How to prove uniform distribution of $m\oplus k$ if $k$ is uniformly distributed? All values $m, k, c$ are $n$-bit strings. $\oplus$ stands for the bitwise modulo-2 addition.
How to prove uniform distribution of $c=m\oplus k$ if $k$ is uniformly distributed? $m$ may be of any distribution and statistically independant of $k$.
For example you have a $m$-bit string that is with probability p=1 always '1111...111'. Adding it bitwise to a random $k$-bit string which is uniformly distributed makes the result also uniformly distributed. Why?

This is not true. For example, if $m = k$, $c$ is not uniformly distributed.
