TY - JOUR
T1 - On Two Problems About Isogenies of Elliptic Curves over Finite Fields
AU - Luo , Lixia
AU - Xiao , Guanju
AU - Deng , Yingpu
JO - Communications in Mathematical Research 
VL - 4
SP - 460
EP - 488
PY - 2020
DA - 2020/11
SN - 36
DO - http://doi.org/10.4208/cmr.2020-0071
UR - https://global-sci.org/intro/article_detail/cmr/18362.html
KW - Elliptic curve, isogeny, kernel ideal, minimal degree.
AB - <p style="text-align: justify;">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.</p>