Abstract

Free space optical (FSO) communications are enabling high-speed global wireless networks. Thanks to the highly directional nature of laser beam, they also yield a greater security advantage over radio frequency counterparts. When combined with a scheme of secret key agreement (SKA), FSO-SKA can establish at high speed a symmetric secret key which cannot be decrypted even by unbounded computer resources. Although there have been many theoretical studies on SKA, experimental investigations have been quite lacking, especially on quantifying eavesdropping risks and secret key rates in realistic environment. Here, we report the first full-field implementations of FSO-SKA in a 7.8-km terrestrial link with a probing station, enabling the estimation of eavesdropping risks. We attain the final key rates from 100 kbps to 7.77 Mbps under various atmospheric and beaming conditions even with total losses of 55dB or higher, in which known quantum key distribution schemes attain impractically low key rates.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Information security of wireless networks is becoming a serious and urgent concern as diversified wireless devices, automotive cars, unmanned aerial vehicles, and low-cost satellites are connected, producing large amounts of critical and valuable data. The inherent broadcast nature of radio frequency (RF) wireless communications allows transmitted data to be easily eavesdropped. An eavesdropper (Eve) can store data now, and decrypt them later on when powerful computers are available. Cryptographic schemes based on the computational hardness of mathematical problems cannot be provably secure against such a “store now, decrypt later” attack [1].

Cryptographic schemes secure even against Eve who has unbounded computational resources have been known for a long time. The first example is Vernam’s one-time pad proposed in 1926 [2], which is based on a symmetric secret key no shorter than the size of a message to be sent. The perfect secrecy of this scheme was proven by Shannon based upon information theory [3], and hence is referred to as information-theoretic security.

Since then the notion of information-theoretic security has been extended into many scenarios. Wyner considered in 1975 secure message transmission over a wiretap channel (WTC), i.e. a channel between a sender (Alice) and a legitimate receiver (Bob) in the presence of Eve, and formulated the secrecy capacity when Eve can tap the channel only via its physically degraded version [4]. Csiszár and Körner soon extended Wyner’s model to relax the degradedness assumption on Eve, and derived generalized secrecy capacity formula [5]. This secure message transmission scheme is referred to as WTC coding.

Quantum key distribution (QKD) proposed by Bennett and Brassard in 1984 [6] and by Ekert in 1991 [7] allows one to share symmetric secret keys even when Eve eavesdrops the channel with any physically implementable attacks and has unbounded computational ability. The laws of quantum mechanics are employed to detect any attempts on the quantum channel launched by Eve.

In 1993, Maurer [8] and Ahlswede and Csiszár [9] formulated theories of secret key agreement (SKA) from common randomness over a WTC to share symmetric secret keys. In this model, by virtue of an error-free authenticated classical channel (public channel), Alice and Bob can generate secret keys even if Eve enjoys better channel conditions than Bob in the WTC.

WTC coding and SKA are referred to as physical layer security (PLS) schemes [10, 11], attracting much attention in wireless communications. In particular, SKA has been investigated experimentally in the RF domain during the last decade [12, 13]. RF-SKA exploits common randomness acquired by probing the channel subject to multipath scattering and fading. The key rate of RF-SKA is very low, e.g. at about 100 bps (bits per second) even in a short range [13].

QKD has been studied extensively since the 1990s [14], and deployed in metropolitan-scale [15,16] and continental-scale fiber networks [17]. QKD has been commercialized [18], and found a new application to long-term storage solutions [19]. QKD-related technologies have been studied and tested in free space optical (FSO) links [20–35]. Recently, QKD was successfully demonstrated in space over an intercontinental scale [36, 37], in which a key rate at around 1 kbps over a distance of up to 1200 km was reported.

A next important challenge is to increase the key rate. Unfortunately, however, recent theoretical studies clarify an upper bound of the maximum achievable key rate of QKD, which is just a few times larger than the theoretical key rates of known QKD protocols [38,39]. Even if it would be reached, it is insufficient to protect increasing confidential data traffic in wireless networks.

One attractive option to overcome this difficulty is SKA in a FSO channel. FSO communication is a promising scheme to realize high-capacity wireless links [40–45], which cannot be possible using RF communications only. FSO quantum communication was also demonstrated over a geostationary-orbit-ground link recently [46]. The highly directional nature of laser beam in FSO communications yields a greater security advantage over RF counterparts. A FSO channel must be in the line-of-sight (LoS) from Alice and Bob. This makes Eve much harder to eavesdrop the signals. This situation can be contrasted to the case of optical fibers, in which even Alice and Bob can hardly know precise details of installation routes and environment, and hence the channel is completely out of their control. In the case of LoS links, on the other hand, it is too much to assume that the whole channel is at Eve’s hand and any changes of channel characteristics are attributed to Eve’s attempts. It is reasonable to restrict Eve’s physical ability to access the FSO channel, and to model it as a FSO-WTC [47–50]. In return one can explore a possibility to improve the key rate and the distance.

Mathematically, a WTC is described by the joint probability distribution P(y, z|x) of observations at Bob and Eve, y and z, respectively, given Alice’s input x. The secret key rate is then expressed in terms of the mutual information based on this probability distribution. One then immediately faces the non-trivial question of how one can know such a distribution which includes the observation z of malicious Eve, whose attempts are usually hard to know. There have been tremendous amounts of theoretical studies on SKA. However, almost all literatures simply assumed that P(y, z|x) is just given, but did not ask how the model can be certified. In practical applications, however, some method to certify the model is indispensable. Although there have been many experimental works on RF-SKA [51–54], this issue has not been solved yet. As for FSO-SKA, it was only recently, to our best knowledge, that a first step experiment was reported in the specific case of nearly atmospheric-turbulence-free condition [55]. Thorough investigations under various atmospheric conditions are left open.

In this paper, we demonstrate the full-field implementation of FSO-SKA with WTC certifications by using a terrestrial 7.8 km FSO link testbed (Tokyo FSO Testbed), which consists of one sender terminal and two receiver terminals, one for Bob and the other for a probing station to evaluate WTC characteristics. Using this practical configuration, we demonstrate high-speed FSO-SKA for various channel realizations.

2. Concept of FSO-SKA

2.1. Secret key rate of SKA

The fundamental metric of SKA is the secret key capacity CK, i.e., the maximum achievable rate of secret key generation [8, 9]. To derive an explicit expression of CK is generally very hard. Instead, an achievable lower bound is known as [10,56]

CKRK=I(X;Y)min[I(X;Z),I(Y;Z)],
where I(·; ·) is the mutual information between two random variables. The secret key rate RK is the difference between the information rate between Alice and Bob and that leaked to Eve.

The mutual information are given by

I(X;Y)=xyPA(x)PAB(y|x)log2[PAB(y|x)xPA(x)PAB(y|x)],
I(X;Z)=xzPA(x)PAE(z|x)log2[PAE(z|x)xPA(x)PAE(z|x)],
I(Y;Z)=yzPB(y)PBE(z|y)log2[PBE(z|y)yPB(y)PBE(z|y)],
where PA(x) and PB(y) are the probability distributions of Alice’s and Bob’s random variables X and Y with observations x and y, respectively, and PAB(y|x) is the conditional probability distribution that Bob observes y given Alice’s input x, which corresponds to the marginal distribution of the joint probability distribution P(y, z|x) by the relation PAB(y|x)=zP(y,z|x). The similar convention applies for the other conditional probability distributions.

Thus, the secret key rate RK directly depends on Eve’s random variable Z and related probability distributions, which are hard to know in practice. This is in sharp contrast to QKD. In QKD, one can estimate the leaked information to Eve solely from the measured quantum bit error rate at Bob along with a prescribed theoretical formula based on the laws of quantum mechanics — the no-cloning theorem. In SKA, on the other hand, no single principle exists, but one should employ elaborate measurement on the WTC characteristics to estimate the leaked information to Eve.

2.2. Case-by-case risk analysis on FSO communications

Because a FSO channel is in the LoS between Alice and Bob, if Eve gets into the channel or is located around them, they have a stronger chance to become aware of the presence of Eve. This aspect should be fully employed for FSO-SKA. In the case of short-distance inter-building links, such as the last one-mile link from the fiber backbone to the client premises, surveillance cameras suffice to monitor the channel [57], however, channel probing with surveillance cameras becomes more difficult (detailed analysis is presented in Section 3.1 later on). In that case, one should introduce other means. Today, space debris surveillance systems with radar and lidar (light detection and ranging) can detect and monitor flying objects as small as 10-cm long [58]. Although implementing such systems is out of scope of this paper, it is understood that it is quite hard for Eve to tap the main lobe of the channel in the middle of Alice and Bob in principle.

If in the future, stealth drones against both radar and lidar were available, the Eve-in-the-middle attack could be likely risks. In that case, PLS schemes based on the laws of classical physics cannot provide mechanisms to detect stealth drones, and hence only the option is QKD. In the following, we restrict ourselves to more realistic scenarios of the two kinds;

  • Eve-in-the-far-side scenario: Eve locates far away from the main lobe and tries to collect scattered light.
  • Eve-behind-Bob scenario: Eve locates behind Bob and tries to receive comparable beam power with that at Bob by using a large telescope.

In any case, we can reasonably assume that Eve’s capability to access the channel is physically limited. However, certifying her physical capability in a quantitative way is an involved task. So, we give further consideration on these two scenarios for typical use cases of FSO communications.

Satellite-to-ground/drone-to-ground links

It is reasonable to assume that the ground station and the beam footprint can be within a physically protected area against Eve. Obviously, the Eve-behind-Bob scenario is not the case. The problem is actually to evaluate the characteristics of the WTC in the Eve-in-the-far-side scenario.

Satellite-to-drone/drone-to-drone/car-to-car/car-to-infrastructure/inter-building links

In these cases, the Eve-behind-Bob scenario is very likely. It is not easy to detect Eve by radar, lidar, and surveillance cameras because in a sight from Bob in the opposite direction to Alice there are usually various objects and atmospheric structures, and hence Eve can slip away in such a complex background and receive comparable beam power with that at Bob by using a large telescope. Only the exception may be inter-building links on a rooftop in which a shielding wall twice as wide as the diameter of the FSO beam can be installed ensuring that the entire FSO signal will be absorbed and cannot be intercepted beyond Bob’s building [59]. This countermeasure will not work well for FSO links between mobile terminals. In this Eve-behind-Bob scenario, the options may include both SKA and QKD. If FSO-SKA can be applicable, then the key rate and the transmission distance can be greatly improved with lower-cost devices than those of QKD.

