Let $E$ be an elliptic curve over $\mathbb{F}_p$. The n-th division polynomial is $\psi_n$.
Given points $P=(x_P,y_P)$, $Q=k P$ (where $k$ is unknown) and $\psi_k(x_P,y_P)$, can one efficiently find $\psi_{k+1}(x_P,y_P)$?
For some $P$ can find non-trivial $\psi_k(x_P,y_P)$ and finding $\psi_{k+1}(x_P,y_P)$ might have some implications to EC discrete logarithm, so I suppose this problem is hard.
I am working with curves not of the form $y^2=x^3+Ax+B$ but even for them will be interesting.
Tried working with unknowns, computed the next terms via the recurrence, equated with $(k+1)P \ldots (k+i)P$. The groebner basis didn't give solution.