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211
Symbolic Analysis for Parallelizing Compilers
, 1994
"... Symbolic Domain The objects in our abstract symbolic domain are canonical symbolic expressions. A canonical symbolic expression is a lexicographically ordered sequence of symbolic terms. Each symbolic term is in turn a pair of an integer coefficient and a sequence of pairs of pointers to program va ..."
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Cited by 105 (4 self)
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Symbolic Domain The objects in our abstract symbolic domain are canonical symbolic expressions. A canonical symbolic expression is a lexicographically ordered sequence of symbolic terms. Each symbolic term is in turn a pair of an integer coefficient and a sequence of pairs of pointers to program variables in the program symbol table and their exponents. The latter sequence is also lexicographically ordered. For example, the abstract value of the symbolic expression 2ij+3jk in an environment that i is bound to (1; (( " i ; 1))), j is bound to (1; (( " j ; 1))), and k is bound to (1; (( " k ; 1))) is ((2; (( " i ; 1); ( " j ; 1))); (3; (( " j ; 1); ( " k ; 1)))). In our framework, environment is the abstract analogous of state concept; an environment is a function from program variables to abstract symbolic values. Each environment e associates a canonical symbolic value e x for each variable x 2 V ; it is said that x is bound to e x. An environment might be represented by...
A Gröbner free alternative for polynomial system solving
 Journal of Complexity
, 2001
"... Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic ..."
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Cited by 82 (16 self)
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Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton’s iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation
Analytic Combinatorics of Noncrossing Configurations
, 1997
"... This paper describes a systematic approach to the enumeration of "noncrossing" geometric configurations built on vertices of a convex ngon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. A consequence is exact and asymptotic c ..."
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Cited by 55 (8 self)
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This paper describes a systematic approach to the enumeration of "noncrossing" geometric configurations built on vertices of a convex ngon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. A consequence is exact and asymptotic counting results for trees, forests, graphs, connected graphs, dissections, and partitions. Limit laws of the Gaussian type are also established in this framework; they concern a variety of parameters like number of leaves in trees, number of components or edges in graphs, etc.
A uniform approach for the fast computation of Matrixtype Padé approximants
, 1996
"... Recently, a uniform approach was given [5] for different concepts of matrixtype Pad'e approximants, such as descriptions of vector and matrix Pad'e approximants along with generalizations of simultaneous and Hermite Pad'e approximants. The considerations in this paper are based on this generalized ..."
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Cited by 52 (10 self)
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Recently, a uniform approach was given [5] for different concepts of matrixtype Pad'e approximants, such as descriptions of vector and matrix Pad'e approximants along with generalizations of simultaneous and Hermite Pad'e approximants. The considerations in this paper are based on this generalized form of the classical scalar Hermite Pad'e approximation problem, power Hermite Pad'e approximation. In particular we study the problem of computing these new approximants. A recurrence relation is presented for the computation of a basis for the corresponding linear solution space of these approximants. This recurrence also provides bases for particular subproblems. This generalizes previous work by Van Barel and Bultheel and, in a more general form, by Beckermann. The computation of the bases has complexity O(oe 2 ) where oe is the order of the desired approximant, and requires no conditions on the input data. A second algorithm using the same recurrence relation along with divideand...
Reduction of Systems of Nonlinear Partial Differential Equations to Simplified Involutive Forms
, 1996
"... We describe the rif algorithm which uses a finite number of differentiations and algebraic operations to simplify analytic nonlinear systems of partial differential equations to what we call reduced involutive form. This form includes the integrability conditions of the system and satisfies a consta ..."
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Cited by 43 (14 self)
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We describe the rif algorithm which uses a finite number of differentiations and algebraic operations to simplify analytic nonlinear systems of partial differential equations to what we call reduced involutive form. This form includes the integrability conditions of the system and satisfies a constant rank condition. The algorithm is useful for classifying initial value problems for determined pde systems and can yield dramatic simplifications of complex overdetermined nonlinear pde systems. Such overdetermined systems arise in analysis of physical pdes for reductions to odes using the Nonclassical Method, the search for exact solutions of Einstein's field equations and the determination of discrete symmetries of differential equations. Application of the algorithm to the associated nonlinear overdetermined system of 856 pdes arising when the Nonclassical Method is applied to a cubic nonlinear Schrodinger system yields new results. Our algorithm combines features of geometric involutiv...
