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Indiscreet logarithms in finite fields of small characteristic

The first author is supported by the Swiss National Science Foundation via grant number 200021-156420

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  • Recently, several striking advances have taken place regarding the discrete logarithm problem (DLP) in finite fields of small characteristic, despite progress having remained essentially static for nearly thirty years, with the best known algorithms being of subexponential complexity. In this expository article we describe the key insights and constructions which culminated in two independent quasi-polynomial algorithms. To put these developments into both a historical and a mathematical context, as well as to provide a comparison with the cases of so-called large and medium characteristic fields, we give an overview of the state-of-the-art algorithms for computing discrete logarithms in all finite fields. Our presentation aims to guide the reader through the algorithms and their complexity analyses ab initio.

    Mathematics Subject Classification: Primary: 11Y16, 11T71.


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  • Table 1.  Discrete logarithm record computations in finite fields of small or medium characteristic. Details, as well as further announcements, can be found in the number theory mailing list (https://listserv.nodak.edu/cgi-bin/wa.exe?A0=NMBRTHRY)

    bitlengthcharact.Kummerwho/whenrunning time
    1272noCoppersmith 1984 [16] $L(1/3\, , \, 1.526..1.587)$
    4012noGordon, McCurley 1992 [26] $L(1/3\, , \, 1.526..1.587)$
    5212noJoux, Lercier 2001 [36] $L(1/3\, , \, 1.526)$
    6072noThomé 2002 $L(1/3\, , \, 1.526..1.587)$
    6132noJoux, Lercier 2005 $L(1/3\, , \, 1.526)$
    556mediumyesJoux, Lercier 2006 [38] $L(1/3\, , \, 1.442)$
    6763noHayashi et al. 2010 [31] $L(1/3\, , \, 1.442)$
    9233noHayashi et al. 2012 [30] $L(1/3\, , \, 1.442)$
    1175mediumyesJoux 24 Dec 2012 [34] $L(1/3\, , \, 1.260)$
    1425mediumyesJoux 6 Jan 2013 [34] $L(1/3\, , \, 1.260)$
    17782yesJoux 11 Feb 2013 [35] $L(1/4 + o(1))$
    19712yesGGMZ 19 Feb 2013 [23] $L(1/3\, , \, 0.763)$
    40802yesJoux 22 Mar 2013 [35] $L(1/4 + o(1))$
    8092noCARAMEL 6 Apr 2013 [7] $L(1/3\, , \, 1.526)$
    61202yesGGMZ 11 Apr 2013 [24] $L(1/4)$
    61682yesJoux 21 May 2013 $L(1/4 + o(1))$
    13033noAMOR 27 Jan 2014 [2] $L(1/4 + o(1))$
    44042noGKZ 30 Jan 2014 [27] $L(1/4 + o(1))$
    92342yesGKZ 31 Jan 2014 $L(1/4 + o(1))$
    15513noAMOR 26 Feb 2014 [2] $L(1/4 + o(1))$
    37963noJoux, Pierrot 15 Sep 2014 [41] $L(0 + o(1))$
    12792noKleinjung 17 Oct 2014 $L(0 + o(1))$
    48413noACCMORR, 18 Jul 2016 [3] $L(0 + o(1))$
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