I have to use the following proposition, but since I'm not that into statistics, I don't know how to prove it formally.
If there are two independent random variables $A$ and $B$ over $\{0,1,...,m-1\}$, with $A$ uniformly distributed, the random variable $C = A + B \text{ mod }m$ is also uniformly distributed (the distribution of $B$ is arbitrary).
I think you can argue that if $B$ has a certain value $b$, then $A + b \text{ mod }m$ is uniformly distributed. Can anyone help me to write this down correctly?