Complexity of finding solutions of trapdoored polynomial?

[Georgi Guninski]

Cryptography based on hardness of finding solutions of diophantine equations.

This is related to crypto and there is money in crypto,
so someone may profit :)

With latex on mathoverflow [1].
For the general approach, check [2]

Working over $K=\mathbb{Q}[x_1,...,x_n,y_1,...y_m]$.

Let $f_i=g_i \cdot (h_i(y_i)+l_i(x_i))$ where $l_i$ is linear
and depends on only $x$ variables and $h_i$ depends on only $y$
variables.
There are no restrictions on $g_i$.

Let $F=\sum_{i=1}^k f_i$ be given as sum of monomials.

If we know the set of $f_i$ (the secret trapdoor) and we are given $y_i$
, for sufficiently general $l_i$, we can find the solutions
of $F=0$ as the solutions of the linear in $x_i$ system
of linear equations $(h_i(y_i)+l_i(x_i))=0$

>Q1 Can we find $f_i$ such that solving $F=0$ is hard
without knowing the trapdoor?

>Q2 Assume an oracle gives many solutions to $F=0$,
can we still get hardness results for new solutions?


[1] Complexity of finding solutions of trapdoored polynomial
https://mathoverflow.net/questions/445899/complexity-of-finding-solutions-of-trapdoored-polynomial

[2] Cryptography signature scheme based on hardness of finding points
on varieties?
https://mathoverflow.net/questions/445898/cryptography-signature-scheme-based-on-hardness-of-finding-points-on-varieties