Quantum key distribution with quantum walks

  • Chrysoula Vlachou
  • Walter Krawec
  • Paulo Mateus
  • Nikola Paunković
  • André SoutoEmail author


Quantum key distribution is one of the most fundamental cryptographic protocols. Quantum walks are important primitives for computing. In this paper, we take advantage of the properties of quantum walks to design new secure quantum key distribution schemes. In particular, we introduce a secure quantum key distribution protocol equipped with verification procedures against full man-in-the-middle attacks. Furthermore, we present a one-way protocol and prove its security. Finally, we propose a semi-quantum variation and prove its robustness against eavesdropping.


Quantum walks Quantum key distribution Semi-quantum key distribution 



WK would like to acknowledge the hospitality of SQIG–Security and Quantum Information Group in IT—Instituto de Telecomunicações, in Lisbon, during his visit while working on this project. CV acknowledges the support from DP-PMI and FCT (Portugal) through the Grant PD/BD/ 52652/2014. CV, PM, NP and AS acknowledge the support of SQIG-Security and Quantum Information Group. PM, NP and AS also acknowledge the support from UID/EEA/50008/2013 and the support of the project QuantumMining POCI-01-0145-FEDER-031826 funded by FCT. NP acknowledges the IT project QbigD funded by FCT PEst-OE/ EEI/LA0008/2013. PM and AS acknowledges the FCT project Confident PTDC/EEI-CTP/4503/2014. A.S. also acknowledges the support of LASIGE Research Unit, ref. UID/CEC/00408/2013.


  1. 1.
    Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of ACM Symposium on Theory of Computation (STOC ’01), pp. 50–59 (2001)Google Scholar
  2. 2.
    Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993). ADSCrossRefGoogle Scholar
  3. 3.
    Ambainis, A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 01(04), 507–518 (2003). CrossRefzbMATHGoogle Scholar
  4. 4.
    Ashwin, N., Ashvin, V.: Quantum walk on the line. Tech. Rep. arXiv:quant-ph/0010117 (2000)
  5. 5.
    Bae, J., Acín, A.: Key distillation from quantum channels using two-way communication protocols. Phys. Rev. A 75, 012,334 (2007). CrossRefGoogle Scholar
  6. 6.
    Bavaresco, J., Herrera Valencia, N., Klckl, C., Pivoluska, M., Friis, N., Malik, M., Huber, M.: Two measurements are sufficient for certifying high-dimensional entanglement. arXiv:1709.07344 (2017)
  7. 7.
    Beaudry, N.J., Lucamarini, M., Mancini, S., Renner, R.: Security of two-way quantum key distribution. Phys. Rev. A 88(6), 062,302 (2013)CrossRefGoogle Scholar
  8. 8.
    Bechmann-Pasquinucci, H., Tittel, W.: Quantum cryptography using larger alphabets. Phys. Rev. A 61, 062,308 (2000). MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bedington, R., Arrazola, J.M., Ling, A.: Progress in satellite quantum key distribution. Npj Quantum Inf. 3, 30 (2017). ADSCrossRefGoogle Scholar
  10. 10.
    Bennett, C., Brassard, G.: Quantum cryptography: public key distribution and coin tossing. In: Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, pp. 175–179. IEEE Press, New York (1984)Google Scholar
  11. 11.
    Bennett, C.H., Brassard, G., Mermin, N.D.: Quantum cryptography without bell’s theorem. Phys. Rev. Lett. 68, 557–559 (1992). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Berta, M., Christandl, M., Colbeck, R., Renes, J.M., Renner, R.: The uncertainty principle in the presence of quantum memory. Nat. Phys. 6, 659–662 (2010). CrossRefGoogle Scholar
  13. 13.
    Boyer, M., Gelles, R., Kenigsberg, D., Mor, T.: Semiquantum key distribution. Phys. Rev. A 79, 032,341 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Boyer, M., Kenigsberg, D., Mor, T.: Quantum key distribution with classical bob. Phys. Rev. Lett. 99, 140,501 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Brassard, G., Lütkenhaus, N., Mor, T., Sanders, B.: Security aspects of practical quantum cryptography. In: In Advances in Cryptology? EUROCRYPT’2000, pp. 289–299 (2000)CrossRefGoogle Scholar
  16. 16.
