Quantum Non-malleability and Authentication

  • Gorjan Alagic
  • Christian MajenzEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10402)


In encryption, non-malleability is a highly desirable property: it ensures that adversaries cannot manipulate the plaintext by acting on the ciphertext. In [6], Ambainis et al. gave a definition of non-malleability for the encryption of quantum data. In this work, we show that this definition is too weak, as it allows adversaries to “inject” plaintexts of their choice into the ciphertext. We give a new definition of quantum non-malleability which resolves this problem. Our definition is expressed in terms of entropic quantities, considers stronger adversaries, and does not assume secrecy. Rather, we prove that quantum non-malleability implies secrecy; this is in stark contrast to the classical setting, where the two properties are completely independent. For unitary schemes, our notion of non-malleability is equivalent to encryption with a two-design (and hence also to the definition of [6]).

Our techniques also yield new results regarding the closely-related task of quantum authentication. We show that “total authentication” (a notion recently proposed by Garg et al. [18]) can be satisfied with two-designs, a significant improvement over the eight-design construction of [18]. We also show that, under a mild adaptation of the rejection procedure, both total authentication and our notion of non-malleability yield quantum authentication as defined by Dupuis et al. [16].



The authors would like to thank Anne Broadbent, Alexander Müller-Hermes, Frédéric Dupuis and Christopher Portmann for helpful discussions. G.A. and C.M. acknowledge financial support from the European Research Council (ERC Grant Agreement 337603), the Danish Council for Independent Research (Sapere Aude) and VILLUM FONDEN via the QMATH Centre of Excellence (Grant 10059).


  1. 1.
    Aaronson, S., Gottesman, D.: Improved simulation of stabilizer circuits. Phys. Rev. A 70, 052328 (2004). doi: 10.1103/PhysRevA.70.052328 CrossRefGoogle Scholar
  2. 2.
    Aharonov, D., Ben-Or, M., Eban, E.: Interactive proofs for quantum computations. In: Innovations in Computer Science - ICS 2010, Proceedings, Tsinghua University, Beijing, China, 5–7 January 2010, pp. 453–469 (2010)Google Scholar
  3. 3.
    Alagic, G., Majenz, C.: Quantum non-malleability and authentication. CoRR, abs/1610.04214 (2016).
  4. 4.
    Alicki, R., Fannes, M.: Continuity of quantum conditional information. J. Phys. A: Math. Gen. 37(5), L55 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ambainis, A., Mosca, M., Tapp, A., De Wolf, R.: Private quantum channels. In: Proceedings of the FOCS 2000, pp. 547–553 (2000)Google Scholar
  6. 6.
    Ambainis, A., Bouda, J., Winter, A.: Nonmalleable encryption of quantum information. J. Math. Phys. 50(4), 042106 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Barnum, H., Crépeau, C., Gottesman, D., Smith, A., Tapp, A.: Authentication of quantum messages. In: The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002, Proceedings, pp. 449–458. IEEE (2002)Google Scholar
  8. 8.
    Berta, M., Christandl, M., Renner, R.: The quantum reverse shannon theorem based on one-shot information theory. Commun. Math. Phys. 306(3), 579–615 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Berta, M., Brandao, F.G.S.L., Majenz, C., Wilde, M.M.: Deconstruction and conditional erasure of quantum correlations. arXiv preprint arXiv:1609.06994 (2016)
  10. 10.
    Brandao, F.G.S.L., Harrow, A.W., Horodecki, M.: Local random quantum circuits are approximate polynomial-designs. arXiv preprint arXiv:1208.0692 (2012)
  11. 11.
    Broadbent, A., Wainewright, E.: Efficient simulation for quantum message authentication. arXiv preprint arXiv:1607.03075 (2016)
  12. 12.
    Choi, M.-D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10(3), 285–290 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cleve, R., Leung, D., Liu, L., Wang, C.: Near-linear constructions of exact unitary 2-designs. Quantum Inf. Comput. 16(9&10), 0721–0756 (2016)MathSciNetGoogle Scholar
  14. 14.
    Dankert, C., Cleve, R., Emerson, J., Livine, E.: Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A 80(1), 012304 (2009)CrossRefGoogle Scholar
  15. 15.
    Dupuis, F., Nielsen, J.B., Salvail, L.: Secure two-party quantum evaluation of unitaries against specious adversaries. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 685–706. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14623-7_37 CrossRefGoogle Scholar
  16. 16.
    Dupuis, F., Nielsen, J.B., Salvail, L.: Actively secure two-party evaluation of any quantum operation. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 794–811. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-32009-5_46 CrossRefGoogle Scholar
  17. 17.
    Dupuis, F., Berta, M., Wullschleger, J., Renner, R.: One-shot decoupling. Commun. Math. Phys. 328(1), 251–284 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Garg, S., Yuen, H., Zhandry, M.: New security notions and feasibility results for authentication of quantum data. arXiv preprint arXiv:1607.07759 (2016)
  19. 19.
    Jamiołkowski, A.: Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys. 3(4), 275–278 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kawachi, A., Portmann, C., Tanaka, K.: Characterization of the relations between information-theoretic non-malleability, secrecy, and authenticity. In: Fehr, S. (ed.) ICITS 2011. LNCS, vol. 6673, pp. 6–24. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-20728-0_2 CrossRefGoogle Scholar
  21. 21.
    Lieb, E.H., Ruskai, M.B.: A fundamental property of quantum-mechanical entropy. Phy. Rev. Lett. 30(10), 434 (1973a)Google Scholar
  22. 22.
    Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phy. 14(12), 1938–1941 (1973b)Google Scholar
  23. 23.
    Low, R.A.: Pseudo-randomness and learning in quantum computation. arXiv preprint arXiv:1006.5227 (2010)
  24. 24.
    Majenz, C., Berta, M., Dupuis, F., Renner, R., Christandl, M.: Catalytic decoupling of quantum information. arXiv preprint arXiv:1605.00514 (2016)
  25. 25.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, New York (2010)CrossRefzbMATHGoogle Scholar
  26. 26.
    Portmann, C.: Quantum authentication with key recycling. ArXiv e-prints, October 2016Google Scholar
  27. 27.
    Stinespring, W.F.: Positive functions on c*-algebras. Proc. Am. Math. Soc. 6(2), 211–216 (1955)MathSciNetzbMATHGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2017

Authors and Affiliations

  1. 1.QMATH, Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark

Personalised recommendations