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We explore algorithmic aspects of a simply transitive commutative groupaction coming from the class field theory of imaginary hyperelliptic functionfields. Namely, the Jacobian of an imaginary hyperelliptic curve defined over$\mathbb F_q$ acts on a subset of isomorphism classes of Drinfeld modules. Wedescribe an algorithm to compute the group action efficiently. This is afunction field analog of the Couveignes-Rostovtsev-Stolbunov group action. Wereport on an explicit computation done with our proof-of-concept C++/NTLimplementation; it took a fraction of a second on a standard computer. We provethat the problem of inverting the group action reduces to the problem offinding isogenies of fixed $\tau$-degree between Drinfeld $\mathbbF_q[X]$-modules, which is solvable in polynomial time thanks to an algorithm byWesolowski. We give asymptotic complexity bounds for all algorithms presentedin this paper.

Computing a Group Action from the Class Field Theory of Imaginary Hyperelliptic Function Fields