TY - GEN

T1 - Improved algorithms for solving polynomial systems over GF(2) by multiple parity-counting

AU - Dinur, Itai

N1 - Publisher Copyright:
© 2021 by SIAM

PY - 2021/1/1

Y1 - 2021/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85105267262&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:85105267262

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 2550

EP - 2564

BT - ACM-SIAM Symposium on Discrete Algorithms, SODA 2021

A2 - Marx, Daniel

PB - Association for Computing Machinery

T2 - 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021

Y2 - 10 January 2021 through 13 January 2021

ER -