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 - Publisher Copyright:
© 2019, The Author(s) and The Association for Women in Mathematics.
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 -