Security of Binary-Modulated Optical Key Distribution Against Quantum-Enhanced Coherent Eavesdropping

Abstract

Optical key distribution (OKD) protects the physical layer of communication links by taking advantage of the inherent noise present in the photodetection process. It allows for efficient generation of a shared random key between two distant users which can subsequently be used for cryptographic purposes secure against passive eavesdropping. Moreover, it can be straightforwardly implemented over standard intensity modulation and direct detection links, making it an attractive alternative to quantum key distribution. Here we present a comprehensive security analysis against more powerful eavesdroppers possessing either the ability to perform coherent detection, or even quantum-optimal measurements on the intercepted transmission.

Figures (2)

• Figure 1: Schematic view of the OKD protocol and different eavesdropping scenarios considered in this work. In binary-modulated OKD, Alice prepares one of two macroscopic pulses of similar energy values $n_0$ and $n_1$. She transmits them to Bob via a channel with transmission $\tau_B$. Bob implements direct detection and records normally distributed outputs $k_B$. The eavesdropper Eve passively captures fraction $\tau_E$ of Alice's signal. In the article, four strategies of Eve are considered: direct detection (Sec. \ref{['sec:dd']}), coherent detection (Sec. \ref{['sec:coh']}), error-minimizing Helstrom measurements (Sec. \ref{['sec:helstrom']}, here pictured in reference to the Helstrom-achieving displacement receivers Dolinar1973), and optimal measurements saturating the Holevo bound for accessible information (Sec. \ref{['sec:holevo']}).
• Figure 2: a) Key generation rates as a function of the eavesdropper's advantage $\mathcal{E}$ for the case of passive eavesdropping with: direct/coherent detection (blue); optimal state discrimination via Helstrom measurements (purple); optimal measurements saturating the Holevo bound (red). Dotted lines indicate approximations derived in text for the strong eavesdropping limit, $\mathcal{E} \gg 1$, which are seen to converge to the exact values. The upper axis indicates the ratio of Eve's and Bob's signal strength if Bob's detection is shot-noise limited, in which case $\mathcal{E} = \tau_E / \tau_B$ according to Eq. \ref{['eq:eavesdropper_adv']}. b) Values of the modulation depth $\delta_E$ that maximize the key rate for the three considered passive eavesdropping scenarios as a function of the eavesdropper's advantage. Arrows indicate the optimal values that maximize the corresponding asymptotic key rate expressions in Eq. \ref{['eq:KDDapprox']}, \ref{['eq:K_Helstrom_approx_opt']}, and \ref{['eq:KHolevoapprox']}.