    Broome, M.A., Fedrizzi, A., Lanyon, B.P., Kassal, I., Aspuru-Guzik, A., White, A.G.: Discrete single-photon quantum walks with tunable decoherence. Phys. Rev. Lett. 104, 153,602 (2010). CrossRefGoogle Scholar
  17. 17.
    Brun, T.A., Carteret, H.A., Ambainis, A.: Quantum walks driven by many coins. Phys. Rev. A 67, 052,317 (2003). MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bruss, D., Christandl, M., Ekert, A., Englert, B.G., Kaszlikowski, D., Macchiavello, C.: Tomographic quantum cryptography: equivalence of quantum and classical key distillation. Phys. Rev. Lett. 91, 097,901 (2003). CrossRefGoogle Scholar
  19. 19.
    Cardano, F., Massa, F., Qassim, H., Karimi, E., Slussarenko, S., Paparo, D., de Lisio, C., Sciarrino, F., Santamato, E., Boyd, R.W., Marrucci, L.: Quantum walks and wavepacket dynamics on a lattice with twisted photons. Sci. Adv. 1, e1500,087 (2015). CrossRefGoogle Scholar
  20. 20.
    Cerf, N.J., Bourennane, M., Karlsson, A., Gisin, N.: Security of quantum key distribution using d-level systems. Phys. Rev. Lett. 88, 127,902 (2002). CrossRefGoogle Scholar
  21. 21.
    Chau, H.F.: Unconditionally secure key distribution in higher dimensions by depolarization. IEEE Trans. Inf. Theory 51, 1451–1468 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Chau, H.F.: Quantum key distribution using qudits that each encode one bit of raw key. Phys. Rev. A 92, 062,324 (2015). CrossRefGoogle Scholar
  23. 23.
    Childs, A., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.: Exponential algorithmic speedup by a quantum walk. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, STOC ’03, pp. 59–68. ACM, New York, NY, USA (2003)Google Scholar
  24. 24.
    Childs, A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180,501 (2009). MathSciNetCrossRefGoogle Scholar
  25. 25.
    Childs, A.M., Gosset, D., Webb, Z.: Universal computation by multiparticle quantum walk. Science 339(6121), 791–794 (2013). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Christandl, M., König, R., Renner, R.: Postselection technique for quantum channels with applications to quantum cryptography. Phys. Rev. Lett. 102, 020,504 (2009). CrossRefGoogle Scholar
  27. 27.
    Curty, M., Guhne, O., Lewenstein, M., Lutkenhaus, N.: Detecting two-party quantum correlations in quantum-key-distribution protocols. Phys. Rev. A 71, 022,306 (2005). CrossRefGoogle Scholar
  28. 28.
    Curty, M., Lewenstein, M., Lutkenhaus, N.: Entanglement as a precondition for secure quantum key distribution. Phys. Rev. Lett. 92, 217,903 (2004). CrossRefGoogle Scholar
  29. 29.
    Dada, A.C., Leach, J., Buller, G.S., Padgett, M.J., Andersson, E.: Experimental high-dimensional two-photon entanglement and violations of generalized bell inequalities. Nat. Phys. 7, 677–680 (2011). CrossRefGoogle Scholar
  30. 30.
    Devetak, I., Winter, A.: Distillation of secret key and entanglement from quantum states. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 461(2053), 207–235 (2005). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Diamanti, E., Lo, H.K., Qi, B., Yuan, Z.: Practical challenges in quantum key distribution. Npj Quantum Inf. 2, 16,025 (2016). CrossRefGoogle Scholar
  32. 32.
    Dixon, A.R., Dynes, J.F., Lucamarini, M., Frhlich, B., Sharpe, A.W., Plews, A., Tam, W., Yuan, Z.L., Tanizawa, Y., Sato, H., Kawamura, S., Fujiwara, M., Sasaki, M., Shields, A.J.: Quantum key distribution with hacking countermeasures and long term field trial. Sci. Rep. 7, 1978 (2017). ADSCrossRefGoogle Scholar
  33. 33.
    Ekert, A.K.: Quantum cryptography based on bell’s theorem. Phys. Rev. Lett. 67, 661–663 (1991). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Elezov, M., Ozhegov, R., Kurochkin, Y., Goltsman, G., Makarov, V.: Countermeasures against blinding attack on superconducting nanowire detectors for QKD. EPJ Web Conf. 103, 10,002 (2015). CrossRefGoogle Scholar
  35. 35.
