Welcome to the resource topic for
**2002/181**

**Title:**

Counting Points for Hyperelliptic Curves of type y^2=x^5+ax over Finite Prime Fields

**Authors:**
Eisaku Furukawa, Mitsuru Kawazoe, Tetsuya Takahashi

**Abstract:**

Counting rational points on Jacobian varieties of hyperelliptic curves

over finite fields is very important for constructing

hyperelliptic curve cryptosystems (HCC),

but known algorithms for general curves over given large prime

fields need very long running times.

In this article, we propose an extremely fast point counting algorithm for

hyperelliptic curves of type y^2=x^5+ax over given large

prime fields \Fp, e.g. 80-bit fields.

For these curves, we also determine the necessary condition

to be suitable for HCC, that is, to satisfy that the order

of the Jacobian group is of the form l\cdot c where l

is a prime number greater than about 2^{160} and

c is a very small integer.

We show some examples of suitable curves for HCC obtained by

using our algorithm.

We also treat curves of type y^2=x^5+a where a is not

square in \Fp.

**ePrint:**
https://eprint.iacr.org/2002/181

See all topics related to this paper.

Feel free to post resources that are related to this paper below.

**Example resources include:**
implementations, explanation materials, talks, slides, links to previous discussions on other websites.

For more information, see the rules for Resource Topics .