I have read some texts concerning fully homomorphic encryption but didn't find the following simple idea.
Let $p$ be a prime. We choose secret key $x_0$ in the interval $1\le x_0<p$. Let $\mathrm{Enc}(m)$ be a polynomial $P(x)=a_{p-2}x^{p-2}+\ldots+a_1x+a_0$ with random $a_{p-2},\ldots,a_1$ and $a_0$ such that $P(x_0)=m$; $\mathrm{Dec}(P):=P(x_0)$ (everything is modulo $p$). This construction gives FHE because $(P_1P_2)(x_0)=P_1(x_0)P_2(x_0) \bmod(x^{p-1}-1)$ and $(P_1+P_2)(x_0)=P_1(x_0)+P_2(x_0).$
Is this idea reasonable or I missed some pitfalls?