    Erhard, M., Malik, M., Krenn, M., Zeilinger, A.: Experimental ghz entanglement beyond qubits. arXiv:1708.03881 (2017)
  36. 36.
    Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998). ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Gettrick, M., Miszczak, J.A.: Quantum walks with memory on cycles. Phys. A Stat. Mech. Its Appl. 399, 163–170 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Gisin, N., Fasel, S., Kraus, B., Zbinden, H., Ribordy, G.: Trojan-horse attacks on quantum-key-distribution systems. Phys. Rev. A 73, 022,320 (2006). CrossRefGoogle Scholar
  39. 39.
    Goyal, S.K., Roux, F.S., Forbes, A., Konrad, T.: Implementing quantum walks using orbital angular momentum of classical light. Phys. Rev. Lett. 110, 263,602 (2013). CrossRefGoogle Scholar
  40. 40.
    Goyal, S.K., Roux, F.S., Forbes, A., Konrad, T.: Implementation of multidimensional quantum walks using linear optics and classical light. Phys. Rev. A 92, 040,302 (2015). MathSciNetCrossRefGoogle Scholar
  41. 41.
    Harrington, J.W., Ettinger, J.M., Hughes, R.J., Nordholt, J.E.: Enhancing practical security of quantum key distribution with a few decoy states. arXiv:quant-ph/0503002 (2005)
  42. 42.
    Hiesmayr, B.C., de Dood, M.J.A., Löffler, W.: Observation of four-photon orbital angular momentum entanglement. Phys. Rev. Lett. 116, 073,601 (2016). CrossRefGoogle Scholar
  43. 43.
    Hwang, W.Y.: Quantum key distribution with high loss: toward global secure communication. Phys. Rev. Lett. 91, 057,901 (2003). CrossRefGoogle Scholar
  44. 44.
    Islam, N.T., Cahall, C., Aragoneses, A., Lezama, A., Kim, J., Gauthier, D.J.: Robust and stable delay interferometers with application to \(d\)-dimensional time-frequency quantum key distribution. Phys. Rev. Appl. 7, 044,010 (2017). CrossRefGoogle Scholar
  45. 45.
    Jain, N., Stiller, B., Khan, I., Elser, D., Marquardt, C., Leuchs, G.: Attacks on practical quantum key distribution systems (and how to prevent them). Contemp. Phys. 57(3), 366–387 (2016). ADSCrossRefGoogle Scholar
  46. 46.
    Jha, A.K., Malik, M., Boyd, R.W.: Exploring energy-time entanglement using geometric phase. Phys. Rev. Lett. 101, 180,405 (2008). CrossRefGoogle Scholar
  47. 47.
    Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44(4), 307–327 (2003). ADSCrossRefGoogle Scholar
  48. 48.
    Knight, P.L., Roldan, E., Sipe, J.E.: Quantum walk on the line as an interference phenomenon. Phys. Rev. A 68, 020,301 (2003). CrossRefGoogle Scholar
  49. 49.
    Krawec, W.O.: Restricted attacks on semi-quantum key distribution protocols. Quantum Inf. Process. 13(11), 2417–2436 (2014). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Krawec, W.O.: History dependent quantum walk on the cycle with an unbalanced coin. Phys. A Stat. Mech. Its Appl. 428, 319–331 (2015). ADSMathSciNetCrossRefGoogle Scholar
  51. 51.
    Krawec, W.O.: Security proof of a semi-quantum key distribution protocol. In: 2015 IEEE International Symposium on Information Theory (ISIT), pp. 686–690. IEEE (2015)Google Scholar
  52. 52.
    Krawec, W.O.: Quantum key distribution with mismatched measurements over arbitrary channels. arXiv:1608.07728 (2016)
  53. 53.
    Krawec, W.O.: Security of a semi-quantum protocol where reflections contribute to the secret key. Quantum Inf. Process. 15(5), 2067–2090 (2016)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    Krenn, M., Huber, M., Fickler, R., Lapkiewicz, R., Ramelow, S., Zeilinger, A.: Generation and confirmation of a (\(100 \times 100\))-dimensional entangled quantum system. Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014). ADSCrossRefGoogle Scholar
  55. 55.
    Krenn, M., Malik, M., Fickler, R., Lapkiewicz, R., Zeilinger, A.: Automated search for new quantum experiments. Phys. Rev. Lett. 116, 090,405 (2016). CrossRefGoogle Scholar
  56. 56.
