{"title":"构造超奇异椭圆曲线等根图中的环","authors":"Guanju Xiao, Lixia Luo, Yingpu Deng","doi":"10.1515/jmc-2020-0029","DOIUrl":null,"url":null,"abstract":"Abstract Loops and cycles play an important role in computing endomorphism rings of supersingular elliptic curves and related cryptosystems. For a supersingular elliptic curve E defined over 𝔽p2, if an imaginary quadratic order O can be embedded in End(E) and a prime L splits into two principal ideals in O, we construct loops or cycles in the supersingular L-isogeny graph at the vertices which are next to j(E) in the supersingular ℓ-isogeny graph where ℓ is a prime different from L. Next, we discuss the lengths of these cycles especially for j(E) = 1728 and 0. Finally, we also determine an upper bound on primes p for which there are unexpected 2-cycles if ℓ doesn’t split in O.","PeriodicalId":43866,"journal":{"name":"Journal of Mathematical Cryptology","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/jmc-2020-0029","citationCount":"1","resultStr":"{\"title\":\"Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves\",\"authors\":\"Guanju Xiao, Lixia Luo, Yingpu Deng\",\"doi\":\"10.1515/jmc-2020-0029\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Loops and cycles play an important role in computing endomorphism rings of supersingular elliptic curves and related cryptosystems. For a supersingular elliptic curve E defined over 𝔽p2, if an imaginary quadratic order O can be embedded in End(E) and a prime L splits into two principal ideals in O, we construct loops or cycles in the supersingular L-isogeny graph at the vertices which are next to j(E) in the supersingular ℓ-isogeny graph where ℓ is a prime different from L. Next, we discuss the lengths of these cycles especially for j(E) = 1728 and 0. Finally, we also determine an upper bound on primes p for which there are unexpected 2-cycles if ℓ doesn’t split in O.\",\"PeriodicalId\":43866,\"journal\":{\"name\":\"Journal of Mathematical Cryptology\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2019-12-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1515/jmc-2020-0029\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematical Cryptology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1515/jmc-2020-0029\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Cryptology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/jmc-2020-0029","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves
Abstract Loops and cycles play an important role in computing endomorphism rings of supersingular elliptic curves and related cryptosystems. For a supersingular elliptic curve E defined over 𝔽p2, if an imaginary quadratic order O can be embedded in End(E) and a prime L splits into two principal ideals in O, we construct loops or cycles in the supersingular L-isogeny graph at the vertices which are next to j(E) in the supersingular ℓ-isogeny graph where ℓ is a prime different from L. Next, we discuss the lengths of these cycles especially for j(E) = 1728 and 0. Finally, we also determine an upper bound on primes p for which there are unexpected 2-cycles if ℓ doesn’t split in O.