Integer factorization as subset-sum problem

TL;DR

This paper presents a polynomial-time reduction from integer factorization to a subset-sum problem, improving specialized factorization algorithms and refining Fermat's method with deterministic and heuristic procedures.

Contribution

It generalizes a sieving technique to reduce integer factorization to subset-sum, enhancing existing algorithms and providing new deterministic and heuristic methods.

Findings

Polynomial-time reduction from factorization to subset-sum.

Improved runtime complexity for Fermat's factorization.

Deterministic and heuristic procedures with different space complexities.

Abstract

This paper elaborates on a sieving technique that has first been applied in 2018 for improving bounds on deterministic integer factorization. We will generalize the sieve in order to obtain a polynomial-time reduction from integer factorization to a specific instance of the multiple-choice subset-sum problem. As an application, we will improve upon special purpose factorization algorithms for integers composed of divisors with small difference. In particular, we will refine the runtime complexity of Fermat's factorization algorithm by a large subexponential factor. Our first procedure is deterministic, rigorous, easy to implement and has negligible space complexity. Our second procedure is heuristically faster than the first, but has non-negligible space complexity.