On Two Problems About Isogenies of Elliptic Curves Over Finite Fields

Lixia Luo
2020 Communications in Mathematical Research  
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
more » ... eals. In addition, some results about the non-trivial minimal degree of isogenies between two elliptic curves are also provided. Proposition 2.2. Let I be a left ideal of the endomorphism ring End k (E), ϕ I : E → E/H(I)= E ′ be the isogeny with kernel H(I). Let J be a left ideal of End k (E ′ ), ϕ J : E ′ → E ′ /H(J) = E ′′ . Then ϕ J ϕ I is the isogeny from E to E ′′ with kernel H(I J). Proof. See [21, Proposition 3.12].
doi:10.4208/cmr.2020-0071 fatcat:qqurztll5zda5bhou4qldqyfty