- Email: [email protected]
New estimates of single photon parameters for satellite-based QKD
Optics Communications 282 (2009) 3185–3189
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
New estimates of single photon parameters for satellite-based QKD Sellami Ali *, M.R.B. Wahiddin Faculty of Science, International Islamic University of Malaysia (IIUM), P.O. Box 141, 25710 Kuantan, Pahang Darul Makmur, Malaysia Information Security Cluster, MIMOS Berhad, Technology Park Malaysia, 57000 Kuala Lumpur, Malaysia
a r t i c l e
i n f o
Article history: Received 10 March 2009 Received in revised form 30 April 2009 Accepted 4 May 2009
Keywords: Optical communication Quantum cryptography Quantum key distribution Decoy state protocol
a b s t r a c t A new decoy state method has been presented to tighten the lower bound of the key generation rate for BB84 using one decoy state and one signal state. It can give us different lower and upper bounds of the fraction of single-photon counts and single-photon QBER, respectively, for one decoy state protocol. We have also analyzed the feasibility of performing quantum key distribution (QKD), with different exiting protocols, in earth-satellite and intersatellite links. Our simulation shows the choice of intensity of signal state and the effect of choosing the number of decoy states on key generation rate. The final key rate over transmission distance has been simulated, which shows that security proofs give a zero key generation rate at long distances (larger than 16,000 km). It has been shown that the practical QKD can be established with low earth orbit and medium earth orbit satellites. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Quantum key distribution (QKD) is a method of generating a secret key between two parties (Alice, the sender, and Bob, the receiver) that is guaranteed by the fundamental laws of quantum mechanics and therefore QKD provides an unconditionally secure communication system. Introduced in 1984, the BB84 protocol for QKD is based on the use of single photons for encoding the quantum information [1]. In the practical setting of optical communication, however, it is necessary to substitute qubits in the original BB84 QKD protocol with heavily attenuated laser pulses because perfect single photon emitting devices are not available currently. Such laser pulses – the phase randomized weak coherent states – contain inevitably multiphoton states although small but finite probability, which give a malicious eavesdropper (Eve) a chance to obtain some amount of information on the shared keys by a photon-number-splitting attack [2–4]. Gottesman–Lo–Lütkenhaus–Preskill (GLLP) showed, however, that it is still possible to obtain unconditionally secret key by BB84 protocol with such imperfect light sources, although the key generation rate and distances are very limited [5]. These problems have been solved using the decoy state method introduced by Hwang [6]. The decoy state method achieves unconditional security based on quantum mechanics as well as improves dramatically the performance of the QKD. Also it faithfully estimates the upper bound of multi-photon counting rate through decoy-pulses regardless of the type of
* Corresponding author. Address: Faculty of Science, International Islamic University of Malaysia (IIUM), P.O. Box 141, 25710 Kuantan, Pahang Darul Makmur, Malaysia. E-mail address: [email protected] (S. Ali). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.05.005
attack. The basic idea of the decoy state QKD is: in addition to the signal state with the specific average photon number, one introduces some decoy states with some other average photon numbers and blends signal states with decoy states randomly on Alice’s sides. The decoy state QKD can be used to calculate the lower bound of counting rate of single-photon pulses and upper bound of quantum bit error rate (QBER) of bits generated by single-photon pulses. Many methods have been developed to improve the performance of the decoy states QKD, including more decoy states [7], nonorthogonal decoy-state method [8], photon numberresolving method [9], herald single photon source method [10,11], modified coherent state source method [12], the intensity fluctuations of the laser pulses [13,14]. Some prototypes of decoy state QKD have already been implemented [15–20]. Due to the limitations of the propagation along optical fibres, QKD over fibers can only reach a few hundred of kilometers [21,22]. Free-space links permit to increase this distance [23] thanks to the low absorption of the atmosphere in certain wavelength ranges and to its nonbirefringent character, which guarantees the conservation of the polarization. However, terrestrial free space links suffer from attenuation caused by the atmosphere and objects in the line of view. In order to totally exploit the potential of free space communications, satellites should be used. Thus, significant improvements in the QKD range could be obtained since, in an earth-satellite link, only around 30 km of the path (depending on the satellite elevation) are inside the atmosphere. In this paper, we propose and derive a new decoy state method with a one decoy state and one signal state. The method is simple and further improves key generation rate. The main idea of this work is security, we get a tight verification of the fraction of single–photon state and verify the upper bound of quantum bit
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error rate for quantum key distribution (QKD).We then analyze the feasibility of performing quantum key distribution (QKD), with our decoy state method, for earth-satellite and intersatellite links. This paper is prepared in the following order. In Section 2, we propose a tight verification of the fraction of the single photon state and the quantum bit error rate (QBER) for practical decoy method with one decoy state. In Section 3, we simulate the key generation rate over transmission distance. Also, we investigate the maximization of the key generation rate by controlling the average photon number of signal state, decoy states and number of decoy state pulses. The main conclusions are summarized in Section 4.
