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Abstract
Due to the absorption of water, communication between two parties submersed below the water is normally performed with acoustic waves. However, with the need for higher data rates, the use of RF or optical frequencies is needed. Currently, optical wavelengths have been demonstrated for classical communication over short distances. For these short distances, if a large amount of data needs to be transmitted securely, it is not feasible for both parties to return to the surface to communicate. Additionally, it can be assumed that a third party (Eve) is located in the channel trying to gather information. The solution is to use quantum key distribution (QKD) to generate the secure key, allowing the parties to continuously encrypt and transmit the data. It is assumed the BB84 protocol using pairs of polarization entangled photons generated from a spontaneous parametric down conversion (SPDC) source of Type-II. By using entangled photons, Eve is not able to gain information without being detected. In this work, horizontal oceanic channel is studied for various distances ranging from 10 m to 100 m, depth ranging from 100 m to 200 m, and surface chlorophyll-a concentrations at a wavelength of 532 nm. The secure key rates are calculated, assuming that a low-density parity check (LDPC) error correction code is used for information reconciliation. The maximum secure key rate and optimal number of average entangled photons transmitted are then studied for the various channels.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
To perform communication between two parties underwater there are several options depending on the distance between the parties as well as the desired communication rate. Possible solutions for underwater communication are acoustic or electromagnetic waves. Using acoustic waves, data rates of ∼ 600 bps can be achieved at distances of 1000 km, while data rates of ∼ 30 kbps can be achieved for distances between 0.1 and 1 km. For the electromagnetic waves in the radio frequency (RF) range, as the frequency is increased the attenuation also increases. This results in only being able to achieve data rates on the order of Mbps at distances of ≈ 10 m. Using optical frequencies, data rates on the order of Gbps can be achieve for distances up to 100 m [1]. When using acoustic waves and RF frequencies, the transmission is not highly directional introducing a security vulnerability. Optical frequencies are highly directional, reducing the security vulnerability and requiring a lower transmission power. In underwater optical communication, links are also not limited to direct line of sight (LOS) but can also include reflective LOS by using total internal reflection at the water-air boundary [1].
In any link the ability to transmit data securely between two parties is a desirable property, however conventional methods for encrypting data are designed using computational complexity. If an adversary is provided enough time or adequate computational power, the encrypted data is no longer secure [2]. The solution to this security loophole is through the use of one-timepad keys for encryption, where the key is discarded after use. For continuous encryption, the secrete key needs to constantly be generated between the two authenticated parties, Alice and Bob, at a distance. This can be accomplished by using quantum key distribution (QKD) [3–5]. QKD can be implemented using either single photons or entangled photons, of which this work considers the use of entangled photons [6,7]. It has been shown that an underwater quantum channel has very limited depolarization and disentanglement, implying that QKD is feasible [8]. Additionally, the use of twisted photons to increase the dimensionality of QKD protocols has been demonstrated in a 3 m underwater channel [9]. The consideration of a QKD system in an underwater channel has also been considered for a BB84 system using single photons and excluding the effects of oceanic turbulence [10,11].
Considering secure communication between two submarines, it is not feasible for them to return to the ocean surface to generate a new shared key as this will be time consuming and may reveal their position to an adversary. This work is a continuation of the conference proceedings [12], where a submarine to submarine QKD system employing the BB84 with entangled photons was studied. The system under consideration is located in the South Atlantic ocean at a latitude of 35° South, with the channel distances ranging from 10 m to 100 m and at depths of 100, 150 and 200 m. It was found that increasing the depth allowed for higher secure key rates, and the background photons prevented any secure key from being generated at a depth of 100 m. In this work a larger range of depths and two concentration of surface concentrations of Chlorophyll-a have been considered. This paper is organized as follows: first a description of the submarine to submarine QKD system that is implemented, then the various channels being considered are discussed, and finally show the resulting secure key rates that were able to be achieved for each channel.
2. Submarine to submarine QKD system
To implement an entanglement assisted QKD system that is capable of operating in an underwater channel, photons need to be generated in the blue-green wavelength region, the reason for which will be explained in section 3. In the work of Sciarrino et al, the generation of polarization entangled photons at a wavelength of 532 nm with high purity and fidelity using the SPDC process is described [13]. Additionally, the work of Li et al, showed that superconducting nanowire single photon detectors (SNSPDs) operating at a 532 nm, with detection efficiency of 75% and having a dark count rate of <0.1 Hz [14]. The SNSPDs have a sensitive area diameter of 42 μm and a full recovery time of 300 ns, thus the system will be assumed to operate at a rate of 3 MHz to ensure the detectors can fully recover. This also implies that the probability of a detector clicking due to a dark count is PDark = 3 × 10−7.
