Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves

TL;DR

This paper explores the construction of cycles in supersingular elliptic curve isogeny graphs, which are crucial for cryptographic applications and understanding endomorphism rings, especially focusing on specific cases and bounds.

Contribution

It introduces methods to construct cycles in supersingular isogeny graphs based on embedding quadratic orders and prime splitting, providing new insights into cycle lengths and bounds.

Findings

Constructed cycles in supersingular isogeny graphs using quadratic order embeddings.

Analyzed cycle lengths for specific j-invariants like 1728 and 0.

Established upper bounds on primes for unexpected 2-cycles when certain splitting conditions are not met.

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 $\mathbb{F}_{p^2}$, if an imaginary quadratic order $O$ can be embedded in $\text{End}(E)$ 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 $\ell$-isogeny graph where $\ell$ 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 $\ell$ doesn't split in $O$.