One of the best arithmetics on elliptic curves involves Montgomery xz-coordinates, which are used in several cryptographic protocols such as ECDSA or ECDH. These coordinates also offer fast computations of 2- and 4-isogenies, used in several protocols like it was the case with SIDH, both for doubling and images of points, so improving isogeny formulas also improves scalar products on an elliptic curve. We realized there was a more general theory of Kummer lines under which xz-coordinates fall.

In this talk, we will describe the general framework of Kummer lines, based on two families of examples: Montgomery xz-coordinates and theta models. We will then explain how to find 2-isogeny formulas, whether they were already known or new, and how we mixed them to improve elliptic curve arithmetic.