In Fig. 1, a system model is presented showing the channel configuration broken into multiple segments; the source to Eve (SE), and Eve to Bob (EB) and detection apparatus. The source is located at Alice’s side of the channel and an eavesdropper, Eve, is assumed to be located in the channel between Alice and Bob. Eve has the ability to correct all distortions to the wavefront caused by the channel and the ability to detect how many photons were transmitted without measuring the photons. By using entangled photons, Eve is not able to gain any information due to the generation of multiple entangled photon pairs [15]. Eve is not actively simulated to perform the intercept-resend attack, however all errors are assumed to be generated by Eve. Eve however is passively listening to all of the communication on the public channel, and gains knowledge of the final key when Alice and Bob perform error reconciliation. To implement the BB84 protocol using polarization entangled photon, Alice and Bob will both have similar detection apparatuses. A 50:50 beam splitter (BS) will sort the photons into one of two arms. Using a half wave-plate (HWP) the photons are put into mutually unbiased polarization basis, horizontal/vertical (HV) or ±45° (AD), then a polarizing beam splitter (PBS) determines the measurement orientation for each photon.
Fig. 1 a) QKD System model with an entangled photon source located at Alice’s side of the channel. b) The detection apparatus used to implement the BB84 protocol.
The source is assumed to be a Type-II periodically poled LiNbO3 (PPLN) nonlinear crystal, and the probability that n entangled photon pairs are generated is given by
where μ is the mean number of entangled photon pairs generated by one pump pulse [16]. In this work only μ will be considered, however by definition μ is dependent on the coupling efficiency. As the channel from the source to Alice has significantly lower loss, the probability that multiple detectors click increases as μ increases. Alice can choose to select only one of the detector clicks to treat as the true detection, and treat all others as noise. This will lead to an increase in the bit error rate δ. To calculate the total amount of information that Eve gains, the security proof by Koashi et al is used [17]. For a given δ, the rate of generating a secure key is given as where PKey is the probability of a time slot being included in the unsecured key, H2(δ) is the binary entropy function, and f(δ) is the error reconciliation efficiency, where ideal error correction efficiency is f(δ) = 1. It is assumed that the practical LDPC codes that were presented by Elkouss with code rates ranging from 0.5 to 0.9 and thresholds BER ranging from 0.1071 to 0.0108 respectively were used [18]. For bit error rates in excess of 0.1071, it is assumed that the cascade protocol is used. The error reconciliation of an LDPC with a code rate, R, and threshold BER, δT can be calculated as [19] where δT is the lowest threshold that satisfies δ ≤ δT. The described QKD system is modeled using the Monte Carlo simulation described in [20]. By providing μ, the loss of the channel and the background noise/dark count noise probabilities the simulation can calculate δ and RSKR. The SKR can then be calculated by dividing RSKR by the number of time slots used.3. Oceanic underwater optical channel
Similar to a free-space optical channel, an underwater channels has attenuation, from absorption and scattering, as well as beam wander and degradation due to turbulence. For pure water the absorption spectrum is known and dependent on the wavelength used, where the minimum absorption is aw(λ) = 0.00442 m−1 for λ = 417.5 nm [21]. Additionally the size and type of particles in the water can also impact the channel loss. In an oceanic underwater channel, scattering from particles can be classified into either small or large particles, each with a different statistical distribution. For absorption, factors such as the concentration of chlorophyll-a, and fulvic/humic acids are relevant as well [22]. Chlorophyll-a is the main substance of phytoplankton (photosynthesizing microorganisms), while fulvic/humic acids act as nutrients for the phytoplankton. The concentration of the particles has been modeled using the chlorophyll-a concentration at a depth d, Cc(d), and the total channel attenuation is given by [22,23]:
