Large-Alphabet Encoding Schemes for Floodlight Quantum Key Distribution

TL;DR

This paper demonstrates that using 32-ary phase-shift keying in floodlight quantum key distribution significantly doubles the secret-key rate compared to binary phase-shift keying, enhancing secure communication speeds.

Contribution

The study introduces a novel encoding scheme with 32-ary PSK for FL-QKD, achieving higher secret-key rates over metropolitan distances.

Findings

32-ary PSK doubles the SKR compared to BPSK in FL-QKD

Higher alphabet encoding improves secure communication rates

Experimental validation confirms theoretical predictions

Abstract

Floodlight quantum key distribution (FL-QKD) uses binary phase-shift keying (BPSK) of multiple optical modes to achieve Gbps secret-key rates (SKRs) at metropolitan-area distances. We show that FL-QKD's SKR can be doubled by using 32-ary PSK.

Full text

Quantum key distribution [1] (QKD) allows remote parties (Alice and Bob) to create a shared random bit string with unconditional security. Later, they can employ their shared string for one-time-pad (OTP) encryption of messages they wish to keep entirely private from any eavesdropper (Eve). Unfortunately, current QKD systems’ secret-key rates (SKRs) fall far short of what is needed to make high-speed (Gbps) transmission with OTP encryption ready for widespread deployment. Floodlight QKD (FL-QKD) [2, 3] is a new protocol that uses binary phase-shift keying (BPSK) of multiple optical modes and homodyne detection to achieve security against the optimum frequency-domain collective attack. It is predicted to permit Gbps SKRs at metropolitan-area distances in a single-wavelength implementation without the need to develop any new technology. In this paper we extend FL-QKD’s security analysis to $K$-ary phase-shift keying (KPSK), and show that the increased alphabet size affords SKR increases by up to a factor of two. Thus, over a 50-km-long fiber, going from BPSK to 32-ary PSK increases FL-QKD’s SKR from 2.0 Gbps to 4.5 Gbps.

In KPSK FL-QKD (schematic shown in Fig. 1), Alice splits the $W$-Hz-bandwidth, flat-top spectrum, high-brightness output from an amplified spontaneous emission (ASE) source into a low-brightness signal and a high-brightness reference. To enable channel monitoring, Alice combines her low-brightness ASE in an $n\gg 1$ ASE-to-SPDC-ratio with the signal output from a spontaneous parametric downconverter (SPDC) of the same $W$-Hz-bandwidth flat-top spectrum. Alice uses a single-photon detector to monitor her SPDC’s idler and another single-photon detector to monitor a $\kappa_{A}\ll 1$ fraction of her combined ASE-SPDC light, while sending the remainder of that light—whose brightness is $N_{S}\ll 1$ photons/mode—to Bob. Alice retains her bright reference beam in an optical-fiber delay line—using amplifiers as needed—for use as her dual-homodyne receiver’s $N_{\rm LO}\gg 1$ photons/mode brightness local oscillator (LO).

In the absence of Eve, the fiber link from Alice to Bob is a pure-loss channel with transmissivity $\kappa_{S}\ll 1$. Eve’s presence, however, allows her control that channel, hence Alice and Bob must perform channel monitoring to bound Eve’s information gain. So, prior to his KPSK encoding operation, Bob routes a small fraction $\kappa_{B}\ll 1$ of the light he receives to a single-photon detector. The outputs from Alice and Bob’s single-photon detectors enable them to determine the single rates $S_{I}$ for Alice’s idler and $S_{A}\,\left(S_{B}\right)$ for Alice’s (Bob’s) tap, as well as $C_{IA}\,\left(C_{IB}\right)$ and $\widetilde{C}_{IA}\,\left(\widetilde{C}_{IB}\right)$, the time-aligned and time-shifted coincidence rates between Alice’s idler and Alice’s (Bob’s) tap. They use their measurements to: (1) verify that Bob receives the photon flux he would get were Eve absent; and (2) determine Eve’s intrusion parameter $f_{E}=1-[(C_{IB}-\widetilde{C}_{IB})/S_{B}]/[(C_{IA}-\widetilde{C}_{IA})/S_{A}],$ which quantifies the integrity of the Alice-to-Bob channel and allows them to place an upper bound, $\chi_{EB}^{\rm UB}$, on Eve’s Holevo-information rate for her optimized frequency-domain collective attack, which she can realize in the form of an SPDC light-injection attack [2].

Bob’s KPSK modulation works as follows. In each $T$-s-duration symbol interval (symbol rate $R=1/T$), Bob applies a $2\pi k/K$ phase shift to the light remaining after his monitor tap, where $k$ is equally likely to be any integer between 0 and $K-1$ and the $k$ values for different symbol intervals are statistically independent. He then amplifies his modulated light with a gain $G_{B}\gg 1$ amplifier whose output ASE has brightness $N_{B}=G_{B}-1$, and sends the amplified and modulated light back to Alice through what, in Eve’s absence, is a $\kappa_{S}$-transmissivity fiber. The amplifier’s gain will overcome the return-path loss insofar as Alice is concerned, while its output ASE will mask Bob’s modulation from Eve. To decode Bob’s symbols, Alice uses dual-homodyne reception, i.e., she 50–50 beam splits both the light returned from Bob and her LO, and then makes homodyne measurements of the $I$ (0 phase shift) and $Q$ ($\pi/2$ phase shift) in-phase and quadrature components of the returned light as in classical KPSK fiber-optic communication.

When $M=TW$, the number of optical modes per symbol, is high ($M\gg 1$), the joint statistics of $I$ and $Q$ conditioned on knowledge of $k$ can be well approximated by a Gaussian distribution whose symmetric behavior for $0\leq k\leq K-1$ is shown, schematically, in Fig. 2. This symmetry, plus all $k$ values being equally likely, makes Alice’s minimum error-probability decision rule choosing her decoded symbol to be the one whose signal location in the $I$-$Q$ plane is closest to her measured $(I,Q)$ value. See Fig. 2 for the resulting decision regions.

Once Alice has decoded Bob’s string of transmitted symbols the two of them use a tamper-proof classical channel (not shown in Fig. 1) to perform reconciliation (error correction) and privacy amplification. During reconciliation, Alice and Bob obtain values for the conditional probabilities $\Pr(\,\tilde{k}\mid k\,)$, i.e., the probabilities that Alice decoded $\tilde{k}$ given Bob sent $k$, from which they calculate their Shannon-information rate $I_{AB}$ via $I_{AB}=R\{\sum_{k=0}^{K-1}\sum_{\tilde{k}=0}^{K-1}[\Pr(\,\tilde{k}\mid k\,)/K]\log_{2}[K\Pr(\,\tilde{k}\mid k\,)/\sum_{k^{\prime}=0}^{K-1}\Pr(\,\tilde{k}\mid k^{\prime}\,)]\}.$ Then, using their upper bound on Eve’s Holevo-information rate, they know that their achievable SKR is bounded from below by $\Delta I_{AB}^{\rm LB}=\beta I_{AB}-\chi_{EB}^{\rm UB},$ where $\beta$ is the efficiency of their reconciliation algorithm, and, because of FL-QKD’s extraordinarily high SKR, finite-key effects have been neglected.

To explore the SKR behavior of KPSK FL-QKD we performed numerical maximization of $\Delta I_{AB}^{\rm LB}$ over Alice’s source brightness, $N_{S}$, for one-way path lengths up to 150 km using parameter values similar to those employed in Ref. [2]: $W=2$ THz source bandwidth; $n=99$ ASE-to-SPDC ratio; $\kappa_{A}=\kappa_{B}=0.01$ monitor taps; $0.2$ dB/km fiber loss; $R=10$ Gbaud symbol rate; $G_{B}=10^{6}$ amplifier gain; $N_{\rm LO}=10^{4}$ LO brightness; 0.9 homodyne-detection efficiency; and $\beta=0.94$ reconciliation efficiency. The maximum SKRs we obtained for $K=2$, 4, 8, and 32 are shown in Fig. 2. We see that going from BPSK to 32-ary PSK approximately doubles the achievable SKR over all the distances shown, with BPSK providing 2.0 Gbps SKR and 32-ary PSK giving 4.5 Gbps SKR at 50 km.

It is interesting to note how FL-QKD’s KPSK performance differs from that seen in fiber-optic communication using high-order signal constellations and coherent detection [4]. In fiber-optic communication, high-order signal constellations can enormously improve spectral efficiency (bits/sec-Hz = bits/mode), and such systems are now moving beyond KPSK to quadrature amplitude modulation (QAM). Our work shows that FL-QKD benefits from the increased spectral efficiency of KPSK, but we have found that there is no value to conventional (square-lattice) QAM, because that format’s amplitude modulation gives away too much information to Eve.