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Deterministic factorization of sums and differences of powers


Author: Markus Hittmeir
Journal: Math. Comp.
MSC (2010): Primary 11A51
DOI: https://doi.org/10.1090/mcom/3197
Published electronically: April 7, 2017
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Abstract: Choose $ a,b \in \mathbb{N}$ and let $ N$ be a number of the form $ a^n\pm b^n$, $ n\in \mathbb{N}$. We will generalize a result of Bostan, Gaudry and Schost (2007) and prove that we may compute deterministically the prime factorization of $ N$ in

$\displaystyle \mathcal {O}\Big {(}\textsf {M}_{\textup {int}}\Big {(}N^{1/4}\sqrt {\log N}\Big {)}\Big {)},$

$ \textsf {M}_{\textup {int}}(k)$ denoting the cost for multiplying two $ \lceil k\rceil $-bit integers. This result is better than the currently best known general bound for the runtime complexity for deterministic integer factorization.

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Additional Information

Markus Hittmeir
Affiliation: Hellbrunnerstraße 34, A-5020 Salzburg, Austria
Email: markus.hittmeir@sbg.ac.at

DOI: https://doi.org/10.1090/mcom/3197
Received by editor(s): December 22, 2015
Received by editor(s) in revised form: July 2, 2016, and July 21, 2016
Published electronically: April 7, 2017
Additional Notes: The author was supported by the Austrian Science Fund (FWF): Project F5504-N26, which is a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.
Article copyright: © Copyright 2017 American Mathematical Society