    Kurtsiefer, C., Zarda, P., Mayer, S., Weinfurter, H.: The breakdown flash of silicon avalanche photodiodes-back door for eavesdropper attacks? J. Mod. Opt. 48(13), 2039–2047 (2001). ADSCrossRefGoogle Scholar
  57. 57.
    Lamas-Linares, A., Kurtsiefer, C.: Breaking a quantum key distribution system through a timing side channel. Opt. Express 15, 9388–9393 (2007). ADSCrossRefGoogle Scholar
  58. 58.
    Lavery, M.P.J., Robertson, D.J., Berkhout, G.C.G., Love, G.D., Padgett, M.J., Courtial, J.: Refractive elements for the measurement of the orbital angular momentum of a single photon. Opt. Express 20, 2110–2115 (2012). ADSCrossRefGoogle Scholar
  59. 59.
    Lee, M.S., Park, B.K., Woo, M.K., Park, C.H., Kim, Y.S., Han, S.W., Moon, S.: Countermeasure against blinding attacks on low-noise detectors with a background-noise-cancellation scheme. Phys. Rev. A 94, 062,321 (2016). CrossRefGoogle Scholar
  60. 60.
    Liu, Q., Lamas-Linares, A., Kurtsiefer, C., Skaar, J., Makarov, V., Gerhardt, I.: A universal setup for active control of a single-photon detector. Rev. Sci. Instrum. 85(1), 013,108 (2014). CrossRefGoogle Scholar
  61. 61.
    Lizama-Prez, L.A., Lpez, J.M., De Carlos Lpez, E.: Quantum key distribution in the presence of the intercept-resend with faked states attack. Entropy 19(1), 4 (2016). ADSCrossRefGoogle Scholar
  62. 62.
    Lo, H.K., Chau, H.: Unconditional security of quantum key distribution over arbitrarily long distances. Science 283(5410), 2050–2056 (1999). Please check and confirm the inserted publication year is correct for the references [61, 68]ADSCrossRefGoogle Scholar
  63. 63.
    Lo, H.K., Chau, H.F., Ardehali, M.: Efficient quantum key distribution scheme and a proof of its unconditional security. J. Cryptol. 18(2), 133–165 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Lo, H.K., Ma, X., Chen, K.: Decoy state quantum key distribution. Phys. Rev. Lett. 94, 230,504 (2005). CrossRefGoogle Scholar
  65. 65.
    Lovett, N.B., Cooper, S., Everitt, M., Trevers, M., Kendon, V.: Universal quantum computation using the discrete-time quantum walk. Phys. Rev. A 81, 042,330 (2010). MathSciNetCrossRefGoogle Scholar
  66. 66.
    Lu, H., Fung, C.H.F., Ma, X., Cai, Q.Y.: Unconditional security proof of a deterministic quantum key distribution with a two-way quantum channel. Phys. Rev. A 84, 042,344 (2011). CrossRefGoogle Scholar
  67. 67.
    Lucamarini, M., Choi, I., Ward, M.B., Dynes, J.F., Yuan, Z.L., Shields, A.J.: Practical security bounds against the trojan-horse attack in quantum key distribution. Phys. Rev. X 5, 031,030 (2015). CrossRefGoogle Scholar
  68. 68.
    Lucamarini, M., Dynes, J.F., Frohlich, B., Yuan, Z., Shields, A.J.: Security bounds for efficient decoy-state quantum key distribution. IEEE J. Sel. Top. Quantum Electron. 21(3), 1–8 (2015). CrossRefGoogle Scholar
  69. 69.
    Lutkenhaus, N.: Security against individual attacks for realistic quantum key distribution. Phys. Rev. A 61, 052,304 (2000). CrossRefGoogle Scholar
  70. 70.
    Lutkenhaus, N., Jahma, M.: Quantum key distribution with realistic states: photon-number statistics in the photon-number splitting attack. New J. Phys. 4(1), 44 (2002)ADSCrossRefGoogle Scholar
  71. 71.
    Lydersen, L., Wiechers, C., Wittmann, C., Elser, D., Skaar, J., Makarov, V.: Hacking commercial quantum cryptography systems by tailored bright illumination. Nat. Photon. 4, 686 (2010). ADSCrossRefGoogle Scholar
  72. 72.
