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207
On the complexity of Gröbner basis computation of semiregular overdetermined . . .
, 2004
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Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
, 2001
"... In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposi ..."
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Cited by 56 (26 self)
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In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As byproducts, the computation also determines the degree of each component and an upper bound on itsmultiplicity. The bound issharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroesof a finite number of polynomials.
Cube Attacks on Tweakable Black Box Polynomials
"... Abstract. Almost any cryptographic scheme can be described by tweakable polynomials over GF (2), which contain both secret variables (e.g., key bits) and public variables (e.g., plaintext bits or IV bits). The cryptanalyst is allowed to tweak the polynomials by choosing arbitrary values for the publ ..."
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Cited by 46 (4 self)
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Abstract. Almost any cryptographic scheme can be described by tweakable polynomials over GF (2), which contain both secret variables (e.g., key bits) and public variables (e.g., plaintext bits or IV bits). The cryptanalyst is allowed to tweak the polynomials by choosing arbitrary values for the public variables, and his goal is to solve the resultant system of polynomial equations in terms of their common secret variables. In this paper we develop a new technique (called a cube attack) for solving such tweakable polynomials, which is a major improvement over several previously published attacks of the same type. For example, on the stream cipher Trivium with a reduced number of initialization rounds, the best previous attack (due to Fischer, Khazaei, and Meier) requires a barely practical complexity of 2 55 to attack 672 initialization rounds, whereas a cube attack can find the complete key of the same variant in 2 19 bit operations (which take less than a second on a single PC). Trivium with 735 initialization rounds (which could not be attacked by any previous technique) can now be broken with 2 30 bit operations, and by extrapolating our experimentally verified complexities for various sizes, we have reasons to believe that cube attacks will remain faster than exhaustive search even for 1024 initialization rounds. Whereas previous attacks were heuristic, had to be adapted to each cryptosystem, had no general complexity bounds,
Minimizing polynomial functions
 Proceedings of the DIMACS Workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science
, 2003
"... Abstract. We compare algorithms for global optimization of polynomial functions in many variables. It is demonstrated that existing algebraic methods (Gröbner bases, resultants, homotopy methods) are dramatically outperformed by a relaxation technique, due to N.Z. Shor and the first author, which in ..."
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Cited by 44 (3 self)
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Abstract. We compare algorithms for global optimization of polynomial functions in many variables. It is demonstrated that existing algebraic methods (Gröbner bases, resultants, homotopy methods) are dramatically outperformed by a relaxation technique, due to N.Z. Shor and the first author, which involves sums of squares and semidefinite programming. This opens up the possibility of using semidefinite programming relaxations arising from the Positivstellensatz for a wide range of computational problems in real algebraic geometry. 1.
Asymptotic Behaviour of the Degree of Regularity of SemiRegular Polynomial Systems
 In MEGA’05, 2005. Eighth International Symposium on Effective Methods in Algebraic Geometry
"... We compute the asymptotic expansion of the degree of regularity for overdetermined semiregular sequences of algebraic equations. This degree implies bounds for the generic complexity of Gröbner bases algorithms, in particular the F5 [Fau02] algorithm. Bounds can also be derived for the XL [SPCK00] ..."
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Cited by 41 (24 self)
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We compute the asymptotic expansion of the degree of regularity for overdetermined semiregular sequences of algebraic equations. This degree implies bounds for the generic complexity of Gröbner bases algorithms, in particular the F5 [Fau02] algorithm. Bounds can also be derived for the XL [SPCK00] family of algorithms used by the cryptographic community. 1 Motivations and Results The worstcase complexity of Gröbner bases has been the object of extensive studies. In the most general case, it is well known after work by Mayr and Meyer that the complexity is doubly exponential in the number of variables. For subclasses of polynomial systems, the complexity may be much smaller. Of particular importance is the class of regular sequences of polynomials. There, it is known that after a generic linear change of variables the complexity of the computation for the degreereverselexicographic order is simply exponential in the number of variables. Moreover, in characteristic 0, these systems are generic. Our goal is to give similar complexity bounds for overdetermined systems, for a class of systems that we
Using monodromy to decompose solution sets of polynomial systems into irreducible components
 PROCEEDINGS OF A NATO CONFERENCE, FEBRUARY 25  MARCH 1, 2001, EILAT
, 2001
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Solving parametric polynomial systems
 Journal of Symbolic Computation
, 2007
"... We present a new algorithm for solving basic parametric constructible or semialgebraic ..."
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Cited by 34 (2 self)
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We present a new algorithm for solving basic parametric constructible or semialgebraic
A minimal solution to the autocalibration of radial distortion
, 2007
"... Epipolar geometry and relative camera pose computation are examples of tasks which can be formulated as minimal problems and solved from a minimal number of image points. Finding the solution leads to solving systems of algebraic equations. Often, these systems are not trivial and therefore special ..."
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Cited by 31 (11 self)
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Epipolar geometry and relative camera pose computation are examples of tasks which can be formulated as minimal problems and solved from a minimal number of image points. Finding the solution leads to solving systems of algebraic equations. Often, these systems are not trivial and therefore special algorithms have to be designed to achieve numerical robustness and computational efficiency. In this paper we provide a solution to the problem of estimating radial distortion and epipolar geometry from eight correspondences in two images. Unlike previous algorithms, which were able to solve the problem from nine correspondences only, we enforce the determinant of the fundamental matrix be zero. This leads to a system of eight quadratic and one cubic equation in nine variables. We simplify the system by eliminating six of these variables. Then, we solve the system by finding eigenvectors of an action matrix of a suitably chosen polynomial. We show how to construct the action matrix without computing complete Gröbner basis, which provides an efficient and robust solver. The quality of the solver is demonstrated on synthetic and real data. 1.
Using Galois Ideals For Computing Relative Resolvents
 J. Symb. Comp
, 1998
"... . In this paper we establish that some ideals which occur in Galois theory are generated by a triangular set of polynomials. This geometric property seems important for the development of symbolic methods in Galois theory. It may be exploited in order to obtain more efficient algorithms. Actually, i ..."
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Cited by 29 (4 self)
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. In this paper we establish that some ideals which occur in Galois theory are generated by a triangular set of polynomials. This geometric property seems important for the development of symbolic methods in Galois theory. It may be exploited in order to obtain more efficient algorithms. Actually, it enables us to present here a new algebraic method for computing relative resolvents which works with any kind of invariant. 1. Introduction Let k be a perfect field and ¯ k an algebraic closure of k. Let f be a univariate polynomial of k[X] supposed separable with degree n, and\Omega be an ordered set of the n roots of f in ¯ k n . In [25] is introduced the notion of ideal of\Omega\Gamma32615/39 invariant by a subset L of the symmetric group of degree n. It generalizes the notion of ideal of relations and the notion of ideal of symmetric relations. We call them Galois ideals. This paper presents two important results. First, we prove in Theorem 5.5 that a Galois ideal associated w...