论文标题
计算假想过性函数场的类字段理论的小组动作
Computing a Group Action from the Class Field Theory of Imaginary Hyperelliptic Function Fields
论文作者
论文摘要
我们探索了来自假想的高纤维化功能场的阶级字段理论的简单传递群体动作的算法方面。也就是说，在$ \ mathbb f_q $上定义的假想高纤维曲线的雅各比式作用于德林菲尔德模块的同构类别的子集。我们描述了一种有效地计算小组动作的算法。这是COUVEIGNES-ROSTOSTOVTSEV-StolBunov组动作的函数字段类似物。我们报告了通过概念验证C ++/NTL实现进行的明确计算；在标准计算机上花了一秒钟的时间。我们证明，反转小组动作的问题减少了drinfeld $ \ mathbb f_q [x] $ - 模块之间找到固定$τ$ - 数量的固定$τ$ - 数量的问题，这要归功于Wesolowski的算法在多项式时间内解决。我们给出了本文介绍的所有算法的渐近复杂性界限。
We explore algorithmic aspects of a simply transitive commutative group action coming from the class field theory of imaginary hyperelliptic function fields. Namely, the Jacobian of an imaginary hyperelliptic curve defined over $\mathbb F_q$ acts on a subset of isomorphism classes of Drinfeld modules. We describe an algorithm to compute the group action efficiently. This is a function field analog of the Couveignes-Rostovtsev-Stolbunov group action. We report on an explicit computation done with our proof-of-concept C++/NTL implementation; it took a fraction of a second on a standard computer. We prove that the problem of inverting the group action reduces to the problem of finding isogenies of fixed $τ$-degree between Drinfeld $\mathbb F_q[X]$-modules, which is solvable in polynomial time thanks to an algorithm by Wesolowski. We give asymptotic complexity bounds for all algorithms presented in this paper.
