Abstract
A new branch of Gröbner basis algorithm over boolean ring has been presented in an earlier paper. In this paper, the detailed implementation and a rough complexity analysis is given. The branch Gröbner basis algorithm implements a variation of the F5 algorithm and bases on the ZDD data structure, which is also the data structure of the framework PolyBoRi. This branch Gröbner basis algorithm is mainly used to solve algebraic systems and attack multivariable cryptosystems, and its goal is to lower the complexity in each branch and expect better total complexity. An important proposition ensures the two original criteria of the non-branch F5 algorithm could still reject almost all unnecessary computations in this new branch algorithm. The timings show this branch algorithm performs very well for randomly generated systems as well as a class of stream ciphers which is generated by the linear feedback shift register (LFSR).
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References
Wu, W.T.: Basic principles of mechanical theorem-proving in elementary geometries. J. Sys. Sci. Math. Sci. 4, 207–235 (1984)
Wu, W.T.: Basic principles of mechanical theorem-proving in elementary geometries. J. Autom. Reason. 2, 221–252 (1986)
Faugère, J.C.: A new efficient algorithm for computing Gröbner bases (f4). J. Pure Appl. Algebr. 139(1), 61–88 (1999)
Faugère, J.C.: A new efficient algorithm for computing Grönber bases without reduction to zero (F5). Symbolic and Algebraic Computation, Porc. Conferenz ISSAC 2002, 75–83 (2002)
Brickenstein, M., Dreyer, A.: PolyBoRi: A framework for Gröbner basis computations with boolean polynomials. MEGA 2007, Austria (2007)
Gao, X.S., Chai, F.J., Yuan, C.M.: A characteristic set method for equation solving in F2 and applications in cryptanalysis of stream ciphers. J. Syst. Sci. Complex. 21, 191–208 (2008)
Sun, Y., Wang, D.K.: Branch Gröbner bases algorithm over boolean ring (Chinese). Preprint (2009)
Bardet, M., Faugère, J.C., Salvy, B.: Complexity of Grönber basis computation for Semi-regular overdetermined sequences over F\(_2\) with solutions in F\(_2\). In: Proceedings of the ICPPSS International Conference on Polynomial System Solving Paris, November 24–25-26 2004 in honor of Daniel Lazard (2004)
Faugère, J.C., Ars, G.: An Algebraic Cryptanalysis of Nonlinear Filter Generators Using Gröbner Bases. Reserch report 4739, Institut National de Recherche en Informatique et en Automatique, Lorraine (2003)
Acknowledgments
We thank Professor Xiaoshan Gao for his useful suggestions and Zhenyu Huang for discussing the programming codes.
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Sun, Y., Wang, D. (2014). The Implementation and Complexity Analysis of the Branch Gröbner Bases Algorithm Over Boolean Polynomial Rings. In: Feng, R., Lee, Ws., Sato, Y. (eds) Computer Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43799-5_14
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DOI: https://doi.org/10.1007/978-3-662-43799-5_14
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