Factoring with Hints

TL;DR

This paper presents a new deterministic factoring algorithm that leverages partial factorization hints of nearby integers to recover the prime factors of a large composite, revealing relationships between close integers' factorizations.

Contribution

The paper introduces a novel deterministic factoring method using known factorizations of nearby integers, highlighting potential for further research in related algorithms.

Findings

Factorizations of close integers are related.

The new algorithm operates in $O(N^{1/3+psilon})$ time.

This approach suggests new directions in factoring research.

Abstract

We introduce a new deterministic factoring algorithm, which could be described in the cryptographically fashionable term of "factoring with hints": we show that, given the knowledge of the factorisations of $O(N^{1/3+\epsilon})$ terms surrounding $N=pq$ product of two large primes, we can recover deterministically $p$ and $q$ in $O(N^{1/3+\epsilon})$ bit operations. Although this is slower than the current best factoring algorithms, this method shows that the factorisations of close integers are related and that consequently one can expect more results along this line of thought.