Eavesdropper's ability to attack a free-space quantum-key-distribution receiver in atmospheric turbulence

The ability of an eavesdropper (Eve) to perform an intercept-resend attack on a free-space quantum-key-distribution (QKD) receiver by precisely controlling the incidence angle of an attack laser has been previously demonstrated. However, such an attack could be ineffective in the presence of atmospheric turbulence due to beam wander and spatial mode aberrations induced by the air's varying index of refraction. We experimentally investigate the impact turbulence has on Eve's attack on a free-space polarization-encoding QKD receiver by emulating atmospheric turbulence with a spatial light modulator. Our results identify how well Eve would need to compensate for turbulence to perform a successful attack by either reducing her distance to the receiver or using beam wavefront correction via adaptive optics. Furthermore, we use an entanglement-breaking scheme to find a theoretical limit on the turbulence strength that hinders Eve's attack.

Figure 1

Comparison between measured and theoretical far-field intensity distributions of a laser beam corresponding to one of 29 SLM phase holograms per turbulence strength ($r_0$) for a beam with $D=20\mathrm{c}\mathrm{m}$ and $\lambda =532\mathrm{n}\mathrm{m}$. The grayscale in the holograms represents a 0 to $2\pi$ phase range. The results show our SLM setup accurately emulates a range of turbulence strengths.

Figure 2

Turbulence emulator characterization for $r_0=1.00\mathrm{c}\mathrm{m}, D=20\mathrm{c}\mathrm{m}$, and $\lambda =532\mathrm{n}\mathrm{m}$. (a) Simulated centroid displacements corresponding to 500 phase holograms ($\sigma$ is the two-axis standard deviation). The diameter of each data point is proportional to the count frequency. The centroid displacement distribution is normally distributed along both axes in agreement with Eq. (2). (b) Comparison between measured and simulated centroid displacements for a subset of 29 holograms. This subset was chosen to represent the normal statistical distribution of the 500-hologram set. The measured values are within error of most theoretical predictions (error bars for measured data are represented by diameter of data points). (c) Phase hologram and (d) far-field intensity distribution corresponding to one centroid data point.

Figure 3

Scanning setup. (a) Experimental setup of our spatial mode attack in a turbulent channel, top view (drawing not to scale). The green central ray that is parallel to the optical axis denotes normal alignment of Alice's beam into Bob's receiver. The red rays show the optical path of Eve's scanning beam when tilted at an angle ($\theta ,\varphi$) via lens $L_E$. CW, continuous wave; HWP, half-wave plate; QWP, quarter-wave plate; BS, beam splitter; PBS, polarization beam splitter; ND, neutral density filter; SLM, spatial light modulator; L, lens. (b) Photograph of the actual free-space QKD receiver for detecting polarization-encoded light.

Figure 4

Normalized count rates ${\tau }_k$ for each detector $k=\mathbf{H},\mathbf{V},\mathbf{D},\mathrm{or}\mathbf{A}$ at different incoming beam angles ($\theta ,\varphi$), and the corresponding attack angles for different turbulence strengths $r_0$. The attack angles for the four polarization detectors are shown left to right as horizontal H (yellow), vertical V (red), diagonal D (green), and antidiagonal A (light blue). The emulated turbulence corresponds to different $r_0$ values for an initial beam diameter $D=20\mathrm{c}\mathrm{m}$ and $\lambda =532\mathrm{n}\mathrm{m}$. A smaller $r_0$ value corresponds to stronger atmospheric turbulence.

Figure 5

Modeled attack performance. Quantum bit error rate (QBER) as a function of transmission loss for no turbulence (blue solid line) and different turbulence strengths corresponding to $r_0=7.00\mathrm{c}\mathrm{m}$ (pink dashed line), $3.50\mathrm{c}\mathrm{m}$ (green dotted line), $2.21\mathrm{c}\mathrm{m}$ (red dot-dashed line), $1.53\mathrm{c}\mathrm{m}$ (black dashed line), and $1.00\mathrm{c}\mathrm{m}$ (cyan dashed line). The horizontal gray dashed line denotes the $8\%$ threshold where Eve's attack is successful when QBER is below this value in our attack model. The maximum transmission loss where Eve's attack is successful decreases as turbulence strength increases. The mismatch ratios are too small in the case of $1.00\mathrm{c}\mathrm{m}$ (${\delta }_k\le 2$ for all channels), and the optimization program could not find a solution with a QBER below 8% threshold given any transmission loss. The higher QBER at low loss (i.e., $3.5–7\mathrm{d}\mathrm{B}$) is because Eve has to send higher mean photon number states for channels with lower ${\delta }_k$ in order to match the expected detection rate of Bob.