Q^{3}Sat: quantum communications uplink to a 3U CubeSat—feasibility & design
- Sebastian Philipp Neumann†^{1, 2},
- Siddarth Koduru Joshi†^{1, 2},
- Matthias Fink^{1, 2},
- Thomas Scheidl^{1, 2},
- Roland Blach^{1},
- Carsten Scharlemann^{3},
- Sameh Abouagaga^{3},
- Daanish Bambery^{3},
- Erik Kerstel^{4},
- Mathieu Barthelemy^{4} and
- Rupert Ursin^{1, 2}Email author
Received: 11 November 2017
Accepted: 20 March 2018
Published: 27 April 2018
Abstract
Satellites are the most efficient way to achieve global scale quantum communication (Q.Com) because unavoidable losses restrict fiber based Q.Com to a few hundred kilometers. We demonstrate the feasibility of establishing a Q.Com uplink with a 3U CubeSat, measuring only 10 × 10 × 34 cm^{3}, using commercial off-the-shelf components, the majority of which have space heritage. We demonstrate how to leverage the latest advancements in nano-satellite body-pointing to show that our 4 kg CubeSat can generate a quantum-secure key, which has so far only been shown by a much larger 600 kg satellite mission. A comprehensive link budget and simulation was performed to calculate the secure key rates. We discuss design choices and trade-offs to maximize the key rate while minimizing the cost and development needed. Our detailed design and feasibility study can be readily used as a template for global scale Q.Com.
Keywords
1 Introduction
The security of quantum communication (Q.Com) is based on fundamental and immutable laws of physics and not on the assumption that a problem is and always will be too difficult for an adversary to solve. Naturally, this unconditionally secure communication technology has a large impact on global communications. Attempts to overcome the limits imposed by losses, such as Ref. [1], and attempts to create a global satellite based network, are underway [2, 3]. The latter are large and complex satellites which can cost upwards of 100 M€ each. Small CubeSats however can be constructed and launched for 0.5 to 10 M€. We present a simple, small, light-weight and low power-consuming satellite system capable of Q.Com. To achieve this, we considered several possible designs and individual components. The CubeSat performance was evaluated in each instance and design choices were made to minimize the Size, Weight and Power consumption (SWaP). This was done iteratively to create a commercially viable satellite system capable of Q.Com. Our CubeSat mission is called Q^{3}Sat (pronounced Q-CubeSat). Previous long distance implementations via optical fiber such as Ref. [4] and free space terrestrial links like Ref. [5] have approached the limits of terrestrial Q.com in terms of distance. The successful 600 kg class [2] and 50 kg class [6] large satellites have shown that Q.com in space is feasible. By analyzing the results of these proof-of-concept missions and evaluating their performance in both the uplink and the downlink scenario, we find that a downlink scenario offers a larger key rate. In an uplink, only a relatively simple polarization analysis module needs to be on board the satellite and ultra bright state-of-the-art quantum source(s) can be used on ground. Thus, an uplink is more suitable for a low cost CubeSat mission. Additionally, an uplink allows for a larger variety of implementable Q.Com protocols. This is because many different Q.Com protocols (e.g., E91 [7], BB84 [8], decoy state protocol (DSP) [9], BBM [10], B92 [11]) rely on nearly identical detection schemes for the receiver and can thus all be implemented on our CubeSat. Changes would only have to be made to the easily accessible ground module. Previous studies such as Refs. [12–16] have shown that space-based Q.Com is in principle feasible (also with small satellites) and culminated in two successful Q.Com satellites as well as other quantum experiments in space [17]. Recent efforts have evaluated the feasibility of downlinks [18] while others have attempted to solve the technological challenges identified by space-certifying detectors and sources of entanglement [19]. However, no previous works have evaluated the feasibility of Q.Com uplinks to satellites as small as a 3U CubeSat.
The CubeSat design considered here will also be able to perform tasks beyond Q.Com, e.g. measuring light pollution stemming from ground with a narrow field of view (FoV) to establish a global map in unprecedented resolution at single-photon level. This is crucial to finding dark areas near potential Q.Com customers and for other, more general applications. Additionally, the timing resolution of the single photon detectors enables pulse-position-modulation in classical communication from ground to space with exceptionally fast data rates. The extremely sensitive single photon detectors can also be re-purposed for other terrestrial and astronomical observations requiring an exceptional cadence and narrow FoV. In this manuscript we nevertheless focus on Q.Com, since this objective drives the design for the satellite infrastructure.
1.1 Quantum communication protocols
Let us consider the two most common Q.Com protocols—E91 [7] and the decoy state protocol (DSP) [9] which are explained in detail in Refs. [20, 21]. In both, information is encoded in the polarization state of single photons at the ground station (Alice) which then sends these states to the satellite (Bob). Bob measures the polarization of the received photons in a set of randomly chosen bases. The protocol is divided into several individual “trials”. In each trial, one state is sent and received. The techniques used to identify each trial depend on experimental implementation and protocol. To ensure that the key is secure, Alice and Bob perform statistical tests (i.e., compute the Quantum Bit Error Rate (QBER E) [22] and/or perform a Bell test) on the data they measured from several trials. Thus, they also need a form of (insecure but authenticated) classical communication. To obtain the key, Alice and Bob need various post-processing (PP) steps (detailed in [23]) that vary between protocols.^{1} Importantly, the larger the measured QBER, the more information an eavesdropper (Eve) could, in principle, obtain about the raw key. This means that privacy amplification must use up more raw key bits to reach the same level of security, reducing the total number of secure key bits. Thus, the amount of key that can be exchanged per second strongly depends on the QBER.^{2}
