International Association for Cryptologic Research (IACR)
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 2160 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 y2=x2k+1+ax,
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