Quantum encryption and generalized Shannon impossibility

  • Ching-Yi LaiEmail author
  • Kai-Min Chung


The famous Shannon impossibility result says that any encryption scheme with perfect secrecy requires a secret key at least as long as the message. In this paper we provide its quantum analogue with imperfect secrecy and imperfect correctness. We also give a systematic study of information-theoretically secure quantum encryption with two secrecy definitions. We show that the weaker one implies the stronger but with a security loss in d, where d is the dimension of the encrypted quantum system. This is good enough if the target secrecy error is of \(o(d^{-1})\).


Shannon impossibility Information-theoretic security Key length Quantum one-time pad 

Mathematics Subject Classification

94A60 81P94 



We are grateful to anonymous referees for their constructive comments on this manuscript. CYL was was financially supported from the Young Scholar Fellowship Program by Ministry of Science and Technology (MOST) in Taiwan, under Grant MOST107-2636-E-009-005. KMC was partially supported by 2016 Academia Sinica Career Development Award under Grant No. 23-17 and the Ministry of Science and Technology, Taiwan under Grant No. MOST 103-2221-E-001-022-MY3.


  1. 1.
    Ambainis A., Mosca M., Tapp A., Wolf R.D.: Private quantum channels. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pp. 547–553 (2000).Google Scholar
  2. 2.
    Ambainis A., Smith A.: Small pseudo-random families of matrices: derandomizing approximate quantum encryption. In: Proceedings of RANDOM, Series. Lecture Notes in Computer Science, pp. 249–260. Springer, Berlin (2004).Google Scholar
  3. 3.
    Boykin P.O., Roychowdhury V.: Optimal encryption of quantum bits. Phys. Rev. A 67, 042317 (2003).CrossRefGoogle Scholar
  4. 4.
    Desrosiers S.P.: Entropic security in quantum cryptography. Quant. Inf. Process. 8(4), 331–345 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Desrosiers S.P., Dupuis F.: Quantum entropic security and approximate quantum encryption. IEEE Trans. Inf. Theory 56(7), 3455–3464 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dickinson P.A., Nayak A.: Approximate randomization of quantum states with fewer bits of key. AIP Conference Proceedings 864(1), 18–36 (2006).CrossRefzbMATHGoogle Scholar
  7. 7.
    DiVincenzo D.P., Hayden P., Terhal B.M.: Hiding quantum data. Found. Phys. 33(11), 1629–1647 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dodis Y.: Shannon impossibility, revisited. In: A Smith (ed.) Information Theoretic Security: 6th International Conference, ICITS, Montreal, QC, Canada, August 15–17, pp. 100–110, 2012. Springer, Berlin (2012).Google Scholar
  9. 9.
    Dodis Y., Smith A.: Entropic security and the encryption of high entropy messages. In: Proceedings of the Second International Conference on Theory of Cryptography, Series. TCC’05, pp. 556–577. Springer, Berlin (2005).Google Scholar
  10. 10.
    Fuchs C.A., van de Graaf J.: Cryptographic distinguishability measures for quantum-mechanical states. IEEE Trans. Inf. Theory 45(4), 1216–1227 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Goldwasser S., Micali S.: Probabilistic encryption. J. Comput. Syst. Sci. 28(2), 270–299 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hayden P., Leung D., Shor P.W., Winter A.: Randomizing quantum states: constructions and applications. Commun. Math. Phys. 250(2), 371–391 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hughston L.P., Jozsa R., Wootters W.K.: A complete classification of quantum ensembles having a given density matrix. Phys. Lett. A 183(1), 14–18 (1993).MathSciNetCrossRefGoogle Scholar
  14. 14.
    Iwamoto M., Ohta K.: Security notions for information theoretically secure encryptions. In: 2011 IEEE International Symposium on Information Theory Proceedings, pp. 1777–1781 (2011).Google Scholar
  15. 15.
    Iwamoto M., Ohta K., Shikata J.: Security formalizations and their relationships for encryption and key agreement in information-theoretic cryptography. IEEE Trans. Inf. Theory 64(1), 654–685 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jain R.: Resource requirements of private quantum channels and consequences for oblivious remote state preparation. J. Cryptol. 25(1), 1–13 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Konig R., Renner R., Schaffner C.: The operational meaning of min- and max-entropy. IEEE Trans. Inf. Theory 55(9), 4337–4347 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lai C.-Y., Chung K.-M.: On statistically-secure quantum homomorphic encryption. Quant. Inf. Comput. 18(9&10), 0785–0794 (2018).MathSciNetGoogle Scholar
  19. 19.
    Nagaj D., Kerenidis I.: On the optimality of quantum encryption schemes. J. Math. Phys. 47(9), 092102 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nayak A., Sen P.: Invertible quantum operations and perfect encryption of quantum states. Quant. Inf. Comput. 7(1), 103–110 (2007).zbMATHGoogle Scholar
  21. 21.
    Ouyang Y., Tan S.-H., Fitzsimons J.F.: Quantum homomorphic encryption from quantum codes. Phys. Rev. A 98, 042334 (2018).CrossRefGoogle Scholar
  22. 22.
    Russell A., Wang H.: How to fool an unbounded adversary with a short key. IEEE Trans. Inf. Theory 52(3), 1130–1140 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shannon C.: Communication theory of secrecy systems. Bell Syst. Tech. J. 28, 656–719 (1949).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Uhlmann A.: The Transition Probability in the state space of a *-algebra. Rep. Math. Phys. 9, 273–279 (1976).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Communications EngineeringNational Chiao Tung UniversityHsinchuTaiwan
  2. 2.Institute of Information ScienceAcademia SinicaTaipeiTaiwan

Personalised recommendations