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**2004/151**

**Title:**

Suitable Curves for Genus-4 HCC over Prime Fields: Point Counting Formulae for Hyperelliptic Curves of type y^2=x^{2k+1}+ax

**Authors:**
Mitsuhiro Haneda, Mitsuru Kawazoe, Tetsuya Takahashi

**Abstract:**

Computing the order of the Jacobian group of a hyperelliptic curve

over a finite field is very important to construct

a hyperelliptic curve cryptosystem (HCC), because

to construct secure HCC, we need Jacobian groups of order in the form

l(J\(Bcdot c where l is a prime greater than about 2^{160} and

c is a very small integer.

But even in the case of genus two,

known algorithms to compute the order of a Jacobian group for a general curve

need a very long running time over a large prime field.

In the case of genus three, only a few examples of suitable curves for HCC are known.

In the case of genus four, no example has been known over a large prime field.

In this article, we give explicit formulae of the order of Jacobian groups for

hyperelliptic curves over a finite prime field of type y^2=x^{2k+1}+a x,

which allows us to search suitable

curves for HCC. By using these formulae,

we can find many suitable curves for genus-4 HCC and show some examples.

**ePrint:**
https://eprint.iacr.org/2004/151

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