For x = 0
sL1 ¼
1
l
ðSl el s0 Þ:
ð6Þ
For x = 1
sL1 ¼
1 ðSm em Sl el Þ: ðm lÞ
ð7Þ
For x = l
sL1 ¼
1
lðm 1Þ
ðlSm em Sl el ðl 1Þs0 Þ:
ð8Þ
For x = v 2. The decoy state method It is assumed that Alice produces pulses in state leih ; meih , and j0i as class Yl, Ym, and Y0, respectively. In the protocol h is randomized. Also, Alice mixes the positions of all pulses therefore no one but Alice knows which pulse belongs to which class in the protocol. We then observe the counting rates of each class and then verify the lower bounds of counts caused by single-photon pulses and the upper bounds of quantum bit error rates (QBER) from the mixed class Yl, and Ym. Then, we calculate the key generation rate which is given by GLLP [5]. The counting rate of any state q is the probability that Bob’s detector clicks whenever a state q is sent out by Alice. We disregard what state Bob may receive here. This counting rate is called the yield elsewhere [14,15]. Assume each class contains three subclasses y0, y1, and ym for vacuum pulses, single-photon pulses and multi-pulses, respectively. We denote the counting rate of class Yl, Ym by notations Sl, and Sm respectively. These two parameters are observed in the protocol itself: after all the pulses are sent out, Bob announces which pulse has caused a click and which pulse has not caused a click. Since Alice knows which pulse belongs to which class, Alice can calculate the counting rates of each class of pulses. Also, we use the notations s0, s1, and sm for counting rates of subclasses y0, y1, and ym, respectively. Therefore, we shall regard Sl, Sm as known parameters in protocol. The count ing rate of the state leih can be written in the following form [28]:
Sl el ¼ s0 þ ls1 þ csm ; c ¼1e
l
l
le
> 0:
ð1Þ ð2Þ
According to Eq. (1), the counting rate of the state meih can be written as:
ð3Þ
We used the inequality 0 6 m < l and 0 6 x 6 1 to prove the first inequality in Eq. (3). Where x is a factor for determining the lower and upper bounds of the fraction of single-photon counts and single-photon QBER, respectively. By solving inequality (3), the s1 is given by
ð4Þ
1 ðxSm em Sl el ðx 1Þs0 Þ: ðxm lÞ
sL1 ¼
1
lðm2 1Þ
ðmlSm em Sl el ðml 1Þs0 Þ:
ð10Þ
Next, we find the upper bound of e1; the overall QBER is given by [3]
El ¼
1 1 X li ei si el ; Sl i¼0 i!
ð11Þ
El ¼
1 ðe0 s0 þ edet ectorð1 egl ÞÞ: Sl
ð12Þ
where ei is the QBER of an i-photon signal, e0 is the error rate of background, and edetector is the probability that a photon hit the erroneous detector. According to Eq. (11), the QBER of the weak decoy state is given by
El Sl el ¼ e0 s0 þ le1 s1 þ cem sm e0 s0 þ le1 s1 þ xðEm Sm em e0 s0 me1 s1 Þ: Then,
El Sl el ¼ ð1 xÞe0 s0 þ ðl xmÞe1 s1 þ xEm Sm em :
ð13Þ
By solving inequality (13), the e1 is given by
e1
1 ðEl Sl el xEm Sm em ð1 xÞe0 s0 Þ: ðl xmÞs1
ð14Þ
ð5Þ
According to selected values of x we can deduce different lower bounds for s1.
1 ðEl Sl el xEm Sm em ð1 xÞe0 s0 Þ: ðl xmÞs1
ð15Þ
According to selected values of x we can deduce different upper bounds for e1. For x = 0
eU1 ¼
1
ls1
ðEl Sl el e0 s0 Þ:
ð16Þ
For x = 1
eU1 ¼
1 ðEl Sl el Em Sm em Þ: ðl mÞs1
ð17Þ
For x = l
eU1 ¼
Therefore, the lower bound of s1 is given by
sL1 ¼
ð9Þ
For x = vl
eU1 ¼
Then,
1 s1 ðxSm em Sl el ðx 1Þs0 Þ: ðxm lÞ
1 mSm em Sl el ðm 1Þs0 : ðm2 lÞ
Therefore, the upper bound of e1 is
Sm em ¼ s0 þ ms1 þ csm 1 s0 þ ms1 þ ðSl el s0 ls1 Þ: x 1 l 1 Sm em ¼ 1 s0 þ m s1 þ Sl el : x x x
sL1 ¼
1
lð1 mÞs1
ðEl Sl el lEm Sm em ð1 lÞe0 s0 Þ:
ð18Þ
For x = v
eU1 ¼
1 El Sl el mEm Sm em ð1 mÞe0 s0 : ðl m2 Þs1
ð19Þ
S. Ali, M.R.B. Wahiddin / Optics Communications 282 (2009) 3185–3189
For x = vl
eU1 ¼
1
lð1 m2 Þs1
El Sl el lmEm Sm em ð1 lmÞe0 s0 :
the position of the ground station. Finally, the total channel attenuation can be written as
ð20Þ
3. Numerical simulation In an experiment, we need to observe the values of Sl, Sm and El, Em, and then deduce the lower bound of fraction of single-photon counts (s1) and upper bound QBER of single-photon pulses (e1) by theoretical results. One can then distill the secure final key. In order to make a faithful estimation, we need a channel model to forecast what values for Sl, Sm, and El, Em, would be observed, if we did the experiment without Eve in principle. We have considered three different scenarios: a ground-satellite uplink, a satellite-ground downlink and an intersatellite link. We assume that conventional telescope architectures, like the Cassegrain type, are used both in the transmitting and receiving sides. They are reflective telescopes, in which the secondary mirror produces a central obscuration. Moreover, their finite dimensions and the distance between them are responsible for the beam diffraction. The attenuation due to beam diffraction and obscuration can be expressed as
gdiff ¼ gdifft gdiffr "
# " # 2ðDM2t Þ2 2ðDM1t Þ2 exp ; w2 w2 " # " # 2ðDM2r Þ2 2ðDM1r Þ2 exp ; ¼ exp w2 w2
gdifft ¼ exp gdiffr w
