Explicit connections between supersingular isogeny graphs and Bruhat–Tits trees

Laia Amoros Carafi; Annamaria Iezzi; Kristin Lauter; Chloe Martindale; Jana Sotáková

Explicit connections between supersingular isogeny graphs and Bruhat–Tits trees

Laia Amoros Carafi, Annamaria Iezzi, Kristin Lauter, Chloe Martindale, Jana Sotáková

Department of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Foreword / postscript › Scientific › peer-review

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 verticesand edges have a very simple representation as two-by-two matrices, which, as we show, is usefulfor understanding certain aspects of the corresponding elliptic curves and isogenies. MoreoverBruhat–Tits trees can be given an orientation and a notion of depth that we translate into thesetting of supersingular isogeny graphs. We give some suggestions towards using Bruhat–Titstrees as a tool for cryptanalysis of certain cryptosystems based on supersingular isogeny graphs.

Original language	English
Title of host publication	Proceedings of the Women in Numbers Europe 3 Conference
Editors	Alina Carmen Cojocaru, Sorina Ionica, Elisa Lorenzo Garcia
Number of pages	34
Publication status	Accepted/In press - 17 Sep 2020
MoE publication type	A3 Part of a book or another research book
Event	Women in Numbers Europe - La Hublais Center, Rennes, France Duration: 26 Aug 2019 → 30 Aug 2019 Conference number: 3 http://www.claymath.org/events/women-numbers-europe-3

Workshop

Workshop	Women in Numbers Europe
Country/Territory	France
City	Rennes
Period	26/08/2019 → 30/08/2019
Internet address	http://www.claymath.org/events/women-numbers-europe-3

Cite this

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title = "Explicit connections between supersingular isogeny graphs and Bruhat–Tits trees",

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 verticesand edges have a very simple representation as two-by-two matrices, which, as we show, is usefulfor understanding certain aspects of the corresponding elliptic curves and isogenies. MoreoverBruhat–Tits trees can be given an orientation and a notion of depth that we translate into thesetting of supersingular isogeny graphs. We give some suggestions towards using Bruhat–Titstrees as a tool for cryptanalysis of certain cryptosystems based on supersingular isogeny graphs.",

author = "{Amoros Carafi}, Laia and Annamaria Iezzi and Kristin Lauter and Chloe Martindale and Jana Sot{\'a}kov{\'a}",

year = "2020",

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language = "English",

editor = "Cojocaru, {Alina Carmen} and Sorina Ionica and {Lorenzo Garcia}, Elisa",

booktitle = "Proceedings of the Women in Numbers Europe 3 Conference",

note = "Women in Numbers Europe ; Conference date: 26-08-2019 Through 30-08-2019",

url = "http://www.claymath.org/events/women-numbers-europe-3",

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Explicit connections between supersingular isogeny graphs and Bruhat–Tits trees. / Amoros Carafi, Laia; Iezzi, Annamaria; Lauter, Kristin; Martindale, Chloe; Sotáková, Jana.

Proceedings of the Women in Numbers Europe 3 Conference. ed. / Alina Carmen Cojocaru; Sorina Ionica; Elisa Lorenzo Garcia. 2020.

Research output: Chapter in Book/Report/Conference proceeding › Foreword / postscript › Scientific › peer-review

TY - CHAP

T1 - Explicit connections between supersingular isogeny graphs and Bruhat–Tits trees

AU - Amoros Carafi, Laia

AU - Iezzi, Annamaria

AU - Lauter, Kristin

AU - Martindale, Chloe

AU - Sotáková, Jana

N1 - Conference code: 3

PY - 2020/9/17

Y1 - 2020/9/17

N2 - 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 verticesand edges have a very simple representation as two-by-two matrices, which, as we show, is usefulfor understanding certain aspects of the corresponding elliptic curves and isogenies. MoreoverBruhat–Tits trees can be given an orientation and a notion of depth that we translate into thesetting of supersingular isogeny graphs. We give some suggestions towards using Bruhat–Titstrees as a tool for cryptanalysis of certain cryptosystems based on supersingular isogeny graphs.

AB - 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 verticesand edges have a very simple representation as two-by-two matrices, which, as we show, is usefulfor understanding certain aspects of the corresponding elliptic curves and isogenies. MoreoverBruhat–Tits trees can be given an orientation and a notion of depth that we translate into thesetting of supersingular isogeny graphs. We give some suggestions towards using Bruhat–Titstrees as a tool for cryptanalysis of certain cryptosystems based on supersingular isogeny graphs.

M3 - Foreword / postscript

BT - Proceedings of the Women in Numbers Europe 3 Conference

A2 - Cojocaru, Alina Carmen

A2 - Ionica, Sorina

A2 - Lorenzo Garcia, Elisa

T2 - Women in Numbers Europe

Y2 - 26 August 2019 through 30 August 2019

ER -

Explicit connections between supersingular isogeny graphs and Bruhat–Tits trees

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