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Volume 36, Issue 4
On Two Problems About Isogenies of Elliptic Curves over Finite Fields

Lixia Luo, Guanju Xiao & Yingpu Deng

DOI: 10.4208/cmr.2020-0071

Commun. Math. Res., 36 (2020), pp. 460-488.

Published online: 2020-11

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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 $β$ 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.

  • Keywords

Elliptic curve, isogeny, kernel ideal, minimal degree.

  • AMS Subject Headings

11G05, 11G15, 11G20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-36-460, author = {Luo , LixiaXiao , Guanju and Deng , Yingpu}, title = {On Two Problems About Isogenies of Elliptic Curves over Finite Fields}, journal = {Communications in Mathematical Research }, year = {2020}, volume = {36}, number = {4}, pages = {460--488}, 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.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2020-0071}, url = {http://global-sci.org/intro/article_detail/cmr/18362.html} }
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 -

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.

Luo , LixiaXiao , Guanju and Deng , Yingpu. (2020). On Two Problems About Isogenies of Elliptic Curves over Finite Fields. Communications in Mathematical Research . 36 (4). 460-488. doi:10.4208/cmr.2020-0071
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