The Relationship Between Breaking the DiffieHellman Protocol and Computing Discrete Logarithms
, 1998
"... Both uniform and nonuniform results concerning the security of the DiffieHellman keyexchange protocol are proved. First, it is shown that in a cyclic group G of order jGj = Q p e i i , where all the multiple prime factors of jGj are polynomial in log jGj, there exists an algorithm that re ..."
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Cited by 38 (3 self)
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Both uniform and nonuniform results concerning the security of the DiffieHellman keyexchange protocol are proved. First, it is shown that in a cyclic group G of order jGj = Q p e i i , where all the multiple prime factors of jGj are polynomial in log jGj, there exists an algorithm that reduces the computation of discrete logarithms in G to breaking the DiffieHellman protocol in G and has complexity p maxf(p i )g \Delta (log jGj) O(1) , where (p) stands for the minimum of the set of largest prime factors of all the numbers d in the interval [p \Gamma 2 p p+1; p+2 p p+ 1]. Under the unproven but plausible assumption that (p) is polynomial in log p, this reduction implies that the DiffieHellman problem and the discrete logarithm problem are polynomialtime equivalent in G. Second, it is proved that the DiffieHellman problem and the discrete logarithm problem are equivalent in a uniform sense for groups whose orders belong to certain classes: there exists a p...
DiffieHellman Oracles
 ADVANCES IN CRYPTOLOGY  CRYPTO '96 , LECTURE NOTES IN COMPUTER SCIENCE
, 1996
"... This paper consists of three parts. First, various types of DiffieHellman oracles for a cyclic group G and subgroups of G are defined and their equivalence is proved. In particular, the security of using a subgroup of G instead of G in the DiffieHellman protocol is investigated. Second, we derive ..."
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Cited by 34 (3 self)
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This paper consists of three parts. First, various types of DiffieHellman oracles for a cyclic group G and subgroups of G are defined and their equivalence is proved. In particular, the security of using a subgroup of G instead of G in the DiffieHellman protocol is investigated. Second, we derive several new conditions for the polynomialtime equivalence of breaking the DiffieHellman protocol and computing discrete logarithms in G which extend former results by den Boer and Maurer. Finally, efficient constructions of DiffieHellman groups with provable equivalence are described.
Symbolicnumeric sparse interpolation of multivariate polynomials
 In Proc. Ninth Rhine Workshop on Computer Algebra (RWCA’04), University of Nijmegen, the Netherlands (2004
, 2006
"... We consider the problem of sparse interpolation of an approximate multivariate blackbox polynomial in floatingpoint arithmetic. That is, both the inputs and outputs of the blackbox polynomial have some error, and all numbers are represented in standard, fixedprecision, floating point arithmetic. ..."
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Cited by 34 (6 self)
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We consider the problem of sparse interpolation of an approximate multivariate blackbox polynomial in floatingpoint arithmetic. That is, both the inputs and outputs of the blackbox polynomial have some error, and all numbers are represented in standard, fixedprecision, floating point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give efficient and numerically robust solutions. We note the similarity between the exact BenOr/Tiwari sparse interpolation algorithm and the classical Prony’s method for interpolating a sum of exponential functions, and exploit the generalized eigenvalue reformulation of Prony’s method. We analyze the numerical stability of our algorithms and the sensitivity of the solutions, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques in practice through numerical experiments and applications. 1.
Decoding ReedSolomon codes beyond half the minimum distance
, 1998
"... We describe an efficient implementation of M. Sudan's algorithm for decoding ReedSolomon codes beyond half the minimum distance. Furthermore, we calculate an upper bound of the probability of getting more than one codeword as output. 1 Introduction In a recent paper M. Sudan [1] presented an algori ..."
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Cited by 33 (0 self)
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We describe an efficient implementation of M. Sudan's algorithm for decoding ReedSolomon codes beyond half the minimum distance. Furthermore, we calculate an upper bound of the probability of getting more than one codeword as output. 1 Introduction In a recent paper M. Sudan [1] presented an algorithm for correcting more than dmin \Gamma1 2 errors in a ReedSolomon code with low rate and in [2] he extended his algorithm to higher rates. The algorithm produces a list of codewords closest to the received word. In this paper we present an efficient implementation of Sudan's extended algorithm by speeding up the two crucial steps, namely interpolation and factorization. Based on the weight distribution of MDS codes we also calculate an upper bound on the probability that the list contains more than one codeword. The paper is organized as follows: Section 2 contains some basic definitions and in section 3 we present Sudan's extended algorithm and prove that it works. Section 4 calculate...