Can the inverse to a scalar multiplication be found efficiently in elliptic curves?

Question

Let $P$ be a point over an elliptic curve $F_p$, say Curve25519, and let $k$ be a scalar. Assume I want to find an inverse $k^{-1}$ such that: $$k \cdot k^{-1} \cdot P = P$$ Does such an inverse always exists and can it be calculated efficiently?

Answer 1

Well, it amounts in a way to the discrete logarithm problem for EC.

Given two points $P,Q$ on an EC curve where $Q$ is a multiple of $P$. Find $k$ such that $Q=kP$.

This problem is intractable and assures the security of EC cryptography.