    Ma, X., Qi, B., Zhao, Y., Lo, H.K.: Practical decoy state for quantum key distribution. Phys. Rev. A 72, 012,326 (2005). CrossRefGoogle Scholar
  73. 73.
    Makarov, V., Anisimov, A., Skaar, J.: Effects of detector efficiency mismatch on security of quantum cryptosystems. Phys. Rev. A 74, 022,313 (2006). CrossRefGoogle Scholar
  74. 74.
    Malik, M., Erhard, M., Huber, M., Krenn, M., Fickler, R., Zeilinger, A.: Multi-photon entanglement in high dimensions. Nat. Photon. 10, 248–252 (2016). ADSCrossRefGoogle Scholar
  75. 75.
    Manouchehri, K., Wang, J.: Physical Implementation of Quantum Walks. Springer Publishing Company, Incorporated, New York (2013)zbMATHGoogle Scholar
  76. 76.
    Martin, A., Guerreiro, T., Tiranov, A., Designolle, S., Frowis, F., Brunner, N., Huber, M., Gisin, N.: Quantifying photonic high-dimensional entanglement. Phys. Rev. Lett. 118, 110,501 (2017). CrossRefGoogle Scholar
  77. 77.
    McGettrick, M.: One dimensional quantum walks with memory. Quantum Inf. Comput. 10(5), 509–524 (2010)MathSciNetzbMATHGoogle Scholar
  78. 78.
    Mirhosseini, M., Malik, M., Shi, Z., Boyd, R.W.: Efficient separation of the orbital angular momentum eigenstates of light. Nat. Commun. 4, 2781 (2013). ADSCrossRefGoogle Scholar
  79. 79.
    Nikolopoulos, G.: Applications of single-qubit rotations in quantum public-key cryptography. Phys. Rev. A 77, 032,348 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    Nikolopoulos, G.M., Alber, G.: Security bound of two-basis quantum-key-distribution protocols using qudits. Phys. Rev. A 72, 032,320 (2005). CrossRefGoogle Scholar
  81. 81.
    Nikolopoulos, G.M., Ranade, K.S., Alber, G.: Error tolerance of two-basis quantum-key-distribution protocols using qudits and two-way classical communication. Phys. Rev. A 73, 032,325 (2006). CrossRefGoogle Scholar
  82. 82.
    Portugal, R.: Quantum Walks and Search Algorithms. Quantum Science and Technology. Springer, New York, NY (2013). CrossRefzbMATHGoogle Scholar
  83. 83.
    Pugh, C.J., Kaiser, S., Bourgoin, J.P., Jin, J., Sultana, N., Agne, S., Anisimova, E., Makarov, V., Choi, E., Higgins, B.L., Jennewein, T.: Airborne demonstration of a quantum key distribution receiver payload. Quantum Sci. Technol. 2, 024,009 (2017). CrossRefGoogle Scholar
  84. 84.
    Qi, B., Fung, C.H.F., Lo, H.K., Ma, X.: Time-shift attack in practical quantum cryptosystems. Quantum Inf. Comput. 7, 73–82 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  85. 85.
    Quantique, I.: Quantum key distribution record (2017). Accessed 12 Sept 2018
  86. 86.
    Renner, R.: Symmetry of large physical systems implies independence of subsystems. Nat. Phys. 3, 645–649 (2007). CrossRefGoogle Scholar
  87. 87.
    Renner, R., Gisin, N., Kraus, B.: Information-theoretic security proof for quantum-key-distribution protocols. Phys. Rev. A 72, 012,332 (2005). CrossRefGoogle Scholar
  88. 88.
    Rohde, P.P., Brennen, G.K., Gilchrist, A.: Quantum walks with memory provided by recycled coins and a memory of the coin-flip history. Phys. Rev. A 87, 052,302 (2013). CrossRefGoogle Scholar
  89. 89.
    Rohde, P.P., Fitzsimons, J.F., Gilchrist, A.: Quantum walks with encrypted data. Phys. Rev. Lett. 109(15), 150,501 (2012)CrossRefGoogle Scholar
  90. 90.
    Roldan, E., Soriano, J.C.: Optical implementability of the two-dimensional quantum walk. J. Mod. Opt. 52, 2649–2657 (2006). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  91. 91.
