Cycles in the supersingular $\ell$-isogeny graph and corresponding endomorphisms

TL;DR

This paper investigates the structure of supersingular $ ext{elliptic}$ curves' endomorphism rings via cycles in $ ext{l}$-isogeny graphs, providing criteria for independence and maximality, and introduces an efficient algorithm for trace computation.

Contribution

It establishes necessary and sufficient conditions for cycle-induced endomorphisms to be independent and maximal, and generalizes Schoof's algorithm for efficient trace calculation.

Findings

Criteria for linear independence of endomorphisms from cycles

Conditions under which the generated order is not maximal

An efficient generalized Schoof's algorithm for trace computation

Abstract

We study the problem of generating the endomorphism ring of a supersingular elliptic curve by two cycles in $\ell$-isogeny graphs. We prove a necessary and sufficient condition for the two endomorphisms corresponding to two cycles to be linearly independent, expanding on the work in Kohel's thesis. We also give a criterion under which the order generated by two cycles is not a maximal order. We give some examples in which we compute cycles which generate the full endomorphism ring. The most difficult part of these computations is the calculation of the trace of these cycles. We show that a generalization of Schoof's algorithm can accomplish this computation efficiently.