Skip to main content
SpringerLink
Log in

Quantum algorithm to solve function inversion with time–space trade-off

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

In general, it is a difficult problem to solve the inverse of any function. With the inverse implication operation, we present a quantum algorithm for solving the inversion of function via using time–space trade-off in this paper. The details are as follows. Let function \(f(x)=y\) have k solutions, where \(x\in \{0, 1\}^{n}, y\in \{0, 1\}^{m}\) for any integers n, m. We show that an iterative algorithm can be used to solve the inverse of function f(x) with successful probability \(1-\left( 1-\frac{k}{2^{n}}\right) ^{L}\) for \(L\in Z^{+}\). The space complexity of proposed quantum iterative algorithm is O(Ln), where L is the number of iterations. The paper concludes that, via using time–space trade-off strategy, we improve the successful probability of algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. Math. Phys. Sci. 439, 553–558 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM. J. Comput. 26(5), 1411–1473 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Simon, D.R.: On the power of quantum computation. SIAM. J. Comput. 26, 1474–1483 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Grover, L.K.: Quantum mechanics helps in searching for a needle in haystack. Phys. Rev. Lett. 79, 325–328 (1997)

    Article  ADS  Google Scholar 

  5. Amy, M., Matteo, O.D., Gheorghiu, V., et al.: Estimating the Cost of Generic Quantum Pre-Image Attacks on SHA-2 and SHA-3. arXiv:1603.09383v3

  6. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM. J. Comput. 26(5), 1484–1509 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Mosca, M.,Ekert, A.: The hidden subgroup problem and eigenvalue estimation on a quantum computer, Proceedings of the 1st NASA International conference on Quantum Comput. Quantum Commun. vol. 1509 of lecture Notes in computer Science, 1999

  8. Hallgren, S., Russell, A., Ta-Shma, A.: The hidden subgroup problem and quantum computation using group representations. SIAM. J. Comput. 32(4), 916–934 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Vedral, V., Barenco, A., Ekert, A.: Quantum networks for elementary arithmetic operations. Phys. Rev. A 54(1), 147 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  10. Wu, W.Q., Zhang, H.G., Mao, S.W., et al.: Quantum algorithm to find invariant linear structure of MD hash functions. Quantum Inf. Process. 3, 813–829 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Wu, W.Q., Zhang, H.G., Wang, H.Z., et al.: Polynomial-time quantum algorithms for finding the linear structures of Boolean function. Quantum Inf. Process. 4, 1215–1226 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are supported by the Major State Basic Research Development Program of China (No. 2014CB340600), Natural Science Foundation of HeBei Province (No. F2017201199), and Science and technology research project of Hebei higher education (No. NQ2017020).

Author information

Authors and Affiliations

  1. School of Computer Science and Technology, Hebei University, Baoding, 071002, People’s Republic of China

    WanQing Wu

  2. Computer School of Wuhan University, Wuhan, 430072, People’s Republic of China

    HuanGuo Zhang

Authors
  1. WanQing Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. HuanGuo Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to WanQing Wu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, W., Zhang, H. Quantum algorithm to solve function inversion with time–space trade-off. Quantum Inf Process 16, 171 (2017). https://doi.org/10.1007/s11128-017-1622-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-017-1622-y

Keywords

Mathematics Subject Classification

Search

Navigation