Algorithms for solving the isogeny problem with oriented elliptic curves

We introduce WayFinder, a framework for generalizing the Delfs-Galbraith and SuperSolver algorithms for the supersingular isogeny problem. Our framework extends the search for elliptic curves with an orientation by an order containing $\mathbb{Z}[\ell \sqrt{-p}]$ to more general orders, and we derive a cost model for such generalisations. Our cost model not only works in a more general context, but also provides more accurate predictions when applied to SuperSolver. We instantiate WayFinder for