Mathematics > Number Theory
[Submitted on 13 Nov 2022 (v1), last revised 16 Jul 2023 (this version, v2)]
Title:Factoring using multiplicative relations modulo $n$: a subexponential algorithm inspired by the index calculus
View PDFAbstract: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.
Submission history
From: Katherine E. Stange [view email][v1] Sun, 13 Nov 2022 05:28:04 UTC (16 KB)
[v2] Sun, 16 Jul 2023 14:20:32 UTC (16 KB)
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