Satellite-to-satellite links

In space, there are no atmospheric effects, the LoS is very clear, and every satellite flies in a predictable orbit. In addition, a narrow and collimated laser beam is employed with precise pointing, acquisition, and tracking techniques [60]. Therefore, the Eve-in-the-far-side scenario is most unlikely. On the other hand, Eve may be able to acquire and track the optical beam behind Bob if she launches her satellites into an appropriate orbit, or employs drones or ground terminals to detect the beam footprint. Thus the Eve-behind-Bob scenario is still very likely [61].

Thus the likely use cases of FSO-SKA are the Eve-in-the-far-side scenario in satellite-to-ground or drone-to-ground links, and the Eve-behind-Bob scenario in mobile FSO communications or terrestrial inter-building FSO communications in which installing the shield wall at Bob is not easy by some reasons. To characterize the FSO-WTC in these scenarios, a FSO-SKA system should have some probing mechanisms to know information related to Eve’s observation z, which appears in the joint probability distribution P(y, z|x). So, in our experiment, we implemented a probing station near Bob’s receiver [55].

2.3. Configuration and protocol of FSO-SKA

In FSO-SKA, Alice generates initial random bit sequences (RBSs) x0n+l of length n + l by a physical random number generator (PRNG), and transmits them to Bob by a highly directional laser beam (the main channel). Here, l is a length of test bits used for the WTC characterization and n is a length of remaining random bits from which the secret key is distilled (Fig. 1(a)). Bob receives the sequence from Alice, y0n+l, which may contain transmission errors. Hence, x0n+l and y0n+l are not generally identical. Eve can tap a part of FSO beam to get an observed sequence z0n+l. We call the corresponding channel the wiretapper channel. The sequence z0n+l at Eve is unknown to Alice and Bob, but they have to estimate the upper bound on the leaked information to Eve somehow. To this end, we introduce a probing station near Bob’s receiver. Hereafter, we call the probing station virtual Eve (v-Eve). Alice and v-Eve constitute the probing channel which is located in the side lobe of the FSO beam. The outputs from the probing channel are denoted as z^0n+l. Bob and v-Eve are connected to Alice via the public channel to carry out the key-distillation processing, which consists of information reconciliation and privacy amplification (Fig. 1(b)).

 

Fig. 1 (a) Schematic diagram of the configuration of FSO-SKA. (b) Key-distillation processing conducted over the public channel.

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In information reconciliation [62,63], Alice and Bob randomly select length-l test bits xtestl and ytestl from x0n+l and y0n+l, evaluate bit error rate (BER) to select an appropriate error-correction code, and correct errors by communicating necessary information via the public channel to share the common sequence of length n. In particular, we adopt the reverse reconciliation (RR) scheme [64] rather than the direct reconciliation one, because the RR can realize larger key rate, which is given by RK = I(X; Y) − I(Y; Z) (see Appendix A). Eve has the observed sequence z0n+l and can listen to all the information communicated over the public channel. Therefore, Eve may have partial information on the reconciled sequence of Alice and Bob.

In privacy amplification [63,65], to extract a sequence uncorrelated with Eve, Alice and Bob compress the reconciled sequence into a shorter sequence — the secret key. The compression rate of privacy amplification is determined by the estimated upper bound on the leaked information to Eve via v-Eve’s observation z^0n+l, which is actually I(Y; ) in the RR scheme.

3. Experimental setup

3.1. Tokyo FSO Testbed

Figure 2(a) presents pictures of our experimental setup with Tokyo FSO Testbed [48, 49, 55]. Alice’s transmitter is an all-weather telescope dome on a building in the University of Electro-Communications (UEC: 35°39′28.8″N, 139°32′39.5″E) at Chofu. In the transmitter, the initial RBSs from the PRNG are encoded into the 1550-nm-wavelength optical signals by non-return to zero on-off keying at a rate of 10 Mbps, and are then transmitted to Bob. Bob’s receiver is located in the sixth floor of a building of the National Institute of Information and Communications Technology (NICT: 35°42′24.2″N, 139°29′19.3″E) at Koganei. On the rooftop of the NICT building, a container-type terminal is setup as v-Eve, which is 10-m away from Bob’s receiver.

 

Fig. 2 (a) Geographical configuration of Tokyo FSO Testbed. (b) Beam footprint at Bob’s site (the NICT building). (c) Photographs of Skytower West Tokyo at 4.5-km distance from NICT. The right picture is the enlarged view for the antenna and its supporting poles located on the Skytower West Tokyo. (d) Photographs of Alice’s site taken from NICT at 7.8-km distance from NICT.

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Figure 2(b) indicates contour circles of beam footprint at Bob’s site for several different intensity ratios to the peak power at the beam center. For the 20dB- and 30dB-intensity circles, the upper-right-diagonal part propagates further into free space behind Bob’s building, and hence could potentially be received by Eve. When the beam direction wanders due to atmospheric turbulence, such a wiretapping risk increases. V-Eve located on the rooftop can monitor a fraction of the signal beam to estimate the WTC characteristics and to assess eavesdropping risks. The details of this WTC configuration and specifications of the terminals are described in [48] and [49].

To examine the possibility to detect suspicious objects around the middle point of the link, we developed surveillance camera systems in NICT and UEC. However, in the optical link between Alice and Bob, there is no building structure which can serve as a good reference. Instead, we used the Skytower West Tokyo at 4.5-km distance from Alice in a different direction with the link (see Fig. 2(a)) for this purpose.

Figure 2(c) shows photographs of the antenna and its supporting poles of the Skytower West Tokyo taken with our surveillance camera system consisting of a camera sensor with a pixel pitch dpix of 3.9 μm and a field-scope with a focal length ftele of 1.75 m. The spatial resolution of the camera system is quantified with the angle subtended by a single camera pixel — the instantaneous field of view (IFoV) θIFoV — which is calculated as

θIFoV=2tan1(dpix2ftele).
For our camera system, θIFoV becomes 2.23 μrad. Thus, each camera pixel covers an area of 1.004-cm size at 4.5 km distance. The image quality is slightly degraded due to meteorological effects (dusts and atmospheric turbulence), but the supporting poles of about 10-cm width can be identified, indicating the detectability of objects of this size by the surveillance camera.

Figure 2(d) shows photographs of Alice’s site taken from Bob (7.8 km apart). For such a long distance, the surveillance camera cannot resolve the target, especially when wind velocity is fast, disenabling channel probing.

3.2. WTC characterization

The WTC is characterized by evaluating the conditional probability distributions PAB(y|x), PAE(|x), and PBE(|y) using the test bits. The test-bit length l must be sufficiently long such that the length dependence disappears in statistical metrics such as the mutual information I(Y; ) while keeping the block length n of privacy amplification relatively long. In our experiment, we set l to be 64 kbits and n to be roughly 1 Mbits (precisely, 220 = 1048576 bits). For these lengths, the length dependence in the experimentally observed I(Y; ) becomes very small (see Appendix B). Thus the length of the initial RBS sent from Alice is n + l = 1112576 bits.

To establish synchronization between Alice and Bob/v-Eve, a pseudo-random noise-15 (so-called PN15) sequence of length 215 − 1 = 32767 bits and an ID sequence of length 1024 bits are added to the TRBS generated by Alice. The PN15 sequence is generated by a linear feedback shift register and employed to establish frame synchronization. The ID is uniquely assigned to each TRBS in order to identify it in the key-distillation processing. If the ID alters due to transmission errors, the key-distillation processing would fail. Therefore, we concatenate the 64 copies of the ID for error correction. The probability of ID recognition error can reduce to about 10−6 when the BER is 20%. Even in clear atmospheric conditions, FSO links may often experience fading of 20–30dB lasting up to ∼ 1–100 ms due to atmospheric turbulence (scattering, absorption, and scintillation), which could often result in burst error — a loss of long consecutive bits. Therefore, if these sequences are added to the head of the payload, they may be lost due to the burst errors induced by atmospheric fading. Hence, we interleave these sequences into the payload to mitigate such burst errors. The schematic diagram of the frame structure is given in [55].

Photocurrents from Bob’s and v-Eve’s diodes are acquired by an oscilloscope at a sampling rate of 50 MHz, In each acquisition, a recorded waveform with a duration of 1.28 s is sent to the PC via the USB link. The recorded waveform is divided into 230-ms partial waveforms which contain at least one transmitted frame with a duration of 115 ms. The partial waveform is referred to as the output waveform wy(t) (for Bob) or w(t) (for v-Eve), which is processed by a PC in the following way.

First, clock data corresponding to the 10-MHz repetition rate at Alice is recovered at Bob’s and v-Eve’s sites. This processing is called clock data recovery (CDR) and its detail is presented in [41]. CDR determines the sampling times ti to resample the output waveform wy(t) into a sequence synchronizing with the 10-MHz input symbols. The values wy(ti) and w(ti) are recorded as demodulated sequences, denoted as ay(i) and a(i), respectively. Figs. 3(a) and 3(d) show examples of them.

 

Fig. 3 Channel characterization for the data acquired at 14:50 on 23 May 2017. (a) Demodulated sequence ay(i) at Bob, where the index i runs from 1 to 2300000 to over at least one frame in the sequence. Also shown is the threshold based on the two-sided moving average. (b) Diagram of the main channel. Edges represent the conditional probabilities PAB(y|x). (c) Experimentally characterized PAB(y|x) of the main channel. (d) Demodulated sequence a(i) at v-Eve. (e) Diagram of the probing channel. Edges represent the conditional probabilities PAE(|x). (f) Experimentally characterized PAE(|x). For soft decision, bin width Δ is chosen so that the leaked information is appropriately estimated with finite-size data in practical experiment (see Appendix B). In this figure, the total bin number K is taken to be 763.

