摘要翻译:
我们考虑了寻找密码学上合适的雅克比人的问题。通过应用概率泛型算法计算任意族低亏格曲线的zeta函数,我们可以搜索包含一个大的素数阶子群的Jacobian。对于一个合适的曲线分布,在亏格2中复杂度是次指数的,在亏格3中复杂度是O(n^{1/12})。给出了群阶大于180位的素域上的亏格2和亏格3超椭圆曲线的例子,改进了前人的结果。我们的方法在低次可拓域上特别有效,在亏格2中,我们在F_{p^2)上找到雅克比,在F_{p^3}上找到接近素数阶达372位的零簇,对于p=2_{61}-1,在PC上找到接近素数阶达244位的群的平均时间不到一个小时。
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英文标题:
《A Generic Approach to Searching for Jacobians》
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作者:
Andrew V. Sutherland
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最新提交年份:
2008
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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:
We consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing a large subgroup of prime order. For a suitable distribution of curves, the complexity is subexponential in genus 2, and O(N^{1/12}) in genus 3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size, improving previous results. Our approach is particularly effective over low-degree extension fields, where in genus 2 we find Jacobians over F_{p^2) and trace zero varieties over F_{p^3} with near-prime orders up to 372 bits in size. For p = 2^{61}-1, the average time to find a group with 244-bit near-prime order is under an hour on a PC.
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PDF链接:
https://arxiv.org/pdf/0708.3168