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

  • Arthur Schmidt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3995)


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.


Elliptic Curve Quantum Computer Elliptic Curf Quantum Algorithm Discrete Logarithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [Bea]
    Beauregard, S.: Circuit for Shor’s algorithm using 2n+3 qubits,
  2. [Ben77]
    Bennett, C.H.: Logical Reversibility of Computation. IBM Journal of Research and Development, 525–532 (November 1977)Google Scholar
  3. [BW88]
    Buchmann, J., Williams, H.C.: A Key-Exchange System Based on Imaginary Quadratic Fields. Journal of Cryptology 1(2), 107–118 (1988)MathSciNetCrossRefMATHGoogle Scholar
  4. [BW90]
    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)CrossRefGoogle Scholar
  5. [CVZ+98]
    Chuang, I., Vandersypen, L., Zhou, X., Leung, D., Lloyd, S.: Experimental Realization of a Quantum Algorithm. Nature 393, 143–146 (1998)CrossRefGoogle Scholar
  6. [Dra00]
    Draper, T.G.: Addition on a Quantum Computer,
  7. [GMP]
    GNU multiple precision arithmetic library 4.1.4.
  8. [GN]
    Griffiths, R.B., Niu, C.-S.: Semiclassical Fourier Transform for Quantum Computation,
  9. [Ham02]
    Hamdy, S.: Über die Sicherheit und Effizienz kryptografischer Verfahren mit Klassengruppen imaginär-quadratischer Zahlkörper. PhD thesis, Technische Universität Darmstadt, Fachbereich Informatik, Darmstadt, Germany (2002),
  10. [HHR+05]
    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)CrossRefGoogle Scholar
  11. [HM00]
    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)CrossRefGoogle Scholar
  12. [Kit96]
    Kitaev, A.: Quantum Measurements and the Abelian Stabilizer Problem. Electronic Colloquium on Computational Complexity (ECCC) 3(3) (1996)Google Scholar
  13. [Kun]
    Kunihiro, N.: Practical Running Time of Factoring by Quantum Circuits. In: ERATO Workshop on Quantum Information Science 2003 (EQIS 2003) (2003)Google Scholar
  14. [LiD]
    LiDIA — A C++ Library For Computational Number Theory,
  15. [LV01]
    Lenstra, A.K., Verheul, E.R.: Selecting Cryptographic Key Sizes. Journal of Cryptology 14(4), 255–293 (2001)MathSciNetCrossRefMATHGoogle Scholar
  16. [NC00]
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  17. [NTL]
    NTL – a library for doing number theory. version 5.4.
  18. [PZ]
    Proos, J., Zalka, C.: Shor’s Discrete Logarithm Quantum Algorithm for Elliptic Curves,
  19. [Sho94]
    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. [Vol03]
    Vollmer, U.: Invariant and Discrete Logarithm Computation in Quadratic Orders. PhD thesis, Technische Universität Darmstadt, Fachbereich Informatik (2003),
  21. [VSB+00]
    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)CrossRefGoogle Scholar
  22. [VSB+01]
    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)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Arthur Schmidt
    • 1
  1. 1.Fachbereich Informatik, Fachgebiet Kryptographie und ComputeralgebraTechnische Universität DarmstadtDarmstadtGermany

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