Advertisement

Teleportation-based quantum homomorphic encryption scheme with quasi-compactness and perfect security

  • Min LiangEmail author
Article
  • 17 Downloads

Abstract

Quantum homomorphic encryption (QHE) is an important cryptographic technology for delegated quantum computation. It enables remote server to perform quantum computation on encrypted data, and the specific algorithm performed by Server is unnecessarily known by Client. Quantum fully homomorphic encryption (QFHE) is a QHE that satisfies both compactness and \({\mathcal {F}}\)-homomorphism (homomorphic for any quantum circuits). However, Yu et al. (Phys Rev A 90:050303, 2014) proved a negative result: Assume interaction is not allowed, it is impossible to construct perfectly secure QFHE scheme. So this article focuses on non-interactive and perfectly secure QHE scheme with loose requirement, e.g., quasi-compactness. This article defines encrypted gate, which is denoted by \(EG[U]:|\alpha \rangle \rightarrow \left( (a,b),Enc_{a,b}(U|\alpha \rangle )\right) \). We present a gate-teleportation-based two-party computation scheme for EG[U], where one party gives arbitrary quantum state \(|\alpha \rangle \) as input and obtains the encrypted U-computing result \(Enc_{a,b}(U|\alpha \rangle )\), and the other party obtains the random bits ab. Based on \(EG[P^x](x\in \{0,1\})\), we propose a method to remove the P-error generated in the homomorphic evaluation of \(T/T^\dagger \)-gate. Using this method, we design two non-interactive and perfectly secure QHE schemes named GT and VGT. Both of them are \({\mathcal {F}}\)-homomorphic and quasi-compact (the decryption complexity depends on the \(T/T^\dagger \)-gate complexity). Assume \({\mathcal {F}}\)-homomorphism, non-interaction and perfect security are necessary properties, the quasi-compactness is proved to be bounded by \(\varOmega (M)\), where M is the total number of \(T/T^\dagger \)-gates in the evaluated circuit. We prove VGT is M-quasi-compact and reaches the optimal bound. According to our QHE schemes, the decryption would be inefficient when the evaluated circuit contains exponential number of \(T/T^\dagger \)-gates. Thus, our schemes are suitable for homomorphic evaluation of any quantum circuit with low \(T/T^\dagger \)-gate complexity, such as any polynomial-size quantum circuit or any quantum circuit with polynomial number of \(T/T^\dagger \)-gates.

Keywords

Quantum cryptography Quantum encryption Delegated quantum computation Quantum homomorphic encryption 

