KöMaL problem A, 2014

There is given a prime number $p$ and two positive integers, $k$ and $n$. Determine the smallest nonnegative integer $d$ for which there exists a polynomial $f(x_1,\dots,x_n)$ on $n$ variables, with degree $d$ and having integer coefficients that satisfies the following property: for arbitrary $a_1,\dots,a_n\in\{0,1\}$, $p$ divides $f(a_1,\dots,a_n)$ if and only if $p^k$ divides $a_1+\dots+a_n$.
(5 pont)