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Second, Bob binarizes the demodulated sequences by assigning ‘0’ (‘1’) if ay(i) is lower (higher) than the certain threshold. To adapt to atmospheric fading, we set the threshold āy(i) for ay(i) based on the two-sided moving average

a¯y(i)=13001j=i1500i+1500ay(j).
V-Eve also conducts the same procedure. After the binarization, Bob and v-Eve restore the bit sequences by(i) and b(i), respectively.

Third, frame synchronization is carried out to identify the position of the initial RBS from Alice in by(i) and b(i). To this end, Bob and v-Eve calculate the cross-correlation function between the bit sequence and the PN15 sequence. This procedure fails when the BER is over 47%. If the frame synchronization fails at v-Eve, information leaked to Eve is considered to be 0.01 bit/symbol, which corresponds to the mutual information of the binary symmetric channel with a crossover probability of 45%.

Bob extracts the received RBS y0n+l from by(i) by removing the PN15 sequence and the ID sequence in the reverse way to form the frame. This procedure is called de-interleave. Then, length-l test bits xtestl and ytestl are randomly selected from x0n+l and y0n+l, respectively, at the same bit positions. Finally, the number of events in ytestl detecting y given input x, NAB(y|x), and the number NA(x)=y=0,1 NAB(y|x) are counted. The main channel can be experimentally characterized as

PAB(y|x)=NAB(y|x)NA(x)
as shown in Figs. 3(b) and 3(c). The BER can be evaluated by Pe = PA(0)PAB(1|0) + PA(1)PAB(0|1).

At v-Eve, on the other hand, we further exploit another decision rule with multiple output symbols, often referred to as soft decision, because more information can be extracted than by binary decision (hard decision), and hence the leaked information can be estimated more strictly. So, in the de-interleave, we move back to the demodulated sequence again, divide its value range into K bins with an equal width Δ, and decide which of the symbols 1, · · · , K the value a(i) takes. Through the procedure, we obtain the K-valued random number sequence (RNS) z^0n+l. We characterize the probing channel PAE(|x) by a 2-by-K matrix based on length-l test bits xtestl and test samples z^testl as shown in Figs. 3(e) and 3(f).

Once the main and probing channels have been characterized as PAB(y|x) and PAE(|x), the other probability distribution PBE(|y) can be straightforwardly derived. Then the mutual information I(X; Y), I(X; ), and the estimate for the leaked information I(Y; ) can also be obtained experimentally.

4. Asymptotic key rate RKA

4.1. Theoretical calculation of mutual information

Before presenting experimental results of FSO-SKA, we first see how I(X; Y), I(X; ), and I(Y; ) behave as functions of the average received signal powers (ARSPs) at Bob, B, and at v-Eve, E. To find a rough theoretical fitting for the overall performance seen in experimental campaigns under various conditions, we adopt an idealized additive white Gaussian noise (AWGN) model for the demodulated sequences ay(i) and a(i). We further assume that the noise variance in each distribution can be specified by a quadratic function in terms of B or E, whose coefficients are derived by fitting with experimental data of ay(i) and a(i) (see Appendix C).

Figures 4(a) and 4(b) illustrate I(X; Y) and I(X; ) which depend only on B and E, respectively. Their behaviors are very similar; two regions can be clearly identified, i.e., the loss-independent region where the mutual information is unchanged for B or E, and the noise-limited region where the mutual information monotonically decreases as B or E decreases. In the loss-independent region, the received signal power is enough for the signal to be well discriminated without errors, not limited by the noises. Hence, I(X; Y) and I(X; ) are unchanged even B and E decrease such as due to channel losses, respectively. In the noise-limited region, on the other hand, the ARSP becomes comparable with or less than the noise power. Hence, the discrimination of x = 0 or 1 suffers from errors due to the noises.

 

Fig. 4 Theoretical calculations of mutual information (a) I(X; Y), (b) I(X; ), and (c) I(Y; ). The repetition rate and the laser wave length are assumed to be 10 MHz and 1550 nm, respectively. The detailed description of the calculation is given in Appendix C.

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Figure 4(c) illustrates an estimate for an upper-bound on the leaked information I(Y; ) which depends on both B and E, because the bit discrepancies between yn and n are the superposition of the errors occurring in the main and probing channels. This quantity can be again characterized by the loss-independent and noise-limited regions along each axis of B and E.

The theoretical asymptotic key rate RKA=I(X;Y)I(Y;Z^) as a function of B and E is shown in Figure 5(a). Its behavior can be understood based on the features mentioned above. For a fixed value of E, the asymptotic key rate RKA monotonically decreases as B decreases. For a fixed value of B, RKA shows the opposite behavior with I(X; ); RKA starts to drastically decrease for larger E, resulting in the outage of key generation.

 

Fig. 5 (a) Theoretical curve of the asymptotic key rate RKA. (b) Plot of the experimental RKA. The data points of outage, for which RKA0, are indicated by x-marks. (c) Contour plot of the theoretical and experimental RKA.

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4.2. Experimental results of RKA

We performed FSO-SKA experiments in various conditions as summarized in Table 1. In the first and fourth campaigns, the transmission power and the beam center position were set to several values to collect the transmission data under several WTC conditions. Noting that we succeeded to consistently generate the final keys even in the daytime during these campaigns by maintaining the beam direction. In the second and third campaigns, we fixed the transmission power and the beam center position to investigate the long-term behavior of the performance of FSO-SKA protocol.

Tables Icon

Table 1. Summary of experimental configurations of FSO transmission campaigns

Figure 5(b) plots experimentally evaluated RKA for various ARSPs at Bob and v-Eve. These data points were acquired in the four campaigns summarized in Table 1, and include 20550 frames. The experimental data can reproduce the fundamental properties shown in Fig. 5(a). It is worth noting that RKA could remain finite even when v-Eve received much larger power than Bob (E > B), thanks to the RR scheme.

Figure 5(c) summarizes a contour plot of the theoretical and experimental RKA. Three insets at the top show the B-dependence of RKA for E = 2, 5, and 11 nW. Similarly, the right three insets illustrate the E-dependence of RKA for B = 10, 50, and 100 nW. Each inset includes experimental data within the range of ±10% of corresponding B and E.

It can be confirmed again that the overall features of experimental RKA could be roughly fitted by the theoretical curves. Precisely speaking, however, the experimental RKA for smaller B and E deviate downward from the theoretical curve as seen in the left of the three top insets and bottom of the three right insets. In this case, atmospheric turbulence degrades only the main channel since the leaked information to Eve is negligibly small. On the other hand, the experimental RKA for larger B and E deviate upward from the theoretical curve as seen in the right of three top insets. In this case, atmospheric fading affects more the probing channel than the main channel. This result suggests that the deterioration of key rate due to atmospheric fading can be compensated by increasing transmission power in an appropriate way.

4.3. Effects of the atmospheric turbulence

To understand the effects of the atmospheric turbulence on the performance of FSO-SKA, we compare ten typical cases; A ∼ E for E = 1.8 ∼ 2.1 nW and F ∼ J for E =10.3 ∼ 11.7 nW. The histograms of the demodulated sequences ay(i) and a(i) conditioned on input bit symbol x = 0 or 1 are shown as probability density functions (PDFs) in Fig. 6 for cases A ∼ E, and Fig. 7 for cases F ∼ J with meteorological wind velocity and experimental RKA.

 

Fig. 6 Histograms of the demodulated sequences for cases A to E. These cases are shown in the upper left inset of Fig. 5(c). For each case, ARPSs B and E, wind velocity (w.v.), and experimental RKA are shown as well as the time and date.

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Fig. 7 Histograms of the demodulated sequences for cases F to J. These cases are shown in the upper right inset of Fig. 5(c). For each case, ARPSs B and E, wind velocity (w.v.), and experimental RKA are shown as well as the time and date.

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In cases A and B (both B and E are small), the PDFs are Gaussian even under atmospheric turbulence — wind velocity was 4.8 m/s in case A. The effects of atmospheric turbulence cannot be resolved in the histograms when the ARSP is small, because it is masked by the receiver noises whose distribution corresponds to the PDF for x = 0. In cases C and D, B becomes larger (∼27 nW). Then atmospheric fading can be identified as Bob’s non-Gaussian PDF with a large variance as seen in case D. Actually, case D is windier than case C. So in case D, BER gets larger and experimental RKA is degraded below the theoretical curve. In case E, Bob’s PDF suffers from strong atmospheric fading, but experimental RKA attains as high as about 9 Mbps thanks to sufficient ARSP.

For cases F ∼ J, E gets larger (E ∼ 10 nW), more information is leaked to v-Eve, and the rates become lower in general. Case H, however, show higher experimental RKA than case G even though they have the same B and E. This is attributed to the degradation of v-Eve’s probing channel by atmospheric fading (faster wind velocity in case H than case G) as seen in the right middle histogram in Fig. 7. The difference between cases I and J is due to the same reason.

5. Final key rate RKF

5.1. Key-distillation processing

After having the experimental BER Pe and estimate for leaked information I(Y; ), we carry out the key-distillation processing. A flow chart of the processing is shown in Fig. 8.

 

Fig. 8 Flow chart of the key-distillation processing.

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First, the length-n sequences xn and yn, generated by removing the test bits from x0n+l and y0n+l, are input into the processing. The block shuffling is applied on them to mitigate the burst errors occurring in transmission. The resulting sequences are denoted as x′n and y′n.

Second, the RR is performed using low density parity check (LDPC) code. The code rate RLDPC is chosen depending on Pe as follows:

RLDPC={0.85(0%Pe<0.7%)0.80(0.7%Pe<1.7%)0.75(1.7%Pe<2.7%)0.70(2.7%Pe<3.7%)0.65(3.7%Pe<4.7%)0.60(4.7%Pe<5.7%)0.55(5.7%Pe<6.7%)0.50(otherwise).
After RLDPC is determined, Bob generates a m-bit (= nRLDPC) RBS and encodes it into a code word cLDPCn. He sends a bitwise XOR yncLDPCn to Alice via the public channel. Alice then XORs received sequence with x′n and inputs the resulting sequence (i.e., xn(yncLDPCn)) including errors into her LDPC decoder to reproduce cLDPCn.

Third, hash value of the message digest, MD5, with a length of nMD5 (=128) bits is calculated for the reconciled code word at Alice, and is sent to Bob to check whether it coincides with that of cLDPCn. If the hash values are the same, the RR is successfully completed. Otherwise the key-distillation processing for this block is aborted.

