On the discrete logarithm problem in finite fields of fixed characteristic

Granger, Robert; Kleinjung, Thorsten; Zumbrägel, Jens

doi:10.1090/tran/7027

On the discrete logarithm problem in finite fields of fixed characteristic
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by Robert Granger, Thorsten Kleinjung and Jens Zumbrägel PDF

Trans. Amer. Math. Soc. 370 (2018), 3129-3145 Request permission

Abstract:

For $q$ a prime power, the discrete logarithm problem (DLP) in $\mathbb {F}_{q}$ consists of finding, for any $g \in \mathbb {F}_{q}^{\times }$ and $h \in \langle g \rangle$, an integer $x$ such that $g^x = h$. We present an algorithm for computing discrete logarithms with which we prove that for each prime $p$ there exist infinitely many explicit extension fields $\mathbb {F}_{p^n}$ in which the DLP can be solved in expected quasi-polynomial time. Furthermore, subject to a conjecture on the existence of irreducible polynomials of a certain form, the algorithm solves the DLP in all extensions $\mathbb {F}_{p^n}$ in expected quasi-polynomial time.

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