The Isogeny Club: #2.1 Orientations and Supersingular Isogeny Graphs by Sarah Arpin

This is the first talk of the second season of The Isogeny Club, given by Sarah Arpin.

To study supersingular isogeny graphs, one may add to the elliptic curves the information of an orientation, or a particular embedding of an imaginary quadratic field into the endomorphism ring of the curve. Recent cryptographic protocols (Séta, OSIDH) have made use of orientations to define new hard problems on supersingular isogeny graphs. The mathematics of orientations have been studied for a long time, but the algorithmic implications are just now being understood.

As part of a recent Women in Numbers 5 (WIN5) collaboration, Sarah and her collaborators use orientations towards two different goals: 1. path-finding algorithms in the supersingular ℓ-isogeny graph and 2. understanding and counting cycles in the supersingular ℓ-isogeny graph. In this talk, she first introduce the theory of orientations and discuss the relevant hard problems. She then goes on to describe the path-finding algorithms and the theory behind cycle-counting which stem from adding orientations to supersingular elliptic curves.