2 Error budget
List of parameters and values for which we assigned fixed values. Justification of these values is given in Sect. 3
Symbol | Parameter | Value |
---|---|---|
\(d_{B}\) | Diameter of active detector area on CubeSat | 20 μm |
\(D_{A}\) | OGS telescope diameter | 30 cm |
\(D_{B}\) | CubeSat telescope diameter | 10 cm |
\(e_{0}\) | Probability of noise count to be correct | 50% |
\(e_{d}\) | Probability of erroneous detection | 2% |
\(E^{\mathit {max}}_{\mathit {E}91/\mathit {DSP}}\) | Maximum tolerable QBER for E91/DSP | 10.2% /6.2% |
\(\eta_{A}\) | OGS multiplexed SNSPD efficiency (E91 only) | 70% (−1.5 dB) |
\(\eta_{B}\) | CubeSat detector efficiency | 15% (−8.2 dB) |
f | Error correction protocol efficiency | 1.1 |
\(f_{B}\) | Effective focal length CubeSat telescope | 40 cm |
\(f_{\mathit {SYN}}\) | Repetition rate of OGS’s beacon laser | 10 MHz |
FoV | Field of view CubeSat (full angle) | 50 μrad |
λ | Signal photon wavelength | 810 nm |
Λ | Total loss | −62.7 dB (max) |
\(\Lambda_{A}\) | Total loss OGS arm (source to detector) (E91 only) | 60% (−2.3 dB) |
\(\Lambda_{H}\) | Heralding efficiency (E91 only) | 85% (−0.7 dB) |
\(\Lambda_{\mathit {TA}}\) | OGS telescope loss (only E91) | −1.0 dB |
\(\Lambda_{\mathit {TB}}\) | CubeSat telescope loss | −1.5 dB |
\(\Lambda_{\mathit {OB}}\) | CubeSat optical elements loss | −1.0 dB |
\(\Lambda_{\mathit {PB}}\) | CubeSat pointing loss | −2.5 dB |
\(\Lambda_{\mathit {SB}}\) | CubeSat basis switch loss | −0.5 dB |
\(\Lambda_{\mathit {SYN}}\) | Loss due to errors in clock sync. | −0.5 dB |
\(\mu_{\mathit {DSP}}\) | Mean photon number per signal pulse (DSP only) | 0.64 |
\(\mu_{\mathit {E}91}\) | Mean photon number per coincidence window (E91 only) | 0.01 |
\(r_{0}\) | Fried parameter | 5 cm–40 cm |
\(R_{A}\) | OGS count rate (E91 only) | 60 Mcps |
\(R_{B}\) | CubeSat count rate (including noise) | 3 kcps (max) |
\(R_{B}^{\mathit {max}}\) | CubeSat detectors’ maximum count rate | 100 kHz |
\(R_{\mathit {BG}}\) | CubeSat background counts (total) | 80–180 cps |
\(R_{\mathit {DC}}\) | CubeSat dark count rate (per detector) | 200 cps |
\(R_{B+D}\) | CubeSat total noise counts | 480–580 cps |
\(R^{P}_{\mathit {DSP}}\) | Effective signal photon rate (DSP only) | 315 Mcps |
\(R^{P}_{\mathit {E}91}\) | Pair rate of entangled photon source (E91 only) | 100 Mcps |
\(R_{\mathit {rep}}\) | Repetition rate of single photon source (DSP only) | 1 GHz |
\(\sigma_{A}\) | OGS pointing precision (rms, full angle) | 2.4 μrad |
\(\sigma_{B}\) | CubeSat pointing precision (rms, full angle) | 40 μrad |
\(t_{A}\) | Combined OGS detectors + time tagging jitter | 16 ps |
\(t_{B}\) | CubeSat detector + time tagging jitter | 37 ps |
τ | Coincidence window | 80 ps |
\(t_{\mathit {SB}}\) | CubeSat basis switching time | 100 μs |
\(t_{\mathit {TT}}\) | Time tagging resolution (on board CubeSat) | 10 ps |
\(t_{\mathit {MD}}\) | Measurement duration of each chunk for clock sync. | 100 ms |
\(t_{\mathit {QC}}\) | Maximum duration of quantum connection per pass | 220 s |
3 Preliminary design
3.1 The ground segment
To implement different Q.Com protocols, different photon sources have to be deployed within the OGS. E.g. the E91 protocol requires an entangled photon source with a pair production rate \(R^{P}_{\mathit {E}91}=100\mbox{ Mcps}\) [32] and an intrinsic heralding efficiency \(\Lambda_{H}\) of 85% (−0.7 dB) [33]. For Alice to detect these extreme count rates on ground, we suggest using multiplexed arrays of superconducting nanowire single photon detectors (SNSPDs) with a detection efficiency \(\eta _{A}\) of 70% (85% for one single SNSPD without multiplexing) and a total timing jitter (including electronics) \(t_{A}\) of 16 ps (15 ps for the SNSPD alone) [34]. This results in a total \(\Lambda _{A}=\eta_{A}\cdot\Lambda_{H}=60\)% (−2.3 dB) and a ground based detector noise rate of less than 100 cps which we ignore in comparison to the total E91 singles rate of \(R_{A}\approx60\mbox{ Mcps}\). DSP requires a source capable of producing a controllable mean photon number per pulse \(\mu_{\mathit {DSP}} \approx 0.64\) (0.1) for the signal (decoy) pulse where 50% of all pulses carry a signal^{5} with a repetition rate of >1 GHz. This results in an actual signal photon rate \(R^{P}_{\mathit {DSP}}=315\mbox{ Mcps}\) at Alice. The notion of heralding efficiency \(\Lambda_{H}\) is not applicable for DSP and can be set to 1. The same is true for imperfections in the sender optics, since any losses prior to the free-space link itself can be utilized to realize the desired \(\mu_{\mathit {DSP}}\) value [37]. All sources can be designed to produce a quantum signal at wavelength \(\lambda\approx810\mbox{ nm}\), which is a good compromise taking into account atmospheric absorption, Mie scattering effects, diffraction, suitable lasers for producing entanglement and suitable space based detectors (low power consumption, low dark counts and high temporal resolution). All sources also share a common sending telescope with an unobstructed diameter (to ensure a better Gaussian mode and to limit the ground telescope attenuation \(1/\Lambda_{\mathit {TA}}\) to 1.0 dB [38]) of \(D_{A}=30\mbox{ cm}\). The tracking precision \(\sigma_{A}\) and slew rates of modern telescopes (typically \(\sigma_{A} <2.4\mbox{ }\upmu \mbox{rad}\) RMS (full angle) over 5 minutes with 13°/s slew) are an order of magnitude better than necessary to track and maintain an optical link with the CubeSat. For link calculations we assumed the OGS to be located on La Palma, where both experience from previous experiments and weather data were easily available to us. However, our design is not restricted to this location and need only be slightly adapted for areas with e.g. more cloud coverage. A suitable location for a second OGS still has to be fixed (see Sect. 4.3).
