eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-07-04
26:1
26:13
10.4230/LIPIcs.ICALP.2019.26
article
Solving Systems of Polynomial Equations over GF(2) by a Parity-Counting Self-Reduction
Björklund, Andreas
1
Kaski, Petteri
2
Williams, Ryan
3
Department of Computer Science, Lund University, Sweden
Department of Computer Science, Aalto University, Finland
Department of Electrical Engineering and Computer Science & CSAIL, MIT, Cambridge, MA, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol132-icalp2019/LIPIcs.ICALP.2019.26/LIPIcs.ICALP.2019.26.pdf
equation systems
polynomial method