How to calculate all possible values for $m$, where $m=i^k \mod p$, $k,p$ are fixed?

Question

For example, all possible values for $i^{10} \mod 71$ is $1, 20, 30, 32, 37, 45, 48$. Is it possible to directly calculate these values without trying all possible $i$ from 1 to 71?

Answer 1

In this case $71-1=70$, so as the nonzero residues modulo $71$ form a cyclic group of order $70$, the tenth powers form a cyclic group of order $7$. So once you have one non-trivial value, say $10$, then $10^0$, $10^1$, $10^2,\ldots,10^6$ are all of them.