Hard Homogeneous Spaces from the Class Field Theory of Imaginary Hyperelliptic Function Fields

We explore algorithmic aspects of a free and 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. Our proof-of-concept C++/NTL implementation only requires a fraction of a second on a standard computer. Also, we state a conjecture - supported by experiments - which implies that the current fastest algorithm to solve its inverse problem runs in exponential time. This action is therefore a promising candidate for the construction of \emph{Hard Homogeneous Spaces}, which are the building blocks of several post-quantum cryptographic protocols. This demonstrates the relevance of using imaginary hyperelliptic curves and Drinfeld modules as an alternative to the standard setting of imaginary quadratic number fields and elliptic curves for isogeny-based cryptographic applications. Moreover, our function field setting enables the use of Kedlaya's algorithm and its variants for computing the order of the group in polynomial time when $q$ is fixed. No such polynomial-time algorithm for imaginary quadratic number fields is known. For $q=2$ and parameters similar to CSIDH-512, we compute this order more than 8500 times faster than the record computation for CSIDH-512 by Beullens, Kleinjung and Vercauteren.