function) amounts to solving the following path problem in the supersingular $\ell$-isogeny graph:

\[prob:preim\] Let $p$ and $\ell$ be prime numbers, $p\neq\ell$. Let $E_0$ and $E_1$ be two supersingular elliptic curves over ${\mathbb{F}}_{p^2}$ with $|E_0({\mathbb{F}}_{p^2})|=|E_1({\mathbb{F}}_{p^2})|=(p+1)^2$. Find $k\in\mathbb{N}$ and an isogeny of degree $\ell^k$ from $E_0$ to $E_1$.

Similarly, finding collisions requires a solution to the following multiple path problem in the supersingular $\ell$-isogeny graph:

\[prob:coll\] Let $p$ and $\ell$ be prime numbers, $p\neq\ell$. Let $E_0$ be a supersingular elliptic curve over ${\mathbb{F}}_{p^2}$. Find $k_1,k_2\in\mathbb{N}$, a supersingular elliptic curve $E_1$ and two distinct isogenies (i.e. with distinct kernels) of degrees respectively $\ell^{k_1}$ and $\ell^{k_2}$ from $E_0$ to $E_1$.

Setting ${\mathcal{O}}= {\mathrm{End}}(E_0)$, we have a category of left ${\mathcal{O}}$-ideals, with morphisms $I \rightarrow I\alpha \subseteq J$, for $\alpha$ in $B = {\mathcal{O}}\otimes {\mathbb{Q}}$, which is equivalent to the category of supersingular elliptic curves and isogenies. The analog of the path problem in supersingular $\ell$-isogeny graphs is that of finding a representative ideal $J$ for given $I$ of norm $\ell^k$. We call this problem the *quaternion $\ell$-isogeny path problem*, and focus on its effective solution in this article.

Quaternion algebras\[sec:intro:QA\]
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In this work we consider the structure of left ideals of a 