kL
pw0
3187
;
where the subscript t refers to the transmitting telescope and r to the receiving one. k is the wavelength, and DM1 and DM2 are the radius of the primary and secondary mirrors, respectively. w is the waist radius of the Gaussian beam and L is the distance between the telescopes [29]. Since the atmospheric attenuation (gatm) is produced by three phenomena: scattering, absorption and turbulence, it can be expressed as gatm = gscattgabsgturb. The light is absorbed and scattered by the gas molecules and the aerosols when it passes through the atmosphere. However, the most relevant contribution to the atmospheric attenuation is caused by the turbulence, which is due to thermal fluctuations that produce refractive index variations. The turbulence depends basically on the atmospheric conditions and
g ¼ gdiff gatm gdet : where gdet is the detector attenuation. The assumed link parameters are the wavelength k = 650 nm corresponding to an absorption window and 0.65 efficiency peak of the chosen detector (an SPCM-AQR-15 commercial silicon avalanche photodiode detector) with 50 106 counts/pulse dark counts. The satellite telescopes radius of the primary and secondary mirrors are 15 cm and 1 cm, respectively. The ground telescope radius of the primary and secondary mirrors are 50 cm and 5 cm, respectively. The values of telescope radii have been obtained from the SILEX Experiment [24] and the Tenerife’s telescope [25]. The uplink attenuation due to turbulence has been computed considering the Tenerife’s telescope (3 km above sea level) for two conditions: 1 h before sunset (gturb = 5 dB) and a typical clear summer day (gturb = 11 dB) [26]. The turbulence effect on the downlink is negligible. The scattering attenuation is evaluated using a model of clear standard atmosphere [27], which results in gscatt = 1 dB. We choose the intensities, the percentages of signal state and decoy states which could give out the maximum key generation rate. The search for optimal parameters can be obtained by numerical simulation. We take the values of l and m which are mean photon numbers of signal state and decoy states respectively, in the range of [0, 1] with a step 0.001. Similar strategy can be applied to the percentage of each state. With certain combination of intensities and percentages, the gains and QBERs of different states could be simulated. Also, the key generation rate can be estimated for the specified protocols. Therefore, we can find out the optimal combination that can give maximum key generation rate. Furthermore, we can get the maximum secure distance for the protocols which are considered. Fig. 1 illustrates the simulation results of the key generation rate for our protocol against the signal mean photon number l at different transmission distance links (10,000 km, 8000 km, and 6000 km). We have chosen the efficiency of detectors as 65%, detectors dark count rate 50 106 and the number of pulses used as the signal state and the vacuum state are Nl = 0.95 N, and N0 = 0.05 N (sent by Alice) where N = 100 Mbit. The graph shows the mean photon number l values which give the lower and upper key generation rates for different transmission distance links. This shows that the mean photon number increases by increasing the transmission distance link. Also, it helps us to choose the intensities of signal sate that give the maximum key generation rate.
Fig. 1. The key generation rate against the signal mean photon number l for different transmission distance links (10,000 km, 8000 km, and 6000 km).
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S. Ali, M.R.B. Wahiddin / Optics Communications 282 (2009) 3185–3189
Fig. 2. A ground-satellite uplink 1 h before sunset (gturb = 5 dB). The key generation rate against the transmission distance link (km). The dashed line shows our protocol for x = 0. The dotted line shows the protocol for x = 1. The solid line shows the protocol for x = m.
Fig. 3. A ground-satellite uplink during a typical clear summer day (gturb = 11 dB).The key generation rate against the transmission distance link (km). The dashed line shows our protocol for x = 0 .The dotted line shows the protocol for x = 1. The solid line shows the protocol for x = m.
Fig. 4. A satellite-ground downlink. The key generation rate against the transmission distance link (km). The dashed line shows our protocol for x = 0 .The dotted line shows the protocol for x = 1. The solid line shows the protocol for x = m.
S. Ali, M.R.B. Wahiddin / Optics Communications 282 (2009) 3185–3189
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Fig. 5. An inter-satellite link. The key generation rate against the transmission distance link (km). The dashed line shows our protocol for x = 0. The dotted line shows the protocol for x = 1. The solid line shows the protocol for x = m.
