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 $β$ 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 between
isogenies and kernel ideals. In addition, some results about the non-trivial
minimal degree of isogenies between two elliptic curves are also provided. |