is also reduced to an effective strong approximation theorem in Section \[sec:ideals:strongapproximation\]. In particular if the ideal is given by a pair of generators $I = {\mathcal{O}}(N,\alpha)$, the quaternion $\ell$-isogeny path problem is reduced to finding $\lambda \in {\mathbb{Z}}$ coprime to $N$ and $$\beta \equiv \lambda\alpha \bmod N{\mathcal{O}}$$ with ${\mathrm{Nrd}}(\beta) = N\ell^e$ for some positive integer $e$.

Sections \[sec:ideals:ellpow:isom\], \[sec:ideals:ellpow:lift\], and \[sec:ideals:ellpow:results\] describe the core of our approach to solve this problem. Since the index of $R + Rj$ in ${\mathcal{O}}$ is coprime to $N$, we have an isomorphism $$\frac{R + Rj}{N(R + Rj)} \cong \frac{{\mathcal{O}}}{N{\mathcal{O}}}\cdot$$ We can therefore choose representative elements in $R + Rj$ as convenient to simplify the algorithm. Since the index $[{\mathcal{O}}:R + Rj] = |{\mathrm{disc}}(R)|$ is assumed to be small (in $O(\log(p)^2)$ under the GRH), the size of the output might be slightly larger, but the distinction is asymptotically insignificant. A direct approach to the strong approximation problem to solve for $\beta$ seems daunting, so instead we reduce to the following steps:

1.  Solve for a random $\gamma \in {\mathcal{O}}$ of reduced norm $N\ell^{e_0}$.

2.  Solve for $[\mu]$ in $({\mathcal{O}}/N{\mathcal{O}})^*$ such that $({\mathcal{O}}\gamma/N{\mathcal{O}})[\mu] = I/N{\mathcal{O}}$.

3.  Solve for the strong approximation of $[\mu]$ (modulo $N$) 