Counting Points for Hyperelliptic Curves of Type y2= x5 + ax over Finite Prime Fields
Counting Points for Hyperelliptic Curves of Type y2= x5 + ax over Finite Prime Fields
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DOI：
10.1007/978-3-540-24654-1_3
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发表时间：
2003-08
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影响因子：
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通讯作者：
E. Furukawa;M. Kawazoe;Tetsuya Takahashi
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文献类型：
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作者：
E. Furukawa;M. Kawazoe;Tetsuya Takahashi
计算有限域上超椭圆曲线的Jacobian簇上的有理点对于构造超椭圆曲线密码系统（HCC）是非常重要的，但已知的计算大素域上一般曲线的算法需要很长的运行时间.在这篇文章中，我们提出了一个非常快速的点计数算法的类型为y2 =x5+axover给定的大素域，如80位域的超椭圆曲线。对于这些曲线，我们还确定了适用于HCC的必要条件，即满足Jacobian群的阶为l·c，其中l是大于约2160的素数，且是一个很小的整数。我们展示了一些例子，通过使用我们的算法获得的HCC的合适的曲线。本文还讨论了y ~ 2 = x ~ 5 + a型曲线，其中a在中不是平方的。
Counting rational points on Jacobian varieties of hyperelliptic curves over finite fields is very important for constructing hyperelliptic curve cryptosystems (HCC), but known algorithms for general curves over given large prime fields need very long running time. In this article, we propose an extremely fast point counting algorithm for hyperelliptic curves of typey2=x5+axover given large prime fields, e.g. 80-bit fields. For these curves, we also determine the necessary condition to be suitable for HCC, that is, to satisfy that the order of the Jacobian group is of the forml·cwherelis a prime number greater than about 2160andcis a very small integer. We show some examples of suitable curves for HCC obtained by using our algorithm. We also treat curves of typey2=x5+awhereais not square in.