#### Vol. 7, No. 6, 2013

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On the discrete logarithm problem in elliptic curves II

### Claus Diem

Vol. 7 (2013), No. 6, 1281–1323
##### Abstract

We continue our study on the elliptic curve discrete logarithm problem over finite extension fields. We show, among others, the following results:

For sequences of prime powers ${\left({q}_{i}\right)}_{i\in ℕ}$ and natural numbers ${\left({n}_{i}\right)}_{i\in ℕ}$ with ${n}_{i}\to \infty$ and ${n}_{i}∕log{\left({q}_{i}\right)}^{2}\to 0$ for $i\to \infty$, the discrete logarithm problem in the groups of rational points of elliptic curves over the fields ${\mathbb{F}}_{{q}_{i}^{{n}_{i}}}$ can be solved in subexponential expected time ${\left({q}_{i}^{{n}_{i}}\right)}^{o\left(1\right)}$.

Let $a$, $b>0$ be fixed. Then the problem over fields ${\mathbb{F}}_{{q}^{n}}$, where $q$ is a prime power and $n$ a natural number with $a\cdot log{\left(q\right)}^{1∕3}\le n\le b\cdot log\left(q\right)$, can be solved in an expected time of ${e}^{\mathsc{O}\left(log{\left({q}^{n}\right)}^{3∕4}\right)}$.

##### Keywords
elliptic curves, discrete logarithm problem
##### Mathematical Subject Classification 2010
Primary: 11Y16
Secondary: 14H52, 11G20
##### Milestones
Received: 28 July 2011
Revised: 12 June 2012
Accepted: 15 July 2012
Published: 19 September 2013
##### Authors
 Claus Diem Mathematical Institute University of Leipzig Augustusplatz 10 D-04109 Leipzig Germany