\begin{align*}
\sum_{i=0}^{l-1}{2^{i p}x_i} &\equiv \sum_{i=0}^{l-1}{(2^{i p}x_i \bmod m)} \tag{mod m} \\
&\equiv \sum_{i=0}^{l-1}{(2^{i p} \bmod m)(x_i \bmod m)} \tag{mod m}\\
&\equiv \sum_{i=0}^{l-1}{x_i \bmod m} \tag{mod m, $m=2^p-1$}
\end{align*}