All the random varibales are defined in $\mathrm{GF}_2^m$ and $\oplus$ denotes the bitwise exclusive or.
If all the three random variables $x$, $y$ and $x \oplus y$ are (seperately) independent to $z$, does it imply that $(x,y)$ is independent to $z$? Note: $x$ and $y$ do not have to be independent.
We can start by $m=1$ (I believe that the result can easily to be generalized to the case of $m > 1$), that is, all the varibles are sampled from either '1' or '0', and we have $0 \oplus 0 = 0$, $1 \oplus 0 = 1$, $0 \oplus 1 = 1$, $1 \oplus 1 = 0$.
Now I only we that: If $x$ is independent to $z$ and both of them are uniformed distributed, we have that $x$ is independent to $z \oplus x$
Thanks in advance!