Linear algebra to compute syzygies and Gröbner bases

Authors:
Daniel Cabarcas;Jintai Ding
Affiliations:
University of Cincinnati, Cincinnati, OH, USA;University of Cincinnati, Cincinnati, OH, USA
Venue:
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Year:
2011

Citing 13
Cited 0

On an installation of Buchberger's algorithm

Journal of Symbolic Computation
Gröbner bases computation using syzygies

ISSAC '92 Papers from the international symposium on Symbolic and algebraic computation
The MAGMA algebra system I: the user language

Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
A criterion for detecting unnecessary reductions in the construction of Groebner bases

EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations

EUROCAL '83 Proceedings of the European Computer Algebra Conference on Computer Algebra
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)

Proceedings of the 2002 international symposium on Symbolic and algebraic computation
MXL2: Solving Polynomial Equations over GF(2) Using an Improved Mutant Strategy

PQCrypto '08 Proceedings of the 2nd International Workshop on Post-Quantum Cryptography
Hidden fields equations (HFE) and isomorphisms of polynomials (IP): Two new families of asymmetric algorithms

EUROCRYPT'96 Proceedings of the 15th annual international conference on Theory and application of cryptographic techniques
Efficient algorithms for solving overdefined systems of multivariate polynomial equations

EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
A new incremental algorithm for computing Groebner bases

Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
F5C: A variant of Faugère's F5 algorithm with reduced Gröbner bases

Journal of Symbolic Computation
MXL3: an efficient algorithm for computing gröbner bases of zero-dimensional ideals

ICISC'09 Proceedings of the 12th international conference on Information security and cryptology
Flexible partial enlargement to accelerate gröbner basis computation over F2

AFRICACRYPT'10 Proceedings of the Third international conference on Cryptology in Africa

Quantified Score

Hi-index	0.00

Visualization

Abstract

In this paper, we introduce a new method to avoid zero reductions in Gröbner basis computation. We call this method LASyz, which stands for Lineal Algebra to compute Syzygies. LASyz uses exhaustively the information of both principal syzygies and non-trivial syzygies to avoid zero reductions. All computation is done using linear algebra techniques. LASyz is easy to understand and implement. The method does not require to compute Gröbner bases of subsequences of generators incrementally and it imposes no restrictions on the reductions allowed. We provide a complete theoretical foundation for the LASyz method and we describe an algorithm to compute Gröbner bases for zero dimensional ideals based on this foundation. A qualitative comparison with similar algorithms is provided and the performance of the algorithm is illustrated with experimental data.