Cycles in the Supersingular ℓ-Isogeny Graph and Corresponding Endomorphisms
Cycles in the Supersingular ℓ-Isogeny Graph and Corresponding Endomorphisms
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超奇异 ℓ-同构图中的循环和相应的同态
DOI：
10.1007/978-3-030-19478-9_2
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Jennifer Park
中科院分区：
文献类型：
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作者：
Efrat Bank;Catalina Camacho;Kirsten Eisentraeger;T. Morrison;Jennifer Park
我们研究了通过l-同源图中的两个周期生成超奇异椭圆曲线的自同态环的问题。我们证明了对应于两个循环的两个自同态线性独立的充要条件，扩展了 Kohel 在其论文中的工作。我们还给出了一个准则，在该准则下，两个循环生成的环不是最大阶。我们给出了一些例子，其中我们计算生成完整自同态环的循环。这些计算中最困难的部分是计算这些周期的迹线。我们证明了 Schoof 算法的推广可以有效地完成这种计算。
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.