10 000 to 50 000.

Figure 4

Schematic of the CNN-based entanglement-distillation attack for two-mode squeezed state protocol. The entangled states are prepared by interfering two polarization squeezed beams on a 50:50 BS at Alice's side. The prepared states are sent to Bob through a fluctuating lossy channel. A weak measurement on beam $B_1$ is diagnosing the states and, subsequently, the measurement results input into the CNN model to distinguish the non-Gaussianity of the transmitted states. Once the non-Gaussianity is confirmed, the entanglement distillation is available, otherwise the beam $B_1$ is sent to Bob directly. The distillation operation consists of a weak measurement (implemented by a beam splitter with transmittance $T_e$) and domodyne measurement, and the distilled beam $B_2$ is sent to Bob.

Figure 5

Elliptical beam profile on a receiving aperture of radius $r$ after transmitting through the fluctuating channel. $W_1$ and $W_2$ are the half axis of the elliptic beam spot, where $W_1$ rotates on the angle $\varphi$ relating to $x$ axis. $d$ represents the beam wandering, which denotes the distance between the beam centroid with the center of the aperture.

Figure 6

The probability density function of transmittance $T_{\text{ell}}$ at distances 4 km, 8 km, and 12 km. The initial beam-spot radius $W_0=50\text{mm}$, and the receiving aperture radius $r=80\text{mm}$.

Figure 7

Performance comparison of the QKD system based on the elliptical beam case and constant transmittance case. The dotted line denotes the secret key rate under the elliptical beam case and the solid line denotes the secret key rate under constant fading case.

Figure 8

Simulation based on the quantum channel that is represented by the elliptical beam model. The secret key rate as a function of the tap beam splitter transmittivity $T_e$ and threshold value $x_{\text{th}}$. (a), (b), and (c) illustrate the performances at different transmission distances, 8 km, 10 km, 12 km, respectively. (d) Security region bounds of the QKD system under the entanglement-distillation attack at different distances 8 km, 10 km, 12 km, which decrease with the increasing transmission distance. The black solid lines represent the bounds between positive and negative secret key rates.