3.2 The CubeSat
3.2.1 Limiting noise counts
The most challenging aspect of designing a CubeSat is minimizing total noise counts \(R_{B+D}\) which therefore influences many design parameters. Unavoidable stray light collected by the CubeSat’s receiving telescope (i.e., background counts \(R_{\mathit {BG}}\)) and the intrinsic thermal/radiation damage counts of the detectors (i.e., dark counts per detector \(R_{\mathit {DC}}\)) add up to \(R_{B+D}=R_{\mathit {BG}}+2R_{\mathit {DC}}\) and significantly degrade the SNR. \(R_{\mathit {DC}}\), which we assume to be constant, has to be below 200 cps per detector to achieve a reasonable SNR. Firstly, the detector noise is reduced when operating at low temperatures. −30°C diode temperature is desirable. Two 250 cm^{2} radiators on the sun-averted sides of the CubeSat could dissipate the 0.6 W of thermal energy required to cool both detectors. A heating resistor should be used to further regulate the temperature to within ±1°C. While \(R_{\mathit {DC}}\) of such a cooled detector can be less than 5 cps in laboratory conditions [43], it is increased by damage due to energetic particles and ionizing radiation in space. This can be mitigated by using very small active detector areas \(d_{B}\). The smallest commercially available ones have a \(d_{B}\) of 20 μm, which we expect to be small enough to keep \(R_{\mathit {DC}}\) well below the 200 cps limit [44] despite a radiation damage equivalent to a 2 year mission lifetime. Using other satellite components such as high density batteries accounts for additional radiation shielding. Other procedures to further lower the dark count rate, such as annealing the diodes, could also be implemented if necessary [45]. We therefore assume a constant 200 cps of thermal and radiation noise per detector which is, at least for the first months of operation, a conservative estimate.
\(R_{\mathit {BG}}\) are the erroneous measurement clicks due to near-infrared noise photons originating from the ground area which are not blocked by the spectral filters. We estimate the magnitude of this effect by using measurements of earth’s luminous intensity from space [46] considering the spectral response of the Visible Infrared Imaging Radiometer Suite (VIIRS) [47] in use. More than 50% of the European Union’s land area have less than 270 μcd/m^{2} night sky brightness. We divide this background intensity into contributions of artifical (light pollution mainly by high pressure sodium (HPS) lamp based street lights [48] which undergoes absorption through the atmosphere [49])^{6} and natural (earthshine [50]) sources. These calculations are valid for new moon conditions. Additionally, as a worst-case scenario, we account for scattered sunlight from a full moon (brightness: 4000 cd/m^{2} [51]) reflected from earth (mean albedo: 0.3 [52]) into to the CubeSat (we used the solar radiation spectrum). We then translate the luminous intensity into photons [53] per second per m^{2} footprint impinging on the CubeSat telescope with aperture \(D_{B}=10\mbox{ cm}\) and calculate how many of these photons would pass through our 3 nm wide bandpass filters centered at 810 nm. We arrive at values of 0.55 photons s^{−1}m^{−2} in zenith and 0.17 photons s^{−1}m^{−2} for the lowest elevations (because of the larger distance between OGS and satellite). This effect of decreasing background counts per area for low elevations is however less significant than the increase in area because of the larger footprint on ground. The closer the CubeSat is to the horizon, the more ground area is covered by the satellite’s FoV since the circular footprint in zenith changes to a substantially larger elliptical one. Optical losses and detection efficiency of the CubeSat on the other hand reduce the background count value again (see below in this section).
In total this gives us a worst-case estimate of total noise counts which we use for all orbits regardless of the moon phase: \(R_{B+D}\) varies from ≈480 cps in zenith to ≈575 cps at 30° elevation from horizon. This assumption is very conservative, especially when considering the 350 cps total noise counts at full moon of a similar uplink experiment [54].^{7}
3.2.2 Field of view (FoV) and attitude control
To achieve an optimal \(f_{B}\), a Cassegrain-type reflector is a good choice for the receiving telescope despite the decreased telescope transmission \(\Lambda_{\mathit {TB}}\) due to the secondary mirror (which we estimate to be −1.5 dB in total). This is because the overall design is lightweight and the required \(f_{B}\) of 40 cm can be realized with a 10 cm long telescope. The telescope covers the CubeSat’s square Z+ surface of about 9 × 9 cm. For simplicity, our calculations assume a circular telescope with \(D_{B}=10\mbox{ cm}\).
3.2.3 Basis choice and polarization analysis
After passing the LC-HWP, the photons are spatially separated by a PBS, depending on their polarization. As seen in Fig. 3, the receiving telescope focuses the beam through the polarizing optics onto the detectors. To compensate for the angle dependent extinction ratios of the PBS and ensure \(e_{d}\leq2\)%, another polarizer (we suggest a second PBS rotated by 90° due to its high transmission) must be used in the reflected arm of the first PBS. To minimize polarization detection errors due to misalignment between the CubeSat and the ground station, the CubeSat has to rotate around its Z axis to maintain the same frame of reference. Another possibility would be to calculate the CubeSat’s rotation and precompensate on ground via a half-wave plate, analogous to Ref. [3].
3.2.4 Dead time and timing resolution
To ensure that saturation and dead time effects do not cause losses \({>}0.1\mbox{ dB}\), we require a maximum count rate of each CubeSat detector \(R_{B}^{\mathit {max}}\gg R_{B}\) in the order of 100 kHz. The detectors consist of actively quenched silicon-based avalanche photo diodes (APDs) operated in Geiger mode, placed at the output ports of the PBS. The detector diameter \(d_{B}\) of only 20 μm strongly reduces the cross sectional area for harmful radiation. Therefore little to no radiation shielding is required, which has a positive effect on the mass budget (see Table 2).
In addition to the quantum payload, the CubeSat optics should also accommodate an earth-facing beacon diode to aid in the ground station’s tracking of the CubeSat. There should also be a dichroic mirror to separate the quantum signal from the OGS beacon. The latter assists in locating and tracking the OGS and can be detected by a fast quadrant photo diode. The OGS’s beacon signal is pulsed to facilitate clock synchronization, and the detection pulses from the fast photo diode (along with GPS signals) are used to discipline the local clock on board the CubeSat.
3.2.5 Classical communication
In addition to the transmission of photons, classical communication and processing is required to generate a secure key. The amount of processing done on board the CubeSat must be minimized. Thus the CubeSat will need to transmit all detection events to the OGS, which will compute coincidence events and share data identifying these sparse events with the CubeSat. Therefore the amount of data transmitted by the CubeSat far exceeds the amount of data received. We use an S-band transceiver for the actual transmission of data. Additionally, we deploy a slower UHF transceiver for housekeeping communications [63]. For the SWaP budget, we assume peak power consumption during the whole orbit as a worst-case scenario and to account for the use of several successive OGS connections. Details about the data rates can be found in Sect. 4.5 while the processing power and time required for the CubeSat to calculate the secure key is estimated in Sect. 4.6.
3.3 Preliminary SMaP analysis
Using commercially available CubeSat components, we optimized the secure key rate produced by the CubeSat while adhering to the strict SWaP limitations. Our results are shown in Table 2. All systems not described in Sect. 3.2 are based entirely on readily available standard CubeSat components. Further customizing of certain parts would significantly lower the total SWaP consumption. The only component that would have to be modified is the time tagger, which is however within reach of current technology [61].