    Rosenberg, D., Peterson, C.G., Harrington, J.W., Rice, P.R., Dallmann, N., Tyagi, K.T., McCabe, K.P., Nam, S., Baek, B., Hadfield, R.H., Hughes, R.J., Nordholt, J.E.: Practical long-distance quantum key distribution system using decoy levels. New J. Phys. 11(4), 045,009 (2009)CrossRefGoogle Scholar
  92. 92.
    Sajeed, S., Minshull, C., Jain, N., Makarov, V.: Invisible Trojan-horse attack. Sci. Rep. 7, 8403 (2017). ADSCrossRefGoogle Scholar
  93. 93.
    Sansoni, L., Sciarrino, F., Vallone, G., Mataloni, P., Crespi, A., Ramponi, R., Osellame, R.: Two-particle bosonic-fermionic quantum walk via integrated photonics. Phys. Rev. Lett. 108, 010,502 (2012). CrossRefGoogle Scholar
  94. 94.
    Santha, M.: Quantum walk based search algorithms. In: Agrawal, M., Du, D., Duan, Z., Li, A. (eds.) Theory and Applications of Models of Computation, Lecture Notes in Computer Science, vol. 4978, pp. 31–46. Springer, Berlin (2008). CrossRefGoogle Scholar
  95. 95.
    Scarani, V., Bechmann-Pasquinucci, H., Cerf, N., Dušek, M., Lutkenhaus, N., Peev, M.: The security of practical quantum key distribution. Rev. Mod. Phys. 81, 1301–1350 (2009). ADSCrossRefGoogle Scholar
  96. 96.
    Scarani, V., Bechmann-Pasquinucci, H., Cerf, N.J., Dušek, M., Lütkenhaus, N., Peev, M.: The security of practical quantum key distribution. Rev. Mod. Phys. 81, 1301–1350 (2009). ADSCrossRefGoogle Scholar
  97. 97.
    Schreiber, A., Cassemiro, K.N., Potocek, V., Gabris, A., Mosley, P.J., Andersson, E., Jex, I., Silberhorn, C.: Photons walking the line: a quantum walk with adjustable coin operations. Phys. Rev. Lett. 104, 050,502 (2010). CrossRefGoogle Scholar
  98. 98.
    Schreiber, A., Gabris, A., Rohde, P.P., Laiho, K., Stefanak, M., Potocek, V., Hamilton, C., Jex, I., Silberhorn, C.: A 2D quantum walk simulation of two-particle dynamics. Science 336, 55–58 (2012). ADSCrossRefGoogle Scholar
  99. 99.
    Seyfarth, U., Nikolopoulos, G., Alber, G.: Symmetries and security of a quantum-public-key encryption based on single-qubit rotations. Phys. Rev. A 85(2), 022,342 (2012)CrossRefGoogle Scholar
  100. 100.
    Sheridan, L., Scarani, V.: Security proof for quantum key distribution using qudit systems. Phys. Rev. A 82, 030,301 (2010). CrossRefGoogle Scholar
  101. 101.
    Stipcević, M.: Preventing detector blinding attack and other random number generator attacks on quantum cryptography by use of an explicit random number generator. arXiv:1403.0143 (2014)
  102. 102.
    Takesue, H., Sasaki, T., Tamaki, K., Koashi, M.: Experimental quantum key distribution without monitoring signal disturbance. Nat. Photon. 9, 827–831 (2015). ADSCrossRefGoogle Scholar
  103. 103.
    Ursin, R., Tiefenbacher, F., Schmitt-Manderbach, T., Weier, H., Scheidl, T., Lindenthal, M., Blauensteiner, B., Jennewein, T., Perdigues, J., Trojek, P., Omer, B., Furst, M., Meyenburg, M., Rarity, J., Sodnik, Z., Barbieri, C., Weinfurter, H., Zeilinger, A.: Entanglement-based quantum communication over 144 km. Nat. Phys. 3, 481–486 (2007). CrossRefGoogle Scholar
  104. 104.
    Vakhitov, A., Makarov, V., Hjelme, D.R.: Large pulse attack as a method of conventional optical eavesdropping in quantum cryptography. J. Mod. Opt. 48(13), 2023–2038 (2001). ADSCrossRefzbMATHGoogle Scholar
  105. 105.
    Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11(5), 1015–1106 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  106. 106.
