Welcome to the resource topic for 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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