Paper 2022/1588
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
Metadata
 Available format(s)
 Category
 Foundations
 Publication info
 Preprint.
 Keywords
 factoring index calculus
 Contact author(s)
 kstange @ math colorado edu
 History
 20221117: approved
 20221115: received
 See all versions
 Short URL
 https://ia.cr/2022/1588
 License

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} }