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264
LengthBased Attacks for Certain Group Based Encryption Rewriting Systems
, 2002
"... In this note, we describe a probabilistic attack on public key cryptosystems based on the word/conjugacy problems for finitely presented groups of the type proposed recently by Anshel, Anshel and Goldfeld. In such a scheme, one makes use of the property that in the given group the word problem has a ..."
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Cited by 39 (1 self)
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In this note, we describe a probabilistic attack on public key cryptosystems based on the word/conjugacy problems for finitely presented groups of the type proposed recently by Anshel, Anshel and Goldfeld. In such a scheme, one makes use of the property that in the given group the word problem has a polynomial time solution, while the conjugacy problem has no known polynomial solution. An example is the braid group from topology in which the word problem is solvable in polynomial time while the only known solutions to the conjugacy problem are exponential. The attack in this paper is based on having a canonical representative of each string relative to which a length function may be computed. Hence the term length attack. Such canonical representatives are known to exist for the braid group.
Semidefinite Representations for Finite Varieties
 MATHEMATICAL PROGRAMMING
, 2002
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming prob ..."
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Cited by 37 (7 self)
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We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming problem. This semidefinite program involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space R[x 1 , . . . , x n ]/I. Our arguments are elementary and extend known facts for the grid case including 0/1 and polynomial programming. They also relate to known algebraic tools for solving polynomial systems of equations with finitely many complex solutions. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem at optimality.
Twenty combinatorial examples of asymptotics derived from multivariate generating functions
"... Abstract. Let {ar: r ∈ Nd} be a ddimensional array of numbers for which the generating function F (z): = ∑ r arzr is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asym ..."
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Cited by 34 (14 self)
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Abstract. Let {ar: r ∈ Nd} be a ddimensional array of numbers for which the generating function F (z): = ∑ r arzr is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of F. Our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions. Next we describe the Morsetheoretic underpinnings of some new asymptotic techniques. We then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations. The remainder of the survey focuses on combinatorial applications, such as enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, and descents in permutations. After the individual examples, we discuss three broad classes of examples, namely, functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. These methods have
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 32 (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.
Camera pose and calibration from 4 or 5 known 3D points
 In Proc. 7th Int. Conf. on Computer Vision
, 1999
"... We describe two direct quasilinear methods for camera pose (absolute orientation) and calibration from a single image of 4 or 5 known 3D points. They generalize the 6 point ‘Direct Linear Transform ’ method by incorporating partial prior camera knowledge, while still allowing some unknown calibratio ..."
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Cited by 31 (0 self)
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We describe two direct quasilinear methods for camera pose (absolute orientation) and calibration from a single image of 4 or 5 known 3D points. They generalize the 6 point ‘Direct Linear Transform ’ method by incorporating partial prior camera knowledge, while still allowing some unknown calibration parameters to be recovered. Only linear algebra is required, the solution is unique in nondegenerate cases, and additional points can be included for improved stability. Both methods fail for coplanar points, but we give an experimental eigendecomposition based one that handles both planar and nonplanar cases. Our methods use recent polynomial solving technology, and we give a brief summary of this. One of our aims was to try to understand the numerical behaviour of modern polynomial solvers on some relatively simple test cases, with a view to other vision applications.
Computer Algebra Methods for Studying and Computing Molecular Conformations
, 1997
"... A relatively new branch of computational biology has been emerging as an effort to supplement traditional techniques of large scale search in drug design by structurebased methods, in order to improve efficiency and guarantee completeness. This paper studies the geometric structure of cyclic molecu ..."
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Cited by 29 (4 self)
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A relatively new branch of computational biology has been emerging as an effort to supplement traditional techniques of large scale search in drug design by structurebased methods, in order to improve efficiency and guarantee completeness. This paper studies the geometric structure of cyclic molecules, in particular the enumeration of all possible conformations, which is crucial in finding the energetically favorable geometries, and the identification of all degenerate conformations. Recent advances in computational algebra are exploited, including distance geometry, sparse polynomial theory, and matrix methods for numerically solving nonlinear multivariate polynomial systems. Moreover, we propose a complete array of computer algebra and symbolic computational geometry methods for modeling the rigidity constraints, formulating the problems in algebraic terms and, lastly, visualizing the computed conformations. The use of computer algebra systems and of public domain software is illustrated...
Comparison between XL and Gröbner Basis Algorithms
 ASIACRYPT 2004, LECTURE
, 2004
"... This paper compares the XL algorithm with known Gröbner basis algorithms. We show that to solve a system of algebraic equations via the XL algorithm is equivalent to calculate the reduced Gröbner basis of the ideal associated with the system. Moreover we show that the XL algorithm is also a Gröbner ..."
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Cited by 27 (10 self)
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This paper compares the XL algorithm with known Gröbner basis algorithms. We show that to solve a system of algebraic equations via the XL algorithm is equivalent to calculate the reduced Gröbner basis of the ideal associated with the system. Moreover we show that the XL algorithm is also a Gröbner basis algorithm which can be represented as a redundant variant of a Gröbner basis algorithm F4. Then we compare these algorithms on semiregular sequences, which correspond, in conjecture, to almost all polynomial systems in two cases: over the fields F2 and Fq with q ≫ n. We show that the size of the matrix constructed by XL is large compared to the ones of the F5 algorithm. Finally, we give an experimental study between XL and the Buchberger algorithm on the cryptosystem HFE and find that the Buchberger algorithm has a better behavior.
