Improved algorithms for solving polynomial systems over GF(2) by multiple parity-counting

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5 Scopus citations

Abstract

We consider the problem of finding a solution to a multivariate polynomial equation system of degree d in n variables over F2. For d = 2, the best-known algorithm for the problem is by Bardet et al. [J. Complexity, 2013] and was shown to run in time O(20.792n) under assumptions that were experimentally found to hold for random equation systems. The best-known worst-case algorithm for the problem is due to Björklund et al. [ICALP'19]. It runs in time O(20.804n) for d = 2 and O(2(1−1/(2.7d))n) for d > 2. In this paper, we devise a worst-case algorithm that improves the one by Björklund et al. It runs in time O(20.6943n) (or O(1.6181n)) for d = 2 and O(2(1−1/(2d))n) for d > 2. Our algorithm thus outperforms all known worst-case algorithms, as well as ones analyzed for random equation systems. We also devise a second algorithm that outputs all solutions to a polynomial system and has similar complexity to the first (provided that the number of solutions is not too large). A central idea in the work of Björklund et al. was to reduce the problem of finding a solution to a polynomial system over F2 to the problem of counting the parity of all solutions. A parity-counting instance was then reduced to many smaller parity-counting instances. Our main observation is that these smaller instances are related and can be solved more efficiently by a new algorithm to a problem which we call multiple parity-counting.

Original languageEnglish
Title of host publicationACM-SIAM Symposium on Discrete Algorithms, SODA 2021
EditorsDaniel Marx
PublisherAssociation for Computing Machinery
Pages2550-2564
Number of pages15
ISBN (Electronic)9781611976465
StatePublished - 1 Jan 2021
Event32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 - Alexandria, Virtual, United States
Duration: 10 Jan 202113 Jan 2021

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Conference

Conference32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021
Country/TerritoryUnited States
CityAlexandria, Virtual
Period10/01/2113/01/21

ASJC Scopus subject areas

  • Software
  • Mathematics (all)

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