  Let $p\geq 3$ be a prime and $n\geq 1$ be an integer. Let $K\subseteq
{\mathbb{F}_p}$ denote a fixed subset with $0\in K$. Let $A\subseteq
({\mathbb{F}_p})^n$ be an arbitrary subset such that $$\{
\mathbf{a}-\mathbf{b}:~\mathbf{a},\mathbf{b}\in A,\mathbf{a}\neq
\mathbf{b}\}\cap K^n=\emptyset. $$ Then we prove the exponential upper bound $$
|A|\leq ( p-|K|+ 1 )^n. $$
  We use in our proof the linear algebra bound method.
