I'm looking to show that the two terms described at the bottom are identically distributed, but I'm not sure how to show this:
Denoting by $\in_R$ a random uniform choice from a set, let:
$A\in_{R} \mathbb{Z}^{m\times n}_q$
$ r \in_{R} \{0,1\}^m$
$ l, s \in_{R} \{0,1\}^n$
$ k \in_R \{0,1\}$
Then the following are identically distributed:
$$ (r^T A +\frac{q}{2} l, r^T A s+\frac{q}{2}k) $$ $$ (r^T A, r^T A s) $$
How can I accomplish this? (if at all possible -I only have a vague intuition that this should be correct, but cannot formally show this).