No Way Back Home
We are presented a key-exchange protocol where there is some $v$ generated by $p * x$, where $p$ is one of the two primes given, $x$ is a random number generated. We are also given the following:
$$ vka = v \times k_A \mod n $$
$$ vkakb = vka \times k_B \mod n $$
$$ vkb = v \times k_B \mod n $$
Obviously we can calculate $k_A$ by doing $(vkb)^{-1} \times vkakb$, but unfortunately $v$ is a multiple of $p$ that is not possible.