Suitable curves for genus-4 HCC over prime fields: point counting formulae for hyperelliptic curves of type y2 = x2k+1 + ax

Authors:
Mitsuhiro Haneda;Mitsuru Kawazoe;Tetsuya Takahashi
Affiliations:
Sharp Corporation;Faculty of Liberal Arts and Sciences, Osaka Prefecture University, Osaka, Japan;Faculty of Liberal Arts and Sciences, Osaka Prefecture University, Osaka, Japan
Venue:
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Year:
2005

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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 lc 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 this article, we give explicit formulae of the order of Jacobian groups for hyperelliptic curves over a finite prime field of type y2=x$^{\rm 2{\it k}+1}$+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.