## Cryptology ePrint Archive: Report 2008/398

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.

Short URL: ia.cr/2008/398

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