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

N1 - Funding Information:
Acknowledgments. We are deeply grateful to John Voight for several helpful discussions, and for providing us with some code that became a part of the code that generated the computational examples in Section 6. We also thank Sean Hallgren and Rachel Pries for helpful discussions. We thank Andrew Sutherland for many comments on an earlier version that led to a significant improvement in the running time analysis of the generalization of Schoof’s algorithm, and for outlining the proof of Proposition 2.4. Finally, we thank the Women in Numbers 4 conference and BIRS, for enabling us to start this project in a productive environment. K.E. was partially supported by National Science Foundation awards DMS-1056703 and CNS-1617802. T.M. was partially supported by National Science Foundation awards DMS-1056703 and CNS-1617802, and by funding from the Natural Sciences and Engineering Research Council of Canada, the Canada First Research Excellence Fund, CryptoWorks21, Public Works and Government Services Canada, and the Royal Bank of Canada. C.CN. was partially supported by Universidad de Costa Rica.

PY - 2019

Y1 - 2019

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.

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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 -