On Two Problems About Isogenies of Elliptic Curves over Finite Fields (Communications in Mathematical Research, 2020)
Isogenies occur throughout the theory of elliptic curves. Recently,the cryptographic protocols based on isogenies are considered as candidatesof 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 Hom$_k$($E_1$,$E_2$)$β$ as a nonzero left ideal in End$_k$($E_2$) and figures out the correspondence betweenisogenies and kernel ideals. In addition, some results about the non-trivialminimal degree of isogenies between two elliptic curves are also provided.