On Two Problems About Isogenies of Elliptic Curves Over Finite Fields

• 作者: Guanju Xiao Lixia Luo Yingpu Deng
• 来源期刊: 数学研究通讯
• 年，卷（期）: 2020,（4）
• 页码范围: 460-488
• 页数: 29页
• 语种: 英文
• DOI: 10.4208/cmr.2020-0071

摘要

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 E1,E2 defined over a finite field k with the same trace, there is a nonconstant isogenyβfrom E2 to E1 defined over k. This study gives out the index of Homk(E1,E2)βas a nonzero left ideal in Endk(E2) and figures out the correspondence between isogenies and kernel ideals. In addition, some results about the non-trivial minimal degree of isogenies between two elliptic curves are also provided.