## CryptoDB

### Koh-ichi Nagao

#### Publications

Year
Venue
Title
2015
EPRINT
2015
EPRINT
2015
EPRINT
2007
EPRINT
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
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
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)