The CubeSat standard mass limit is 4 kg for a 3U. We can include a 6% mass margin and remain below this value. However this requirement can be relaxed to 5 kg depending on the launch provider [66]. This is useful if an operational lifetime of more than 6 months is desired which necessitates heavier shielding of the APDs (not included in the current SWaP). The type of solar panels [39] and the orbit of the CubeSat (see Sect. 4.3) limit the total power production per orbit to 21 Wh. We consume only 83% of this value. The satellite is within line of sight of the OGS for a maximum of 11 min (if it passes with 0° inclination), of which at most 220 s can be used for key generation. Thus most subsystems only operate for a fraction of each orbit. Together these consume 17.4 Wh^{11} while the always-on systems (attitude control, UHF-band communications, GPS and main computer) consume a further 13.5 Wh per orbit. The CubeSat’s single-photon detector system must operate only at night to avoid excessive background counts. Therefore a large set of batteries are necessary. To preserve battery life and provide a safety margin we assumed that the batteries are never drained by more than 30% . Thus we require a total battery capacity of at least 58 Wh. Our design provides for 60 Wh [67]. The CubeSat consumes a total of 17.4 Wh per orbit while its solar panels can produce a maximum of 21 Wh. The typical performance of this class of triple junction solar panels degrades to ≈85% of the above beginning of life value over 10 to 15 years [68, 69]. Thus with our short 1 to 2 year mission lifetime we can safely ignore this degradation. This means that the CubeSat is capable of one Q.Com connection per orbit. Larger satellites would be needed for continuous operation of the Q.Com link with more than one OGS per orbit, however this drastically increases the cost.
4 Performance analysis
Having specified the key parameters for the design of our CubeSat, we now want to give an estimate on the amount of secret key the satellite could acquire with two sufficiently separated OGS^{12} over one year (Sect. 4.7). To this end, we derive a model for geometric losses due to beam divergence (Sect. 4.1) while incorporating long-time measurements of atmospheric turbulence and weather influences (Sect. 4.2) to calculate different loss scenarios for our uplink. We also carry out an orbit assessment (Sect. 4.3). Lastly, we evaluate the requirements for an on board clock (Sect. 4.4) and estimate the data storage and -transmission needs (Sect. 4.5) as well as the computational requirements of the CubeSat (Sect. 4.6).
4.1 Optical loss model
4.2 Weather considerations
4.3 Orbit considerations
We therefore conclude that altogether, a 30° CO would be the optimal choice in terms of performance and reliability of the CubeSat and use it for our further calculations. Nevertheless, launching the CubeSat into an SSO is more common and can be significantly cheaper.
If we assume that an eavesdropper cannot access the secret key exchanged between the ground and the CubeSat, then the CubeSat can be trusted (i.e., it is a trusted node) to exchange another key with a second ground station and securely relay a message. The second OGS should be situated along the path of the CubeSat. Currently, daytime Q.Com is not possible with our scheme. However, an OGS in e.g. Australia would be able to communicate with the CubeSat during daytime in La Palma (assuming the choice of a 30° CO, Brisbane would have as much link-time per year as La Palma).
4.4 Clock synchronization
Both the OGS and the CubeSat measure the arrival time of photons according to their own local clocks (oscillators). Nevertheless, to identify photon pairs, we must synchronize these two clocks. The precision of this clock synchronization along with the timing jitter of the detectors and electronics determines the coincidence window. Improper synchronization leads to otherwise avoidable losses.
Synchronization can be achieved using various methods such as coarse synchronization to 10 ns using GPS [74], exploiting the intrinsic time correlation of entangled photon pairs [75], or using a pulsed beacon laser [2]. GPS alone is insufficiently precise. In order to exploit the time correlations of photon pairs, we must measure a cross-correlation peak in the arrival times between the OGS and CubeSat. The smallest measurement duration where we can unambiguously identify almost every coincidence peak (with the maximum acceptable total loss calculated above) is 100 ms.^{15} In LEO, the velocity of the CubeSat is so large that the optical path length between the OGS and satellite can change by as much as ≈6 km/s. Naturally, this causes the coincidence peak to broaden significantly. Orbital predictions and measurements can be used to correct for this. However, their typical precision is about 10 cm [76]. This still adds a few hundred picoseconds to the coincidence window needed.
Thus we use a pulsed beacon laser on the OGS and fast photo diodes in the CubeSat to implement a phase-locked loop and make sure that the CubeSat clock oscillates at the same frequency as the OGS’s. A beacon laser pulsed at a repetition rate of \(f_{\mathit {SYN}} = 10\mbox{ MHz}\) coupled with a fast photo diode receiver (≈1 GHz bandwidth) on the satellite can be used to synchronize the two oscillators to within 10 ps. Additionally, turbulence in the atmosphere can account for up to 3 mm (i.e. ≈10 ps) of jitter in the beacon laser’s arrival time [77]. The effects of such phase jitter on the received signal can be mitigated to a large extent using a technique called jitter attenuation [78]. Doppler and relativistic shifts could also affect the oscillator synchronization. The latter is corrected for by precompensating the CubeSat oscillator frequency on the ground similar to Ref. [79]. The former can be addressed by adjusting the repetition frequency of the pulsed beacon. Nevertheless, let us conservatively consider a total clock synchronization jitter of 20 ps. Using our chosen coincidence window of 80 ps, the above results in a synchronization loss \(1/\Lambda _{\mathit {SYN}}<0.5\mbox{ dB}\). Alternatively, we could avoid this additional loss by increasing the coincidence window to accommodate the uncertainty in clock synchronization (i.e. the coincidence window would be 100 ps instead of the chosen 80 ps).
4.5 Data storage and transmission
4.6 On-board computing requirements
The classical post processing required to obtain a secure key is not trivial and dictates the choice of the on-board processing capabilities of the CubeSat. A detailed overview of these requirements can be found in Ref. [80]. We also base our estimates on the equations provided there.