Although the scenarios are different, the curves have similar shapes. They illustrate the simulation results of the key generation rate against the transmission distance at different estimations as shown in Figs. 2–5. The distances in the downlink are significantly larger compared to the uplink thanks to the lack of turbulence attenuation: In fact, MEO satellite downlink communication using our protocol is possible. This increase in distance is not achieved in the intersatellite link due to the reduced telescope dimensions. The most relevant parameters that influence the critical distance are the turbulence attenuation and the telescopes dimensions. Therefore, bidirectional ground-to-LEO satellite communication is possible with our protocol. To compare our method with previous methods [30] we have simulated the QKD system for three different scenarios: a ground-satellite uplink, a satellite-ground downlink and an intersatellite link using the method of [30]. Based on the comparisons made free space QKD systems using our proposed method for BB84 are able to achieve both a higher secret key rate and greater secure distance than previous methods for three different scenarios as seen in Tables 1 and 2.
Table 1 Critical distance (km). Scenarios
BB84 without decoy state protocol
BB84 with decoy state method [30]
BB84 with proposed decoy state method
Downlink Intersatellite Uplink (dturb = 5 dB) Uplink (dturb = 11 dB)
1540 430 460 –
9450 2660 4650 2200
17,395 7476.874 7930 4009
Table 2 Maximum rate (bits/pulse). Scenarios
BB84 without decoy state protocol
BB84 with decoy state method [30]
BB84 with proposed decoy state method
Downlink Intersatellite Uplink (dturb = 5 dB) Uplink (dturb = 11 dB)
1.7 102 2.0 102 1.4 104 –
4.4 102 4.8 102 5.8 103 1.4 103
11.08 102 7.7 102 8.65 102 7.9 103
4. Conclusion We have proposed a new method for estimation which can give different lower and upper bounds of the fraction of single-photon counts and single-photon QBER for one decoy state protocol. It allows us to deduce the value of the key generation rate. The numerical results have shown the effect of choosing the signal and decoy state intensities on key generation rate and the feasibility of performing quantum key distribution (QKD) with the proposed method in earth-satellite and intersatellite link. Therefore, bidirectional ground-to-LEO satellite communication is possible with our method. References [1] C.H. Bennett, G. Brassard, in: Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India, 1984 (IEEE, New York, 1984), p. 175. [2] B. Huttner et al., Phys. Rev. A 51 (1995) 1863. [3] G. Brassard et al., Phys. Rev. Lett. 85 (2000) 1330. [4] N. Lütkenhaus, M. Jahma, New J. Phys. 4 (2002) 44. [5] D. Gottesman et al., Quant. Inf. Comput. 4 (2004) 325. [6] W.-Y. Hwang, Phys. Rev. Lett. 91 (2003) 057901. [7] X.-B. Wang, Phys. Rev. A 72 (2005) 012322. [8] J.-B. Li, X.-M. Fang, Chin. Phys. Lett. 23 (4) (2006). [9] Qing-Yu Cai, Yong-Gang Tan, Phys. Rev. A. 73 (2006) 032305. [10] Tomoyuki Horikiri, Takayoshi Kobayashi, Phys. Rev. A 73 (2006) 032331. [11] Qin Wang, X.-B. Wang, G.-C. Guo, Phys. Rev. A 75 (2007) 012312. [12] Z.-Q. Yin, Z.-F. Han, F.-W. Sun, G.-C. Guo, Phys. Rev. A 76 (2007) 014304. [13] X.-B. Wang, C.-Z. Peng, J.-W. Pan, Appl. Phys. Lett. 90 (2007) 6031110. [14] X.-B. Wang, Phys. Rev. A 75 (2007) 052301. [15] Y. Zhao et al., Phys. Rev. Lett. 96 (2006) 070502. [16] Yi Zhao et al., in: Proceedings of the IEEE International Symposium Information Theory, 2006, pp. 2094–2098. [17] C.-Z. Peng et al., Phys. Rev. Lett. 98 (2007) 010505. [18] D. Rosenberg, J.W. Harrington, P.R. Rice, et al., Phys. Rev. Lett. 98 (2007) 010503. [19] Z.L. Yuan, A.W. Sharpe, A.J. Shields, Appl. Phys. Lett. 90 (2007) 011118. [20] Tobias Schmitt-Manderbach et al., Phys. Rev. Lett. 98 (2007) 010504. [21] E. Waks, A. Zeevi, Y. Yamamoto, Phys. Rev. A. 65 (2002) 52310. [22] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Rev. Mod. Phys. 74 (2002) 145. [23] T. Schmitt-Manderbach, H. Weier, M. Furst, R. Ursin, F. Tiefenbacher, T. Scheid, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J.G. Rarity, et al., Phys. Rev. Lett. 98 (2007) 010504. [24] P.V. Gatenby, M.A. Grant, Electron. Commun. Eng. J. 3 (1991) 280. [25] R. Ursin et al., Nat. Phys. 3 (2007) 481. [26] D.G. Aviv (Ed.), Laser Space Communications, Artech House, 216 pp. [27] L. Elterman, Appl. Opt. 3 (1964) 745. [28] Xiang-Bin Wang, A review on the decoy-state method for practical quantum key distribution. Available from: arXiv:. [29] Tobias Schmitt-Manderbach, Long distance free-space quantum key distribution . [30] X. Ma, B. Qi, Y. Zhao, H.-K. Lo, quant-ph/0503005 (2005).
