On two problems about isogenies of elliptic curves over finite fields

Abstract: 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 $\beta$ from $E_2$ to $E_1$ defined over $k$. This study gives out the index of $\rm{Hom}{\it k}(\it E{\rm 1},E_{\rm 2})\beta$ 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.