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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


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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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