Using $\epsilon$-bias spaces over $F_2$, we show that the Remote Point
Problem (RPP), introduced by Alon et al [APY09], has an $NC^2$ algorithm
(achieving the same parameters as [APY09]). We study a generalization of the
Remote Point Problem to groups: we replace $F^n$ by $G^n$ for an arbitrary
fixed group $G$. When $G$ is Abelian, we give an $NC^2$ algorithm for RPP,
again using $\epsilon$-bias spaces. For nonabelian $G$, we give a deterministic
polynomial-time algorithm for RPP. We also show the connection to construction
of expanding generator sets for the group $G^n$. All our algorithms for the RPP
achieve essentially the same parameters as [APY09].