{"title":"有限域上椭圆曲线的等同源性的两个问题","authors":"Lixia Luo, Guanju Xiao, Yingpu Deng","doi":"10.4208/cmr.2020-0071","DOIUrl":null,"url":null,"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.","PeriodicalId":66427,"journal":{"name":"数学研究通讯","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Two Problems About Isogenies of Elliptic Curves Over Finite Fields\",\"authors\":\"Lixia Luo, Guanju Xiao, Yingpu Deng\",\"doi\":\"10.4208/cmr.2020-0071\",\"DOIUrl\":null,\"url\":null,\"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.\",\"PeriodicalId\":66427,\"journal\":{\"name\":\"数学研究通讯\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"数学研究通讯\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4208/cmr.2020-0071\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"数学研究通讯","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4208/cmr.2020-0071","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On Two Problems About Isogenies of Elliptic Curves Over Finite Fields
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.