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Person# Robert Granger

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Discrete logarithm

In mathematics, for given real numbers a and b, the logarithm logb a is a number x such that bx = a. Analogously, in any group G, powers bk can be defined for all i

Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the op

Finite field arithmetic

In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) contrary to arithmetic in a field with an infinite number of elements, like th

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Robert Granger, Thorsten Kleinjung, Arjen Lenstra, Benjamin Pierre Charles Wesolowski

This paper reports on the computation of a discrete logarithm in the finite field F-230750, breaking by a large margin the previous record, which was set in January 2014 by a computation in F-29234. The present computation made essential use of the elimination step of the quasi-polynomial algorithm due to Granger, Kleinjung and Zumbragel, 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 Grobner basis computations, in contrast to all previous records since early 2013. While such computations are crucial to the L(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.

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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.

2018