The first step is to identify coincidence events. This is commonly done by computing a timing cross-correlation histogram which can be a computationally intensive task.^{17} We recommend that the CubeSat share the timing of all its detection events with the OGS. The OGS can identify coincidence events and notify the CubeSat. This minimizes the amount of data transferred and the amount of calculations the CubeSat needs to perform. The on board processing of all the \(n_{\mathit {tag}}\) time tags should be less than \(18n_{\mathit {tag}}\) operations in the worst case. Calculating a sifted a key of length \(m_{\mathit {key}}\) is estimated to require roughly \(m_{\mathit {key}}\) bits of memory and \(15m_{\mathit {key}}\) operations to complete. Error correction requires additional memory and computational power. About 10 to 20 MB of memory are sufficient for this when using algorithms based on LDPC codes. The necessary LDPC matrices can be agreed upon before the mission and be stored locally for different QBER configurations. The same is true for privacy amplification algorithms. Together they consume less than 100 MB of storage space. Privacy amplification can be very memory efficient when using a linear-feedback shift-register-based matrix implementation and only requires memory equal to the sifted key length (i.e., \(m_{\mathit {key}}\) bits). To estimate the processing power required, we must keep in mind that a lower SNR increases the amount of error correction and privacy amplification necessary. In the worst case we estimate that all these PP steps will require ≈258 million operations per second to calculate the secure key in real time. This can easily be handled by a commercially available space certified on-board computer (OBC) with an ARM9 processor running at 400 MHz with enough spare processing power for other satellite tasks [81]. Considering possible delays and interruptions in the classical communication link, we estimate that PP would require approximately 300 MB of temporary memory. The OBC we consider can provide as much as 4 GB SD card storage space. We note that the on-board operating system, control programs, housekeeping functions etc. will require additional processing power and memory.
4.7 Expected secure key rates
5 Conclusion
Q.Com offers the best security currently possible since it is based on laws of physics as opposed to the difficulty of solving certain problems. However, it is expensive and communication distances are limited. Our complete feasibility study has shown that it is possible to achieve Q.Com over thousands of kilometers, via a single trusted node, using a relatively cheap and easy to construct CubeSat. By miniaturizing the design, optimizing power consumption and minimizing the mass we have shown that full-fledged commercial global Q.Com can be achieved with a simple 3U CubeSat. We have provided an outline for building a Q.Com mission which includes selection guides for the components, trade-offs and optimizations for the secure key rate, choice of orbits etc. We discussed methods to overcome key challenges using currently available technology. We showed that the fine pointing capabilities of CubeSats no longer limit their applicability for Q.Com and optical links.
Using our CubeSat design, a pair of ground stations can exchange \(13\cdot10^{6}\) secure bits a year (ignoring finite key effects). Our CubeSat design consists of commercially available components that cost <200,000 € [82]. A typical launch price is <300,000 € [66]. Naturally, the research/development and manpower costs for the first such satellite would be higher and are not included. Assuming a lifetime of two years, information theoretic security could be bought for ≈20 €/kbit,^{18} provided that an operational OGS is readily available. If deploying the decoy protocol, such an OGS would be about 100,000 €. For E91, there is no serious assessment possible at the moment due to rapid developments in nanowire technology and its strong dependence on the final detection scheme.
A commercially viable Q.Com satellite needs significant classical computation power, data storage and classical communication bandwidth. We have evaluated these requirements and outlined strategies to achieve all this with minimal resources. Our CubeSat is compatible with the widest possible variety of polarization based Q.Com protocols. It can implement the decoy state protocol to minimize client resources or entanglement-based protocols for best verifiable security. We have provided a CAD model of the CubeSat as well as a detailed discussion of the trade-offs involved in selecting components (such as those between: detection efficiency and timing jitter, radiation damage and FoV, erroneous counts and detector size, E91 and DSP, orbit of the satellite and total key etc.).
In the current design, the CubeSat is a trusted node. This is suitable for useage scenarios like communication between many branches of a single organization. The current state-of-the-art Q.com satellites are prohibitively expensive trusted nodes, for communication across the globe, that can only be built by a few select industries. A CubeSat—such as we have shown above—is cheaper and interested organizations can build their own or carefully supervise the building of these trusted nodes for their own use.
The proposed CubeSat can also be used for fundamental experiments such as Bell tests which require a SNR of only 4.8 (as opposed to the SNR of 8.8/15.1 needed by QKD), clock synchronization, light pollution measurements and earth/atmosphere observation at the beacon wavelengths. It can also be used to study the effect of gravity on quantum systems [83].
It is important to note that imperfect implementations of Q.com, such as those with high transmission loss [24] or those where the detectors are susceptible to blinding [25], can be vulnerable to an eavesdropper. However, security can still be guaranteed (and verified in the case of entanglement based protocols) using reasonable assumptions. First we assume that the losses in transmission are reasonably well known and are to a large extent beyond the control of Eve. Second, by maintaining line of sight during communication and securing the area around the Optical Ground Station (OGS) covered by the satellite’s FoV, we can prevent a blinding attack.
Because the information entropy factor (in square brackets) depends on the gains for DSP (see Eq. 9), there is no constant limit for DSP, it depends on losses and on the average photon number per pulse. The value given is a mean value for the loss scenarios considered by us.
In information theory, “rate” is a normalized quantity related to entropy. However, throughout this paper we continue to use the common definition of rate as number of occurrences/instances per second.
Our key rate estimation based on the signal pulse’s \(\mu_{\mathit {DSP}}\) is just an approximation without taking the photon statistics of the decoy states into account, which have a small, but non-negligible effect on the key rate. For simplicity and in order to obtain algorithms compatible with the computing power available to us, we stick to the partial formalism outlined in Ref. [9]. For a more detailed analysis, we refer the reader to [35] and [36].
The recent development of replacing HPS street lights with LEDs positively affects noise counts because the LED spectrum is marginal in NIR and IR. For a conservative estimate, we only considered HPS.
A direct comparison with the results of Ref. [54] is not possible because of differences in the FoV, spectral response etc. of the systems used. Nevertheless, the reported noise count values are in good agreement with our simulations and can be used as an approximate guideline for our design.
It should be noted that the manufacturer’s precision performance claims regarding the quoted XACT system are tentative since this performance has only been shown in-orbit for static (inertial) pointing.
For the sake of completeness it has to be noted that \(1/\Lambda_{\mathit {SB}}\) is the only attenuation which also acts on intrinsic dark counts (\(R_{\mathit {DC}}\)). Since the losses are not very high and we want to avoid underestimating noise counts, this effect is omitted.
This would e.g. be possible by launching the CubeSat from the ISS into an approximately 400 km orbit since the space station has a 4U launcher readily available.
The time each subsystem needs to run is calculated using conservative estimates. The detectors plus cooling are assumed to run at peak power throughout, although they only consume so much during the initial temperature stabilization phase before Q.Com starts. Similarly, fast S-band data transmission is assumed to run continuously, when in reality it need only operate when in line of sight of an OGS.
For our preferred 500 km 30° inclination orbit, e.g. Brisbane on the Australian east coast would be a near choice as second OGS additional to La Palma. If larger global coverage with many OGSs is desired—however leading to less passes over La Palma—, a 97° SSO orbit could be the better choice (see also Sect. 4.3).
