Generating Genus Two Hyperelliptic Curves over Large Characteristic Finite Fields

Authors:
Takakazu Satoh
Affiliations:
Department of Mathematics, Tokyo Institute of Technology, Tokyo, Japan 152-8551
Venue:
EUROCRYPT '09 Proceedings of the 28th Annual International Conference on Advances in Cryptology: the Theory and Applications of Cryptographic Techniques
Year:
2009

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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 2 = X 5 + u X 3 + v X satisfies the condition and, if so, to give 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.