Explicit connections between supersingular isogeny graphs and Bruhat–Tits trees

Laia Amorós, Annamaria Iezzi, Kristin Lauter, Chloe Martindale, and Jana Sotáková

Abstract

We give an exposition of supersingular isogeny graphs, quaternion ideal graphs and Bruhat–Tits trees, and of their connections. Bruhat–Tits trees are combinatorial objects whose vertices and edges have a very simple representation as two-by-two matrices, which, as we show, is useful for understanding certain aspects of the corresponding elliptic curves and isogenies. Moreover Bruhat–Tits trees can be given an orientation and a notion of depth that we translate into the setting of supersingular isogeny graphs. We give some suggestions towards using Bruhat–Tits trees as a tool for cryptanalysis of certain cryptosystems based on supersingular isogeny graphs.

Available format(s)
Category
Public-key cryptography
Publication info
Published elsewhere. MINOR revision.Women in Numbers Europe III: Research Directions in Number Theory.
Keywords
isogenySIKESIDHquaternionsBruhat-Titselliptic curves
Contact author(s)
laia ac @ protonmail com
annamaria iezzi @ gmail com
kristinelauter @ gmail com
chloe martindale @ bristol ac uk
ja sotakova @ gmail com
History
Short URL
https://ia.cr/2021/372

CC BY

BibTeX

@misc{cryptoeprint:2021/372,
author = {Laia Amorós and Annamaria Iezzi and Kristin Lauter and Chloe Martindale and Jana Sotáková},
title = {Explicit connections between supersingular isogeny graphs and Bruhat–Tits trees},
howpublished = {Cryptology ePrint Archive, Paper 2021/372},
year = {2021},
note = {\url{https://eprint.iacr.org/2021/372}},
url = {https://eprint.iacr.org/2021/372}
}

Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.