Decoding a message on elliptic curve
if your $e$ is coprime to $p-1$ where $p=2^{255}-19$ you can calculate the $e$-th of an integer modulo $p$ which means you can recover $E(m)$ given $E(m)^e$. In fact, if the condition holds, there exist $u,v \in \mathbb{Z}$ such that $eu+(p-1)v=1$ (Bezout's identity). Now, we are looking for an integer $x$ such that $x \equiv a^{1/e} \pmod p$ i.e. $x^e \equiv a \pmod p$. It is easy to find it: $a^u \equiv x^{eu} \equiv x^{1-(p-1)v} \equiv x\times (x^{p-1})^{(-v)}\equiv x \pmod p$ because $x^{p-1} \equiv 1 \pmod p$ according to Fermat's little theorem.