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?

I don't think the answer is correct because it relies on you making the assumption and then using that assumption to prove the point.
i.e.
Given that the event {Y=m} occurs, suppose that the conditional distribution of X conditioned on {Y=m} is uniform on the 2n values of X. 

I'm not sure if this is allowed because this is the part you are trying to prove and it creates a circularity problem. It is because you are effectively using two constraints, that it is conditional and the other is uniform. Then you are stating that it works even when there is unconditional probability, but you have already "assumed it is uniform" previously, so you can't just use that again...
The uniformality part just appears from nowhere without justification.
I would just use the empirical proof such as the last two points mentioned in Dilip's answer.