    Vlachou, C., Rodrigues, J., Mateus, P., Paunković, N., Souto, A.: Quantum walk public-key cryptographic system. Int. J. Quantum Inf. 13(07), 1550,050 (2015)MathSciNetCrossRefGoogle Scholar
  107. 107.
    Wang, F., Erhard, M., Babazadeh, A., Malik, M., Krenn, M., Zeilinger, A.: Generation of the complete four-dimensional bell basis. Optica 4, 1462–1467 (2017). CrossRefGoogle Scholar
  108. 108.
    Wang, J., Wang, H., Qin, X., Wei, Z., Zhang, Z.: The countermeasures against the blinding attack in quantum key distribution. Eur. Phys. J. D 70(1), 5 (2016). ADSCrossRefGoogle Scholar
  109. 109.
    Wang, Q., Wang, X.B., Bjrk, G., Karlsson, A.: Improved practical decoy state method in quantum key distribution with parametric down-conversion source. EPL Europhys. Lett. 79(4), 40,001 (2007)MathSciNetCrossRefGoogle Scholar
  110. 110.
    Wang, X.B.: Beating the photon-number-splitting attack in practical quantum cryptography. Phys. Rev. Lett. 94, 230,503 (2005). CrossRefGoogle Scholar
  111. 111.
    Wang, X.B.: Decoy-state protocol for quantum cryptography with four different intensities of coherent light. Phys. Rev. A 72, 012,322 (2005). CrossRefGoogle Scholar
  112. 112.
    Wiechers, C., Lydersen, L., Wittmann, C., Elser, D., Skaar, J., Marquardt, C., Makarov, V., Leuchs, G.: After-gate attack on a quantum cryptosystem. New J. Phys. 13(1), 013043 (2011)ADSCrossRefGoogle Scholar
  113. 113.
    Woodhead, E., Pironio, S.: Secrecy in prepare-and-measure clauser-horne-shimony-holt tests with a qubit bound. Phys. Rev. Lett. 115, 150,501 (2015). CrossRefGoogle Scholar
  114. 114.
    Xu, G., Chen, X.B., Dou, Z., Yang, Y.X., Li, Z.: A novel protocol for multiparty quantum key management. Quantum Inf. Process. 14(8), 2959–2980 (2015). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  115. 115.
    Yao, A.C.: How to generate and exchange secrets. In: 27th Annual Symposium on Foundations of Computer Science, 1986., pp. 162–167. IEEE (1986)Google Scholar
  116. 116.
    Yin, J., Cao, Y., Li, Y.H., Liao, S.K., Zhang, L., Ren, J.G., Cai, W.Q., Liu, W.Y., Li, B., Dai, H., Li, G.B., Lu, Q.M., Gong, Y.H., Xu, Y., Li, S.L., Li, F.Z., Yin, Y.Y., Jiang, Z.Q., Li, M., Jia, J.J., Ren, G., He, D., Zhou, Y.L., Zhang, X.X., Wang, N., Chang, X., Zhu, Z.C., Liu, N.L., Chen, Y.A., Lu, C.Y., Shu, R., Peng, C.Z., Wang, J.Y., Pan, J.W.: Satellite-based entanglement distribution over 1200 kilometers. Science 356, 1140–1144 (2017). CrossRefGoogle Scholar
  117. 117.
    Zhang, W., Qiu, D., Mateus, P.: Security of a single-state semi-quantum key distribution protocol. arXiv:1612.03170 (2016)
  118. 118.
    Zhao, Y., Fung, C.H.F., Qi, B., Chen, C., Lo, H.K.: Quantum hacking: experimental demonstration of time-shift attack against practical quantum-key-distribution systems. Phys. Rev. A 78, 042,333 (2008). CrossRefGoogle Scholar
  119. 119.
    Zou, X., Qiu, D., Li, L., Wu, L., Li, L.: Semiquantum-key distribution using less than four quantum states. Phys. Rev. A 79, 052,312 (2009). CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Instituto de TelecomunicaçõesLisbonPortugal
  2. 2.Dep. de Matemática – Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  3. 3.CeFEMA, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  4. 4.Computer Science and Engineering DepartmentUniversity of ConnecticutStorrsUSA
  5. 5.Dep. de Informática – Faculdade de CiênciasUniversidade de LisboaLisbonPortugal
  6. 6.LASIGE – Faculdade de CiênciasUniversidade de LisboaLisbonPortugal

Personalised recommendations