Reducing the factorization of a semiprime integer to the integration of highly oscillatory functions

We reduce the problem of factoring a semiprime integer to the problem of (numerically) integrating a certain highly oscillatory function. We provide two algorithms to address this problem, one based on the residue theorem and the other in the (extended) Cauchy’s argument principle. In the former algorithm, we show that computing the residue of the function at a certain pole leads to obtaining the factors of a semiprime integer. In the latter, we consider a contour integral for which we are able to obtain an analytical solution with several branches. The computational difficulty reduces to discovering the branch of the solution which gives the precise integral. We address this problem by numerically computing an upper and lower bound of the integral and then considering the branch that fits these bounds. The time-complexity of the algorithms is left as an open problem.