On two problems about isogenies of elliptic curves over finite fields

TL;DR

This paper explores the algebraic structure of isogenies between elliptic curves over finite fields, providing new insights into their indices, kernel ideals, and minimal degrees, with implications for cryptography.

Contribution

It introduces a novel analysis of the index of certain homomorphism modules and establishes a correspondence between isogenies and kernel ideals, advancing understanding of elliptic curve isogenies.

Findings

Determined the index of Hom modules as left ideals in endomorphism rings.

Established a correspondence between isogenies and kernel ideals.

Provided results on the minimal degree of non-trivial isogenies.

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.