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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| Format: | Article |
| Sprog: | English |
| Udgivet: |
De Gruyter
2021-05-01
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| Serier: | Journal of Mathematical Cryptology |
| Fag: | |
| Online adgang: | https://doi.org/10.1515/jmc-2020-0029 |
| Summary: | 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 |