LSB extraction in MPC: Doubts about correctness
The protocol is only designed to work when $a$ is in the range $\{-2^{k-1},\dots,2^{k-1}-1\}$. This ensures that $c$ doesn't overflow mod $p$, as long as $p$ is sufficiently large.
In the paper they ensure this by choosing $p &gt; 2^{k+\kappa+\log n}$, where $\kappa$ is a statistical security parameter and $n$ is the number of parties. See the discussions on p3-5.