Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves

• Main Authors: Xiao Guanju, Luo Lixia, Deng Yingpu
• Format: Article
• Language: English
• Published: De Gruyter 2021-05-01
• Series: Journal of Mathematical Cryptology
• Subjects: elliptic curves; isogeny graphs; loops; cycles; 11g05; 11g15; 14h52; 94a60
• Online Access: https://doi.org/10.1515/jmc-2020-0029

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.