  Assume a polynomial-time algorithm for factoring integers,
Conjecture~\ref{conj}, $d\geq 3,$ and $q$ and $p$ are prime numbers, where
$p\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\log(q)$
that lifts every $\mathbb{Z}/q\mathbb{Z}$ point of $S^{d-2}\subset S^{d}$ to a
$\mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our
algorithm for $d=3 \text{ and }4$. Based on our numerical results, we formulate
a conjecture which can be checked in polynomial-time and gives the optimal
bound on the diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of
$S^{d-2}\subset S^d$.
