Upper and lower bounds on the OBDD-width of a special integer multiplication

We consider the Boolean function ${\mathrm{S}\mathrm{M}\mathrm{u}\mathrm{l}}_{n-1}^n(\mathit{x},\mathit{y})$, which computes the middle bit of the multiplication of two natural numbers represented as $n$-bit binary strings $\mathit{x}$ and $\mathit{y}$, drawn from a restricted domain. We investigate the width of OBDDs computing ${\mathrm{S}\mathrm{M}\mathrm{u}\mathrm{l}}_{n-1}^n$. We introduce a combinatorially defined function $s_{\ast }(n)$ and show that the width of such OBDDs is $\mathrm{\Theta }(2^{s_{\ast }(n)})$.