**Generating genus two hyperelliptic curves over large characteristic finite fields**

*Takakazu Satoh*

**Abstract: **In hyperelliptic curve cryptography, finding a suitable
hyperelliptic curve is an important fundamental problem.
One of necessary conditions is that the order of its Jacobian
is a product of a large prime number and a small number.
In the paper, we give a probabilistic polynomial
time algorithm to test whether the Jacobian of the given hyperelliptic curve of the form $Y sup 2 = X sup 5 + u X sup 3 + v X$
satisfies the condition and, if so, gives the largest prime factor.
Our algorithm enables us to generate random curves of the form
until the order of its Jacobian is almost prime in the above sense.
A key idea is to obtain candidates of its zeta function over the base field from
its zeta function over the extension field where the Jacobian splits.

**Category / Keywords: **public-key cryptography / hyperelliptic curve, point counting

**Date: **received 19 Sep 2008, last revised 31 Oct 2008

**Contact author: **satohaar at mathpc-satoh math titech ac jp

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Note: **Results of numerical experiments with cryptographic size parameters are added. Some new references are added.

**Version: **20081031:112242 (All versions of this report)

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