Computation of a 30750-bit binary field discrete logarithm
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- by Robert Granger, Thorsten Kleinjung, Arjen K. Lenstra, Benjamin Wesolowski and Jens Zumbrägel HTML | PDF
- Math. Comp. 90 (2021), 2997-3022 Request permission
Abstract:
This paper reports on the computation of a discrete logarithm in the finite field $\mathbb {F}_{2^{30750}}$, breaking by a large margin the previous record, which was set in January 2014 by a computation in $\mathbb {F}_{2^{9234}}$. The present computation made essential use of the elimination step of the quasi-polynomial algorithm due to Granger, Kleinjung and Zumbrägel, and is the first large-scale experiment to truly test and successfully demonstrate its potential when applied recursively, which is when it leads to the stated complexity. It required the equivalent of about $2900$ core years on a single core of an Intel Xeon Ivy Bridge processor running at 2.6 GHz, which is comparable to the approximately $3100$ core years expended for the discrete logarithm record for prime fields, set in a field of bit-length $795$, and demonstrates just how much easier the problem is for this level of computational effort. In order to make the computation feasible we introduced several innovative techniques for the elimination of small degree irreducible elements, which meant that we avoided performing any costly Gröbner basis computations, in contrast to all previous records since early 2013. While such computations are crucial to the $L(\frac 1 4 + o(1))$ complexity algorithms, they were simply too slow for our purposes. Finally, this computation should serve as a serious deterrent to cryptographers who are still proposing to rely on the discrete logarithm security of such finite fields in applications, despite the existence of two quasi-polynomial algorithms and the prospect of even faster algorithms being developed.References
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Additional Information
- Robert Granger
- Affiliation: Surrey Centre for Cyber Security, Department of Computer Science, University of Surrey, United Kingdom
- MR Author ID: 744248
- Email: r.granger@surrey.ac.uk
- Thorsten Kleinjung
- Affiliation: Laboratory for Cryptologic Algorithms, School of Computer and Communication Sciences, EPFL, Switzerland
- MR Author ID: 704259
- Arjen K. Lenstra
- Affiliation: Laboratory for Cryptologic Algorithms, School of Computer and Communication Sciences, EPFL, Switzerland
- MR Author ID: 112545
- Benjamin Wesolowski
- Affiliation: Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence, France; and INRIA, IMB, UMR 5251, F-33400 Talence, France
- MR Author ID: 1163085
- ORCID: 0000-0003-1249-6077
- Email: benjamin.wesolowski@math.u-bordeaux.fr
- Jens Zumbrägel
- Affiliation: Faculty of Computer Science and Mathematics, University of Passau, Germany
- MR Author ID: 843678
- Email: jens.zumbraegel@uni-passau.de
- Received by editor(s): August 6, 2020
- Published electronically: July 16, 2021
- Additional Notes: The first and fourth listed authors were supported by the Swiss National Science Foundation via grant no. 200021-156420.
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 2997-3022
- MSC (2020): Primary 11Y16, 11T71
- DOI: https://doi.org/10.1090/mcom/3669
- MathSciNet review: 4305378