Formulae of the order of Jacobians for certain hyperelliptic curves (Algebraic Aspects of Coding Theory and Cryptography)

This article is the summary of our work on the order of some hyperelliptic Jacobian groups.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 $\mathrm{H}\mathrm{C}\mathrm{C}$ , we need Jacobian groups of order in the form $l\cdot$ $c$ where $l$ is aprime 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 $\mathrm{a}$ 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, we do not know any example over $\mathrm{a}$ large prime field.In this note, we give explicit formulae of the order of Jacobian groups for certain hyperelliptic curves of genus three and four, which allows us to search suitable curves for $\mathrm{H}\mathrm{C}\mathrm{C}$ of genus greater than two.By using these formulae, we can find many suitable curves for $\mathrm{H}\mathrm{C}\mathrm{C}$ of genus four.In this article, we have contained the results for the case genus greater than two, which are obtained after the conference.