Bruhat–Tits trees as a tool for exploring supersingular isogeny graphs
Linfoot Number Theory Seminar
15th December 2020, 4:00 pm – 5:00 pm
Virtual Seminar, https://zoom.us/j/94769949859
Supersingular isogeny graphs are the main characters of a young subfield of post-quantum cryptography called isogeny-based cryptography. Thanks to Deuring's correspondence, supersingular isogeny graphs are normally studied from the algebraic point of view of quaternion algebras. In this talk we propose that we take one step further, from quaternion algebras to Bruhat–Tits trees. In particular we show how these infinite trees whose vertices and edges have a very simple representation as two-by-two matrices can be given an orientation and a notion of depth that we translate into the setting of supersingular isogeny graphs. Bruhat–Tits trees may then represent a useful tool for getting additional information on the structure of supersingular isogeny graphs which underlies the security of isogeny-based cryptosystems such as SIKE.
This is a joint work with Laia Amorós, Kristin Lauter, Chloe Martindale and Jana Sotáková.