Interpolating isogenies between elliptic curves: destructive and constructive applications

A degree-$d$ isogeny $\Phi : E \to E'$ between elliptic curves is always uniquely determined by the images of any $4d + 1$ points $P \in E$. In a series of recent(ish) works, this statement was made algorithmically effective: given any point $Q \in E$, we now understand how to efficiently compute its image $\Phi(Q)$ from such interpolation data (over finite fields, and assuming that the interpolation points generate a group of smooth order). We will explain this method, in which higher-dimensional abelian varieties play a surprising role. We will then discuss SIKE, a candidate for post-quantum key exchange that had advanced to round 4 of a standardization effort run by NIST, and show why it is naturally broken by isogeny interpolation. Finally, as time permits, we will discuss various constructive applications.