Suitable Curves for Genus-4 HCC over Prime Fields: Point Counting Formulae for Hyperelliptic Curves of Type y2=x2k+1+ax
Suitable Curves for Genus-4 HCC over Prime Fields: Point Counting Formulae for Hyperelliptic Curves of Type y2=x2k+1+ax
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DOI：
10.1007/11523468_44
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发表时间：
2005-07
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通讯作者：
Mitsuhiro Haneda;M. Kawazoe;Tetsuya Takahashi
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文献类型：
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作者：
Mitsuhiro Haneda;M. Kawazoe;Tetsuya Takahashi
计算有限域上超椭圆曲线的雅可比群的阶对于构造超椭圆曲线密码系统（HCC）是非常重要的，因为要构造安全的HCC，我们需要lc阶的雅可比群，其中l是大于2160的素数，c是非常小的整数.但是，即使在亏格2的情况下，已知的算法来计算一般曲线的雅可比群的阶需要在一个大的素域上非常长的运行时间。本文给出了y2 = x^\rm2\itk +1+ ax型有限素域上超椭圆曲线的Jacobian群阶的显式公式，从而为寻找适合于HCC的曲线提供了条件.利用这些公式，我们可以找到许多适合于亏格4 HCC的曲线，并给出了一些例子。
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 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 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 ^\rm2\itk+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.