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∗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.
Original language | English |
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Title of host publication | 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 |
Editors | Ioannis Chatzigiannakis, Christel Baier, Stefano Leonardi, Paola Flocchini |
Publisher | Schloss Dagstuhl-Leibniz-Zentrum für Informatik |
Pages | 1-13 |
ISBN (Electronic) | 9783959771092 |
DOIs | |
Publication status | Published - 1 Jul 2019 |
MoE publication type | A4 Article in a conference publication |
Event | International Colloquium on Automata, Languages and Programming - Patras, Greece Duration: 8 Jul 2019 → 12 Jul 2019 Conference number: 46 |
Publication series
Name | Leibniz international proceedings in informatics |
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Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |
Volume | 132 |
ISSN (Print) | 1868-8969 |
Conference
Conference | International Colloquium on Automata, Languages and Programming |
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Abbreviated title | ICALP |
Country/Territory | Greece |
City | Patras |
Period | 08/07/2019 → 12/07/2019 |
Keywords
- Equation systems
- Polynomial method