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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board Member: Blackburn, Simon R. / Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Schäge, Sven / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura

2 Issues per year

SCImago Journal Rank (SJR) 2015: 1.208
Source Normalized Impact per Paper (SNIP) 2015: 2.294
Impact per Publication (IPP) 2015: 1.103

Mathematical Citation Quotient (MCQ) 2015: 0.48

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

Christian Schridde
  • Department of Mathematics and Computer Science, University of Marburg, Germany. Current address: Federal Office for Information Security, Bonn, Germany
  • :
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

Received: 2015-09-08

Published Online: 2016-10-11

Published in Print: 2016-11-01

Citation Information: Groups Complexity Cryptology. Volume 8, Issue 2, Pages 91–107, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2016-0009, October 2016

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