Given a Polig-Hellman (is that really the name for that?) system with
$$C = M^k \bmod p$$
where $M$ is the message, $C$ is the ciphertext, $k$ is a (secret) key (any integer relatively prime to $p-1$), and $p$ is a large public prime.
Encrypting twice like this
$$(M^{k_{A}})^{k_{AB}} \bmod p = (M^{k_{B}})^{k_{BA}} \bmod p$$
where the second pair of keys is generated according to https://crypto.stackexchange.com/a/42140/41839, by selecting a random $k_B$ (relatively prime to $p-1$), and then computing
$$k_{BA} = k_{A} k_{AB} k_{B}^{-1} \bmod p-1$$
If the second encryption step with $k_{AB}$ and $k_{BA}$) is done by an untrusted server, so it knows these keys, what are the security aspects of such a system (besides not offering ciphertext indistinguishability as wanted)?
Given more key pairs matching the first one, can you calculate the secret key $k_{A}$ by solving the following equations if you know $k_{AB}$, $k_{AC}$, $k_{BA}$ and $k_{CA}$?
$$k_{BA} = k_{A} k_{AB} k_{B}^{-1} \bmod p-1$$ $$k_{CA} = k_{A} k_{AC} k_{C}^{-1} \bmod p-1$$
