Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves
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,...
| Những tác giả chính: | , , |
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| Định dạng: | Bài viết |
| Ngôn ngữ: | English |
| Được phát hành: |
De Gruyter
2021-05-01
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| Loạt: | Journal of Mathematical Cryptology |
| Những chủ đề: | |
| Truy cập trực tuyến: | https://doi.org/10.1515/jmc-2020-0029 |
| _version_ | 1828152803533520896 |
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| author | Xiao Guanju Luo Lixia Deng Yingpu |
| author_facet | Xiao Guanju Luo Lixia Deng Yingpu |
| author_sort | Xiao Guanju |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-11T22:15:31Z |
| format | Article |
| id | doaj.art-901b77dc661f431985bdcc1c58edd201 |
| institution | Directory Open Access Journal |
| issn | 1862-2976 1862-2984 |
| language | English |
| last_indexed | 2024-04-11T22:15:31Z |
| publishDate | 2021-05-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Journal of Mathematical Cryptology |
| spelling | doaj.art-901b77dc661f431985bdcc1c58edd2012022-12-22T04:00:24ZengDe GruyterJournal of Mathematical Cryptology1862-29761862-29842021-05-0115145446410.1515/jmc-2020-0029Constructing Cycles in Isogeny Graphs of Supersingular Elliptic CurvesXiao Guanju0Luo Lixia1Deng Yingpu2Key Laboratory of Mathematics Mechanization, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing100190, China; School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing100049, ChinaKey Laboratory of Mathematics Mechanization, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing100190, China; School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing100049, ChinaKey Laboratory of Mathematics Mechanization, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing100190, China; School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing100049, ChinaLoops 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.https://doi.org/10.1515/jmc-2020-0029elliptic curvesisogeny graphsloopscycles11g0511g1514h5294a60 |
| spellingShingle | Xiao Guanju Luo Lixia Deng Yingpu Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves Journal of Mathematical Cryptology elliptic curves isogeny graphs loops cycles 11g05 11g15 14h52 94a60 |
| title | Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves |
| title_full | Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves |
| title_fullStr | Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves |
| title_full_unstemmed | Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves |
| title_short | Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves |
| title_sort | constructing cycles in isogeny graphs of supersingular elliptic curves |
| topic | elliptic curves isogeny graphs loops cycles 11g05 11g15 14h52 94a60 |
| url | https://doi.org/10.1515/jmc-2020-0029 |
| work_keys_str_mv | AT xiaoguanju constructingcyclesinisogenygraphsofsupersingularellipticcurves AT luolixia constructingcyclesinisogenygraphsofsupersingularellipticcurves AT dengyingpu constructingcyclesinisogenygraphsofsupersingularellipticcurves |