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摘要翻译：
设k是奇特征的有限域。在Koblitz模型的基础上，给出了k上g属的点超椭圆曲线和非点超椭圆曲线的k-同构类个数的一个封闭公式。这些数字表示为k的基数q中的多项式，具有整数系数（对于有尖曲线）和有理系数（对于非有尖曲线）。系数依赖于g和q-1和q+1的因子集。这些公式证明了g属的超椭圆曲线在原理上适用于密码学应用的个数是渐近的(1-e^{-1}）2q^{2g-1},而不是我们所认为的2q^{2g-1}。G=2和G=3属的曲线更能抵抗DLP的攻击；对于这些g值，曲线数分别为(91/72)q^3+O(q^2)和(3641/2880)q^5+O(q^4)。
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英文标题：
《Counting hyperelliptic curves that admit a Koblitz model》
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作者：
Cevahir Demirkiran and Enric Nart
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最新提交年份：
2007
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分类信息：
一级分类：Mathematics 数学
二级分类：Number Theory 数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
Let k be a finite field of odd characteristic. We find a closed formula for the number of k-isomorphism classes of pointed, and non-pointed, hyperelliptic curves of genus g over k, admitting a Koblitz model. These numbers are expressed as a polynomial in the cardinality q of k, with integer coefficients (for pointed curves) and rational coefficients (for non-pointed curves). The coefficients depend on g and the set of divisors of q-1 and q+1. These formulas show that the number of hyperelliptic curves of genus g suitable (in principle) of cryptographic applications is asymptotically (1-e^{-1})2q^{2g-1}, and not 2q^{2g-1} as it was believed. The curves of genus g=2 and g=3 are more resistant to the attacks to the DLP; for these values of g the number of curves is respectively (91/72)q^3+O(q^2) and (3641/2880)q^5+O(q^4).
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PDF链接：
https://arxiv.org/pdf/0705.1423
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