Let $K:=\mathbb F_{p^{16}}=\mathbb F[x]/(x^{16}-2)$ be a finite field where $p$ is chosen, that this ring-extension is really a field. Now we take an element $A\in K$ and we want to compute
$$A^d \in \mathbb F_{p^{16}}$$
The easiest way to do this, is to split $d$ into a sum to the base $p$, that means $$d = \sum\limits_{k=0}^{15} d_kp^k$$ where $d_k\in\mathbb Z \cap (-\delta, \delta)$ where $\delta \to 0$.
Since my special purpose is not depending on $d$ itselfe, I'm also looking for an integer $d'$, such that $d\mid d'$.
Definition: Ideal integer
I will call an integer $d$ with coefficients $d_k$ ideal, if and only if the sum of the absolute values of all coeffcients is the minimal possible. That means: $$d'' \text{ ideal }\Leftrightarrow \sum\limits_{k=0}^{15} |d''_k| = \min\left\{ \sum\limits_{k=0}^{15} |d'_k| : d'=\sum\limits_{k=0}^{^15} d'_k p^k\text{ and } d\mid d'\right\}$$
Now the Question is: Is there any algorithm, that solves the above problem, of finding an ideal integer?
I would think about interpreting this theory as a vector space and looking for minimal vectors. But I don't know anything about this problem.