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
New estimates of single photon parameters for satellite-based QKD Sellami Ali *, M.R.B. Wahiddin Faculty of Science, International Islamic University of Malaysia (IIUM), P.O. Box 141, 25710 Kuantan, Pahang Darul Makmur, Malaysia Information Security Cluster, MIMOS Berhad, Technology Park Malaysia, 57000 Kuala Lumpur, Malaysia
a r t i c l e
i n f o
Article history: Received 10 March 2009 Received in revised form 30 April 2009 Accepted 4 May 2009
Keywords: Optical communication Quantum cryptography Quantum key distribution Decoy state protocol
a b s t r a c t A new decoy state method has been presented to tighten the lower bound of the key generation rate for BB84 using one decoy state and one signal state. It can give us different lower and upper bounds of the fraction of single-photon counts and single-photon QBER, respectively, for one decoy state protocol. We have also analyzed the feasibility of performing quantum key distribution (QKD), with different exiting protocols, in earth-satellite and intersatellite links. Our simulation shows the choice of intensity of signal state and the effect of choosing the number of decoy states on key generation rate. The final key rate over transmission distance has been simulated, which shows that security proofs give a zero key generation rate at long distances (larger than 16,000 km). It has been shown that the practical QKD can be established with low earth orbit and medium earth orbit satellites. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Quantum key distribution (QKD) is a method of generating a secret key between two parties (Alice, the sender, and Bob, the receiver) that is guaranteed by the fundamental laws of quantum mechanics and therefore QKD provides an unconditionally secure communication system. Introduced in 1984, the BB84 protocol for QKD is based on the use of single photons for encoding the quantum information [1]. In the practical setting of optical communication, however, it is necessary to substitute qubits in the original BB84 QKD protocol with heavily attenuated laser pulses because perfect single photon emitting devices are not available currently. Such laser pulses – the phase randomized weak coherent states – contain inevitably multiphoton states although small but finite probability, which give a malicious eavesdropper (Eve) a chance to obtain some amount of information on the shared keys by a photon-number-splitting attack [2–4]. Gottesman–Lo–Lütkenhaus–Preskill (GLLP) showed, however, that it is still possible to obtain unconditionally secret key by BB84 protocol with such imperfect light sources, although the key generation rate and distances are very limited [5]. These problems have been solved using the decoy state method introduced by Hwang [6]. The decoy state method achieves unconditional security based on quantum mechanics as well as improves dramatically the performance of the QKD. Also it faithfully estimates the upper bound of multi-photon counting rate through decoy-pulses regardless of the type of
* Corresponding author. Address: Faculty of Science, International Islamic University of Malaysia (IIUM), P.O. Box 141, 25710 Kuantan, Pahang Darul Makmur, Malaysia. E-mail address: [email protected] (S. Ali). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.05.005
attack. The basic idea of the decoy state QKD is: in addition to the signal state with the specific average photon number, one introduces some decoy states with some other average photon numbers and blends signal states with decoy states randomly on Alice’s sides. The decoy state QKD can be used to calculate the lower bound of counting rate of single-photon pulses and upper bound of quantum bit error rate (QBER) of bits generated by single-photon pulses. Many methods have been developed to improve the performance of the decoy states QKD, including more decoy states [7], nonorthogonal decoy-state method [8], photon numberresolving method [9], herald single photon source method [10,11], modified coherent state source method [12], the intensity fluctuations of the laser pulses [13,14]. Some prototypes of decoy state QKD have already been implemented [15–20]. Due to the limitations of the propagation along optical fibres, QKD over fibers can only reach a few hundred of kilometers [21,22]. Free-space links permit to increase this distance [23] thanks to the low absorption of the atmosphere in certain wavelength ranges and to its nonbirefringent character, which guarantees the conservation of the polarization. However, terrestrial free space links suffer from attenuation caused by the atmosphere and objects in the line of view. In order to totally exploit the potential of free space communications, satellites should be used. Thus, significant improvements in the QKD range could be obtained since, in an earth-satellite link, only around 30 km of the path (depending on the satellite elevation) are inside the atmosphere. In this paper, we propose and derive a new decoy state method with a one decoy state and one signal state. The method is simple and further improves key generation rate. The main idea of this work is security, we get a tight verification of the fraction of single–photon state and verify the upper bound of quantum bit
3186
S. Ali, M.R.B. Wahiddin / Optics Communications 282 (2009) 3185–3189
error rate for quantum key distribution (QKD).We then analyze the feasibility of performing quantum key distribution (QKD), with our decoy state method, for earth-satellite and intersatellite links. This paper is prepared in the following order. In Section 2, we propose a tight verification of the fraction of the single photon state and the quantum bit error rate (QBER) for practical decoy method with one decoy state. In Section 3, we simulate the key generation rate over transmission distance. Also, we investigate the maximization of the key generation rate by controlling the average photon number of signal state, decoy states and number of decoy state pulses. The main conclusions are summarized in Section 4.
