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

Authors Andreas Björklund, Petteri Kaski, Ryan Williams

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Andreas Björklund
  • Department of Computer Science, Lund University, Sweden
Petteri Kaski
  • Department of Computer Science, Aalto University, Finland
Ryan Williams
  • Department of Electrical Engineering and Computer Science & CSAIL, MIT, Cambridge, MA, USA

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Andreas Björklund, Petteri Kaski, and Ryan Williams. Solving Systems of Polynomial Equations over GF(2) by a Parity-Counting Self-Reduction. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 26:1-26:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • equation systems
  • polynomial method


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