论文标题
关于有限场上椭圆曲线的同学的两个问题
On two problems about isogenies of elliptic curves over finite fields
论文作者
论文摘要
在椭圆曲线的整个理论中都出现了同基因。最近，基于同基因的加密协议被认为是抗量子的加密协议的候选者。给定两条椭圆曲线$ e_1，E_2 $定义在有限的字段$ k $的情况下，具有相同的痕迹，有一个非稳定的等级$β$从$ e_2 $到$ k $定义的$ e_1 $。这项研究给出了$ \ rm {hom} _ {\ it k}（\ it e e _ {\ rm 1}，e _ {\ rm 2}）β$作为$ \ rm {end eend} _ {\ it K}（\ it K}（\ IT k}（\ IT e e e e _ {内核理想。此外，还提供了有关两个椭圆曲线之间非平凡最小程度的同基因的结果。
Isogenies occur throughout the theory of elliptic curves. Recently, the cryptographic protocols based on isogenies are considered as candidates of quantum-resistant cryptographic protocols. Given two elliptic curves $E_1, E_2$ defined over a finite field $k$ with the same trace, there is a nonconstant isogeny $β$ from $E_2$ to $E_1$ defined over $k$. This study gives out the index of $\rm{Hom}_{\it k}(\it E_{\rm 1},E_{\rm 2})β$ as a left ideal in $\rm{End}_{\it k}(\it E_{\rm 2})$ and figures out the correspondence between isogenies and kernel ideals. In addition, some results about the non-trivial minimal degree of isogenies between the two elliptic curves are also provided.