For x = 0
sL1 ¼
1
l
ðSl el s0 Þ:
ð6Þ
For x = 1
sL1 ¼
1 ðSm em Sl el Þ: ðm lÞ
ð7Þ
For x = l
sL1 ¼
1
lðm 1Þ
ðlSm em Sl el ðl 1Þs0 Þ:
ð8Þ
For x = v 2. The decoy state method It is assumed that Alice produces pulses in state leih ; meih , and j0i as class Yl, Ym, and Y0, respectively. In the protocol h is randomized. Also, Alice mixes the positions of all pulses therefore no one but Alice knows which pulse belongs to which class in the protocol. We then observe the counting rates of each class and then verify the lower bounds of counts caused by single-photon pulses and the upper bounds of quantum bit error rates (QBER) from the mixed class Yl, and Ym. Then, we calculate the key generation rate which is given by GLLP [5]. The counting rate of any state q is the probability that Bob’s detector clicks whenever a state q is sent out by Alice. We disregard what state Bob may receive here. This counting rate is called the yield elsewhere [14,15]. Assume each class contains three subclasses y0, y1, and ym for vacuum pulses, single-photon pulses and multi-pulses, respectively. We denote the counting rate of class Yl, Ym by notations Sl, and Sm respectively. These two parameters are observed in the protocol itself: after all the pulses are sent out, Bob announces which pulse has caused a click and which pulse has not caused a click. Since Alice knows which pulse belongs to which class, Alice can calculate the counting rates of each class of pulses. Also, we use the notations s0, s1, and sm for counting rates of subclasses y0, y1, and ym, respectively. Therefore, we shall regard Sl, Sm as known parameters in protocol. The count ing rate of the state leih can be written in the following form [28]:
Sl el ¼ s0 þ ls1 þ csm ; c ¼1e
l
l
le
> 0:
ð1Þ ð2Þ
According to Eq. (1), the counting rate of the state meih can be written as:
ð3Þ
We used the inequality 0 6 m < l and 0 6 x 6 1 to prove the first inequality in Eq. (3). Where x is a factor for determining the lower and upper bounds of the fraction of single-photon counts and single-photon QBER, respectively. By solving inequality (3), the s1 is given by
ð4Þ
1 ðxSm em Sl el ðx 1Þs0 Þ: ðxm lÞ
sL1 ¼
1
lðm2 1Þ
ðmlSm em Sl el ðml 1Þs0 Þ:
ð10Þ
Next, we find the upper bound of e1; the overall QBER is given by [3]
El ¼
1 1 X li ei si el ; Sl i¼0 i!
ð11Þ
El ¼
1 ðe0 s0 þ edet ectorð1 egl ÞÞ: Sl
ð12Þ
where ei is the QBER of an i-photon signal, e0 is the error rate of background, and edetector is the probability that a photon hit the erroneous detector. According to Eq. (11), the QBER of the weak decoy state is given by
El Sl el ¼ e0 s0 þ le1 s1 þ cem sm e0 s0 þ le1 s1 þ xðEm Sm em e0 s0 me1 s1 Þ: Then,
El Sl el ¼ ð1 xÞe0 s0 þ ðl xmÞe1 s1 þ xEm Sm em :
ð13Þ
By solving inequality (13), the e1 is given by
e1
1 ðEl Sl el xEm Sm em ð1 xÞe0 s0 Þ: ðl xmÞs1
ð14Þ
ð5Þ
According to selected values of x we can deduce different lower bounds for s1.
1 ðEl Sl el xEm Sm em ð1 xÞe0 s0 Þ: ðl xmÞs1
ð15Þ
According to selected values of x we can deduce different upper bounds for e1. For x = 0
eU1 ¼
1
ls1
ðEl Sl el e0 s0 Þ:
ð16Þ
For x = 1
eU1 ¼
1 ðEl Sl el Em Sm em Þ: ðl mÞs1
ð17Þ
For x = l
eU1 ¼
Therefore, the lower bound of s1 is given by
sL1 ¼
ð9Þ
For x = vl
eU1 ¼
Then,
1 s1 ðxSm em Sl el ðx 1Þs0 Þ: ðxm lÞ
1 mSm em Sl el ðm 1Þs0 : ðm2 lÞ
Therefore, the upper bound of e1 is
Sm em ¼ s0 þ ms1 þ csm 1 s0 þ ms1 þ ðSl el s0 ls1 Þ: x 1 l 1 Sm em ¼ 1 s0 þ m s1 þ Sl el : x x x
sL1 ¼
1
lð1 mÞs1
ðEl Sl el lEm Sm em ð1 lÞe0 s0 Þ:
ð18Þ
For x = v
eU1 ¼
1 El Sl el mEm Sm em ð1 mÞe0 s0 : ðl m2 Þs1
ð19Þ
S. Ali, M.R.B. Wahiddin / Optics Communications 282 (2009) 3185–3189
For x = vl
eU1 ¼
1
lð1 m2 Þs1
El Sl el lmEm Sm em ð1 lmÞe0 s0 :
the position of the ground station. Finally, the total channel attenuation can be written as
ð20Þ
3. Numerical simulation In an experiment, we need to observe the values of Sl, Sm and El, Em, and then deduce the lower bound of fraction of single-photon counts (s1) and upper bound QBER of single-photon pulses (e1) by theoretical results. One can then distill the secure final key. In order to make a faithful estimation, we need a channel model to forecast what values for Sl, Sm, and El, Em, would be observed, if we did the experiment without Eve in principle. We have considered three different scenarios: a ground-satellite uplink, a satellite-ground downlink and an intersatellite link. We assume that conventional telescope architectures, like the Cassegrain type, are used both in the transmitting and receiving sides. They are reflective telescopes, in which the secondary mirror produces a central obscuration. Moreover, their finite dimensions and the distance between them are responsible for the beam diffraction. The attenuation due to beam diffraction and obscuration can be expressed as
gdiff ¼ gdifft gdiffr "
# " # 2ðDM2t Þ2 2ðDM1t Þ2 exp ; w2 w2 " # " # 2ðDM2r Þ2 2ðDM1r Þ2 exp ; ¼ exp w2 w2
gdifft ¼ exp gdiffr w
