Factoring using multiplicative relations modulo $n$: a subexponential algorithm inspired by the index calculus

TL;DR

This paper presents a modified index calculus algorithm for integer factorization that operates in subexponential time, offering a conceptually simple approach inspired by number field sieve techniques.

Contribution

It introduces a new factoring algorithm based on multiplicative relations modulo n, combining index calculus ideas with a subexponential runtime.

Findings

Achieves subexponential runtime $ ext{exp}(O( oot rac{1}{2}{ ext{log} n ext{log} ext{log} n}))$

Requires a more intensive rational linear algebra phase

Simplifies the index calculus approach for factoring

Abstract

We demonstrate that a modification of the classical index calculus algorithm can be used to factor integers. More generally, we reduce the factoring problem to finding an overdetermined system of multiplicative relations in any factor base modulo $n$, where $n$ is the integer whose factorization is sought. The algorithm has subexponential runtime $\exp(O(\sqrt{\log n \log \log n}))$ (or $\exp(O( (\log n)^{1/3} (\log \log n)^{2/3} ))$ with the addition of a number field sieve), but requires a rational linear algebra phase, which is more intensive than the linear algebra phase of the classical index calculus algorithm. The algorithm is certainly slower than the best known factoring algorithms, but is perhaps somewhat notable for its simplicity and its similarity to the index calculus.