  We describe a novel analogue algorithm that allows the simultaneous
factorization of an exponential number of large integers with a polynomial
number of experimental runs. It is the interference-induced periodicity of
"factoring" interferograms measured at the output of an analogue computer that
allows the selection of the factors of each integer [1,2,3,4]. At the present
stage the algorithm manifests an exponential scaling which may be overcome by
an extension of this method to correlated qubits emerging from n-order quantum
correlations measurements. We describe the conditions for a generic physical
system to compute such an analogue algorithm. A particular example given by an
"optical computer" based on optical interference will be addressed in the
second paper of this series [5].