The divisor 0.316 results from the fact that any aperture passed by a real beam results in an Airy disk pattern. We consider only the innermost disk since all others’ divergence is too great to hit the satellite. Now 0.316 \(D_{A}\) is the beam waist that an ideal Gaussian beam of the same intensity distribution as the innermost airy disk would have at the sending aperture, which allows us in good approximation to stick to Gaussian optics instead of having to apply Bessel functions.
If a more detailed analysis shows that stray sunlight is still a problem, further reducing the allowable time for contact is an option. For example, limiting the time from midnight to 5:00 am reduces the total contact time for one year to 57,700 s which is still acceptable.
Our minimum expected pair rate is about 52 pairs s^{−1}. To be able to correctly identify a peak, we must have more coincidence events than accidentals. With 5 coincidences we can correctly identify the peak >95% of the time. We choose 100 ms as the minimum chunk duration in order to be able to obtain the required number of coincidences.
These data transfer calculations assume an error free S-band link. Additional time or bandwidth will be needed to avoid garbled data.
The computational complexity of this task depends on the range of time delays that need to be scanned. Poor clock synchronization, low count rates and ever changing delays due to the satellite’s motion increase the range of delays over which the cross correlation function must be computed.
There are several ways to improve the cost per kbit. First, better radiation resistance and shielding would increase the lifetime of the CubeSat and proportionally decrease costs. Second, the current cost estimate is for the interaction of one CubeSat with a pair of OGSs on opposite sides of the globe. However with careful selection, one can use multiple OGSs with the same satellite during a single orbit provided we can increase the battery capacity of the CubeSat. Third, a mass produced constellation of satellites could reduce the cost by a further order of magnitude. Fourth, key expansion protocols can be used to grow the key with only marginal security implications. And finally, the deployment of detectors with other characteristics can help improve the key rate at cost of the SNR (see Fig. 4).
Notes
Declarations
Acknowledgements
We would like to acknowledge valuable discussions with Johannes Handsteiner, Bo Liu and Dominik Rauch of IQOQI Vienna.
Availability of data and materials
Simulations and scripts are made available upon request by the corresponding author RU.
Funding
FFG/ASAP11 Grant Number: 4927524 / 847964 (QubeSat), FFG Grant Nr. 6238191 / 854022, ESA/ESTEC Grant Nr. 4000112591/14/NL/US.
Authors’ contributions
The satellite design, feasibility study and simulations were done by SN, MF, and SKJ. The mechanical CAD design was created by RB, DB and SN. Orbital calculations were made by CS and SA. EK and MB handled the systems engineering and provided valuable feedback. The effort was conceived and supervised by RU and co-supervised by SKJ. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Azuma K, Mizutani A, Lo HK. Fundamental rate-loss trade-off for the quantum Internet. Nat Commun. 2016;7:13523. ADSView ArticleGoogle Scholar
- Yin J, Cao Y, Li YH, Liao SK, Zhang L, Ren JG, et al.. Satellite-based entanglement distribution over 1200 kilometers. Science. 2017;356(6343):1140–4. View ArticleGoogle Scholar
- Liao SK, Cai WQ, Liu WY, Zhang L, Li Y, Ren JG, et al.. Satellite-to-ground quantum key distribution. Nature. 2017;549:43–7. ADSView ArticleGoogle Scholar
- Yin HL, Chen TY, Yu ZW, Liu H, You LX, Zhou YH, et al.. Measurement-device-independent quantum key distribution over a 404 km optical fiber. Phys Rev Lett. 2016;117(19):190501. ADSView ArticleGoogle Scholar
- Ursin R, Tiefenbacher F, Schmitt-Manderbach T, Weier H, Scheidl T, Lindenthal M, et al.. Free-space distribution of entanglement and single photons over 144 km. Nat Phys. 2007;3:481–6. View ArticleGoogle Scholar
- Takenaka H, Carrasco-Casado A, Fujiwara M, Kitamura M, Sasaki M, Toyoshima M. Satellite-to-ground quantum-limited communication using a 50-kg-class microsatellite. Nat Photonics. 2017;11:502–8. View ArticleGoogle Scholar
- Ekert AK. Quantum cryptography based on Bell’s theorem. Phys Rev Lett. 1991;67(6):661. ADSMathSciNetView ArticleMATHGoogle Scholar
- Bennett ChH, Brassard G. Quantum cryptography: public key distribution and coin tossing. In: Int. conf. on computers, systems and signal processing. Bangalore, India. Dec. 1984. 1984. p. 175–9. Google Scholar
- Lo HK, Ma X, Chen K. Decoy state quantum key distribution. Phys Rev Lett. 2005;94:230504. ADSView ArticleGoogle Scholar
- Bennett CH, Brassard G, Mermin ND. Quantum cryptography without Bell’s theorem. Phys Rev Lett. 1992;68(5):557. ADSMathSciNetView ArticleMATHGoogle Scholar
- Bennett CH. Quantum cryptography using any two nonorthogonal states. Phys Rev Lett. 1992;68(21):3121. ADSMathSciNetView ArticleMATHGoogle Scholar
- Ursin R, Jennewein T, Kofler J, Perdigues JM, Cacciapuoti L, de Matos CJ, et al.. Space-quest, experiments with quantum entanglement in space. Europhys News. 2009;40(3):26–9. View ArticleGoogle Scholar
- Scheidl T, Wille E, Ursin R. Quantum optics experiments using the International Space Station: a proposal. New J Phys. 2013;15(4):043008. View ArticleGoogle Scholar
- Günthner K, Khan I, Elser D, Stiller B, Bayraktar Ö, Müller CR, et al.. Quantum-limited measurements of optical signals from a geostationary satellite. Optica. 2017;4(6):611–6. View ArticleGoogle Scholar
- Jennewein T, Grant C, Choi E, Pugh C, Holloway C, Bourgoin J, et al. The NanoQEY mission: ground to space quantum key and entanglement distribution using a nanosatellite. In: Emerging technologies in security and defence II; and quantum-physics-based information security III. vol. 9254. International Society for Optics and Photonics; 2014. 925402. Google Scholar
- Kerstel E, Gardelein A, Barthelemy M, Team TC, Fink M, Joshi SK, et al. Nanobob: a Cubesat mission concept for quantum communication experiments in an uplink configuration. 2017. arXiv:1711.01886.
