Does $h_{a,b}(x) = ((ax + b) \bmod 2^w)/2^{w-M}$ satisfy the Uniform Difference Property: given constant integers $w, M: w >= M$, and for any choice of distinct $x$ and $y$ in $[0, 2^w]$, $h(x) - h(y)$ is uniformly distributed over $[0, 2^M]$, given that $a, b$ are chosen uniformly at random?
If not, does it satisfy the weaker property that the probability that $h(x)-h(y) = z < k/2^M$ for some constant $k$?