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

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.

Note: 7 pages

Available format(s)
Category
Foundations
Publication info
Preprint.
Keywords
factoring index calculus
Contact author(s)
History
2022-11-17: approved
See all versions
Short URL
https://ia.cr/2022/1588

CC BY

BibTeX

@misc{cryptoeprint:2022/1588,
author = {Katherine E. Stange},
title = {Factoring using multiplicative relations modulo n: a subexponential algorithm inspired by the index calculus},
howpublished = {Cryptology ePrint Archive, Paper 2022/1588},
year = {2022},
note = {\url{https://eprint.iacr.org/2022/1588}},
url = {https://eprint.iacr.org/2022/1588}
}

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