kL
pw0
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;
where the subscript t refers to the transmitting telescope and r to the receiving one. k is the wavelength, and DM1 and DM2 are the radius of the primary and secondary mirrors, respectively. w is the waist radius of the Gaussian beam and L is the distance between the telescopes [29]. Since the atmospheric attenuation (gatm) is produced by three phenomena: scattering, absorption and turbulence, it can be expressed as gatm = gscattgabsgturb. The light is absorbed and scattered by the gas molecules and the aerosols when it passes through the atmosphere. However, the most relevant contribution to the atmospheric attenuation is caused by the turbulence, which is due to thermal fluctuations that produce refractive index variations. The turbulence depends basically on the atmospheric conditions and
g ¼ gdiff gatm gdet : where gdet is the detector attenuation. The assumed link parameters are the wavelength k = 650 nm corresponding to an absorption window and 0.65 efficiency peak of the chosen detector (an SPCM-AQR-15 commercial silicon avalanche photodiode detector) with 50 106 counts/pulse dark counts. The satellite telescopes radius of the primary and secondary mirrors are 15 cm and 1 cm, respectively. The ground telescope radius of the primary and secondary mirrors are 50 cm and 5 cm, respectively. The values of telescope radii have been obtained from the SILEX Experiment [24] and the Tenerife’s telescope [25]. The uplink attenuation due to turbulence has been computed considering the Tenerife’s telescope (3 km above sea level) for two conditions: 1 h before sunset (gturb = 5 dB) and a typical clear summer day (gturb = 11 dB) [26]. The turbulence effect on the downlink is negligible. The scattering attenuation is evaluated using a model of clear standard atmosphere [27], which results in gscatt = 1 dB. We choose the intensities, the percentages of signal state and decoy states which could give out the maximum key generation rate. The search for optimal parameters can be obtained by numerical simulation. We take the values of l and m which are mean photon numbers of signal state and decoy states respectively, in the range of [0, 1] with a step 0.001. Similar strategy can be applied to the percentage of each state. With certain combination of intensities and percentages, the gains and QBERs of different states could be simulated. Also, the key generation rate can be estimated for the specified protocols. Therefore, we can find out the optimal combination that can give maximum key generation rate. Furthermore, we can get the maximum secure distance for the protocols which are considered. Fig. 1 illustrates the simulation results of the key generation rate for our protocol against the signal mean photon number l at different transmission distance links (10,000 km, 8000 km, and 6000 km). We have chosen the efficiency of detectors as 65%, detectors dark count rate 50 106 and the number of pulses used as the signal state and the vacuum state are Nl = 0.95 N, and N0 = 0.05 N (sent by Alice) where N = 100 Mbit. The graph shows the mean photon number l values which give the lower and upper key generation rates for different transmission distance links. This shows that the mean photon number increases by increasing the transmission distance link. Also, it helps us to choose the intensities of signal sate that give the maximum key generation rate.
Fig. 1. The key generation rate against the signal mean photon number l for different transmission distance links (10,000 km, 8000 km, and 6000 km).
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Fig. 2. A ground-satellite uplink 1 h before sunset (gturb = 5 dB). The key generation rate against the transmission distance link (km). The dashed line shows our protocol for x = 0. The dotted line shows the protocol for x = 1. The solid line shows the protocol for x = m.
Fig. 3. A ground-satellite uplink during a typical clear summer day (gturb = 11 dB).The key generation rate against the transmission distance link (km). The dashed line shows our protocol for x = 0 .The dotted line shows the protocol for x = 1. The solid line shows the protocol for x = m.
Fig. 4. A satellite-ground downlink. The key generation rate against the transmission distance link (km). The dashed line shows our protocol for x = 0 .The dotted line shows the protocol for x = 1. The solid line shows the protocol for x = m.
S. Ali, M.R.B. Wahiddin / Optics Communications 282 (2009) 3185–3189
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Fig. 5. An inter-satellite link. The key generation rate against the transmission distance link (km). The dashed line shows our protocol for x = 0. The dotted line shows the protocol for x = 1. The solid line shows the protocol for x = m.
