Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
Journal of Symbolic Computation
Extension of the Berlekamp-Massey algorithm to N dimensions
Information and Computation
Gro¨bner bases: a computational approach to commutative algebra
Gro¨bner bases: a computational approach to commutative algebra
Efficient computation of zero-dimensional Gro¨bner bases by change of ordering
Journal of Symbolic Computation
ISSAC '94 Proceedings of the international symposium on Symbolic and algebraic computation
Converting bases with the Gröbner walk
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Changing the ordering of Gröbner bases with LLL: case of two variables
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Change of order for bivariate triangular sets
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Change of order for regular chains in positive dimension
Theoretical Computer Science
Computing loci of rank defects of linear matrices using Gröbner bases and applications to cryptology
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
A zero-dimensional gröbner basis for AES-128
FSE'06 Proceedings of the 13th international conference on Fast Software Encryption
IEEE Transactions on Information Theory - Part 1
Fast change of ordering with exponent ω
ACM Communications in Computer Algebra
Critical points and Gröbner bases: the unmixed case
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Algorithms for the universal decomposition algebra
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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Let I in K[x1,...,xn] be a 0-dimensional ideal of degree D where K is a field. It is well-known that obtaining efficient algorithms for change of ordering of Gröbner bases of I is crucial in polynomial system solving. Through the algorithm FGLM, this task is classically tackled by linear algebra operations in K[x1,...,n]/I. With recent progress on Gröbner bases computations, this step turns out to be the bottleneck of the whole solving process. Our contribution is an algorithm that takes advantage of the sparsity structure of multiplication matrices appearing during the change of ordering. This sparsity structure arises even when the input polynomial system defining I is dense. As a by-product, we obtain an implementation which is able to manipulate 0-dimensional ideals over a prime field of degree greater than 30000. It outperforms the Magma/Singular/FGb implementations of FGLM. First, we investigate the particular but important shape position case. The obtained algorithm performs the change of ordering within a complexity O(D(Ni1+nlog(D))), where N1 is the number of nonzero entries of a multiplication matrix. This almost matches the complexity of computing the minimal polynomial of one multiplication matrix. Then, we address the general case and give corresponding complexity results. Our algorithm is dynamic in the sense that it selects automatically which strategy to use depending on the input. Its key ingredients are the Wiedemann algorithm to handle 1-dimensional linear recurrence (for the shape position case), and the Berlekamp-Massey-Sakata algorithm from Coding Theory to handle multi-dimensional linearly recurring sequences in the general case.