Fourth and finally, Alice and Bob perform the privacy amplification. They compress the reconciled code word cLDPCn into the secret key sk with a length of k = n[RLDPCI(Y; )] − nMD5 with a Toeplitz matrix [66]. The ratio k/nf of the secret key length k to the frame length nf (=1146367) is referred to as the final key rate RKF.

The theoretical curve of RKF is shown in Fig. 9(a). The curve rapidly falls down to zero when B or E exceeds certain threshold. The cut-off along B, which is not seen in the theoretical asymptotic key rate RKA, occurs at around B =20 nW. This is caused by the limitation of our LDPC code; error correction fails when the BER is over 6.7% (see Eq. (8)). The cut-off along E occurs at around E =10 nW, which is much steeper than that of RKA shown in Fig. 5(a). This is due to the fact that RLDPC can be smaller than not only I(X; Y) but also I(Y; ). For the same reason, RKF can no longer be non-zero finite for EB.

 

Fig. 9 (a) Theoretical characteristics of the final key rate RKF. (b) Plot of the experimental RKF. The data points of outage, for which RKF0, are indicated by x-marks. (c) Contour plot of the theoretical and experimental RKF.

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5.2. Experimental results of RKF

Figure 9(b) shows a plot of experimental RKF. Although the number of black x-marks indicating the outage of key distillation increases compared with Fig. 5(b), the experimental RKF can stay larger than 1 Mbps in the region of B > 20 nW and E < 10 nW, and reach 7.77 Mbps at higher B even under total channel-loss of 55dB.

Figure 9(c) summarizes a contour plot of the theoretical and experimental RKF. As with Fig. 5(c), three insets at the top show the B-dependence of RKF for E = 2, 5, and 11 nW, and right three insets illustrate the E-dependence of RKF for B = 10, 50, and 100 nW. The experimental key rates in most cases can be fitted by the theoretical model, except in the region of E ≥ 10 nW and B ≥ 20 nW (the right of three top insets), in which the experimental RKF attain 100 kbps–5 Mbps for larger B while the theoretical rates result in the outage. This discrepancy is due to the same reason as in Fig. 5(c).

To see how meteorological conditions affect the SKA performance, we conducted a transmission campaign for about a week, fixing the transmission power and the beam pointing (Campaign No. 3 in Table 1), once the beaming from Alice to Bob was optimized at the evening time. Figure 10 shows temporal variations of the one-hour averaged final key rate, the failure rate (the ratio of the number of frames for which key generation fails to the total number of frames transmitted per hour), Bob’s and v-Eve’s ARSPs, the solar radiation, and the cloud ceiling. The one-hour averaged final key rate and the failure rate vary periodically day by day (Fig. 10(a)). The key rate reaches about 7 Mbps or higher in the night time, whereas it becomes zero almost during the day time.

 

Fig. 10 Summary of Campaign No. 3. (a) Variations of the one-hour averaged final key rate RKF (blue solid line) and the failure rate (red chain line). During experiment, 10 frames were recorded every 5 minutes. Thus, 120 frames were transmitted in each one hour. Gray shades indicate the night time between sunset and sunrise and dashed lines show the 00:00 AM in JST. (b) Variations of the one-hour averaged B (solid line) and E (chain line). (c) Variations of the one-hour averaged solar radiation and cloud ceiling. These meteorological data were provided by OBSOC [67], the compound meteorological sensor at NICT premises. For the interested reader, we provide Data File 1 [68] which includes the original data sets for this figure (RKF, B, and E) as well as the histograms of the demodulated sequences. In Visualization 1, we present a movie of the variations of RKF and the histograms of the demodulated sequences during 27 hours from 18:00 on 11 June.

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The final key rate correlates with Bob’s and v-Eve’s ARSPs as seen in Fig. 10(b), and, more remarkably, directly anti-correlates with the solar radiation as seen in Fig. 10(c). These periodic changes may stem from the thermal drifts of the buildings and instruments. Empirically these drifts deviate the beam direction with a rate of 0.2 mrad/hour in the daytime [47]. The deterioration of key rate in the day time is not the defect of the protocol but due to the misalignment. If the beam alignment is made in the day time, we can generate the final keys even in the daytime. The periodic behaviors observed for about a week prove the stability of our instruments. In the midnight on 12 June to the early morning on 13 June, the key rate drastically dropped even in the night time. In the same time, the decrease of the cloud ceiling was observed, implying that the rainfall shut the FSO link.

Finally, the 15-test battery proposed by NIST [69] was applied on the generated key. Our keys successfully passed all the tests as shown in Fig. 11.

 

Fig. 11 Summary of the NIST randomness test suites [69] for keys generated on 9 June 2017. We applied the tests on 1000 instances with a length of 106 bits. The acceptable uniformity should be larger than 10−4. As shown by the black dotted lines, the acceptable success rate range for excluding Random Excursions (Variant) test is 0.99 ± 0.00943: The acceptable success rate range is 0.99 ± 0.0121 for Random Excursions (Variant) test. For the items with multiple tests (Cumulative Sums, Non Overlapping Template, Random Excursions, and Random Excursions Variant), we show the range of the lowest and highest scores by the black solid lines.

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6. Conclusion

We demonstrated high-speed generation of information-theoretically secure keys over the FSO-WTC. The final key rates from 100 kbps to the maximum of 7.77 Mbps were attained for the 10-MHz repetition rate even under the main channel losses of 55dB or higher, in which known QKD schemes attain impractically low key rates. If the repetition rate is increased, the final key rate would increase further. Our results show that a FSO communication system, if supplemented with an appropriate probing station, promises secure high-speed key establishment for wireless networks. Several important challenges remain. Efficient WTC probing methods need to be investigated further, and their data should be accumulated for practical security certification. Information reconciliation performance should be improved to get close to the asymptotic performance shown in Fig. 5, such as by revising error-correction code and by introducing appropriate filtering for the input sequence into the key distillation processing. Key distillation performances should be investigated with various block sizes to study finite size effects.

Appendix

A. Secret key rates for reverse reconciliation

In information reconciliation, the one-way transmission over the public channel suffices to achieve the lower bound (1) [8, 9]. Then, we have two options; direct reconciliation (DR) and reverse reconciliation (RR). In DR, Alice sends the correction information to Bob, and Bob reconstructs xn. The leaked information to Eve is I(X; Z) if the process is perfect. In RR, Bob sends Alice the correction information to share yn between them. The leaked information to Eve is I(Y; Z). Because the bit discrepancies between yn and zn are the superposition of the errors occurred in the main and wiretapper channels if they are statistically independent, the following inequalities

I(X;Z)I(Y;Z),I(X;Y)I(Y;Z)
hold. Consequently,
I(X;Y)I(Y;Z)max[I(X;Y)I(X;Z),0],
implying that even if Eve enjoys a better channel condition than Bob, RR allows SKA. Therefore, in the present experiment, we adopt RR and simply set RK = I(X; Y) − I(Y; Z) where the mutual information between Bob and v-Eve, I(Y; ), is substituted for I(Y; Z).

We can experimentally confirm this point as well. Figure 12(a) shows a comparison of the mutual information I(X; ) and I(Y; ) leaked to Eve in DR and RR, respectively, which are evaluated from the experimental data presented in Fig. 5(b). Every point locates below the dashed line of the boundary I(X; ) = I(Y; ), meaning I(X; ) ≥ I(Y; ). This result yields the superiority of RR over DR. The difference of I(X; ) and I(Y; ) becomes larger as the BER in the main channel increases, because the error in the main channel influences the v-Eve’s observation in the RR.

 

Fig. 12 (a) Comparison between I(X; ) and I(Y; ). The color of each point corresponds to the BER in the main channel. (b) Comparison between I(X; Y)−I(X; ) and I(X; Y)−I(Y; ). The color of each point corresponds to the BER in the probing channel.

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Figure 12(b) shows a comparison of I(X; Y) − I(X; ) and I(X; Y) − I(Y; ) corresponding to the secret key rates based on DR and RR, respectively. Every point locates above the dashed line of the boundary I(X; Y) − I(X; ) = I(X; Y) − I(Y; ), also yielding the superiority of RR over DR. Since v-Eve employs soft decision, I(X; Y) − I(Y; ) can be lower than 0.

B. Sample-size dependences of statistical metrics

Ideally, the block length n for key-distillation and the test-bit length l should be as long as possible. In practice, however, they remain finite. Theories on finite length analysis on SKA are quite lacking at present. As for the block length, a standard often adopted in fiber-based QKD systems is n ∼ 1 Mbits. In the case of FSO-SKA, a longer length would be preferred because the channels are affected by atmospheric fading. Unfortunately, however, our current setup with the repetition rate of 10 MHz and the interface specification of the 50-MHz oscilloscope limits the block length at around 1 Mbits. Therefore we investigate test-bit length dependences of statistical metrics by varying its length l for an experimental data of fixed length n + l = 1112576 bits. More precisely, we randomly select the length-l test bits from the RBS/RNS of length n + l obtained in a certain FSO transmission campaign (actually the one for 14:28:30 on 22 May), and see the changes of BER and mutual information for various random selections of the RBS/RNS by simulation.

First, we investigate the length dependence of BER, by evaluating the difference Pe,lPe,n, where the BER Pe,l is calculated based on the test bits ytestl and the BER Pe,n is calculated based on the remaining RBS yn. The random selection of ytestl is repeated 10000 times.

Figure 13 shows a result of the simulation. The average — E[Pe,lPe,n], where E[x] denotes the sample average — shown in Fig. 13(a) is independent of the test-bit length l and remains at the value 0. On the other hand, the standard deviation — var[Pe,lPe,n], where var[x] = E[x2] − (E[x])2 denotes the sample variance— shown in Fig. 13(b) decreases as the test-bit length increases. Fig. 13(c) shows the histogram of Pe,lPe,n for l = 10000 bits. The histogram is well fitted with the Gaussian distribution with an average of 4.79 × 105% and a standard deviation of 0.0964%. Hence, the 99.7% confident interval (or 3σ-interval, where σ denotes the standard deviation) is between ± 0.289%, which reaches one third of the BER interval (1%) for the LDPC code selection rule of Eq. (8). Thus, the probability to fail in the RR due to the estimation error cannot be negligible, and l should be longer. The same analysis is done for l = 64000 bits as shown in Fig. 13(d). Since the standard deviation becomes three times smaller than that for l = 10000 bits, the 99.7% confident area also shrinks in one third, between ± 0.118%. In this case, the failure probability of RR are suppressed to a much smaller value. Therefore, the test-bit length of 64000 bits is sufficient to estimate the BER.

 

Fig. 13 (a) Average E[Pe,lPe,n] for 10000 trials. (b) Standard deviation var[Pe,lPe,n] for 10000 trials. (c) Histogram of Pe,lPe,n for l = 10000 bits. The solid line represents the PDF of the Gaussian distribution with an average of 4.79 × 10−5% and a standard deviation of 0.0964%. (d) Histogram of Pe,lPe,n for l = 64000 bits. The solid line is the PDF of the Gaussian distribution with an average of 1.39 × 10−4% and a standard deviation of 0.0392%.

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The estimation of the leaked information is more directly related to the security certification of FSO-SKA. In our key-distillation processing, the estimation of Il(Y; ) also depends on the bin width Δ of soft decision made by v-Eve. So we next show a bin-width dependence of Il(Y; ) in Fig. 14(a) for l = 64000. The characteristics of the curve is the same as that shown in our previous work [48]. If Δ is too large, soft decision is completely equivalent to hard decision. If Δ is too small, on the other hand, the mutual information is overestimated due to the short of sample size against bin number, as shown above. Therefore, Δ should carefully be chosen not to over- and underestimate the leaked information. This curve has a flat region within a range of Δ = 0.08 mV to Δ = 1 mV. In the present experiment, we adopted Δ = 0.08 mV at the left end of the flat region because we are to estimate the leaked information Il(Y; ) in the worst case. We note that the optimum bin width changes according to the dynamic range of the oscilloscope which is adjusted to cover the whole range of the waveform, which varies depending on the atmospheric conditions.

 

Fig. 14 (a) Bin-width dependence of Il(Y; ) when test-sample length l is 64000 bits. (b) Average E[Il(Y; )−In(Y; )] for 10000 trials. (c) Standard deviation var[Il(Y;Z^)In(Y;Z^)] for 10000 trials. (d) Histogram of Il(Y; ) − In(Y; ) for 10000 test samples. The solid line represents the PDF of the Gaussian distribution with an average of 2.78 × 10−2 bits/symbol and a standard deviation of 6.61 × 10−3 bits/symbol. (e) Histogram of Il(Y; ) − In(Y; ) for 64000 test samples. The solid line represents the PDF of the Gaussian distribution with an average of 6.57 × 10−3 bits/symbol and a standard deviation of 2.86 × 10−3 bits/symbol. For these two histograms, the bin width Δ is 0.08 mV.

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Subsequently, we evaluate the test-sample-length dependence of Il(Y; ) − In(Y; ), where the mutual information Il(Y; ) is calculated based on the test samples z^testl and the mutual information In(Y; ) is calculated based on the remaining RNS n. The test-sample length l should be chosen such that the average E[Il(Y; ) − In(Y; )] is as close to 0 as possible, and does not change as the test-sample length l varies. As shown in Fig. 14(b), the average decreases as l increases. At the point of l = 64000, the three curves get close to 0, and l-dependence becomes small.

The standard deviation var[Il(Y;Z^)In(Y;Z^)] shown in Fig. 14(c) behaves similarly among the different Δ. To evaluate the 99.7% confident interval, the histogram of Il(Y; ) − In(Y; ) for l = 10000 bits and Δ = 0.08 mV is fitted by the Gaussian PDF as shown in Fig. 14(d). If the test-bit length l increases to 64000 bits, the 99.7% confident interval shrinks by almost one third, in which the probability of fatal estimation error is very small.

C. Theoretical calculation of mutual information

The theoretical calculation of mutual information shown in Fig. 4 was done as follows.

First, as shown in (2), the mutual information I(X; Y) between Alice and Bob is evaluated as

I(X;Y)=x=0,1y=0,1PA(x)PAB(y|x)log2[PAB(y|x)xPA(x)PAB(y|x)].
In the OOK scheme, bits x = 0 and x = 1 are generated with equal probability, PA(x = 0) = PA(x = 1) = 1/2. In hard decision, Bob decides that y = 1 is received when the signal intensity vB exceeds a certain threshold vth, otherwise y = 0 is received. Hence, the conditional probability distribution PAB(y|x) is described as
PAB(y=1|x)=vth0QAB(vB|x)dvB,PAB(y=0|x)=vthQAB(vB|x)dvB,
where QAB(vB|x) is the conditional PDF of Bob’s signal intensity vB given x. We assume that the signal intensity vB can be represented with infinite precision and the signal suffers only from AWGN. Thus, vB for x = 1 and x = 0 follow the Gaussian PDF of means 2 B and 0, respectively, where B is the ARSP at Bob with the unit of nW. We further assume that the noise variance is the function of B. Precise characterization of the AWGN which may include the thermal noise, the background noise, the electrical noise and so on, is generally difficult. So, we estimated the noise variance as a function of given received power S from the experimental data as
σB2(S)=1.57×102S2+7.75×1019S+1.44×102,
by using the least square method. Therefore, the conditional PDF QAB(vB|x) is given as
QAB(vB|x)=12πσB2(2xS¯B)exp((vB2xS¯B)2σB2(2xS¯B)).
Substituting (14) into (12), we obtain the conditional probability distribution as
PAB(1|0)=S¯BQAB(vB|0)dvB=12erfc(S¯B2σB2(0)),
PAB(0|1)=S¯BQAB(vB|1)dvB=12erfc(S¯B2σB2(2S¯B)),
where the threshold is set to be B and erfc(·) is the complementary error function.

Next, for the mutual information I(X; ) between Alice and v-Eve, we assume that the PDF of signal intensity E at v-Eve can be estimated with infinite precision. Thus, I(X; ) is calculated as

I(X;Z^)=x=0,1PA(x)QAE(v^E|x)log2[QAE(v^E|x)xPA(x)QAE(v^E|x)]dv^E,
where QAE(E|x) is the conditional PDF of E given x. Assuming that the signal intensity E can be represented with infinite precision and the signal suffers only from the AWGN, like I(X; Y), QAE(E|x) is given as
QAE(v^E|x)=12πσE2(2xS¯E)exp((v^E2xS¯E)2σE2(2xS¯E)),
where E denotes the ARSP at v-Eve and the variance σE2(S) as a function of given received power S is determined from the experimental data as
σE2(S)=1.92×102S2+1.65×1019S+2.68×101.
Since deriving a closed form expression of this integral is difficult, we evaluated I(X; ) by numerical integration.

Finally, the mutual information I(Y; ) between Bob and v-Eve is calculated as

I(Y;Z^)=x=0,1PB(y)QBE(v^E|y)log2[QBE(v^E|y)yPB(y)QBE(v^E|y)]dv^B,
where PB(y) is the input probability of bit y and QBE(E| y) is the conditional PDF of E given y. These probabilities can be evaluated using the probabilities already derived. The probability PB(y) is calculated as
PB(y)=x=0,1PA(x)PAB(y|x)=12x=0,1PAB(y|x).
The conditional PDF QBE(E|y) is also calculated as
QBE(v^E|y)=x=0,1QAE(v^E|x)PBA(x|y)
=x=0,1QAE(v^E|x)PAB(y|x)PA(x)PB(y)
=PAB(y|0)QAE(v^E|0)+PAB(y|1)QAE(v^E|1)2PB(y),
where PBA(x|y) is the conditional probability distribution of x given y. In the second equality, we use the Bayes rule
PBA(x|y)PB(y)=PAB(y|x)PA(x).
Like I(X; ), we evaluated I(Y; ) by numerical integration.

Funding

ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan).

Acknowledgments

The authors thank R. Matsumoto for theoretical discussion. They also thank K. Suzuki for providing meteorological data from OBSOC.

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2018 (4)

S.-K. Liao, W.-Q. Cai, J. Handsteiner, B. Liu, J. Yin, L. Zhang, D. Rauch, M. Fink, J.-G. Ren, W.-Y. Liu, Y. Li, Q. Shen, Y. Cao, F.-Z. Li, J.-F. Wang, Y.-M. Huang, L. Deng, T. Xi, L. Ma, T. Hu, L. Li, N.-L. Liu, F. Koidl, P. Wang, Y.-A. Chen, X.-B. Wang, M. Steindorfer, G. Kirchner, C.-Y. Lu, R. Shu, R. Ursin, T. Scheidl, C.-Z. Peng, J.-Y. Wang, A. Zeilinger, and J.-W. Pan, “Satellite-relayed intercontinental quantum network,” Phys. Rev. Lett. 120(3), 030501 (2018).
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C. Chen, A. Grier, M. Malfa, E. Booen, H. Harding, C. Xia, M. Hunwardsen, J. Demers, K. Kudinov, G. Mak, B. Smith, A. Sahasrabudhe, F. Patawaran, T. Wang, A. Wang, C. Zhao, D. Leang, J. Gin, M. Lewis, B. Zhang, D. Nguyen, D. Jandrain, F. Haque, and K. Quirk, “Demonstration of a bidirectional coherent air-to-ground optical link,” Proc. SPIE 10524, 105240G (2018).

M. Fujiwara, T. Ito, M. Kitamura, H. Endo, O. Tsuzuki, M. Toyoshima, H. Takenaka, Y. Takayama, R. Shimizu, M. Takeoka, R. Matsumoto, and M. Sasaki, “Free-space optical wiretap channel and experimental secret key agreement in 7.8 km terrestrial link,” Opt. Express 26(15), 19513–19523 (2018).
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K. Suzuki, D. Kolev, A. C. Casado, and M. Toyoshima, “Environmental-data collection system for satellite-to-ground optical communications,” Trans. JSASS Aerosp. Tech. Jpn. 16(1), 35–39 (2018). The data from OBSOC are available on http://sstg.nict.go.jp/OBSOC/?lang=e .
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2017 (10)

M. Sasaki, H. Endo, M. Fujiwara, M. Kitamura, T. Ito, R. Shimizu, and M. Toyoshima, “Quantum photonic network and physical layer security,” Philos. Trans. R. Soc. Lond. A 375(2099), 20160243 (2017).
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L. Li, R. Zhang, Z. Zhao, G. Xie, P. Liao, K. Pang, H. Song, C. Liu, Y. Ren, G. Labroille, P. Jian, D. Starodubov, B. Lynn, R. Bock, M. Tur, and A. E. Willner, “High-capacity free-space optical communications between a ground transmitter and a ground receiver via a UAV using multiplexing of multiple orbital-angular-momentum beams,” Sci. Rep. 7(1), 17427 (2017).
[Crossref] [PubMed]

H. Takenaka, A. Carrasco-Casado, M. Fujiwara, M. Kitamura, M. Sasaki, and M. Toyoshima, “Satellite-to-ground quantum-limited communication using a 50-kg-class microsatellite,” Nat. Photon. 11502 (2017).
[Crossref]

K. Günthner, I. Khan, D. Elser, B. Stiller, Ömer Bayraktar, C. R. Müller, K. Saucke, D. Tröndle, F. Heine, S. Seel, P. Greulich, H. Zech, B. Gütlich, S. Philipp-May, C. Marquardt, and G. Leuchs, “Quantum-limited measurements of optical signals from a geostationary satellite,” Optica 4(6), 611–616 (2017).
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S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8, 15043 (2017).
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C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
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J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356(6343), 1140–1144 (2017).
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S.-K. Liao, H.-L. Yong, C. Liu, G.-L. Shentu, D.-D. Li, J. Lin, H. Dai, S.-Q. Zhao, B. Li, J.-Y. Guan, W. Chen, Y.-H. Gong, Y. Li, Z.-H. Lin, G.-S. Pan, J. S. Pelc, M. M. Fejer, W.-Z. Zhang, W.-Y. Liu, J. Yin, J.-G. Ren, X.-B. Wang, Q. Zhang, C.-Z. Peng, and J.-W. Pan, “Long-distance free-space quantum key distribution in daylight towards inter-satellite communication,” Nat. Photon. 11, 509 (2017).
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J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, M. Li, Y.-M. Huang, L. Deng, L. Li, Q. Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground entanglement-based quantum key distribution,” Phys. Rev. Lett. 119(20), 200501 (2017).
[Crossref] [PubMed]

S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground quantum key distribution,” Nature 549, 43 (2017).
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2016 (5)

J. Zhang, T. Q. Duong, A. Marshall, and R. Woods, “Key generation from wireless channels: A review,” IEEE Access 4, 614–626 (2016).
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R. Courtland, “China’s 2,000-km quantum link is almost complete [news],” IEEE Spectr. 53(11), 11–12 (2016).

M. Fujiwara, A. Waseda, R. Nojima, S. Moriai, W. Ogata, and M. Sasaki, “Unbreakable distributed storage with quantum key distribution network and password-authenticated secret sharing,” Sci. Rep. 6, 28988 (2016).
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H. Endo, M. Fujiwara, M. Kitamura, T. Ito, M. Toyoshima, Y. Takayama, H. Takenaka, R. Shimizu, N. Laurenti, G. Vallone, P. Villoresi, T. Aoki, and M. Sasaki, “Free-space optical channel estimation for physical layer security,” Opt. Express 24(8), 8940–8955 (2016).
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J. Zhang, R. Woods, T. Q. Duong, A. Marshall, Y. Ding, Y. Huang, and Q. Xu, “Experimental study on key generation for physical layer security in wireless communications,” IEEE Access 4, 4464–4477 (2016).
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2015 (6)

F. J. Lopez-Martinez, G. Gomez, and J. M. Garrido-Balsells, “Physical-layer security in free-space optical communications,” IEEE Photon. J. 7(2), 7901014 (2015).
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W. S. Rabinovich, C. I. Moore, R. Mahon, P. G. Goetz, H. R. Burris, M. S. Ferraro, J. L. Murphy, L. M. Thomas, G. C. Gilbreath, M. Vilcheck, and M. R. Suite, “Free-space optical communications research and demonstrations at the U.S Naval Research Laboratory,” Appl. Opt. 54(31), F189–F200 (2015).
[Crossref] [PubMed]

H. Endo, T. S. Han, T. Aoki, and M. Sasaki, “Numerical study on secrecy capacity and code length dependence of the performances in optical wiretap channels,” IEEE Photon. J. 7(5), 7903418 (2015).
[Crossref]

K. Zeng, “Physical layer key generation in wireless networks: challenges and opportunities,” IEEE Commun. Mag. 53(6), 33–39 (2015).
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J.-P. Bourgoin, B. L. Higgins, N. Gigov, C. Holloway, C. J. Pugh, S. Kaiser, M. Cranmer, and T. Jennewein, “Free-space quantum key distribution to a moving receiver,” Opt. Express 23(26), 33437–33447 (2015).
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G. Vallone, D. Bacco, D. Dequal, S. Gaiarin, V. Luceri, G. Bianco, and P. Villoresi, “Experimental satellite quantum communications,” Phys. Rev. Lett. 115(4), 040502 (2015).
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2014 (1)

M. Takeoka, S. Guha, and M. M. Wilde, “Fundamental rate-loss tradeoff for optical quantum key distribution,” Nat. Commun. 5, 5235 (2014).
[Crossref] [PubMed]

2013 (4)

J. Yin, Y. Cao, S.-B. Liu, G.-S. Pan, J.-H. Wang, T. Yang, Z.-P. Zhang, F.-M. Yang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Experimental quasi-single-photon transmission from satellite to earth,” Opt. Express 21(17), 20032–20040 (2013).
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J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photon. 7, 387 (2013).
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S. Nauerth, F. Moll, M. Rau, C. Fuchs, J. Horwath, S. Frick, and H. Weinfurter, “Air-to-ground quantum communication,” Nat. Photon. 7, 382 (2013).
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S. N. Premnath, S. Jana, J. Croft, P. L. Gowda, M. Clark, S. K. Kasera, N. Patwari, and S. V. Krishnamurthy, “Secret key extraction from wireless signal strength in real environments,” IEEE Trans. Mob. Comput. 12(5), 917–930 (2013).
[Crossref]

2012 (2)

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185 (2012).
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M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, Y. Koyama, and H. Kunimori, “Results of Kirari optical communication demonstration experiments with NICT optical ground station (KODEN) aiming for future classical and quantum communications in space,” Acta. Astronaut. 74, 40–49 (2012).
[Crossref]

2011 (1)

2009 (1)

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11, 075001 (2009).
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2008 (1)

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, A. Zeilinger, and C. Barbieri, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10(3), 033038 (2008).
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2007 (3)

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481 (2007).
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T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett. 98(1), 010504 (2007).
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T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and the orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
[Crossref]

2005 (2)

T. Aono, K. Higuchi, T. Ohira, B. Komiyama, and H. Sasaoka, “Wireless secret key generation exploiting reactance-domain scalar response of multipath fading channels,” IEEE Trans. Antennas Propag. 53(11), 3776–3784 (2005).
[Crossref]

C.-Z. Peng, T. Yang, X.-H. Bao, J. Zhang, X.-M. Jin, F.-Y. Feng, B. Yang, J. Yang, J. Yin, Q. Zhang, N. Li, B.-L. Tian, and J.-W. Pan, “Experimental free-space distribution of entangled photon pairs over 13 km: Towards satellite-based global quantum communication,” Phys. Rev. Lett. 94(15), 150501 (2005).
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2003 (1)

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–242 (2003).
[Crossref] [PubMed]

2002 (5)

V. Ramasarma, “Free space optics: A viable last-mile solution,” Bechtel Telecommun. Tech. J. 1, 22–30 (2002).

A. Acampora, “Last mile by laser,” Sci. Am. 287(1), 32–37 (2002).
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N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
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J. E. Nordholt, R. J. Hughes, G. L. Morgan, C. G. Peterson, and C. C. Wipf, “Present and future free-space quantum key distribution,” Proc. SPIE 4635, 116–126 (2002).
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1998 (1)

R. Sridharan and A. F. Pa, “U. S. space surveillance network capabilities,” Proc. SPIE 3434, 88–100 (1998).
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1995 (1)

C. H. Bennett, G. Brassard, C. Crepeau, and U. M. Maurer, “Generalized privacy amplification,” IEEE Trans. Inf. Theory 41(6), 1915–1923 (1995).
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1994 (1)

G. Brassard and L. Salvail, “Secret key reconciliation by public discussion,” Lect. Notes Comput. Sci. 765, 410–423 (1994).
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1993 (2)

U. M. Maurer, “Secret key agreement by public discussion from common information,” IEEE Trans. Inf. Theory 39(3), 733–742 (1993).
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1991 (1)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
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1988 (1)

C. H. Bennett, G. Brassard, and J. M. Robert, “Privacy amplification by public discussion,” SIAM J. Comput. 17(2), 210–229 (1988).
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1979 (1)

J. L. Carter and M. N. Wegman, “Universal classes of hash functions,” J. Comput. Syst. Sci. 18(2), 143–154 (1979).
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1978 (1)

I. Csiszár and J. Körner, “Broadcast channels with confidential messages,” IEEE Trans. Inf. Theory 24(3), 339–348 (1978).
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1975 (1)

A. D. Shannon, “The wire-tap channel,” Bell Syst. Tech. J. 54(8), 1355–1387 (1975).
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C. E. Shannon, “Communication theory of secrecy systems,” Bell Labs Tech. J. 28(4), 656–715 (1949).
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A. Acampora, “Last mile by laser,” Sci. Am. 287(1), 32–37 (2002).
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C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
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Ahlswede, R.

R. Ahlswede and I. Csiszar, “Common randomness in information theory and cryptography. I. secret sharing,” IEEE Trans. Inf. Theory 39(4), 1121–1132 (1993).
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Allacher, A.

Alléaume, R.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11, 075001 (2009).
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Anisimova, E.

C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
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Aoki, T.

H. Endo, M. Fujiwara, M. Kitamura, T. Ito, M. Toyoshima, Y. Takayama, H. Takenaka, R. Shimizu, N. Laurenti, G. Vallone, P. Villoresi, T. Aoki, and M. Sasaki, “Free-space optical channel estimation for physical layer security,” Opt. Express 24(8), 8940–8955 (2016).
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H. Endo, T. S. Han, T. Aoki, and M. Sasaki, “Numerical study on secrecy capacity and code length dependence of the performances in optical wiretap channels,” IEEE Photon. J. 7(5), 7903418 (2015).
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Aono, T.