Although the scenarios are different, the curves have similar shapes. They illustrate the simulation results of the key generation rate against the transmission distance at different estimations as shown in Figs. 2–5. The distances in the downlink are significantly larger compared to the uplink thanks to the lack of turbulence attenuation: In fact, MEO satellite downlink communication using our protocol is possible. This increase in distance is not achieved in the intersatellite link due to the reduced telescope dimensions. The most relevant parameters that influence the critical distance are the turbulence attenuation and the telescopes dimensions. Therefore, bidirectional ground-to-LEO satellite communication is possible with our protocol. To compare our method with previous methods [30] we have simulated the QKD system for three different scenarios: a ground-satellite uplink, a satellite-ground downlink and an intersatellite link using the method of [30]. Based on the comparisons made free space QKD systems using our proposed method for BB84 are able to achieve both a higher secret key rate and greater secure distance than previous methods for three different scenarios as seen in Tables 1 and 2.
Table 1 Critical distance (km). Scenarios
BB84 without decoy state protocol
BB84 with decoy state method [30]
BB84 with proposed decoy state method
Downlink Intersatellite Uplink (dturb = 5 dB) Uplink (dturb = 11 dB)
1540 430 460 –
9450 2660 4650 2200
17,395 7476.874 7930 4009
Table 2 Maximum rate (bits/pulse). Scenarios
BB84 without decoy state protocol
BB84 with decoy state method [30]
BB84 with proposed decoy state method
Downlink Intersatellite Uplink (dturb = 5 dB) Uplink (dturb = 11 dB)
1.7 102 2.0 102 1.4 104 –
4.4 102 4.8 102 5.8 103 1.4 103
11.08 102 7.7 102 8.65 102 7.9 103
4. Conclusion We have proposed a new method for estimation which can give different lower and upper bounds of the fraction of single-photon counts and single-photon QBER for one decoy state protocol. It allows us to deduce the value of the key generation rate. The numerical results have shown the effect of choosing the signal and decoy state intensities on key generation rate and the feasibility of performing quantum key distribution (QKD) with the proposed method in earth-satellite and intersatellite link. Therefore, bidirectional ground-to-LEO satellite communication is possible with our method. References [1] C.H. Bennett, G. Brassard, in: Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India, 1984 (IEEE, New York, 1984), p. 175. [2] B. Huttner et al., Phys. Rev. A 51 (1995) 1863. [3] G. Brassard et al., Phys. Rev. Lett. 85 (2000) 1330. [4] N. Lütkenhaus, M. Jahma, New J. Phys. 4 (2002) 44. [5] D. Gottesman et al., Quant. Inf. Comput. 4 (2004) 325. [6] W.-Y. Hwang, Phys. Rev. Lett. 91 (2003) 057901. [7] X.-B. Wang, Phys. Rev. A 72 (2005) 012322. [8] J.-B. Li, X.-M. Fang, Chin. Phys. Lett. 23 (4) (2006). [9] Qing-Yu Cai, Yong-Gang Tan, Phys. Rev. A. 73 (2006) 032305. [10] Tomoyuki Horikiri, Takayoshi Kobayashi, Phys. Rev. A 73 (2006) 032331. [11] Qin Wang, X.-B. Wang, G.-C. Guo, Phys. Rev. A 75 (2007) 012312. [12] Z.-Q. Yin, Z.-F. Han, F.-W. Sun, G.-C. Guo, Phys. Rev. A 76 (2007) 014304. [13] X.-B. Wang, C.-Z. Peng, J.-W. Pan, Appl. Phys. Lett. 90 (2007) 6031110. [14] X.-B. Wang, Phys. Rev. A 75 (2007) 052301. [15] Y. Zhao et al., Phys. Rev. Lett. 96 (2006) 070502. [16] Yi Zhao et al., in: Proceedings of the IEEE International Symposium Information Theory, 2006, pp. 2094–2098. [17] C.-Z. Peng et al., Phys. Rev. Lett. 98 (2007) 010505. [18] D. Rosenberg, J.W. Harrington, P.R. Rice, et al., Phys. Rev. Lett. 98 (2007) 010503. [19] Z.L. Yuan, A.W. Sharpe, A.J. Shields, Appl. Phys. Lett. 90 (2007) 011118. [20] Tobias Schmitt-Manderbach et al., Phys. Rev. Lett. 98 (2007) 010504. [21] E. Waks, A. Zeevi, Y. Yamamoto, Phys. Rev. A. 65 (2002) 52310. [22] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Rev. Mod. Phys. 74 (2002) 145. [23] T. Schmitt-Manderbach, H. Weier, M. Furst, R. Ursin, F. Tiefenbacher, T. Scheid, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J.G. Rarity, et al., Phys. Rev. Lett. 98 (2007) 010504. [24] P.V. Gatenby, M.A. Grant, Electron. Commun. Eng. J. 3 (1991) 280. [25] R. Ursin et al., Nat. Phys. 3 (2007) 481. [26] D.G. Aviv (Ed.), Laser Space Communications, Artech House, 216 pp. [27] L. Elterman, Appl. Opt. 3 (1964) 745. [28] Xiang-Bin Wang, A review on the decoy-state method for practical quantum key distribution. Available from: arXiv: