On the complexity of computing gröbner bases for quasi-homogeneous systems

Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

Volumn , Issue , 2013, Pages 189-196

  Faugère, Jean Charles ^a   El Din, Mohab Safey ^a,b   Verron, Thibaut ^a,c  

^a INRIA   (France)

^b INSTITUT UNIVERSITAIRE DE FRANCE   (France)

^c ECOLE NORMALE SUPÉRIEURE   (France)

Author keywords

Gr bner bases; Polynomial system solving; Quasi homogeneous polynomials

Indexed keywords

DESIGN STRATEGIES; GENERIC SYSTEM; GENERICITY; HOMOGENEOUS SYSTEM; POLYNOMIAL SYSTEM SOLVING; WEIGHTED DEGREE; ZERO-DIMENSIONAL;

CRYPTOGRAPHY;

POLYNOMIALS;

EID: 84882260108     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/2465506.2465943     Document Type: Conference Paper

References (21)

1. G. Agnarsson. On the Sylvester denumerants for general restricted partitions. In Proceedings of the Thirty-third Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 2002), volume 154, pages 49-60, 2002.
2. M. Bardet. Étude des systèmes algébriques surdéterminés. Applications aux codes correcteurs et à la cryptographie. Thesis, Université Pierre et Marie Curie - Paris VI, Dec. 2004.
3. M. Bardet, J.-C. Faugère, and B. Salvy. On the complexity of the F5 Gröbner basis algorithm. Private communication, 2012.
4. T. Becker and V. Weispfenning. Gröbner bases, volume 141 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1993. A computational approach to commutative algebra, In cooperation with Heinz Kredel.
5. W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3-4):235-265, 1997. Computational algebra and number theory (London, 1993).
6. B. Buchberger. A theoretical basis for the reduction of polynomials to canonical forms. ACM SIGSAM Bull., 10(3):19-29, 1976.
7. A. Dickenstein and I. Z. Emiris. Multihomogeneous resultant matrices. In Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, pages 46-54, New York, 2002. ACM.
8. D. Eisenbud. Commutative algebra, volume 150 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. With a view toward algebraic geometry.
9. J.-C. Faugére. A new efficient algorithm for computing Gröbner bases (F4). J. Pure Appl. Algebra, 139(1-3):61-88, 1999. Effective methods in algebraic geometry (Saint-Malo, 1998).
10. J.-C. Faugère. A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). In Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, pages 75-83 (electronic), New York, 2002. ACM.
11. J. C. Faugère, P. Gianni, D. Lazard, and T. Mora. Efficient computation of zero-dimensional Gröbner bases by change of ordering. J. Symbolic Comput., 16(4):329-344, 1993.
12. J.-C. Faugère and A. Joux. Algebraic cryptanalysis of hidden field equation (HFE) cryptosystems using Gröbner bases. In Advances in cryptology-CRYPTO 2003, volume 2729 of Lecture Notes in Comput. Sci., pages 44-60. Springer, Berlin, 2003.
13. J.-C. Faugère and C. Mou. Sparse FGLM algorithms. Preprint available at http://hal.inria.fr/hal-00807540.
14. J.-C. Faugère and S. Rahmany. Solving systems of polynomial equations with symmetries using SAGBI-Gröbner bases. In ISSAC 2009-Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, pages 151-158. ACM, New York, 2009.
15. J.-C. Faugère, M. Safey El Din, and P.-J. Spaenlehauer. Gröbner bases of bihomogeneous ideals generated by polynomials of bidegree (1;1): Algorithms and complexity. J. Symbolic Comput., 46(4):406-437, 2011.
16. J.-C. Faugère. FGb: A Library for Computing Gröbner Bases. In K. Fukuda, J. Hoeven, M. Joswig, and N. Takayama, editors, Mathematical Software - ICMS 2010, volume 6327 of Lecture Notes in Computer Science, pages 84-87, Berlin, Heidelberg, September 2010. Springer Berlin/Heidelberg.
17. J.-C. Faugère, P. Gaudry, L. Huot, and G. Renault. Using symmetries in the index calculus for elliptic curves discrete logarithm. Cryptology ePrint Archive, Report 2012/199, 2012.
18. R. Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.
19. J. S. Milne. Algebraic geometry (v5.22), 2012. Available at www.jmilne.org/math/.
20. L. Robbiano. On the theory of graded structures. J. Symbolic Comput., 2(2):139-170, 1986.
21. R. P. Stanley. Hilbert functions of graded algebras. Advances in Math., 28(1):57-83, 1978.