My question is to some extent related to cryptography, but I'd like the mathematicians answer my question, please (as their answers are usualy more clearer than cryptographers).
Consider I have a polynomial $f$ defined in $R[x]$, polynomial ring. And I have $n$ points, $\textbf{x}:(x_0,...,x_n)$. I evaluate polynomial $f$ at $\textbf{x}$ points and it results $\textbf{y}:(y_0,...,y_n)$ values. I pick a $g$ generator of the field (defined below) and raise $g$ to the power of elements in $\textbf{y}$:
$v_0=g^{y_0} \bmod p$
$v_1=g^{y_1} \bmod p$
. . .
$v_n=g^{y_n} \bmod p$
My question is: given $v_i$'s and $\textbf{x}$ points (but not polynomial $f$), can anybody evaluate any new points on the polynomial to check whether it's a root of the polynomial?
let $p$ be a prime number, $R$ be $\mathbb{Z}_p$ and $g$ be a generator of the field $\mathbb{Z}_p$. These values are public.