Some Remarks on Real and Complex Output Feedback
, 1998
"... . We provide some new necessary and sufficient conditions which guarantee arbitrary pole placement of a particular linear system over the complex numbers. We exhibit a nontrivial real linear system which is not controllable by real static output feedback and discuss a conjecture from algebraic geom ..."
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Cited by 26 (12 self)
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. We provide some new necessary and sufficient conditions which guarantee arbitrary pole placement of a particular linear system over the complex numbers. We exhibit a nontrivial real linear system which is not controllable by real static output feedback and discuss a conjecture from algebraic geometry concerning the existence of real linear systems for which all static feedback laws are real. 1. Preliminaries Let F be an arbitrary field and let m; p; n be fixed positive integers. Let A; B; C be matrices with entries in F of sizes n \Theta n, n \Theta m, and p \Theta n respectively. Identify the space of monic polynomials having degree n, s n + a n\Gamma1 s n\Gamma1 + \Delta \Delta \Delta + a 1 s + a 0 2 F[s]; with the vector space F n . In its simplest form, the static output pole placement problem asks for conditions on the matrices A; B; C which guarantee that the pole placement map Ø (A;B;C) : F mp \Gamma! F n ; K 7\Gamma! det(sI \Gamma A \Gamma BKC) (1) is surjectiv...
Eigenvalues of a real supersymmetric tensor
 J. Symbolic Comput
"... In this paper, we define the symmetric hyperdeterminant, eigenvalues and Eeigenvalues of a real supersymmetric tensor. We show that eigenvalues are roots of a onedimensional polynomial, and when the order of the tensor is even, Eeigenvalues are roots of another onedimensional polynomial. These t ..."
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Cited by 26 (7 self)
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In this paper, we define the symmetric hyperdeterminant, eigenvalues and Eeigenvalues of a real supersymmetric tensor. We show that eigenvalues are roots of a onedimensional polynomial, and when the order of the tensor is even, Eeigenvalues are roots of another onedimensional polynomial. These two onedimensional polynomials are associated with the symmetric hyperdeterminant. We call them the characteristic polynomial and the Echaracteristic polynomial of that supersymmetric tensor. Real eigenvalues (Eeigenvalues) with real eigenvectors (Eeigenvectors) are called Heigenvalues (Zeigenvalues). When the order of the supersymmetric tensor is even, Heigenvalues (Zeigenvalues) exist and the supersymmetric tensor is positive definite if and only if all of its Heigenvalues (Zeigenvalues) are positive. An mthorder ndimensional supersymmetric tensor where m is even has exactly n(m − 1) n−1 eigenvalues, and the number of its Eeigenvalues is strictly less than n(m − 1) n−1 when m ≥ 4. We show that the product of all the eigenvalues is equal to the value of the symmetric hyperdeterminant, while the sum of all the eigenvalues is equal to the sum of the diagonal elements of that supersymmetric tensor, multiplied by (m − 1) n−1.The n(m −1) n−1 eigenvalues are distributed in n disks in C.Thecenters and radii of these n disks are the diagonal elements, and the sums of the absolute values of the corresponding offdiagonal elements, of that supersymmetric tensor. On the other hand, Eeigenvalues are invariant under orthogonal transformations.
Decomposition plans for geometric constraint systems
 J. Symbolic Computation
, 2001
"... A central issue in dealing with geometric constraint systems for CAD/CAM/CAE is the generation of an optimal decomposition plan that not only aids efficient solution, but also captures design intent and supports conceptual design. Though complex, this issue has evolved and crystallized over the past ..."
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Cited by 24 (0 self)
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A central issue in dealing with geometric constraint systems for CAD/CAM/CAE is the generation of an optimal decomposition plan that not only aids efficient solution, but also captures design intent and supports conceptual design. Though complex, this issue has evolved and crystallized over the past few years, permitting us to take the next important step: in this paper, we formalize, motivate and explain the decomposition–recombination (DR)planning problem as well as several performance measures by which DRplanning algorithms can be analyzed and compared. These measures include: generality, validity, completeness, Church–Rosser property, complexity, best and worstchoice approximation factors, (strict) solvability preservation, ability to deal with underconstrained systems, and ability to incorporate conceptual design decompositions specified by the designer. The problem and several of the performance measures are formally defined here for the first time—they closely reflect specific requirements of CAD/CAM applications. The clear formulation of the problem and performance measures allow us to precisely analyze and compare existing DRplanners that use two wellknown types of decomposition methods: SR (constraint shape recognition) and MM (generalized maximum matching) on constraint graphs. This analysis additionally serves to illustrate and provide intuitive substance to the newly formalized measures. In Part II of this article, we use the new performance measures to guide the development of a new DRplanning algorithm which excels with respect to these performance measures. c ○ 2001 Academic Press 1.