Let $c_0,\dots,c_k$ be some known nonnegative integers.
Consider the following system of equations in the unknown bits $a_i,b_i$, i.e. $a_i,b_i\in \{0,1\}$:
$$ \left\{\begin{array}{l} a_0b_0& =& c_0,\\ a_0b_1+a_1b_0& =& c_1,\\ a_0b_2+a_1b_1+a_2b_0& =& c_2,\\ \vdots && \\ \sum_{i+j=k}a_ib_j &=& c_k \\ \end{array}\right.$$
I am looking for an upper bound of the number of solutions of this equation. For instance, if $k=0$, there are at most $2$ solutions (achieved when $c_0\leq 1$).
In addition, I would like to restrict the solutions to a subspace and count the number of solutions there (for instance, restricting $\sum_{i=0}^ka_k=k/4$).
I know there are some nice interpretations with lattices, as in Coppersmith method, but I don't know if that applies here. Any reference or direction will be appreciated.