Cycles in the Supersingular ℓ-Isogeny Graph and Corresponding Endomorphisms

Efrat Bank, Catalina Camacho-Navarro, Kirsten Eisenträger, Travis Morrison, Jennifer Park

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

We study the problem of generating the endomorphism ring of a supersingular elliptic curve by two cycles in ℓ-isogeny graphs. We prove a necessary and sufficient condition for the two endomorphisms corresponding to two cycles to be linearly independent, expanding on the work by Kohel in his thesis. We also give a criterion under which the ring generated by two cycles is not a maximal order. We give some examples in which we compute cycles which generate the full endomorphism ring. The most difficult part of these computations is the calculation of the trace of these cycles. We show that a generalization of Schoof’s algorithm can accomplish this computation efficiently.

Original languageEnglish (US)
Title of host publicationAssociation for Women in Mathematics Series
PublisherSpringer
Pages41-66
Number of pages26
DOIs
StatePublished - Jan 1 2019

Publication series

NameAssociation for Women in Mathematics Series
Volume19
ISSN (Print)2364-5733
ISSN (Electronic)2364-5741

Fingerprint

Isogeny
Endomorphisms
Cycle
Graph in graph theory
Endomorphism Ring
Maximal Order
Elliptic Curves
Linearly
Trace
Ring
Necessary Conditions
Sufficient Conditions

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Gender Studies

Cite this

Bank, E., Camacho-Navarro, C., Eisenträger, K., Morrison, T., & Park, J. (2019). Cycles in the Supersingular ℓ-Isogeny Graph and Corresponding Endomorphisms. In Association for Women in Mathematics Series (pp. 41-66). (Association for Women in Mathematics Series; Vol. 19). Springer. https://doi.org/10.1007/978-3-030-19478-9_2
Bank, Efrat ; Camacho-Navarro, Catalina ; Eisenträger, Kirsten ; Morrison, Travis ; Park, Jennifer. / Cycles in the Supersingular ℓ-Isogeny Graph and Corresponding Endomorphisms. Association for Women in Mathematics Series. Springer, 2019. pp. 41-66 (Association for Women in Mathematics Series).
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Bank, E, Camacho-Navarro, C, Eisenträger, K, Morrison, T & Park, J 2019, Cycles in the Supersingular ℓ-Isogeny Graph and Corresponding Endomorphisms. in Association for Women in Mathematics Series. Association for Women in Mathematics Series, vol. 19, Springer, pp. 41-66. https://doi.org/10.1007/978-3-030-19478-9_2

Cycles in the Supersingular ℓ-Isogeny Graph and Corresponding Endomorphisms. / Bank, Efrat; Camacho-Navarro, Catalina; Eisenträger, Kirsten; Morrison, Travis; Park, Jennifer.

Association for Women in Mathematics Series. Springer, 2019. p. 41-66 (Association for Women in Mathematics Series; Vol. 19).

Research output: Chapter in Book/Report/Conference proceedingChapter

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Bank E, Camacho-Navarro C, Eisenträger K, Morrison T, Park J. Cycles in the Supersingular ℓ-Isogeny Graph and Corresponding Endomorphisms. In Association for Women in Mathematics Series. Springer. 2019. p. 41-66. (Association for Women in Mathematics Series). https://doi.org/10.1007/978-3-030-19478-9_2