License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2019.26
URN: urn:nbn:de:0030-drops-106023
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10602/
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Björklund, Andreas ; Kaski, Petteri ; Williams, Ryan

Solving Systems of Polynomial Equations over GF(2) by a Parity-Counting Self-Reduction

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Abstract

We consider the problem of finding solutions to systems of polynomial equations over a finite field. Lokshtanov et al. [SODA'17] recently obtained the first worst-case algorithms that beat exhaustive search for this problem. In particular for degree-d equations modulo two in n variables, they gave an O^*(2^{(1-1/(5d))n}) time algorithm, and for the special case d=2 they gave an O^*(2^{0.876n}) time algorithm.
We modify their approach in a way that improves these running times to O^*(2^{(1-1/(2.7d))n}) and O^*{2^{0.804n}), respectively. In particular, our latter bound - that holds for all systems of quadratic equations modulo 2 - comes close to the O^*(2^{0.792n}) expected time bound of an algorithm empirically found to hold for random equation systems in Bardet et al. [J. Complexity, 2013]. Our improvement involves three observations:
1) The Valiant-Vazirani lemma can be used to reduce the solution-finding problem to that of counting solutions modulo 2.
2) The monomials in the probabilistic polynomials used in this solution-counting modulo 2 have a special form that we exploit to obtain better bounds on their number than in Lokshtanov et al. [SODA'17].
3) The problem of solution-counting modulo 2 can be "embedded" in a smaller instance of the original problem, which enables us to apply the algorithm as a subroutine to itself.

BibTeX - Entry

@InProceedings{bjrklund_et_al:LIPIcs:2019:10602,
  author =	{Andreas Bj{\"o}rklund and Petteri Kaski and Ryan Williams},
  title =	{{Solving Systems of Polynomial Equations over GF(2) by a Parity-Counting Self-Reduction}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{26:1--26:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10602},
  URN =		{urn:nbn:de:0030-drops-106023},
  doi =		{10.4230/LIPIcs.ICALP.2019.26},
  annote =	{Keywords: equation systems, polynomial method}
}

Keywords: equation systems, polynomial method
Collection: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 04.07.2019

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