Does $(u_1=r_1\cdot x,\ \ w_1=r_1\cdot z, \ \ u_2=r_2\cdot x,\ \ w_2=r_2\cdot z)$ Leak Information? Consider we have a fixed value $x$. We pick values $ r_1,r_2,z\leftarrow \mathbb{F}_p$, where $p$ is a large prime number, and it is public. So $r_1,r_2,z$ are uniformly random values.

We compute four values as below:


$u_1=r_1\cdot x,\ \ w_1=r_1\cdot z$
$u_2=r_2\cdot x,\ \ w_2=r_2\cdot z$


Question: Do $u_1,u_2,w_1,w_2$ leak information about values $x,r_1,z,r_2$?

In other words: can an adversary learn about the values $x,r_1,z,r_2$ when given $u_1,u_2,w_1,w_2$?

Edit: $u_i, w_i\neq 0$
Yes, this does leak some information.  In fact, just knowing $u_1$ and $w_1$ will allow the attacker to calculate $u_1^{-1} w_1 = x^{-1} z$.*  In particular, knowing this value will let the attacker calculate $x$ if they know $z$, or vice versa.

Similarly, knowing $u_1$ and $u_2$ will let the attacker calculate $u_1^{-1} u_2 = r_1^{-1} r_2$.  With $u_1$, $w_1$ and $u_2$, they can calculate both of these.  (Knowing $w_2$ will not actually add any further information, since it can be calculated from the other values as $u_2 u_1^{-1} w_1 = u_2 x^{-1} z = r_2 z = w_2$.)

Without additional information, however, you cannot work out the exact value of any single parameter.  In particular, given any (non-zero) $x$, we can choose $r_1$ and $r_2$ so that $u_1 = r_1x$ and $u_2 = r_2x$ take any values we like, and then further choose $z$ so that $w_1 = r_1z$ takes any value we like; $w_2 = r_2z$ will then equal $u_2 u_1^{-1} w_1$, as it always must.

*) I'm assuming here that all the values are non-zero, and therefore invertible, which they most likely are if they're chosen at random modulo a large prime.  You can work out the special cases where some of the values are zero as an exercise yourself, if you like.