find $x$, given $\{c_ix = k_i + y_i\}_{i=[1,n]} $ Given 

$$c_1x = k_1 + y_1 $$
$$c_2x = k_2 + y_2 $$
$$\vdots $$
$$c_nx = k_n + y_n $$

where the values of $\{c_1 \ldots c_n \}$ and $\{ k_1 \ldots k_n \}$ are known, and $x, \{y_1 \ldots y_n \}$ are unknown. $y_i$ is chosen uniformly at random.

We are working in a field of $GF(p)$, so all $c_i, k_i, x, y_i ∈ \mathbb{Z}_p^*$. $p$ is some prime. 

From here, how can we recover the value of $x$ ?

(suppose $c_i = c_j$ OR  $\ k_i = k_j$ for some $i,j$ that we know )
There is no way to recover $x$ in this scenario; there is not even a way to say "this value of $x$ is more likely than that value"

One demonstration is that if we attempt to test a particular value $x'$; we can compute the necessary values:

$$y_i = c_i x - k_i$$

However, if all the values $y_i$ are randomly distributed, and given that we have no further restriction of what the values $y_i$ might be, this distribution is no less probable than any other distribution; hence these relations give us no information about $x$.

Since those relations are the only thing we know about $x$, that means we have no criteria to say which values are more or less probable.

This holds true for any sets of values $c_i, k_i$, and specifically if we have $c_i = c_j$ or $k_i = k_j$.