Notes

Supplementary material

References

  1. 1.
    Gentry, C.: A fully homomorphic encryption scheme. Ph.D. thesis, Stanford University (2009)Google Scholar
  2. 2.
    Childs, A.M.: Secure assisted quantum computation. Quantum Inf. Comput. 5(6), 456–466 (2005)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Aharonov, D., Ben-Or, M., Eban, E.: Interactive proofs for quantum computations. In: Proceedings of Innovations in Computer Science, ICS 2010, pp. 453–469. Tsinghua University Press (2010)Google Scholar
  4. 4.
    Broadbent, A.J., Fitzsimons, F., Kashefi, E.: Universal blind quantum computation. In: Proceedings of the 50th Annual Symposium on Foundations of Computer Science, pp. 517–526. IEEE (2009)Google Scholar
  5. 5.
    Sueki, T., Koshiba, T., Morimae, T.: Ancilla-driven universal blind quantum computation. Phys. Rev. A 87, 060301 (2013)ADSCrossRefGoogle Scholar
  6. 6.
    Morimae, T., Fujii, K.: Blind topological measurement-based quantum computation. Nat. Commun. 3, 1036 (2012)ADSCrossRefGoogle Scholar
  7. 7.
    Morimae, T.: Continuous-variable blind quantum computation. Phys. Rev. Lett. 109, 230502 (2012)ADSCrossRefGoogle Scholar
  8. 8.
    Giovannetti, V., Lloyd, S., Maccone, L.: Efficient universal blind quantum computing. Phys. Rev. Lett. 111(23), 230501 (2013)ADSCrossRefGoogle Scholar
  9. 9.
    Liang, M.: Quantum fully homomorphic encryption scheme based on universal quantum circuit. Quantum Inf. Process. 14, 2749–2759 (2015)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Rohde, P.P., Fitzsimons, J.F., Gilchrist, A.: Quantum walks with encrypted data. Phys. Rev. Lett. 109(15), 150501 (2012)ADSCrossRefGoogle Scholar
  11. 11.
    Liang, M.: Symmetric quantum fully homomorphic encryption with perfect security. Quantum Inf. Process. 12, 3675–3687 (2013)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Tan, S.H., Kettlewell, J.A., Ouyang, Y.K., Chen, L., Fitzsimons, J.F.: A quantum approach to fully homomorphic encryption. Sci. Rep. 6, 33467 (2016)ADSCrossRefGoogle Scholar
  13. 13.
    Yu, L., Perez-Delgado, C.A., Fitzsimons, J.F.: Limitations on information theoretically secure quantum homomorphic encryption. Phys. Rev. A 90, 050303 (2014)ADSCrossRefGoogle Scholar
  14. 14.
    Fisher, K., Broadbent, A., Shalm, L.K., Yan, Z., Lavoie, J., Prevedel, R., Jennewein, T., Resch, K.J.: Quantum computing on encrypted data. Nat. Commun. 5, 3074 (2014)ADSCrossRefGoogle Scholar
  15. 15.
    Broadbent, A., Jeffery, S.: Quantum homomorphic encryption for circuits of low T-gate complexity. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015 Part II. LNCS, vol. 9216, pp. 609–629. Springer, Heidelberg (2015)Google Scholar
  16. 16.
    Dulek, Y., Schaffner, C., Speelman, F.: Quantum homomorphic encryption for polynomial-sized circuits. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016 Part III. LNCS, vol. 9816, pp. 3–32. Springer, Heidelberg (2016)Google Scholar
  17. 17.
    Liang, M., Yang, L.: Quantum fully homomorphic encryption scheme based on quantum fault-tolerant construction (2015). arXiv:1503.04061
  18. 18.
    Ouyang, Y., Tan, S.-H., Fitzsimons, J.: Quantum homomorphic encryption from quantum codes (2015). arXiv:1508.00938
  19. 19.
    Newman, M., Shi, Y.: Limitationson transversal computation through quantum homomorphic encryption. Quantum Inf. Comput. 18, 927–948 (2018)MathSciNetGoogle Scholar
  20. 20.
    Lai, C.-Y., Chung, K.-M.: On statistically-secure quantum homomorphic encryption. Quantum Inf. Comput. 18, 785–794 (2018)MathSciNetGoogle Scholar
  21. 21.
    Alagic, G., Dulek, Y., Schaffner, C., Speelman, F.: Quantum fully homomorphic encryption with verification. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017 Part I. LNCS, vol. 10624, pp. 438–467. Springer, Cham (2017)Google Scholar
  22. 22.
    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  23. 23.
    Boykin, P., Roychowdhury, V.: Optimal encryption of quantum bits. Phys. Rev. A 67(4), 42317 (2003)ADSCrossRefGoogle Scholar
  24. 24.
    Ambainis, A., Mosca, M., Tapp, A., De Wolf, R.: Private quantum channels. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pp. 547–553. Redondo Beach, CA, USA (2000)Google Scholar
  25. 25.
    Gottesman, D., Chuang, I.L.: Quantum teleportation is a universal computational primitive. Nature 402, 390–393 (1999)ADSCrossRefGoogle Scholar
  26. 26.
    Jozsa, R.: An introduction to measurement based quantum computation (2005). arXiv:quant-ph/0508124v2
  27. 27.
    Shor, P. W.: Algorithms for quantum computation: discrete logarithm and factoring. In: Proceedings of the 35th Annual Symposium on the Theory of Computer Science, pp. 124–134. IEEE Computer Society Press, Los Alamitos (1994)Google Scholar
  28. 28.
    Harrow, A.W., Hassidim, A., Lloyd, S.: Quantum algorithm for solving linear systems of equations. Phys. Rev. Lett. 15(103), 150502 (2009)CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Data Communication Science and Technology Research InstituteBeijingChina

Personalised recommendations