  When considered as orthogonal bases in distinct vector spaces, the unit
vectors of polarization directions and the Laguerre-Gaussian modes of
polarization amplitude are inseparable, constituting a so-called classical
entangled light beam. We apply this classical entanglement to demonstrate
theoretically the execution of Shor's factoring algorithm on a classical light
beam. The demonstration comprises light-path designs for the key algorithmic
steps of modular exponentiation and Fourier transform on the target integer 15.
The computed multiplicative order that eventually leads to the integer factors
is identified through a four-hole diffraction interference from sources
obtained from the entangled beam profile. We show that the fringe patterns
resulted from the interference are uniquely mapped to the sought-after order,
thereby emulating the factoring process originally rooted in the quantum
regime.
