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 𝔽
p
2
, 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.
Supersingular Isogeny Graphs and Endomorphism Rings: Reductions and Solutions