Cryptographic one-way function based on boson sampling

  • Georgios M. NikolopoulosEmail author


The quest for practical cryptographic primitives that are robust against quantum computers is of vital importance for the field of cryptography. Among the abundance of different cryptographic primitives one may consider, one-way functions stand out as fundamental building blocks of more complex cryptographic protocols, and they play a central role in modern asymmetric cryptography. We propose a mathematical one-way function, which relies on coarse-grained boson sampling. The evaluation and the inversion of the function are discussed in the context of classical and quantum computers. The present results suggest that the scope and power of boson sampling may go beyond the proof of quantum supremacy and pave the way toward cryptographic applications.


Quantum public-key cryptography Quantum one-way function Boson sampling 



The author thanks S. Aaronson and J. J. Renema for useful comments on the coarse-grained boson sampling, as well as T. Garefalakis and T. Brougham, for helpful discussions. This work has been co-funded by the Deutsche Forschungsgemeinschaft as part of the project S4 within CRC 1119 CROSSING.


  1. 1.
    Menezes, A., van Oorschot, P., Vanstone, S.: Handbook of Applied Cryptography. CRC Press, Cambridge (1996)zbMATHGoogle Scholar
  2. 2.
    Goldreich, O.: Foundations of Cryptography: Basic Techniques. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  3. 3.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  4. 4.
    Bernstein, D.J., Buchmann, J., Dahmen, E. (eds.): Post-Quantum Cryptography. Springer, Berlin (2009)zbMATHGoogle Scholar
  5. 5.
    Bernstein, D.J., Lange, T.: Post-quantum cryptography. Nature 549, 188 (2017)ADSCrossRefGoogle Scholar
  6. 6.
    Kabashima, Y., Murayama, T., Saad, D.: Cryptographical properties of Ising spin systems. Phys. Rev. Lett. 84, 2030 (2000)ADSCrossRefGoogle Scholar
  7. 7.
    Buhrman, H., Cleve, R., Watrous, J., de Wolf, R.: Quantum fingerprinting. Phys. Rev. Lett. 87, 167902 (2001)ADSCrossRefGoogle Scholar
  8. 8.
    Gottesman, D., Chuang, I.L.: Quantum digital signatures. e-print arXiv:qunt-ph/0105032
  9. 9.
    Curty, M., Santos, D.J.: Quantum authentication of classical messages. Phys. Rev. A 64, 062309 (2001)ADSCrossRefGoogle Scholar
  10. 10.
    Kawachi, A., Koshiba, T., Nishimura, H., Yamakami, T.: Computational indistinguishability between quantum states and its cryptographic application. Lect. Notes Comput. Sci. 3494, 268 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Andersson, E., Curty, M., Jex, I.: Experimentally realizable quantum comparison of coherent states and its applications. Phys. Rev. A 74, 022304 (2006)ADSCrossRefGoogle Scholar
  12. 12.
    Dunjko, V., Wallden, P., Andersson, E.: Quantum digital signatures with quantum-key-distribution components. Phys. Rev. Lett. 112, 040502 (2014)ADSCrossRefGoogle Scholar
  13. 13.
    Nikolopoulos, G.M.: Applications of single-qubit rotations in quantum public-key cryptography. Phys. Rev. A 77, 032348 (2008)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Nikolopoulos, G.M., Ioannou, L.M.: Deterministic quantum-public-key encryption: forward search attack and randomization. Phys. Rev. A 79, 042327 (2008)ADSCrossRefGoogle Scholar
  15. 15.
    Seyfarth, U., Nikolopoulos, G.M., Alber, G.: Symmetries and security of a quantum-public-key encryption based on single-qubit rotations. Phys. Rev. A 85, 022342 (2012)ADSCrossRefGoogle Scholar
  16. 16.
    Goorden, S.A., Horstmann, M., Mosk, A.P., Skoríc, B., Pinkse, P.W.H.: Quantum-secure authentication of a physical unclonable key. Optica 1, 421 (2014)CrossRefGoogle Scholar
  17. 17.
    Nikolopoulos, G.M., Diamanti, E.: Continuous-variable quantum authentication of physical unclonable keys. Sci. Rep. 7, 46047 (2017)ADSCrossRefGoogle Scholar
  18. 18.
    Nikolopoulos, G.M.: Continuous-variable quantum authentication of physical unclonable keys: security against an emulation attack. Phys. Rev. A 97, 012324 (2018)ADSCrossRefGoogle Scholar
  19. 19.
    Uppu, R., Wolterink, T.A.W., Goorden, S.A., Chen, B., Skoríc, B., Mosk, A.P., Pinkse, P.W.H.: Asymmetric cryptography with physical unclonable keys. arXiv:1802.07573
  20. 20.
    Wu, C., Yang, L.: Bit-oriented quantum public-key encryption based on quantum perfect encryption. Quantum Inf. Process. 15, 3285 (2016)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Chen, F.-L., et al.: Public-key quantum digital signature scheme with one-time pad private-key. Quantum Inf. Process. 17, 10 (2018)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Vlachou, C., et al.: Quantum walk public-key cryptographic system. Int. J. Quantum Inf. 13, 1550050 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ioannou, L.M., Mosca, M.: Public-key cryptography based on bounded quantum reference frames. Theor. Comput. Sci. 560, 33 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Fujita, H.: Quantum McEliece public-key cryptosystem. Quantum Inf. Comput. 12, 181 (2012)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Diamanti, E., Lo, H.-K., Qi, B., Yuan, Z.: Practical challenges in quantum key distribution. NPJ Quantum Inf. 2, 16025 (2016)CrossRefGoogle Scholar