where a(λ, d) and b(λ, d) are the absorption and scattering coefficients. Cc(d) is modeled using a Gaussian curve that is dependent upon the depth, depth of the euphotic layer Zeu, average column-integrated content of total chlorophyll-a within the euphotic layer (mg m−3), background chlorophyll-a concentration at the surface Cb, and the vertical gradient s (m−1) as follows [24] where CMax is the maximum concentration, Δζ is the width of the Gaussian peak, and ζMax = ZMax/Zeu is the dimensionless depth of the maximum concentration. This leads to the concentrations of the fulvic acid Cf, humic acid Ch, small particles Cs and large particles Cl at depths d as follows,Now using the chlorophyll-a based model, the absorption spectrum is given as
where the specific absorption coefficients and exponential coefficient for the fulvic acid and humic acid and chlorophyll are af = 35.959 m2/mg, ah = 18.828 m2/mg and kf = 0.0189 nm−1, kf = 0.01105 nm−1. The absorption coefficient for chlorophyll-a can be found in the work of Bricaudi et al [25]. The scattering spectrum can similarly be defined as where the scattering spectrum for pure water bw(λ), small particles bs(λ), and large particles bl(λ) areSimilar to the concentration of chlorophyll-a, the oceanic turbulence strength can also vary with depth and geographic location. In general the ocean can be broken into the following layers; the mixed surface layer extending up to 200m, the upper waters ranging from 200m to 1000m, and deep water which is below 1000m. The surface layer is well mixed by the wave, currents, and tides. The upper waters is a layer where there are rapid changes in the density, temperature, and salinity. The deep water is cold and very dense. In this work, beam propagation in the surface layer of the Southern Atlantic ocean around 35°S is considered. The oceanic turbulence induce changes in the index of refraction resulting in beam wander, intensity fluctuations, and increased dispersion. The power spectrum of the oceanic turbulence is defined by Yi et al [26]
where ∊ is the dissipation rate of turbulent kinetic energy per mass of fluid [Wkg−1], χT is the diffusive dissipation rate of the mean-square temperature fluctuations, ω is the relative strength of temperature and salinity fluctuations, A is the thermal expansion coefficient [1/K] and dr is the ratio of saline eddy diffusivity to thermal eddy diffusivity. κ is the magnitude of the wavenumber vector, and β is the Oboukhov-Corrsin constant. g(κη, Pr) is a universal function defined for the Prandtl number, Pr, which relates the Kolmogorov and Batchelor length scales, as where the constant values for the universal function temperature with PrT = 7, salinity with PrS = 700 and the coupled temperature-salinity with PrTS = 13.86 are defined in [26]. ∊ and χT are found by considering the wind stress at the surface, heat flux at the surface, density, temperature (T), salinity (S) and specific heat at constant pressure. Both ∊ [m2s−3] and χT [k−2s−1] are inversely proportional to the depth [27]. For the South Atlantic ocean around 35°S in June, these constants values can be found in the works of Klinger et al, Dong et al, Jamieson et al, and Goes et al [28–31]. Generally when describing atmospheric turbulence the index of refraction structure function, , is used to describe the turbulence strength where (units m−2/3).To model the effect of oceanic turbulence on an optical beam, beam propagation simulations are ran using the split step beam propagation method [32,33]. The simulation space consisted of 4096 × 4096 grid, with side lengths of 1 m. Using the turbulence power spectrum in Eq. (15), phase screens are generated corresponding to a 1m thick slab of turbulence. The transmit and receive apertures are assumed to be 5cm in diameter with focal lengths of 10 cm, and a plane wave is transmitted.The region that is being considered is a subtropical region, and the parameter for the turbulence profile are shown in Table 1. When only considering the effects of oceanic turbulence and beam propagation, the probability that a photon lands on a circular detector with a diameter of 42 μm is shown in Fig. 2(a) for channels at a depth of 100, 150 and 200 m. With the increase in the depth, the strength of the turbulence decreases, resulting in a higher probability that a photon will land on the detector. However, as the distance increases the beam spreads and more phase distortions are added to the wavefront causing the focused spot to blur and the probability that a photon lands on the detector to decrease.
Fig. 2 a) Probability that a photon lands on a detector for different propagation distances b) The attenuation profile for different surface chlorophyll-a concentrations as a function as underwater depth.
