Related to this question [Complexity of finding solutions of trapdoored polynomial](https://mathoverflow.net/questions/445899/complexity-of-finding-solutions-of-trapdoored-polynomial).

I am trying to build signature scheme based on hardness
of finding points on varieties.

Let $K$ be field and $M=K[x_1,...x_n,y_1,...,y_m]$.

Let $V$ be the variety $f_1(x_i,y_i)=...f_k(x_i,y_i)=0$.

Alice constructs $V$ with some secret $S$ (trapdoor) such that
for all $y_i$, Alice can efficiently find $X_i$ such
that $F(X_i,y_i=Y_i)=0$. Correctness of the signature
of $Y_i$ is the vanishing of $F$.

So $V$ must satisfy the following properties:

1. For all $y_i$, Alice can find efficiently $X_i$ such
that $F$ vanishes.
2. Without knowing the secret $S$, adversary can't efficiently find
points on $V$ given $Y_i$.
3. (optional, feel free to skip) Given many points on $V$, resulting from valid signatures,
adversary can't find the secret $S$ or otherwise forge signatures.

Potential approach is to start from "basis" and then obfuscate
them.

I am looking for hardness results and even high degree polynomial time
would be of interest.

>Q1 What complexity of hardness can we get for the above construction?