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We demonstrate that a modification of the classical index calculus algorithmcan be used to factor integers. More generally, we reduce the factoring problemto finding an overdetermined system of multiplicative relations in any factorbase modulo $n$, where $n$ is the integer whose factorization is sought. Thealgorithm 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 numberfield sieve), but requires a rational linear algebra phase, which is moreintensive than the linear algebra phase of the classical index calculusalgorithm. The algorithm is certainly slower than the best known factoringalgorithms, but is perhaps somewhat notable for its simplicity and itssimilarity to the index calculus.

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