### 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 language | English (US) |
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Title of host publication | Association for Women in Mathematics Series |

Publisher | Springer |

Pages | 41-66 |

Number of pages | 26 |

DOIs | |

State | Published - Jan 1 2019 |

### Publication series

Name | Association for Women in Mathematics Series |
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Volume | 19 |

ISSN (Print) | 2364-5733 |

ISSN (Electronic) | 2364-5741 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Gender Studies

### Cite this

*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

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*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.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - Cycles in the Supersingular ℓ-Isogeny Graph and Corresponding Endomorphisms

AU - Bank, Efrat

AU - Camacho-Navarro, Catalina

AU - Eisenträger, Kirsten

AU - Morrison, Travis

AU - Park, Jennifer

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85071423374&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85071423374&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-19478-9_2

DO - 10.1007/978-3-030-19478-9_2

M3 - Chapter

AN - SCOPUS:85071423374

T3 - Association for Women in Mathematics Series

SP - 41

EP - 66

BT - Association for Women in Mathematics Series

PB - Springer

ER -