- Bedington R, Arrazola JM, Ling A. Progress in satellite quantum key distribution. npj Quantum Inf. 2017;3(1):30. https://www.nature.com/articles/s41534-017-0031-5. ADSView ArticleGoogle Scholar
- Oi DK, Ling A, Vallone G, Villoresi P, Greenland S, Kerr E, et al.. CubeSat quantum communications mission. EPJ Quantum Technol. 2017;4(1):6. View ArticleGoogle Scholar
- Tang Z, Chandrasekara R, Tan YC, Cheng C, Sha L, Hiang GC, et al.. Generation and analysis of correlated pairs of photons aboard a nanosatellite. Phys Rev Appl. 2016;5(5):054022. ADSView ArticleGoogle Scholar
- Gisin N, Ribordy G, Tittel W, Zbinden H. Quantum cryptography. Rev Mod Phys. 2002;74(1):145–95. ADSView ArticleMATHGoogle Scholar
- Scarani V, Bechmann-Pasquinucci H, Cerf NJ, Dušek M, Lütkenhaus N, Peev M. The security of practical quantum key distribution. Rev Mod Phys. 2009;81(3):1301. ADSView ArticleGoogle Scholar
- Sergienko AV. Quantum communications and cryptography. Boca Raton: CRC Press; 2005. View ArticleMATHGoogle Scholar
- Fung CHF, Ma X, Chau H. Practical issues in quantum-key-distribution postprocessing. Phys Rev A. 2010;81(1):012318. ADSView ArticleGoogle Scholar
- Valivarthi R, Lucio-Martinez I, Chan P, Rubenok A, John C, Korchinski D, et al.. Measurement-device-independent quantum key distribution: from idea towards application. J Mod Opt. 2015;62(14):1141–50. https://doi.org/10.1080/09500340.2015.1021725. ADSGoogle Scholar
- Lydersen L, Wiechers C, Wittmann C, Elser D, Skaar J, Makarov V. Hacking commercial quantum cryptography systems by tailored bright illumination. Nat Photonics. 2010;4(10):686–9. ADSView ArticleGoogle Scholar
- Kim T, Fiorentino M, Wong FN. Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer. Phys Rev A. 2006;73(1):012316. ADSView ArticleGoogle Scholar
- Zhao Y, Qi B, Ma X, Lo HK, Qian L. Experimental quantum key distribution with decoy states. Phys Rev Lett. 2006;96:070502. ADSView ArticleGoogle Scholar
- Shor PW, Simple PJ. Proof of security of the BB84 quantum key distribution protocol. Phys Rev Lett. 2000;85:441–4. ADSView ArticleGoogle Scholar
- Elkouss D, Leverrier A, Alléaume R, Boutros JJ. Efficient reconciliation protocol for discrete-variable quantum key distribution. In: Information theory, 2009. ISIT 2009. IEEE international symposium on. IEEE; 2009. p. 1879–83. View ArticleGoogle Scholar
- Ma X, Fung CHF, Lo HK. Quantum key distribution with entangled photon sources. Phys Rev A. 2007;76(1):012307. ADSView ArticleGoogle Scholar
- Ali S, Saharudin S, Wahiddin M. Quantum key distribution using decoy state protocol. Am J Eng Appl Sci. 2009;2(4):694–8. View ArticleGoogle Scholar
- Steinlechner F, Trojek P, Jofre M, Weier H, Perez D, Jennewein T, et al.. A high-brightness source of polarization-entangled photons optimized for applications in free space. Opt Express. 2012;20(9):9640–9. ADSView ArticleGoogle Scholar
- Giustina M, Versteegh MAM, Wengerowsky S, Handsteiner J, Hochrainer A, Phelan K, et al.. Significant-loophole-free test of Bell’s theorem with entangled photons. Phys Rev Lett. 2015;115:250401. ADSView ArticleGoogle Scholar
- Single quantum SNSPD spec sheet. 2017. http://www.singlequantum.com/wp-content/uploads/2017/07/Single-Quantum-Eos.pdf. Accessed 2017-10-17.
- Ma X, Qi B, Zhao Y, Lo HK. Practical decoy state for quantum key distribution. Phys Rev A. 2005;72(1):012326. ADSView ArticleGoogle Scholar
- Achilles D, Rogacheva E, Trifonov A. Fast quantum key distribution with decoy number states. J Mod Opt. 2008;55(3):361–73. ADSMATHGoogle Scholar
- Bourgoin J, Meyer-Scott E, Higgins BL, Helou B, Erven C, Huebel H, et al.. A comprehensive design and performance analysis of low Earth orbit satellite quantum communication. New J Phys. 2013;15(2):023006. View ArticleGoogle Scholar
- Personal communication with Dr. Zoran Sodnik. ESA/ESTEC. 2017. Google Scholar
- Brochure for deployable solar panels from cubesatshop.com. 2016. http://www.cubesatshop.com/wp-content/uploads/2016/07/EXA-DSA-Brochure-1.pdf. Accessed 2017-10-17.
- Aperture Optical Systems CubeSat telescope. 2017. http://www.apertureos.com/products/cube-sat. Accessed 2017-10-17.
- Thorlabs dielectric filters. 2017. https://www.thorlabs.com/navigation.cfm?guide_id=2210. Accessed 2017-10-17.
- Thorlabs polarizing beam splitters. 2017. https://www.thorlabs.de/newgrouppage9.cfm?objectgroup_id=739. Accessed 2017-10-17.
- MPD PDM series data sheet. 2017. http://www.micro-photon-devices.com/Docs/Datasheet/PDM.pdf. Accessed 2017-11-04.
- Anisimova E, Higgins BL, Bourgoin JP, Cranmer M, Choi E, Hudson D, et al.. Mitigating radiation damage of single photon detectors for space applications. EPJ Quantum Technol. 2017;4(1):10. View ArticleGoogle Scholar
- Lim JG, Anisimova E, Higgins BL, Bourgoin JP, Jennewein T, Makarov V. Laser annealing heals radiation damage in avalanche photodiodes. EPJ Quantum Technol. 2017;4(1):11. View ArticleGoogle Scholar
- Falchi F, Cinzano P, Duriscoe D, Kyba CC, Elvidge CD, Baugh K, et al.. The new world atlas of artificial night sky brightness. Sci Adv. 2016;2(6):e1600377. ADSView ArticleGoogle Scholar
- Official site of the Visible Infrared Imaging Radiometer Suite (VIIRS) run by NASA and NOAA. 2017. https://jointmission.gsfc.nasa.gov/VIIRS.html. Accessed 2017-10-18.
- Lamphar HAS, Kocifaj M. Light pollution in ultraviolet and visible spectrum: effect on different visual perceptions. PLoS ONE. 2013;8(2):e56563. ADSView ArticleGoogle Scholar
- MODTRAN WebApp. 2017. http://modtran.spectral.com/modtran_home. Accessed 2017-11-07.
