$k_0$ is randomly generated and subsequently $k_{i+1} = k_{i}+a$, where $a$ is a large unknown constant
I've tried formulating an attack as such:
Given 3 Messages $M_0$, $M_1$ $M_2$and with $(r_0,s_0)$, $(r_1,s_1)$ and $(r_2,s_2)$ respectively. We can write $r_0$, $r_1$ and $r_2$ as such:
$$r_0 = g^{k_0} {\ }mod{\ } p $$ $$r_i = g^{k_i} {\ }mod{\ } p = g^{k_0}g^{a*i} {\ }mod{\ } p = r_0g^{a*i} {\ }mod{\ } p$$
$s_0$, $s_1$ and $s_2$ can then be rewritten as the following:
$$s_0k_0 = M_0-xr_0{\ } mod{\ } p-1$$ $$s_1(k_0+a) = M_1-xr_0g^{a}{\ }mod{\ } p-1$$ $$s_2(k_0+2a) = M_2-xr_0g^{2a}{\ }mod{\ } p-1$$
and making $xr_0$ the subject we get the following: $$xr_0 = M_0 - s_0k_0$$ $$xr_0 = g^{-a}(M_1 - s_1(k_0+a))$$ $$xr_0 = g^{-2a}(M_2 - s_2(k_0+2a))$$
But I do not think we can solve this using linear equations because we do not know what is $g^{-a}$ and $g^{-2a}$.