Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph

Enric Florit; Benjamin Smith

doi:10.1090/conm/779/15672

Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph

Enric Florit, Benjamin Smith

University of Barcelona

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We investigate special structures due to automorphisms in isogeny graphs of principally polarized abelian varieties, and abelian surfaces in particular. We give theoretical and experimental results on the spectral and statistical properties of (2, 2)-isogeny graphs of superspecial abelian surfaces, including stationary distributions for random walks, bounds on eigenvalues and diameters, and a proof of the connectivity of the Jacobian subgraph of the (2, 2)-isogeny graph. Our results improve our understanding of the performance and security of some recently-proposed cryptosystems, and are also a concrete step towards a better understanding of general superspecial isogeny graphs in arbitrary dimension.

Original language	English
Title of host publication	Arithmetic, Geometry, Cryptography, and Coding Theory - 18th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, 2021
Editors	Samuele Anni, Valentijn Karemaker, Elisa Lorenzo García
Publisher	American Mathematical Society
Pages	103-132
Number of pages	30
ISBN (Print)	9781470467944
DOIs	https://doi.org/10.1090/conm/779/15672
Publication status	Published - 1 Jan 2022
Event	18th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, AGC2T 2021 - Virtual, Online Duration: 31 May 2021 → 4 Jun 2021

Publication series

Name	Contemporary Mathematics
Volume	779
ISSN (Print)	0271-4132
ISSN (Electronic)	1098-3627

Conference

Conference	18th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, AGC2T 2021
City	Virtual, Online
Period	31/05/21 → 4/06/21

Keywords

Superspecial abelian varieties
isogeny graphs
isogeny-based cryptography

Access to Document

10.1090/conm/779/15672

Cite this

Florit, E., & Smith, B. (2022). Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph. In S. Anni, V. Karemaker, & E. L. García (Eds.), Arithmetic, Geometry, Cryptography, and Coding Theory - 18th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, 2021 (pp. 103-132). (Contemporary Mathematics; Vol. 779). American Mathematical Society. https://doi.org/10.1090/conm/779/15672

Florit, Enric ; Smith, Benjamin. / Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph. Arithmetic, Geometry, Cryptography, and Coding Theory - 18th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, 2021. editor / Samuele Anni ; Valentijn Karemaker ; Elisa Lorenzo García. American Mathematical Society, 2022. pp. 103-132 (Contemporary Mathematics).

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title = "Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph",

abstract = "We investigate special structures due to automorphisms in isogeny graphs of principally polarized abelian varieties, and abelian surfaces in particular. We give theoretical and experimental results on the spectral and statistical properties of (2, 2)-isogeny graphs of superspecial abelian surfaces, including stationary distributions for random walks, bounds on eigenvalues and diameters, and a proof of the connectivity of the Jacobian subgraph of the (2, 2)-isogeny graph. Our results improve our understanding of the performance and security of some recently-proposed cryptosystems, and are also a concrete step towards a better understanding of general superspecial isogeny graphs in arbitrary dimension.",

keywords = "Superspecial abelian varieties, isogeny graphs, isogeny-based cryptography",

author = "Enric Florit and Benjamin Smith",

note = "Publisher Copyright: {\textcopyright} 2022 Copyright by the authors.; 18th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, AGC2T 2021 ; Conference date: 31-05-2021 Through 04-06-2021",

year = "2022",

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doi = "10.1090/conm/779/15672",

language = "English",

isbn = "9781470467944",

series = "Contemporary Mathematics",

publisher = "American Mathematical Society",

pages = "103--132",

editor = "Samuele Anni and Valentijn Karemaker and Garc{\'i}a, \{Elisa Lorenzo\}",

booktitle = "Arithmetic, Geometry, Cryptography, and Coding Theory - 18th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, 2021",

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Florit, E & Smith, B 2022, Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph. in S Anni, V Karemaker & EL García (eds), Arithmetic, Geometry, Cryptography, and Coding Theory - 18th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, 2021. Contemporary Mathematics, vol. 779, American Mathematical Society, pp. 103-132, 18th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, AGC2T 2021, Virtual, Online, 31/05/21. https://doi.org/10.1090/conm/779/15672

Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph. / Florit, Enric; Smith, Benjamin.
Arithmetic, Geometry, Cryptography, and Coding Theory - 18th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, 2021. ed. / Samuele Anni; Valentijn Karemaker; Elisa Lorenzo García. American Mathematical Society, 2022. p. 103-132 (Contemporary Mathematics; Vol. 779).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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T1 - Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph

AU - Florit, Enric

AU - Smith, Benjamin

PY - 2022/1/1

Y1 - 2022/1/1

N2 - We investigate special structures due to automorphisms in isogeny graphs of principally polarized abelian varieties, and abelian surfaces in particular. We give theoretical and experimental results on the spectral and statistical properties of (2, 2)-isogeny graphs of superspecial abelian surfaces, including stationary distributions for random walks, bounds on eigenvalues and diameters, and a proof of the connectivity of the Jacobian subgraph of the (2, 2)-isogeny graph. Our results improve our understanding of the performance and security of some recently-proposed cryptosystems, and are also a concrete step towards a better understanding of general superspecial isogeny graphs in arbitrary dimension.

AB - We investigate special structures due to automorphisms in isogeny graphs of principally polarized abelian varieties, and abelian surfaces in particular. We give theoretical and experimental results on the spectral and statistical properties of (2, 2)-isogeny graphs of superspecial abelian surfaces, including stationary distributions for random walks, bounds on eigenvalues and diameters, and a proof of the connectivity of the Jacobian subgraph of the (2, 2)-isogeny graph. Our results improve our understanding of the performance and security of some recently-proposed cryptosystems, and are also a concrete step towards a better understanding of general superspecial isogeny graphs in arbitrary dimension.

KW - Superspecial abelian varieties

KW - isogeny graphs

KW - isogeny-based cryptography

U2 - 10.1090/conm/779/15672

DO - 10.1090/conm/779/15672

M3 - Conference contribution

AN - SCOPUS:85130498835

SN - 9781470467944

T3 - Contemporary Mathematics

SP - 103

EP - 132

BT - Arithmetic, Geometry, Cryptography, and Coding Theory - 18th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, 2021

A2 - Anni, Samuele

A2 - Karemaker, Valentijn

A2 - García, Elisa Lorenzo

PB - American Mathematical Society

T2 - 18th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, AGC2T 2021

Y2 - 31 May 2021 through 4 June 2021

ER -

Florit E, Smith B. Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph. In Anni S, Karemaker V, García EL, editors, Arithmetic, Geometry, Cryptography, and Coding Theory - 18th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, 2021. American Mathematical Society. 2022. p. 103-132. (Contemporary Mathematics). doi: 10.1090/conm/779/15672

Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph

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