This paper describes several speedups for computation in the order $p+1$ subgroup of $\GaF{p^2}^*$ and the order $p^2-p+1$ subgroup of~$\GaF{p^6}^*$. These results are in a way complementary to LUC and XTR, where computations in these groups are sped up using trace maps. As a side result, we present an efficient method for XTR with $p\equiv3\bmod 4$.