Skip to main content
SpringerLink
Log in
Book cover

International Conference on Emerging Trends in Information and Communication Security

ETRICS 2006: Emerging Trends in Information and Communication Security pp 481–493Cite as

Quantum Algorithm for Solving the Discrete Logarithm Problem in the Class Group of an Imaginary Quadratic Field and Security Comparison of Current Cryptosystems at the Beginning of Quantum Computer Age

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 3995))

Abstract

In this paper, we present a quantum algorithm which solves the discrete logarithm problem in the class group of an imaginary quadratic number field. We give an accurate estimation of the qubit complexity for this algorithm. Based on this result and analog results for the factoring and the discrete logarithm problem in the point group of an elliptic curve, we compare the run-times of cryptosystems which are based on problems above. Assuming that the size of quantum computers will grow slowly, we give proposals which cryptosystem should be used if middle-size quantum computers will be built.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Beauregard, S.: Circuit for Shor’s algorithm using 2n+3 qubits, http://xxx.lanl.gov/abs/quant-ph/0205095

  2. Bennett, C.H.: Logical Reversibility of Computation. IBM Journal of Research and Development, 525–532 (November 1977)

    Google Scholar 

  3. Buchmann, J., Williams, H.C.: A Key-Exchange System Based on Imaginary Quadratic Fields. Journal of Cryptology 1(2), 107–118 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buchmann, J., Williams, H.C.: Quadratic Fields and Cryptography. In: Loxton, J.H. (ed.) Number Theory and Cryptography. London Mathematical Society Lecture Note Series, vol. 154, pp. 9–25. Cambridge University Press, Cambridge (1990)

    Chapter  Google Scholar 

  5. Chuang, I., Vandersypen, L., Zhou, X., Leung, D., Lloyd, S.: Experimental Realization of a Quantum Algorithm. Nature 393, 143–146 (1998)

    Article  Google Scholar 

  6. Draper, T.G.: Addition on a Quantum Computer, http://xxx.lanl.gov/abs/quant-ph/0008033

  7. GNU multiple precision arithmetic library 4.1.4. http://swox.com/gmp/

  8. Griffiths, R.B., Niu, C.-S.: Semiclassical Fourier Transform for Quantum Computation, http://xxx.lanl.gov/abs/quant-ph/9511007

  9. Hamdy, S.: Über die Sicherheit und Effizienz kryptografischer Verfahren mit Klassengruppen imaginär-quadratischer Zahlkörper. PhD thesis, Technische Universität Darmstadt, Fachbereich Informatik, Darmstadt, Germany (2002), http://www.cdc.informatik.tu-darmstadt.de/reports/reports/hamdy.diss.pdf

  10. Hïfner, H., Hïsel, W., Roos, C.F., Benhelm, J., Chek al kar, D., Chwalla, M., Kïber, T., Rapol, U.D., Riebe, M., Schmidt, P.O., Becher, C., Ghne, O., Dr, W., Blatt., R.: Scalable Multiparticle Entanglement of Trapped Ions p643. Nature 438, 643–646 (2005)

    Article  Google Scholar 

  11. Hamdy, S., Möller, B.: Security of Cryptosystems Based on Class Groups of Imaginary Quadratic Orders. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 234–247. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  12. Kitaev, A.: Quantum Measurements and the Abelian Stabilizer Problem. Electronic Colloquium on Computational Complexity (ECCC) 3(3) (1996)

    Google Scholar 

  13. Kunihiro, N.: Practical Running Time of Factoring by Quantum Circuits. In: ERATO Workshop on Quantum Information Science 2003 (EQIS 2003) (2003)

    Google Scholar 

  14. LiDIA — A C++ Library For Computational Number Theory, http://www.informatik.tu-darmstadt.de/TI/LiDIA/

  15. Lenstra, A.K., Verheul, E.R.: Selecting Cryptographic Key Sizes. Journal of Cryptology 14(4), 255–293 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  17. NTL – a library for doing number theory. version 5.4. http://shoup.net/ntl/

  18. Proos, J., Zalka, C.: Shor’s Discrete Logarithm Quantum Algorithm for Elliptic Curves, http://xxx.lanl.gov/abs/quant-ph/0301141

  19. Shor, P.W.: Algorithms for Quantum Computation: Discrete Logarithms and Factoring. In: IEEE Symposium on Foundations of Computer Science, pp. 124–134 (1994)

    Google Scholar 

  20. Vollmer, U.: Invariant and Discrete Logarithm Computation in Quadratic Orders. PhD thesis, Technische Universität Darmstadt, Fachbereich Informatik (2003), http://elib.tu-darmstadt.de/diss/000494/

  21. Vandersypen, L.M.K., Steffen, M., Breyta, G., Yannoni, C.S., Cleve, R., Chuang, I.L.: Experimental Realization of an Order-Finding Algorithm with an NMR Quantum Computer. Physical Review Letters 85, 5452–5455 (2000)

    Article  Google Scholar 

  22. Vandersypen, L.M.K., Steffen, M., Breyta, G., Yannoni, C.S., Sherwood, M.H., Chuang, I.L.: Experimental Realization of Shor’s Quantum Factoring Algorithm using Nuclear Magnetic Resonance. Nature 414, 883–887 (2001)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Informatik, Fachgebiet Kryptographie und Computeralgebra, Technische Universität Darmstadt, Hochschulstr. 10, 64289, Darmstadt, Germany

    Arthur Schmidt

Authors
  1. Arthur Schmidt
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik und Gesellschaft, Abt. Telematik, Albert-Ludwigs-Universität Freiburg, Friedrichstr. 50, 79098, Freiburg, Germany

    Günter Müller

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Schmidt, A. (2006). Quantum Algorithm for Solving the Discrete Logarithm Problem in the Class Group of an Imaginary Quadratic Field and Security Comparison of Current Cryptosystems at the Beginning of Quantum Computer Age. In: Müller, G. (eds) Emerging Trends in Information and Communication Security. ETRICS 2006. Lecture Notes in Computer Science, vol 3995. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11766155_34

Download citation

  • DOI: https://doi.org/10.1007/11766155_34

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-34640-1

  • Online ISBN: 978-3-540-34642-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation