Can big numbers multiplication be a valid form of encryption?
Given a vector $[S_0 S_1 x_1, S_0 S_1 x_2, ...]$, it is quite easy to recover $S_0 S_1$ (by computing the GCD of the various elements).  With that information, the attacker can then recover the values $x_1, x_2, ...$, and so yes, a semihonest adversary could easily recover the $X_{reg}$ values.