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 $\tau$-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.

翻译：我们探究了虚超椭圆函数域类域理论中一个简单传递交换群作用的算法方面。具体地，定义在$\mathbb F_q$上的虚超椭圆曲线的雅可比簇作用于Drinfeld模的同构类子集。我们描述了一种高效计算该群作用的算法。这是Couveignes-Rostovtsev-Stolbunov群作用的函数域类比。我们报告了使用概念验证型C++/NTL实现完成的一个显式计算结果——在标准计算机上仅需不到一秒。我们证明，该群作用的求逆问题可归约为寻找固定$\tau$-度的Drinfeld $\mathbb F_q[X]$-模同源问题，而借助Wesolowski的算法，该问题可在多项式时间内求解。我们给出了本文所有算法的渐近复杂度界。