Solving systems of polynomial equations over GF(2) by a parity-counting self-reduction
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LIPIcs-ICALP-2019-26.pdf
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Published version
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442.49 KB
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Adobe PDF
Checksum (MD5)
140fc8d9b954ebb5bb1f6a08a3182966
Author(s)
Williams, Richard Ryan
Björklund, Andreas
Kaski, Petteri
Date Issued
2019
Journal
Leibniz International Proceedings in Informatics, LIPIcs
Citation
Williams, Richard Ryan, Björklund, Andreas and Kaski, Petteri. 2019. "Solving systems of polynomial equations over GF(2) by a parity-counting self-reduction." Leibniz International Proceedings in Informatics, LIPIcs, 132.
Version
Final published version
Abstract
© Andreas Björklund, Petteri Kaski, and Ryan Williams; licensed under Creative Commons License CC-BY 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∗20.876n time algorithm. We modify their approach in a way that improves these running times to O∗2(1−1/(27d))n and O∗20.804n, respectively. In particular, our latter bound - that holds for all systems of quadratic equations modulo 2 - comes close to the O∗20.792n 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.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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Creative Commons Attribution 4.0 International license
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DOI of Published Version
https://doi.org/10.4230/LIPIcs.ICALP.2019.26
