Abstract
For a parametric algebraic system in finite fields, this paper presents a method for computing the cover and the refined cover based on the characteristic set method. From the cover, the author knows for what parametric values the system has solutions and at the same time presents the solutions in the form of proper chains. By the refined cover, the author gives a complete classification of the number of solutions for this system, that is, the author divides the parameter space into several disjoint components, and on every component the system has a fix number of solutions. Moreover, the author develops a method of quantifier elimination for first order formulas in finite fields.
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D. Larzard, Resolution of polynomial systems, X. S. Gao, D. Wang, eds. Computer Mathematics, Proc. of ASCM 2000, 1–8. World Scientific, Singapore, 2000.
W. Boege, R. Gebauer, and H. Kredel, Some examples for solving systems of algebraic equations by calculating Gröbner Bases, Journal of Symbolic Computation, 1986, 2(1): 83–98.
X. S. Gao, X. Hou, J. Tang, and H. Cheng, Complete solution classification for the perspective-three-point problem, IEEE Tran. on PAMI, 2003, 25(8): 930–943.
X. S. Gao and S. C. Cho, Solving parametric algebraic systems, Proc. ISSAC’92, 1992: 335–341.
X. S. Gao and D. K. Wang, Zero decomposition theorems for counting the number of solutions for parametric equation systems, Proc. of ASCM 2003, Lecture Notes Series on Computing, World Scientific, Singapore, 2003, 10: 129–144.
D. Kapur, An approach for solving systems of parametric polynomials equations, Principles, Practice of Constraint Programming, Saraswart, Van Hentenryck, MIT Press, 1995.
D. Lazard, A new method for solving algebraic systems of positive dimension, Discrete Appl. Math., 1991, 33: 147–160.
W. Y. Sit, An algorithm for solving parametric linear systems, Journal of Symbolic Computation, 1992, 13: 353–394.
V. Weispfenning, Comprehensive Gröbner bases, Journal of Symbolic Computation, 1992, 14: 1–29.
W. T. Wu, Basic principles of mechanical theorem-proving in elementary geometries, Journal Automated Reasoning, 1986, 2: 221–252.
P. Aubry, D. Lazard, and M. M. Maza, On the theory of triangular sets, Journal of Symbolic Computation, 1999, 25: 105–124.
S. C. Chou and X. S. Gao, Ritt-Wu’s decomposition algorithm and geometry theorem proving, Proc. of CADE-10, Springer, LNAI, 1990, 449: 207–220.
M. Kalkbrener, A generalized Euclidean algorithm for computing triangular representations of algebraic varieties, Journal of Symbolic Computation, 1993, 15: 143–167.
D. Lin and Z. Liu, Some results on theorem proving in geometry over finite fields, Proc. ISSAC’93, 292–300, ACM Press, New York, 1993.
J. S. Coron and B. de Weger, ECRYPT: Hardness of the main computational problems used in cryptography, European Network of Excellence in Cryptology, 2007.
R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, 1997.
X. S. Gao and Z. Huang, Characteristic set algorithms for equation solving in finite fields, Journal of Symbolic Computation, 2012, 47(6): 655–679.
Z. Huang and D. Lin, Attacking bivium and trivium with the characteristic set method, progress in cryptology-africacrypt 2011, LNCS, 2011, 6737: 77–91.
F. Chai, X. S. Gao, and C. Yuan, A Characteristic set method for solving Boolean equations and applications in cryptanalysis of stream ciphers, Journal of Systems Science and Complexity, 2008, 21(2): 191–208.
N. Jacobson, Basic Algebra, vol. 1, W. H. Freeman and Company, 1974.
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This work is supported by the National 973 Program of China under Grant No. 2011CB302400 and the National Natural Science Foundation of China under Grant No. 60970152.
This paper was recommended for publication by Editor Ziming LI.
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Huang, Z. Parametric equation solving and quantifier elimination in finite fields with the characteristic set method. J Syst Sci Complex 25, 778–791 (2012). https://doi.org/10.1007/s11424-012-0168-4
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DOI: https://doi.org/10.1007/s11424-012-0168-4