  26. 26.
    Aaronson, S., Arkhipov, A.: The computational complexity of linear optics. Theory Comput. 9, 143 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Gard, B.T., Motes, K.R., Olson, J.P., Rohde, P.P., Dowling, J.P.: An introduction to Boson sampling. In: Malinovskaya S. A., Novikova, I. (eds.) From Atomic to Mesoscale: The Role of Quantum Coherence in Systems of Various Complexities, Chap. 8, pp. 167–192. World Scientific Publishing Co (2015). CrossRefGoogle Scholar
  28. 28.
    Lund, A.P., Bremner, M.J., Ralph, T.C.: Quantum sampling problems, bosonsampling and quantum supremacy. NPJ Quantum Inf. 3, 15 (2017)ADSCrossRefGoogle Scholar
  29. 29.
    Tillmann, M., Dakić, B., Heilmann, R., Nolte, S., Szameit, A., Walther, P.: Experimental boson sampling. Nat. Photonics 7, 540 (2013)ADSCrossRefGoogle Scholar
  30. 30.
    Broome, M.A., Fedrizzi, A., Rahimi-Keshari, S., Dove, J., Aaronson, S., Ralph, T.C., White, A.G.: Photonic boson sampling in a tunable circuit. Science 339, 794 (2013)ADSCrossRefGoogle Scholar
  31. 31.
    Spring, B., Metcalf, B.J., Humphreys, P.C., Kolthammer, W.S., Jin, X.-M., Barbieri, M., Datta, A., Thomas-Peter, N., Langford, N.K., Kundys, D., Gates, J.C., Smith, B.J., Smith, P.G.R., Walmsley, I.A.: Boson sampling on a photonic chip. Science 339, 798 (2013)ADSCrossRefGoogle Scholar
  32. 32.
    Crespi, A., Osellame, R., Ramponi, R., Brod, D.J., Galvao, E.F., Spagnolo, N., Vitelli, C., Maiorino, E., Mataloni, P., Sciarrino, F.: Integrated multimode interferometers with arbitrary designs for photonic boson sampling. Nat. Photonics 7, 545 (2013)ADSCrossRefGoogle Scholar
  33. 33.
    Spagnolo, N., Vitelli, C., Bentivegna, M., Brod, D.J., Crespi, A., Flamini, F., Giacomini, S., Milani, G., Ramponi, R., Mataloni, P., Osellame, R., Galvao, E.F., Sciarrino, F.: Efficient experimental validation of photonic boson sampling against the uniform distribution. Nat. Photonics 8, 615 (2014)ADSCrossRefGoogle Scholar
  34. 34.
    Carolan, J., Meinecke, J.D.A., Shadbolt, P.J., Russell, N.J., Ismail, N., Wörhoff, K., Rudolph, T., Thompson, M.G., O’Brien, J.L., Matthews, J.C.F., Laing, A.: On the experimental verification of quantum complexity in linear optics. Nat. Photonics 8, 621 (2014)ADSCrossRefGoogle Scholar
  35. 35.
    Wang, H., et al.: High-efficiency multiphoton boson sampling. Nat. Photonics 11, 361 (2017)ADSCrossRefGoogle Scholar
  36. 36.
    Neville, A., Sparrow, C., Clifford, R., Johnston, E., Birchall, P.M., Montanaro, A., Laing, A.: Classical boson sampling algorithms with superior performance to near-term experiments. Nat. Phys. 13, 1153 (2017)CrossRefGoogle Scholar
  37. 37.
    Clifford, P., Clifford, R.: The classical complexity of boson sampling. arXiv:1706.01260
  38. 38.
    Wu, J., Liu, Y., Zhang, B., Jin, X., Wang, Y., Wang, H., Yang, X.: A benchmark test of boson sampling on Tianhe-2 supercomputer. arXiv:1606.05836
  39. 39.
    Wang, S.-T., Duan, L.-M.: Certification of boson sampling devices with coarse-grained measurements. arXiv:1601.02627
  40. 40.
    Shchesnovich, V.S.: Universality of generalized bunching and efficient assessment of boson sampling. Phys. Rev. Lett. 116, 123601 (2016)ADSCrossRefGoogle Scholar
  41. 41.
    Agresti, I., Viggianiello, N., Flamini, F., Spagnolo, N., Crespi, A., Osellame, R., Wiebe, N., Sciarrino, F.: Pattern recognition techniques for boson sampling validation. Phys. Rev. X 9, 011013 (2019)Google Scholar
  42. 42.
    Nikolopoulos, G.M., Brougham, T.: Decision and function problems based on boson sampling. Phys. Rev. A 94, 012315 (2016)ADSCrossRefGoogle Scholar
  43. 43.
    Montgomery, D.C., Runger, G.C.: Applied Statistics and Probability of Engineers. Wiley, Hoboken (2005)zbMATHGoogle Scholar
  44. 44.
    Baron, M.: Probability and Statistics for Computer Scientists. CRC Press, Cambridge (2014)zbMATHGoogle Scholar
  45. 45.
    MacWilliams, F.J., Slone, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1997)Google Scholar
  46. 46.
    Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschr. Phys. 46, 493 (1998)CrossRefGoogle Scholar
  47. 47.
    Stinson, D.R., Paterson, M.B.: Cryptography: Theory and Practice. CRC Press, Taylor & Francis Group, Boca Raton (2018)CrossRefGoogle Scholar
  48. 48.
    Martin, K.M.: Everyday Cryptography: Fundamental Principles and Applications. Oxford University Press, New York (2012)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Institute of Electronic Structure and LaserFORTHHeraklionGreece
  2. 2.Institut für Angewandte PhysikTechnische Universität DarmstadtDarmstadtGermany

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