T. Aono, K. Higuchi, T. Ohira, B. Komiyama, and H. Sasaoka, “Wireless secret key generation exploiting reactance-domain scalar response of multipath fading channels,” IEEE Trans. Antennas Propag. 53(11), 3776–3784 (2005).
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Arai, K.

T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and the orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
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Asai, T.

Aspelmeyer, M.

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, A. Zeilinger, and C. Barbieri, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10(3), 033038 (2008).
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Bacco, D.

G. Vallone, D. Bacco, D. Dequal, S. Gaiarin, V. Luceri, G. Bianco, and P. Villoresi, “Experimental satellite quantum communications,” Phys. Rev. Lett. 115(4), 040502 (2015).
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Banchi, L.

S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8, 15043 (2017).
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Banks, D. L.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, S. D. Leigh, M. Levenson, M. Vangel, N. A. Heckert, and D. L. Banks, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 Rev. 1a.

Bao, X.-H.

C.-Z. Peng, T. Yang, X.-H. Bao, J. Zhang, X.-M. Jin, F.-Y. Feng, B. Yang, J. Yang, J. Yin, Q. Zhang, N. Li, B.-L. Tian, and J.-W. Pan, “Experimental free-space distribution of entangled photon pairs over 13 km: Towards satellite-based global quantum communication,” Phys. Rev. Lett. 94(15), 150501 (2005).
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Barbieri, C.

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, A. Zeilinger, and C. Barbieri, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10(3), 033038 (2008).
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R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481 (2007).
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Barreiro, C.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11, 075001 (2009).
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Barros, J.

M. Bloch and J. Barros, Physical-Layer Security: From Information Theory to Security Engineering (Cambridge University, 2011).
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Bassham, L. E.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, S. D. Leigh, M. Levenson, M. Vangel, N. A. Heckert, and D. L. Banks, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 Rev. 1a.

Bayraktar, Ömer

Bennett, C. H.

C. H. Bennett, G. Brassard, C. Crepeau, and U. M. Maurer, “Generalized privacy amplification,” IEEE Trans. Inf. Theory 41(6), 1915–1923 (1995).
[Crossref]

C. H. Bennett, G. Brassard, and J. M. Robert, “Privacy amplification by public discussion,” SIAM J. Comput. 17(2), 210–229 (1988).
[Crossref]

C. H. Bennett and G. Brassard, “Quantum cryptography: public-key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, (IEEE, 1984), pp. 175–179.

Bianco, G.

G. Vallone, D. Bacco, D. Dequal, S. Gaiarin, V. Luceri, G. Bianco, and P. Villoresi, “Experimental satellite quantum communications,” Phys. Rev. Lett. 115(4), 040502 (2015).
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P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, A. Zeilinger, and C. Barbieri, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10(3), 033038 (2008).
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Bird, A.

T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and the orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
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Blauensteiner, B.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481 (2007).
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Bloch, M.

M. Bloch and J. Barros, Physical-Layer Security: From Information Theory to Security Engineering (Cambridge University, 2011).
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L. Li, R. Zhang, Z. Zhao, G. Xie, P. Liao, K. Pang, H. Song, C. Liu, Y. Ren, G. Labroille, P. Jian, D. Starodubov, B. Lynn, R. Bock, M. Tur, and A. E. Willner, “High-capacity free-space optical communications between a ground transmitter and a ground receiver via a UAV using multiplexing of multiple orbital-angular-momentum beams,” Sci. Rep. 7(1), 17427 (2017).
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Bonato, C.

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, A. Zeilinger, and C. Barbieri, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10(3), 033038 (2008).
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Booen, E.

C. Chen, A. Grier, M. Malfa, E. Booen, H. Harding, C. Xia, M. Hunwardsen, J. Demers, K. Kudinov, G. Mak, B. Smith, A. Sahasrabudhe, F. Patawaran, T. Wang, A. Wang, C. Zhao, D. Leang, J. Gin, M. Lewis, B. Zhang, D. Nguyen, D. Jandrain, F. Haque, and K. Quirk, “Demonstration of a bidirectional coherent air-to-ground optical link,” Proc. SPIE 10524, 105240G (2018).

Bouda, J.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11, 075001 (2009).
[Crossref]

Bourgoin, J.-P.

C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
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J.-P. Bourgoin, B. L. Higgins, N. Gigov, C. Holloway, C. J. Pugh, S. Kaiser, M. Cranmer, and T. Jennewein, “Free-space quantum key distribution to a moving receiver,” Opt. Express 23(26), 33437–33447 (2015).
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Boxleitner, W.

M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11, 075001 (2009).
[Crossref]

Brassard, G.

C. H. Bennett, G. Brassard, C. Crepeau, and U. M. Maurer, “Generalized privacy amplification,” IEEE Trans. Inf. Theory 41(6), 1915–1923 (1995).
[Crossref]

G. Brassard and L. Salvail, “Secret key reconciliation by public discussion,” Lect. Notes Comput. Sci. 765, 410–423 (1994).
[Crossref]

C. H. Bennett, G. Brassard, and J. M. Robert, “Privacy amplification by public discussion,” SIAM J. Comput. 17(2), 210–229 (1988).
[Crossref]

C. H. Bennett and G. Brassard, “Quantum cryptography: public-key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, (IEEE, 1984), pp. 175–179.

Braun, J.

J. Braun, J. Buchmann, D. Demirel, M. Geihs, M. Fujiwara, S. Moriai, M. Sasaki, and A. Waseda, “LINCOS: A storage system providing long-term integrity, authenticity, and confidentiality,” in Proceedings on Asia Conference on Computer and Communications Security, (ACM, 2017), pp. 461–468.

Brouri, R.

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–242 (2003).
[Crossref] [PubMed]

Buchmann, J.

J. Braun, J. Buchmann, D. Demirel, M. Geihs, M. Fujiwara, S. Moriai, M. Sasaki, and A. Waseda, “LINCOS: A storage system providing long-term integrity, authenticity, and confidentiality,” in Proceedings on Asia Conference on Computer and Communications Security, (ACM, 2017), pp. 461–468.

Burris, H. R.

Cai, W.-Q.

S.-K. Liao, W.-Q. Cai, J. Handsteiner, B. Liu, J. Yin, L. Zhang, D. Rauch, M. Fink, J.-G. Ren, W.-Y. Liu, Y. Li, Q. Shen, Y. Cao, F.-Z. Li, J.-F. Wang, Y.-M. Huang, L. Deng, T. Xi, L. Ma, T. Hu, L. Li, N.-L. Liu, F. Koidl, P. Wang, Y.-A. Chen, X.-B. Wang, M. Steindorfer, G. Kirchner, C.-Y. Lu, R. Shu, R. Ursin, T. Scheidl, C.-Z. Peng, J.-Y. Wang, A. Zeilinger, and J.-W. Pan, “Satellite-relayed intercontinental quantum network,” Phys. Rev. Lett. 120(3), 030501 (2018).
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S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground quantum key distribution,” Nature 549, 43 (2017).
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J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, M. Li, Y.-M. Huang, L. Deng, L. Li, Q. Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground entanglement-based quantum key distribution,” Phys. Rev. Lett. 119(20), 200501 (2017).
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J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356(6343), 1140–1144 (2017).
[Crossref] [PubMed]

Cai, X.-D.

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185 (2012).
[Crossref] [PubMed]

Cao, Y.

S.-K. Liao, W.-Q. Cai, J. Handsteiner, B. Liu, J. Yin, L. Zhang, D. Rauch, M. Fink, J.-G. Ren, W.-Y. Liu, Y. Li, Q. Shen, Y. Cao, F.-Z. Li, J.-F. Wang, Y.-M. Huang, L. Deng, T. Xi, L. Ma, T. Hu, L. Li, N.-L. Liu, F. Koidl, P. Wang, Y.-A. Chen, X.-B. Wang, M. Steindorfer, G. Kirchner, C.-Y. Lu, R. Shu, R. Ursin, T. Scheidl, C.-Z. Peng, J.-Y. Wang, A. Zeilinger, and J.-W. Pan, “Satellite-relayed intercontinental quantum network,” Phys. Rev. Lett. 120(3), 030501 (2018).
[Crossref] [PubMed]

S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground quantum key distribution,” Nature 549, 43 (2017).
[Crossref] [PubMed]

J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, M. Li, Y.-M. Huang, L. Deng, L. Li, Q. Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground entanglement-based quantum key distribution,” Phys. Rev. Lett. 119(20), 200501 (2017).
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J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356(6343), 1140–1144 (2017).
[Crossref] [PubMed]

J. Yin, Y. Cao, S.-B. Liu, G.-S. Pan, J.-H. Wang, T. Yang, Z.-P. Zhang, F.-M. Yang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Experimental quasi-single-photon transmission from satellite to earth,” Opt. Express 21(17), 20032–20040 (2013).
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J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185 (2012).
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Carrasco-Casado, A.

H. Takenaka, A. Carrasco-Casado, M. Fujiwara, M. Kitamura, M. Sasaki, and M. Toyoshima, “Satellite-to-ground quantum-limited communication using a 50-kg-class microsatellite,” Nat. Photon. 11502 (2017).
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Carter, J. L.

J. L. Carter and M. N. Wegman, “Universal classes of hash functions,” J. Comput. Syst. Sci. 18(2), 143–154 (1979).
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Casado, A. C.

K. Suzuki, D. Kolev, A. C. Casado, and M. Toyoshima, “Environmental-data collection system for satellite-to-ground optical communications,” Trans. JSASS Aerosp. Tech. Jpn. 16(1), 35–39 (2018). The data from OBSOC are available on http://sstg.nict.go.jp/OBSOC/?lang=e .
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Supplementary Material (2)

NameDescription
» Data File 1       This zip contains the data sets of free space optical transmission conducted from 10:00 on 7 Jun. 2017 to 10:35 on 13 Jun. 2017.
» Visualization 1       This movie presents the variations of experimental final key rate and the histograms of the demodulated sequences during 27 hours from 18:00 on 11 June.

