On the discrete logarithm problem in elliptic curves II

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 ${\left(q_i\right)}_{i\in \mathbb{N}}$ and natural numbers ${\left(n_i\right)}_{i\in \mathbb{N}}$ 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 $e^{\mathcal{O}\left(log{\left(q^n\right)}^{3∕4}\right)}$.