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Groups Complexity Cryptology

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Computing discrete logarithms using 𝒪((log q)2) operations from {+,-,×,÷,&}

Christian Schridde
  • Corresponding author
  • Department of Mathematics and Computer Science, University of Marburg, Germany. Current address: Federal Office for Information Security, Bonn, Germany
  • Email:
Published Online: 2016-10-11 | DOI: https://doi.org/10.1515/gcc-2016-0009


Given a computational model with registers of unlimited size that is equipped with the set {+,-,×,÷,&}=:𝖮𝖯 of unit cost operations, and given a safe prime number q, we present the first explicit algorithm that computes discrete logarithms in q* to a base g using only 𝒪((logq)2) operations from 𝖮𝖯. For a random n-bit prime number q, the algorithm is successful as long as the subgroup of q* generated by g and the subgroup generated by the element p=2log2(q) share a subgroup of size at least 2(1-𝒪(logn/n))n.

Keywords: Cryptography; discrete logarithm problem; Fermat quotient; cyclicintegers

MSC 2010: 68Q25; 68W40; 11Y16


  • [1]

    Allender E., Bürgisser P., Kjeldgaard-Pedersen J. and Miltersen P. B., On the complexity of numerical analysis, Proceeding of the 21st Annual IEEE Conference on Computational Complexity (CCC 2006), IEEE Press, Piscataway (2006), 331–339.

  • [2]

    Baran I., Demaine E. D. and Pǎtraşcu M., Subquadratic algorithms for 3SUM, Algorithms and Data Structures, Lecture Notes in Comput. Sci. 3608, Springer, Berlin (2005), 409–421.

  • [3]

    Du X., Klapper A. and Chen Z., Linear complexity of pseudorandom sequences generated by Fermat quotients and their generalizations, Inf. Process. Lett. 112 (2012), no. 6, 233–237.

  • [4]

    Goresky M., Klapper A., Murty R. and Shparlinski I. E., On decimations of l-sequences, SIAM J. Discrete Math. 18 (2004), no. 1, 130–140.

  • [5]

    van Lint J. H., Introduction to Coding Theory, 3rd ed., Springer, Berlin, 1999.

  • [6]

    Lürwer-Brüggemeier K. and Ziegler M., On faster integer calculations using non-arithmetic primitives, Proceedings of the 7th International Conference on Unconventional Computation (UC’08), Lecture Notes in Comput. Sci. 5204, Springer, Berlin (2008), 111–128.

  • [7]

    Pomerance C. and Shparlinski I., Smooth orders and cryptographic applications, ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory, University of Sydney, Sydney (2002), 338–348.

  • [8]

    Pratt V. R., Rabin M. O. and Stockmeyer L. J., A characterization of the power of vector machines, STOC – Sixth Annual ACM Symposium on Theory of Computing, ACM, San Diego (1974), 122–134.

  • [9]

    Shamir A., Factoring numbers in 𝒪(log(n)) arithmetic steps, Inf. Process. Lett. 8 (1979), no. 1, 28–31.

About the article

Received: 2015-09-08

Published Online: 2016-10-11

Published in Print: 2016-11-01

Citation Information: Groups Complexity Cryptology, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2016-0009. Export Citation

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