Homomorphic mapping between elliptic curve point and Zq
I think that you can use a bilinear pairing map for the function $F$. This map is defined from $G_1 \times G_2$ to $\mu_n$. This means that $F(x)=e(x,T)$ that $T \in G_2$.
The feature of this map is as:
$e(g^a,T^b)=e(g,T)^{ab}$
$e(g^bh^r,T)=e(g^b,T).e(h^r,T)=F(g^b).F(h^r)$
The amount of $\mu_n$  is in $F_{q^k}^*$. There is a map from $F_{q^k}$ to $F_q$ that is called Trace map. You can find more information about this map at &quot;Pairing for beginners&quot; book or http://www.m-hikari.com/ija/ija-2011/ija-21-24-2011/yadavIJA21-24-2011.pdf