Classical polynomial-time algorithm for integer factorization

Determine whether there exists a deterministic polynomial‑time algorithm on a standard discrete model of computation (e.g., a Turing machine or word RAM) that, given a positive integer n in binary, outputs a nontrivial factor of n.

Background

The paper shows that allowing rounding on a real RAM lets one solve the integer factoring problem in polynomial time, illustrating why access to the bit representation of reals is prohibited in their real-number model. In contrast, for standard discrete models, the classical complexity status of integer factorization remains unresolved.

Although Shor’s algorithm gives a polynomial‑time quantum algorithm for factoring, there is no known classical deterministic polynomial‑time algorithm on Turing machines or word RAMs. The authors explicitly note that this remains unknown.