  A scheme to encode arbitrarily long integer pairs on degenerate optical
parametric oscillations multiplexed in time is proposed. The classical
entanglement between the polarization directions and the phases of the
oscillating pulses, regarded as two computational registers, furnishes the
integer correlations within each pair. We show the major algorithmic steps,
modular exponentiation and discrete Fourier transform, of Shor's quantum
factoring algorithm can be executed in the registers as pulse interferences
under the assistance of external logics. The factoring algorithm is thus
rendered equivalent to a semi-classical optical-path implementation that is
scalable and decoherence-free. The sought-after multiplicative order, from
which the prime factors are deduced, is identified from a two-dimensional
fringe image generated by four-hole interference measured at the end of the
path.
