On the Search for Supersingular Elliptic Curves and Their Applications

Elliptic curves with the special quality known as supersingularity have gained much popularity in the rapidly developing field of cryptography. The conventional method of employing random search is quite ineffective in finding these curves. This paper analyzes the search of supersingular elliptic cu...

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Detalles Bibliográficos
Autores: Martínez-Díaz, Ismel, Ali, Rashad, Kamran Jamil, Muhammad
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2025
País:España
Institución:Universitat de Lleida (UdL)
Repositorio:Repositori Obert UdL
OAI Identifier:oai:repositori.udl.cat:10459.1/467417
Acceso en línea:https://doi.org/10.3390/math13020188
https://hdl.handle.net/10459.1/467417
Access Level:acceso abierto
Palabra clave:Supersingular elliptic curves
Metaheuristic
Combinatorial optimization
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spelling On the Search for Supersingular Elliptic Curves and Their ApplicationsMartínez-Díaz, IsmelAli, RashadKamran Jamil, MuhammadSupersingular elliptic curvesMetaheuristicCombinatorial optimizationElliptic curves with the special quality known as supersingularity have gained much popularity in the rapidly developing field of cryptography. The conventional method of employing random search is quite ineffective in finding these curves. This paper analyzes the search of supersingular elliptic curves in the space of curves over 2. We show that naive random search is unsuitable to easily find any supersingular elliptic curves when the space size is greater than 1013. We improve the random search using a necessary condition for supersingularity. As our main result, we define for the first time an objective function to measure the supersingularity in ordinary curves, and we apply local search and a genetic algorithm using that function. The study not only finds these supersingular elliptic curves but also investigates possible uses for them. These curves were used to create cycles inside the isogeny graph in one particular application. The research shows how the design of S-boxes may strategically use these supersingular elliptic curves. The key components of replacement, which is a fundamental step in the encryption process that shuffles and encrypts the data inside images, are S-boxes. This work represents a major advancement in effectively identifying these useful elliptic curves, eventually leading to their wider application and influence in the rapidly expanding field of cryptography.MDPI2025info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttps://doi.org/10.3390/math13020188https://hdl.handle.net/10459.1/467417reponame:Repositori Obert UdL instname:Universitat de Lleida (UdL)InglésReproducció del document publicat a https://doi.org/10.3390/math13020188Mathematics, 2025, vol. 13, núm. 2, 188cc-by (c) Ismel Martinez-Diaz, Rashad Ali, Muhammad Kamran Jamil, 2025info:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by/4.0/oai:repositori.udl.cat:10459.1/4674172026-06-24T12:42:17Z
dc.title.none.fl_str_mv On the Search for Supersingular Elliptic Curves and Their Applications
title On the Search for Supersingular Elliptic Curves and Their Applications
spellingShingle On the Search for Supersingular Elliptic Curves and Their Applications
Martínez-Díaz, Ismel
Supersingular elliptic curves
Metaheuristic
Combinatorial optimization
title_short On the Search for Supersingular Elliptic Curves and Their Applications
title_full On the Search for Supersingular Elliptic Curves and Their Applications
title_fullStr On the Search for Supersingular Elliptic Curves and Their Applications
title_full_unstemmed On the Search for Supersingular Elliptic Curves and Their Applications
title_sort On the Search for Supersingular Elliptic Curves and Their Applications
dc.creator.none.fl_str_mv Martínez-Díaz, Ismel
Ali, Rashad
Kamran Jamil, Muhammad
author Martínez-Díaz, Ismel
author_facet Martínez-Díaz, Ismel
Ali, Rashad
Kamran Jamil, Muhammad
author_role author
author2 Ali, Rashad
Kamran Jamil, Muhammad
author2_role author
author
dc.subject.none.fl_str_mv Supersingular elliptic curves
Metaheuristic
Combinatorial optimization
topic Supersingular elliptic curves
Metaheuristic
Combinatorial optimization
description Elliptic curves with the special quality known as supersingularity have gained much popularity in the rapidly developing field of cryptography. The conventional method of employing random search is quite ineffective in finding these curves. This paper analyzes the search of supersingular elliptic curves in the space of curves over 2. We show that naive random search is unsuitable to easily find any supersingular elliptic curves when the space size is greater than 1013. We improve the random search using a necessary condition for supersingularity. As our main result, we define for the first time an objective function to measure the supersingularity in ordinary curves, and we apply local search and a genetic algorithm using that function. The study not only finds these supersingular elliptic curves but also investigates possible uses for them. These curves were used to create cycles inside the isogeny graph in one particular application. The research shows how the design of S-boxes may strategically use these supersingular elliptic curves. The key components of replacement, which is a fundamental step in the encryption process that shuffles and encrypts the data inside images, are S-boxes. This work represents a major advancement in effectively identifying these useful elliptic curves, eventually leading to their wider application and influence in the rapidly expanding field of cryptography.
publishDate 2025
dc.date.none.fl_str_mv 2025
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://doi.org/10.3390/math13020188
https://hdl.handle.net/10459.1/467417
url https://doi.org/10.3390/math13020188
https://hdl.handle.net/10459.1/467417
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Reproducció del document publicat a https://doi.org/10.3390/math13020188
Mathematics, 2025, vol. 13, núm. 2, 188
dc.rights.none.fl_str_mv cc-by (c) Ismel Martinez-Diaz, Rashad Ali, Muhammad Kamran Jamil, 2025
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by/4.0/
rights_invalid_str_mv cc-by (c) Ismel Martinez-Diaz, Rashad Ali, Muhammad Kamran Jamil, 2025
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.publisher.none.fl_str_mv MDPI
publisher.none.fl_str_mv MDPI
dc.source.none.fl_str_mv reponame:Repositori Obert UdL
instname:Universitat de Lleida (UdL)
instname_str Universitat de Lleida (UdL)
reponame_str Repositori Obert UdL
collection Repositori Obert UdL
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repository.mail.fl_str_mv
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