Paper 2019/903
Probabilistic analysis on Macaulay matrices over finite fields and complexity of constructing Gröbner bases
Igor Semaev and Andrea Tenti
Abstract
Gröbner basis methods are used to solve systems of polynomial equations over finite fields, but their complexity is poorly understood. In this work an upper bound on the time complexity of constructing a Gröbner basis and finding a solutions of a system is proved. A key parameter in this estimate is the degree of regularity of the leading forms of the polynomials. Therefore, we provide an upper bound on the degree of regularity for a sufficiently overdetermined system of forms over any finite field. The bound holds with probability tending to 1 and depends only on the number of variables, the number of polynomials, and their degrees. Our results imply that sufficiently overdetermined systems of polynomial equations are solvable in polynomial time with high probability.
Metadata
- Available format(s)
-
PDF
- Category
- Foundations
- Publication info
- Published elsewhere. Journal of Algebra
- DOI
- 10.1016/j.jalgebra.2020.08.035
- Keywords
- Polynomial equation systemsfinite fieldsMacaulay matricesGroebner basesmultisetscomplexity
- Contact author(s)
- andrea tenti @ uib no
- History
- 2020-10-02: last of 2 revisions
- 2019-08-08: received
- See all versions
- Short URL
- https://ia.cr/2019/903
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2019/903,
author = {Igor Semaev and Andrea Tenti},
title = {Probabilistic analysis on Macaulay matrices over finite fields and complexity of constructing Gröbner bases},
howpublished = {Cryptology {ePrint} Archive, Paper 2019/903},
year = {2019},
doi = {10.1016/j.jalgebra.2020.08.035},
url = {https://eprint.iacr.org/2019/903}
}
