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,...
| Main Authors: | , , |
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| Formato: | Artigo |
| Idioma: | English |
| Publicado em: |
De Gruyter
2021-05-01
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| Colecção: | Journal of Mathematical Cryptology |
| Assuntos: | |
| Acesso em linha: | https://doi.org/10.1515/jmc-2020-0029 |
| Resumo: | 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. |
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| ISSN: | 1862-2976 1862-2984 |