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The isogeny-based cryptographic protocol CSIDH is based on the group action of the class group of a particular imaginary quadratic order on certain elliptic curves. The efficiency and security of CSIDH has inspired variant protocols, such as SCALLOP and OSIDH. Such protocols are based on the vectorization problem: Given elliptic curves E, E', find the ideal class whose action on E yields E'. In this work, we provide a general understanding of this question in the context of orientations and level structure on elliptic curves.
In particular, we study a large family of generalized class groups of imaginary quadratic orders O and prove that they act freely and (essentially) transitively on the set of primitively O-oriented elliptic curves over a field k (assuming this set is non-empty) equipped with appropriate level structure. This is joint work with Wouter Castryck, Jonathan Komada Eriksen, Gioella Lorenzon, and Frederik Vercauteren.
Generalized class group actions on oriented elliptic curves with level structure
Generalized class group actions on oriented elliptic curves with level structure