## CryptoDB

### Koh-ichi Nagao

#### Publications

**Year**

**Venue**

**Title**

2015

EPRINT

2007

EPRINT

On the Decomposition of an Element of Jacobian of a Hyperelliptic Curve
Abstract

In this manuscript, if a reduced divisor $D_0$ of hyperelliptic curve of genus
$g$ over an extension field $F_{q^n}$ is written by a linear sum of $ng$
lements of $F_{q^n}$-rational points of the hyperelliptic curve whose
$x$-coordinates are in the base field $F_q$, $D_0$ is noted by a decomposed
divisor and the set of such $F_{q^n}$-rational points is noted by the
decomposed factor of $D_0$.
We propose an algorithm which checks whether a reduced divisor is decomposed
or not, and compute the decomposed factor, if it is decomposed. This
algorithm needs a process for solving equations system of degree $2$,
$(n^2-n)g$ variables, and $(n^2-n)g$ equations over $F_q$.
Further, for the cases $(g,n)=(1,3),(2,2),$ and $(3,2)$, the concrete
computations of decomposed factors are done by computer experiments.

2004

EPRINT

Improvement of Th?Leriault Algorithm of Index Calculus for Jacobian of Hyperelliptic Curves of Small Genus
Abstract

Gaudry present a variation of index calculus attack for solving the
DLP in the Jacobian of hyperelliptic curves. Harley and Th?Lerialut
improve these kind of algorithm. Here, we will present a variation of
these kind of algorithm, which is faster than previous ones.
Its complexity is $O(2-\frac{2}{g}+\epsilon)$.
Recently, P. Gaudry and E. Thom'e
http://eprint.iacr.org/2004/153/
present the algorithm, whose complexity is same as our results.
So I submit my manuscript to this eprint archive.

2004

EPRINT

A Weil Descent Attack against Elliptic Curve Cryptosystems over Quartic Extension Fields
Abstract

This paper shows that
many of elliptic curve cryptosystems over quartic extension fields of odd characteristics
are reduced to genus two hyperelliptic curve cryptosystems over quadratic extension fields.
Moreover, it shows that almost all of the genus two hyperelliptic curve cryptosystems over quadratic extension fields
of odd characteristics come under Weil descent attack.
This means that many of elliptic curve cryptosystems over quartic extension fields of odd characteristics
can be attacked by Weil descent uniformly.

#### Coauthors

- Seigo Arita (1)
- Kazuto Matsuo (1)
- Mahoro Shimura (1)