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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

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## Volume 1 (2009)

### Volume 8, Issue 2 (Nov 2016)

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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

## Abstract

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

MSC 2010: 68Q25; 68W40; 11Y16