Explicit Connections Between Supersingular Isogeny Graphs and Bruhat–Tits Trees

Laia Amoros; Annamaria Iezzi; Kristin Lauter; Chloe Martindale; Jana Sotakova

doi:10.1007/978-3-030-77700-5_2

Explicit Connections Between Supersingular Isogeny Graphs and Bruhat–Tits Trees

Laia Amoros, Annamaria Iezzi, Kristin Lauter, Chloe Martindale^*, Jana Sotakova

^*Corresponding author for this work

School of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference Contribution (Conference Proceeding)

1 Citation (Scopus)

Abstract

In this paper 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.

Original language	English
Title of host publication	Women in Numbers Europe III
Publisher	Springer, Cham
Pages	39-73
Number of pages	35
ISBN (Electronic)	9783030777005
ISBN (Print)	9783030776992, 9783030777029
DOIs	https://doi.org/10.1007/978-3-030-77700-5_2
Publication status	Published - 26 May 2021
Event	Women in Numbers Europe 3 - Rennes, France Duration: 26 Aug 2019 → 30 Aug 2019 https://www.dpmms.cam.ac.uk/~hk439/wine3.html

Publication series

Name	Association for Women in Mathematics Series (AWMS)
Publisher	Springer, Cham
Volume	24
ISSN (Print)	2364-5733
ISSN (Electronic)	2364-5741

Conference

Conference	Women in Numbers Europe 3
Abbreviated title	WIN-E3
Country/Territory	France
City	Rennes
Period	26/08/19 → 30/08/19
Internet address	https://www.dpmms.cam.ac.uk/~hk439/wine3.html

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@inproceedings{d52f40dbd08545bf95df8bfaca7bf5e4,

title = "Explicit Connections Between Supersingular Isogeny Graphs and Bruhat–Tits Trees",

abstract = "In this paper 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.",

author = "Laia Amoros and Annamaria Iezzi and Kristin Lauter and Chloe Martindale and Jana Sotakova",

year = "2021",

month = may,

day = "26",

doi = "10.1007/978-3-030-77700-5\_2",

language = "English",

isbn = "9783030776992",

series = "Association for Women in Mathematics Series (AWMS)",

publisher = "Springer, Cham",

pages = "39--73",

booktitle = "Women in Numbers Europe III",

address = "Switzerland",

note = "Women in Numbers Europe 3, WIN-E3 ; Conference date: 26-08-2019 Through 30-08-2019",

url = "https://www.dpmms.cam.ac.uk/\textasciitilde{}hk439/wine3.html",

}

Explicit Connections Between Supersingular Isogeny Graphs and Bruhat–Tits Trees. / Amoros, Laia; Iezzi, Annamaria; Lauter, Kristin et al.
Women in Numbers Europe III. Springer, Cham, 2021. p. 39-73 (Association for Women in Mathematics Series (AWMS); Vol. 24).

Research output: Chapter in Book/Report/Conference proceeding › Conference Contribution (Conference Proceeding)

TY - GEN

T1 - Explicit Connections Between Supersingular Isogeny Graphs and Bruhat–Tits Trees

AU - Amoros, Laia

AU - Iezzi, Annamaria

AU - Lauter, Kristin

AU - Martindale, Chloe

AU - Sotakova, Jana

PY - 2021/5/26

Y1 - 2021/5/26

N2 - In this paper 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.

AB - In this paper 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.

UR - https://www.dpmms.cam.ac.uk/~hk439/wine3.html

U2 - 10.1007/978-3-030-77700-5_2

DO - 10.1007/978-3-030-77700-5_2

M3 - Conference Contribution (Conference Proceeding)

SN - 9783030776992

SN - 9783030777029

T3 - Association for Women in Mathematics Series (AWMS)

SP - 39

EP - 73

BT - Women in Numbers Europe III

PB - Springer, Cham

T2 - Women in Numbers Europe 3

Y2 - 26 August 2019 through 30 August 2019

ER -

Explicit Connections Between Supersingular Isogeny Graphs and Bruhat–Tits Trees

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