This is related to cryptography and this question and another question.

In short, we are asking about decomposing multivariate polynomial as sum of perfect powers of linear polynomials.

Working over $\mathbb{Q}[x_1,...,x_n]$

Let $l_i(x_i)$ be $k$ linear polynomials such that the linear system $l_1=l_2=...l_k=0$ has solution $S_0$.

Let $D$ be positive integer and define $F=\sum_{i=1}^k l_i^D$.

$F$ is given as sum of monomials and $F(S_0)=0$.

$l_i$ are kept secret as trapdoor.

Q1 Are the there choices of $n,k,D,l_i$ such that finding solution of $F=0$ is infeasible?

Q2 Same as Q1, but in addition we are given one solution $S_1$ such that $F(S_1)=0$.

Q3 Same as Q1, but in addition we are given many solutions $S_i$ such that $F(S_i)=0$.

This is related to cryptography and this question and another question.

In short, we are asking about decomposing multivariate polynomial as sum of perfect powers of linear polynomials.

Working over $\mathbb{Q}[x_1,\ldots,x_n]$

Let $\ell_i(x_i)$ be $k$ linear polynomials such that the linear system $\ell_1=\ell_2=\cdots =\ell_k=0$ has solution $S_0$.

Let $D$ be positive integer and define $F=\sum_{i=1}^k \ell_i^D$.

$F$ is given as sum of monomials and $F(S_0)=0$.

$\ell_i$ are kept secret as trapdoor.

Q1 Are the there choices of $n,k,D,\ell_i$ such that finding solution of $F=0$ is infeasible?

Q2 Same as Q1, but in addition we are given one solution $S_1$ such that $F(S_1)=0$.

Q3 Same as Q1, but in addition we are given many solutions $S_i$ such that $F(S_i)=0$.