We study the problem of strongly coprime factorization over H-infinity of the unit disc. We give a necessary and sufficient condition for the existence of such a coprime factorization in terms of an optimal control problem over the doubly infinite discrete-time axis. In particular, we show that an equivalent condition for the existence of such a coprime factorization is that both the control and filter algebraic Riccati equation (of an arbitrary realization) have a solution (in general unbounded and even nondensely defined) and that a coupling condition involving these solutions is satisfied.