**Complexity of ECDLP under the First Fall Degree Assumption**

*Koh-ichi Nagao*

**Abstract: **Semaev shows that under the first fall degree assumption, the complexity
of ECDLP over $\bF_{2^n}$, where $n$ is the input size, is
$O(2^{n^{1/2+o(1)}})$.
In his manuscript, the cost for solving equations system is $O((nm)^{4w})$,
where $m$ ($2 \le m \le n$) is the number of decomposition
and $w \sim 2.7$ is the linear algebra constant.
It is remarkable that the cost for solving equations system under the
first fall degree assumption, is poly in input size $n$.
He uses normal factor base and the revalance of "Probability that
the decomposition success" and "size of factor base" is done.
%So that the result is induced.
Here, using disjoint factor base to his method,
"Probability that the decomposition success becomes $ \sim 1$ and
taking the very small size factor
base is useful for complexity point of view.
Thus we have the result that states \\
"Under the first fall degree assumption,
the cost of ECDLP over $\bF_{2^n}$, where $n$ is the input size, is $O(n^{8w+1})$."
Moreover, using the authors results,
in the case of the field characteristic $\ge 3$, the first fall
degree of desired equation system is estimated by $\le 3p+1$.
(In $p=2$ case, Semaev shows it is $\le 4$. But it is exceptional.)
So we have similar result that states \\
"Under the first fall degree assumption,
the cost of ECDLP over $\bF_{p^n}$, where $n$ is the input size and (small) $p$ is a constant, is $O(n^{(6p+2)w+1})$.

**Category / Keywords: **foundations / ECDLP, First fall degree assumption, polynomial time algorithm

**Date: **received 11 Oct 2015

**Contact author: **nagao at kanto-gakuin ac jp

**Available format(s): **PDF | BibTeX Citation

**Version: **20151012:211704 (All versions of this report)

**Short URL: **ia.cr/2015/984

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