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

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 · c where l is a prime greater than about 2 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 = x + 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.