NP‑hardness of integer factorization

Determine whether the integer factorization problem (given a positive integer n in binary, output a nontrivial factor of n) is NP‑hard under standard polynomial‑time reductions from decision problems.

Background

While factoring is efficiently solvable on a real RAM with rounding and on quantum computers via Shor’s algorithm, its placement within classical NP‑hardness theory is unresolved. Establishing NP‑hardness would have major implications for cryptography and complexity theory.

The authors explicitly state that NP‑hardness of factoring is unknown, highlighting this as a separate open question from polynomial‑time solvability.