Solving Polynomial Systems over Finite Fields: Algorithms, Implementation and Applications

Chenqi Mou 1
1 PolSys - Polynomial Systems
LIP6 - Laboratoire d'Informatique de Paris 6, Inria Paris-Rocquencourt
Abstract : Polynomial system solving over finite fields is of particular interest because of its applications in Cryptography, Coding Theory, and other areas of information science and technologies. In this thesis we study several important theoretical and computational aspects for solving polynomial systems over finite fields, in particular on the two widely used tools Gröbner bases and triangular sets. We propose efficient algorithms for change of ordering of Gröbner bases of zero-dimensional ideals by using the sparsity of multiplication matrices and evaluate such sparsity for generic polynomial systems. Original algorithms are presented for decomposing polynomial sets into simple triangular sets over finite fields. We also define squarefree decomposition and factorization of polynomials over unmixed products of field extensions and propose algorithms for computing them. The effectiveness and efficiency of these algorithms have been verified by experiments with our implementations. Methods for polynomial system solving over finite fields are also applied to solve practical problems arising from Biology and Coding Theory.
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Chenqi Mou. Solving Polynomial Systems over Finite Fields: Algorithms, Implementation and Applications. Symbolic Computation [cs.SC]. Université Pierre et Marie Curie, 2013. English. ⟨tel-01110887⟩

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