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We show how the Weil pairing can be used to evaluate the assigned charactersof an imaginary quadratic order $\mathcal{O}$ in an unknown ideal class$[\mathfrak{a}] \in \mathrm{Cl}(\mathcal{O})$ that connects two given$\mathcal{O}$-oriented elliptic curves $(E, \iota)$ and $(E', \iota') =[\mathfrak{a}](E, \iota)$. When specialized to ordinary elliptic curves overfinite fields, our method is conceptually simpler and often somewhat fasterthan a recent approach due to Castryck, Sot\'akov\'a and Vercauteren, who relyon the Tate pairing instead. The main implication of our work is that it breaksthe decisional Diffie-Hellman problem for practically all oriented ellipticcurves that are acted upon by an even-order class group. It can also be used tobetter handle the worst cases in Wesolowski's recent reduction from thevectorization problem for oriented elliptic curves to the endomorphism ringproblem, leading to a method that always works in sub-exponential time.

On the decisional Diffie-Hellman problem for class group actions on oriented elliptic curves