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Figures (14)

Fig. 1
Fig. 1 (a) Schematic diagram of the configuration of FSO-SKA. (b) Key-distillation processing conducted over the public channel.
Fig. 2
Fig. 2 (a) Geographical configuration of Tokyo FSO Testbed. (b) Beam footprint at Bob’s site (the NICT building). (c) Photographs of Skytower West Tokyo at 4.5-km distance from NICT. The right picture is the enlarged view for the antenna and its supporting poles located on the Skytower West Tokyo. (d) Photographs of Alice’s site taken from NICT at 7.8-km distance from NICT.
Fig. 3
Fig. 3 Channel characterization for the data acquired at 14:50 on 23 May 2017. (a) Demodulated sequence ay(i) at Bob, where the index i runs from 1 to 2300000 to over at least one frame in the sequence. Also shown is the threshold based on the two-sided moving average. (b) Diagram of the main channel. Edges represent the conditional probabilities PAB(y|x). (c) Experimentally characterized PAB(y|x) of the main channel. (d) Demodulated sequence a(i) at v-Eve. (e) Diagram of the probing channel. Edges represent the conditional probabilities PAE(|x). (f) Experimentally characterized PAE(|x). For soft decision, bin width Δ is chosen so that the leaked information is appropriately estimated with finite-size data in practical experiment (see Appendix B). In this figure, the total bin number K is taken to be 763.
Fig. 4
Fig. 4 Theoretical calculations of mutual information (a) I(X; Y), (b) I(X; ), and (c) I(Y; ). The repetition rate and the laser wave length are assumed to be 10 MHz and 1550 nm, respectively. The detailed description of the calculation is given in Appendix C.
Fig. 5
Fig. 5 (a) Theoretical curve of the asymptotic key rate R K A. (b) Plot of the experimental R K A. The data points of outage, for which R K A 0, are indicated by x-marks. (c) Contour plot of the theoretical and experimental R K A.
Fig. 6
Fig. 6 Histograms of the demodulated sequences for cases A to E. These cases are shown in the upper left inset of Fig. 5(c). For each case, ARPSs B and E, wind velocity (w.v.), and experimental R K A are shown as well as the time and date.
Fig. 7
Fig. 7 Histograms of the demodulated sequences for cases F to J. These cases are shown in the upper right inset of Fig. 5(c). For each case, ARPSs B and E, wind velocity (w.v.), and experimental R K A are shown as well as the time and date.
Fig. 8
Fig. 8 Flow chart of the key-distillation processing.
Fig. 9
Fig. 9 (a) Theoretical characteristics of the final key rate R K F. (b) Plot of the experimental R K F. The data points of outage, for which R K F 0, are indicated by x-marks. (c) Contour plot of the theoretical and experimental R K F.
Fig. 10
Fig. 10 Summary of Campaign No. 3. (a) Variations of the one-hour averaged final key rate R K F (blue solid line) and the failure rate (red chain line). During experiment, 10 frames were recorded every 5 minutes. Thus, 120 frames were transmitted in each one hour. Gray shades indicate the night time between sunset and sunrise and dashed lines show the 00:00 AM in JST. (b) Variations of the one-hour averaged B (solid line) and E (chain line). (c) Variations of the one-hour averaged solar radiation and cloud ceiling. These meteorological data were provided by OBSOC [67], the compound meteorological sensor at NICT premises. For the interested reader, we provide Data File 1 [68] which includes the original data sets for this figure ( R K F, B, and E) as well as the histograms of the demodulated sequences. In Visualization 1, we present a movie of the variations of R K F and the histograms of the demodulated sequences during 27 hours from 18:00 on 11 June.
Fig. 11
Fig. 11 Summary of the NIST randomness test suites [69] for keys generated on 9 June 2017. We applied the tests on 1000 instances with a length of 106 bits. The acceptable uniformity should be larger than 10−4. As shown by the black dotted lines, the acceptable success rate range for excluding Random Excursions (Variant) test is 0.99 ± 0.00943: The acceptable success rate range is 0.99 ± 0.0121 for Random Excursions (Variant) test. For the items with multiple tests (Cumulative Sums, Non Overlapping Template, Random Excursions, and Random Excursions Variant), we show the range of the lowest and highest scores by the black solid lines.
Fig. 12
Fig. 12 (a) Comparison between I(X; ) and I(Y; ). The color of each point corresponds to the BER in the main channel. (b) Comparison between I(X; Y)−I(X; ) and I(X; Y)−I(Y; ). The color of each point corresponds to the BER in the probing channel.
Fig. 13
Fig. 13 (a) Average E[Pe,lPe,n] for 10000 trials. (b) Standard deviation var [ P e , l P e , n ] for 10000 trials. (c) Histogram of Pe,lPe,n for l = 10000 bits. The solid line represents the PDF of the Gaussian distribution with an average of 4.79 × 10−5% and a standard deviation of 0.0964%. (d) Histogram of Pe,lPe,n for l = 64000 bits. The solid line is the PDF of the Gaussian distribution with an average of 1.39 × 10−4% and a standard deviation of 0.0392%.
Fig. 14
Fig. 14 (a) Bin-width dependence of Il(Y; ) when test-sample length l is 64000 bits. (b) Average E[Il(Y; )−In(Y; )] for 10000 trials. (c) Standard deviation var [ I l ( Y ; Z ^ ) I n ( Y ; Z ^ ) ] for 10000 trials. (d) Histogram of Il(Y; ) − In(Y; ) for 10000 test samples. The solid line represents the PDF of the Gaussian distribution with an average of 2.78 × 10−2 bits/symbol and a standard deviation of 6.61 × 10−3 bits/symbol. (e) Histogram of Il(Y; ) − In(Y; ) for 64000 test samples. The solid line represents the PDF of the Gaussian distribution with an average of 6.57 × 10−3 bits/symbol and a standard deviation of 2.86 × 10−3 bits/symbol. For these two histograms, the bin width Δ is 0.08 mV.

Tables (1)

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Table 1 Summary of experimental configurations of FSO transmission campaigns

Equations (25)

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C K R K = I ( X ; Y ) min [ I ( X ; Z ) , I ( Y ; Z ) ] ,
I ( X ; Y ) = x y P A ( x ) P AB ( y | x ) log 2 [ P AB ( y | x ) x P A ( x ) P AB ( y | x ) ] ,
I ( X ; Z ) = x z P A ( x ) P AE ( z | x ) log 2 [ P AE ( z | x ) x P A ( x ) P AE ( z | x ) ] ,
I ( Y ; Z ) = y z P B ( y ) P BE ( z | y ) log 2 [ P BE ( z | y ) y P B ( y ) P BE ( z | y ) ] ,
θ IFoV = 2 tan 1 ( d pix 2 f tele ) .
a ¯ y ( i ) = 1 3001 j = i 1500 i + 1500 a y ( j ) .
P AB ( y | x ) = N AB ( y | x ) N A ( x )
R LDPC = { 0.85 ( 0 % P e < 0.7 % ) 0.80 ( 0.7 % P e < 1.7 % ) 0.75 ( 1.7 % P e < 2.7 % ) 0.70 ( 2.7 % P e < 3.7 % ) 0.65 ( 3.7 % P e < 4.7 % ) 0.60 ( 4.7 % P e < 5.7 % ) 0.55 ( 5.7 % P e < 6.7 % ) 0.50 ( otherwise ) .
I ( X ; Z ) I ( Y ; Z ) , I ( X ; Y ) I ( Y ; Z )
I ( X ; Y ) I ( Y ; Z ) max [ I ( X ; Y ) I ( X ; Z ) , 0 ] ,
I ( X ; Y ) = x = 0 , 1 y = 0 , 1 P A ( x ) P AB ( y | x ) log 2 [ P AB ( y | x ) x P A ( x ) P AB ( y | x ) ] .
P AB ( y = 1 | x ) = v th 0 Q AB ( v B | x ) d v B , P AB ( y = 0 | x ) = v th Q AB ( v B | x ) d v B ,
σ B 2 ( S ) = 1.57 × 10 2 S 2 + 7.75 × 10 19 S + 1.44 × 10 2 ,
Q AB ( v B | x ) = 1 2 π σ B 2 ( 2 x S ¯ B ) exp ( ( v B 2 x S ¯ B ) 2 σ B 2 ( 2 x S ¯ B ) ) .
P AB ( 1 | 0 ) = S ¯ B Q AB ( v B | 0 ) d v B = 1 2 erfc ( S ¯ B 2 σ B 2 ( 0 ) ) ,
P AB ( 0 | 1 ) = S ¯ B Q AB ( v B | 1 ) d v B = 1 2 erfc ( S ¯ B 2 σ B 2 ( 2 S ¯ B ) ) ,
I ( X ; Z ^ ) = x = 0 , 1 P A ( x ) Q AE ( v ^ E | x ) log 2 [ Q AE ( v ^ E | x ) x P A ( x ) Q AE ( v ^ E | x ) ] d v ^ E ,
Q AE ( v ^ E | x ) = 1 2 π σ E 2 ( 2 x S ¯ E ) exp ( ( v ^ E 2 x S ¯ E ) 2 σ E 2 ( 2 x S ¯ E ) ) ,
σ E 2 ( S ) = 1.92 × 10 2 S 2 + 1.65 × 10 19 S + 2.68 × 10 1 .
I ( Y ; Z ^ ) = x = 0 , 1 P B ( y ) Q BE ( v ^ E | y ) log 2 [ Q BE ( v ^ E | y ) y P B ( y ) Q BE ( v ^ E | y ) ] d v ^ B ,
P B ( y ) = x = 0 , 1 P A ( x ) P AB ( y | x ) = 1 2 x = 0 , 1 P AB ( y | x ) .
Q BE ( v ^ E | y ) = x = 0 , 1 Q AE ( v ^ E | x ) P BA ( x | y )
= x = 0 , 1 Q AE ( v ^ E | x ) P AB ( y | x ) P A ( x ) P B ( y )
= P AB ( y | 0 ) Q AE ( v ^ E | 0 ) + P AB ( y | 1 ) Q AE ( v ^ E | 1 ) 2 P B ( y ) ,
P BA ( x | y ) P B ( y ) = P AB ( y | x ) P A ( x ) .

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