Table 1. Channel Conditions
The surface chlorophyll-a concentrations (Chlasurf [mg m−3]) considered are for the month of June at 35° South. Employing the data set S4 (Chlasurf ∈ [0.12, 0.2]) and S5 (Chlasurf ∈ [0.2, 0.3]) from Uitz et al [24, 34]. The constant values for the data set S4 are , Zeu = 80.2, Cb = 0.570, s = 0.173, Cmax = 0.766, and Δζ = 0.814. The data set S5 constants are , Zeu = 70.3, Cb = 0.611, s = 0.2.4, Cmax = 0.676, and Δζ = 0.663 [24]. The resulting attenuation vs depth profiles are shown in Fig. 2(b). In both scenarios the peak attenuation is located higher than the depths considered, resulting in lower attenuation values as the depth increases and allowing transmission at longer distances to occur. Additionally, the attenuation profile as a function of depth allows us to calculate the total attenuation of the background light that enters the receive aperture. At the surface of the ocean, the background irradiance is Esol = 1440 W/m2, the number of background photons landing on the detector, NBack is calculated as follows [35,36]
where the factor describing the direction of underwater irradiance Lfac = 2.9, the underwater reflectance R = 1.25%, and the average attenuation of the water column from the surface to a depth D is αDav in m−1. To calculate αDav, the numerical integral of a(λ, d) is taken with respect to d is calculated from 0 to D, then divided by D. To calculate NBack, the transmittance of the receiver optics ηr = 0.1881, the narrow band filter spectral bandwidth Δλ = 10Å, the narrow band filter transmission ηλ = 1, the detector diametrical field of view Θ, and hc = 1.986 × 10−23 is Planck’s constant times the speed of light.4. QKD simulation results
To simulate the QKD system that has been described above, we use a simulation that steps through values of μ with a step size of 0.005 [20]. The maximum SKR for each channel scenario is shown in Figs. 3(a) and 3(b), while the optimum value of μ producing that SKR is shown in Figs. 3(c) and 3(d). As expected, with increased distance the SKR and optimal value of μ decrease in both scenarios. This decrease is due to the increased loss introduced into the channel. For a constant depth, the noise level remains constant, higher loss results in a larger portion of the un-secure key bits being generated from noise and increasing the bit error rate. For the channels located at deeper depths, background photons from the surface experience higher attenuation allowing longer links to be established.
Fig. 3 a) Optimum SKR at various distances and depths with data set S4. b) Optimum SKR at various distances and depths with data set S5. c) Optimum μ at various distances and depths with data set S4. d) Optimum μ at various distances and depths with data set S5
By comparing the SKR’s for different surface chlorophyll-a concentrations, for higher surface chlorophyll-a concentrations (data set S5) the achievable the maximum attenuation is now located closer to the surface and deeper channels have lower attenuation. Additionally, the higher concentration of chlorophyll increases the attenuation seen by background photons coming from the surface thus reducing the probability a background photon is detected.
The worst channel being considered is located a channel at a depth of 100 m, as it will have the highest attenuation and background irradiance. When the surface chlorophyll-a concentration is lower (data set S4), no positive SKR can be achieved for any distance considered as the channel is dominated by background noise. The minimum depth at which a positive SKR can be achieved is 110 m for a distance of 10 m. If the surface chlorophyll-a concentration is higher (data set S5), the 100 m depth channel produces a positive SKR is achievable only for a distance of 10 m at a depth of 100. As the depth is decreased the maximum distance of the channel extends as the background irradiance at the receiver decreases and channel attenuation decrease. When a higher surface chlorophyll-a concentration is present, the channel distance at each depth is able to be extended by at least 10 m for all depths considered. The SKR was found to increase by approximately one order of magnitude for the 100 m channel at a depth of 200 m. When the distance between submarines is reduced, the increase in SKR is approximately two times larger.
The affect of decreasing the attenuation and background irradiance at the receiver can be seen easier by considering the optimal value for μ. As the channel is attenuated more, fewer of the transmitted photons reach Bob’s detectors, while all the transmitted photons reach Alice’s detectors. In the time slots where a photon has been lost due to channel attenuation, if Bob detects a click due to noise in the correct basis, the probability of error is 0.5. By reducing μ, Alice’s detector will click in fewer time slots, and reducing the probability that both a click due to noise is recorded by Bob and that Alice has detected a photon that was transmitted. Comparing the optimal value of μ for the different chlorophyll concentration levels, it is found that a higher surface chlorophyll concentration leads to higher optimum value of μ. In addition to being higher, as the channel length changes, the decrease is also slower. With a lower attenuation per meter, an increase in distance results in a smaller change in total channel attenuation. Thus the probability that Alice and Bob’s detectors click in the same timeslot is decreased at a slower rate.
5. Concluding remarks
As underwater optical communication starts to be implemented the ability to communicate securely will also be needed. It has been shown that quantum key distribution operating at a wavelength of 532 nm can generate a secure key between two users. Factors that will impact the generation rate of the secure key are depth, distance, turbulence strength, and concentration of chlorophyll at the ocean surface. As the concentration of chlorophyll at the ocean surface increase, the peak attenuation moves closer to the surface. In addition to lower depths having lower attenuation, the background irradiance at the receiver decreases with depth. As the distance increases, the secure key rates decrease limiting channels at higher depths to shorter distances. In addition to the attenuation in the channel oceanic turbulence causes distortions in the wavefront, further increasing the loss in the channel. In the surface layer, as the depth increase the turbulence strength decreases, resulting in more reliable channels for optical communication. As the channel loss and the probability of detecting a background photon increase, the optimal value of μ decreases.
Funding
ONR MURI program (N00014-13-1-0627)
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