We prove that isogenies between Drinfeld F[x]-modules over a finite field can be computed in polynomial time. This breaks Drinfeld analogs of isogeny-based cryptosystems.

In this work we study algebraic and generic models for group actions, and extend them to the universal composability (UC) framework of Canetti (FOCS 2001). We revisit the constructions of Duman et al. (PKC 2023) integrating the type-safe model by Zhandry (Crypto 2022), adapted to the group action setting, and formally define an algebraic action model (AAM). This model restricts the power of the adversary in a similar fashion to the algebraic group model (AGM). By imposing algebraic behaviour to the adversary and environment of the UC framework, we construct the UC-AAM. Finally, we instantiate UC-AAM with isogeny-based assumptions, in particular the CSIDH action with twists, obtaining the explicit isogeny model, UC-EI; we observe that, under certain assumptions, this model is "closer" to standard UC than the UC-AGM, even though there still exists an important separation. We demonstrate the utility of our definitions by proving UC-EI security for the passive-secure oblivious transfer protocol described by Lai et al. (Eurocrypt 2021), hence providing the first concretely efficient two-message isogeny-based OT protocol in the random oracle model against malicious adversaries.

We use theta groups to study $2$-isogenies between Kummer lines, with a particular focus on the Montgomery model. This allows us to recover known formulas, along with more efficient forms for translated isogenies, which require only $2S+2m_0$ for evaluation. We leverage these translated isogenies to build a hybrid ladder for scalar multiplication on Montgomery curves with rational $2$-torsion, which cost $3M+6S+2m_0$ per bit, compared to $5M+4S+1m_0$ for the standard Montgomery ladder.