How is information disclosed by modular multiplication?
Can an attacker learn some bits of a using this information?


No.  In the case of multiplication modulo a prime, we have, for any possible value of $a$, there is a unique value of $b$ that makes $a \cdot b \bmod p$ give any particular value of $c$ in the range $(1, p-1)$.

That is, even if we knew all the bits of $c$, no particular value of $a$ are any more likely than any other (assuming we have no information about $b$).  And, in particular, we have no information on any of the bits of $a$ (other than what's implied by $1 \le a &lt; p$, which we already knew).

Any observation in the non-modular case (where this observation does not apply) doesn't apply to the modular case.