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Mathematics > Number Theory

arXiv:2001.00126 (math)
[Submitted on 1 Jan 2020]

Title:On two problems about isogenies of elliptic curves over finite fields

Authors:Lixia Luo, Guanju Xiao, Yingpu Deng
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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.
Subjects: Number Theory (math.NT)
Cite as: arXiv:2001.00126 [math.NT]
  (or arXiv:2001.00126v1 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.2001.00126

Submission history

From: Lixia Luo [view email]
[v1] Wed, 1 Jan 2020 02:34:31 UTC (21 KB)
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