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

03/14/2022

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by   Antoine Leudière, et al.

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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 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 𝔽_q[X]-modules, which is solvable in polynomial time thanks to an algorithm by Wesolowski.