- Yan F, Fosbury RA, Petr-Gotzens MG, Zhao G, Wang W, Wang L, et al.. High-resolution transmission spectrum of the Earth’s atmosphere-seeing Earth as an exoplanet using a lunar eclipse. Int J Astrobiol. 2015;14(2):255–66. View ArticleGoogle Scholar
- Luminance and brightness data for the full moon. 2009. http://spaceweather.com/swpod2009/13jan09/Perigee_moon_2009_01_11_corr.pdf. Accessed 2017-10-20.
- Stephens GL, O’Brien D, Webster PJ, Pilewski P, Kato S, Li Jl. The albedo of Earth. Rev Geophys. 2015;53(1):141–63. ADSView ArticleGoogle Scholar
- Zong Y. From candle to candela. Nat Phys. 2016;12(6):614. View ArticleGoogle Scholar
- Ren JG, Xu P, Yong HL, Zhang L, Liao SK, Yin J, et al.. Ground-to-satellite quantum teleportation. Nature. 2017;549(7670):70–3. ADSView ArticleGoogle Scholar
- Blue Canyon XACT data sheet. 2017. http://bluecanyontech.com/wp-content/uploads/2017/07/DataSheet_ADCS_08_F.pdf. Accessed 2017-10-17.
- This value is expected for the Blue Canyon XB-1 spacecraft bus which is entirely compatible with our design. Personal communication with Josh Duncan. Blue Canyon Technologies. For the XB-1 bus specifications see. 2017. http://mstl.atl.calpoly.edu/~bklofas/Presentations/SummerWorkshop2012/Stafford_XB1.pdf. Accessed 2017-10-20.
- Mason JP, Baumgart M, Rogler B, Downs C, Williams M, Woods TN, et al. MinXSS-1 CubeSat on-orbit pointing and power performance: the first flight of the Blue Canyon technologies XACT 3-axis attitude determination and control system. 2017. arXiv:1706.06967.
- Meadowlark ferroelectric liquid crystal data sheet. 2017. http://www.meadowlark.com/store/data_sheet/Liquid%20Crystal%20-%20FLC%20Devices%20NEW.pdf. Accessed 2017-10-17.
- Chandrasekara R, Durak K, Ling A. Tracking capacitance of liquid crystal devices to improve polarization rotation accuracy. Opt Express. 2017;25(17):20363–8. ADSView ArticleGoogle Scholar
- Schindler W, Killmann W. Evaluation criteria for true (physical) random number generators used in cryptographic applications. In: CHES. vol. 2. Berlin: Springer; 2003. p. 431–44. Google Scholar
- Picoquant TimeHarp 260 data sheet. 2017. https://www.picoquant.com/images/uploads/downloads/timeharp260.pdf. Accessed 2017-10-17.
- Personal communication with Dr. Michael Schlagmueller. Swabian Instruments. 2017. Google Scholar
- Endurosat CubeSat S-band/UHF communication module data sheet. 2017. https://www.endurosat.com/modules-datasheets/COMM_User_Manual_Rev1.6.pdf?x65766. Accessed 2017-10-27.
- CubeSat design specification rev. 13 by California Polytechnic State University. 2017. http://www.cubesat.org/s/cds_rev13_final2.pdf. Accessed 2017-10-17.
- NanoRacks CubeSat Deployer (NRCSD interface control document. 2017. http://nanoracks.com/wp-content/uploads/NanoRacks-CubeSat-Interface-Control-Document-CubeSat-Guide.pdf. Accessed 2017-11-02.
- Spaceflight launch company. 2017. http://spaceflight.com/schedule-pricing/#pricing. Accessed 2017-10-17.
- CubeSatShop BA0x high energy density battery array Pegasus Class BA01/D. 2017. https://www.cubesatshop.com/wp-content/uploads/2016/11/EXA-BA0x-Brochure.pdf. Accessed 2017-10-27.
- Bailey S, Raffaelle R. Space solar cells and arrays. In: Handbook of photovoltaic science and engineering. 2nd ed. 2011. p. 365–401. View ArticleGoogle Scholar
- ^{3}SatImaging. 2017. http://propagation.ece.gatech.edu/ECE6390/project/Sum2015/team3/PowerSystem.html. Accessed 2017-10-27.
- Elterman L. UV, visible, and IR attenuation for altitudes to 50 km, 1968: environmental research papers. United States Air Force, Office of Aerospace Research, Air Force Cambridge Research Laboratories, Optical Physics Laboratory; 1968. https://books.google.at/books?id=Vt1VAQAACAAJ.
- RoboDIMM, the ING’s new seeing monitor. 2017. http://www.ing.iac.es/PR/newsletter/news7/ins7.html. Accessed 2017-10-17.
- Oltrogge D, Leveque K. An evaluation of CubeSat orbital decay. In: 25th annual AIAA/USU conference on small satellites. 2011. Google Scholar
- Ginet GP, Madden D, Dichter BK, Brautigam DH. Energetic proton maps for the South Atlantic anomaly. In: Radiation effects data workshop, 2007 IEEE. IEEE; 2007. p. 1–8. Google Scholar
- Montenbruck O, Ramos-Bosch P. Precision real-time navigation of LEO satellites using global positioning system measurements. GPS Solut. 2008;12(3):187–98. View ArticleGoogle Scholar
- Ho C, Lamas-Linares A, Kurtsiefer C. Clock synchronization by remote detection of correlated photon pairs. New J Phys. 2009;11(4):045011. View ArticleGoogle Scholar
- Kirschner M, Weigel M, Kahle R, Kahr E, Choi P, Letsch K, et al. Orbit precision analysis of small man-made space objects in LEO based on radar tracking measurements. DLR website. 2012. Google Scholar
- Prochazka I, Kral L. Atmospheric contribution to the laser ranging jitter. In: Proc. 13th int. laser ranging workshop. 2002. Google Scholar
- AN513 Jitter attenuation: choosing the right phase-locked loop bandwidth. 2017. https://www.silabs.com/documents/public/application-notes/AN513.pdf. Accessed 2017-11-04.
- Ashby N. Relativity in the Global Positioning System. Living Rev Relativ. 2003;6:1. ADSView ArticleMATHGoogle Scholar
- Gigov N. Quantum key distribution data post-processing with limited resources. Towards Satellite-Based Quantum Communication [mathesis]. 2013. Google Scholar
- On-board computer brochure from Cubesatshop. 2017. https://www.cubesatshop.com/wp-content/uploads/2016/06/iOBC-Brochure-v1.pdf. Accessed 2017-10-17.
- CubeSatShop one-stop webshop for CubeSats and Nanosats. 2018. https://www.cubesatshop.com/. Accessed 2018-02-15.
- Joshi SK, Pienaar J, Ralph TC, Cacciapuoti L, McCutcheon W, Rarity J, et al. Space QUEST mission proposal: experimentally testing decoherence due to gravity. 2017. arXiv:1703.08036.