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24
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 80 (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
Security of Signed ElGamal Encryption
 In Asiacrypt ’2000, LNCS 1976
, 2000
"... . Assuming a cryptographically strong cyclic group G of prime order q and a random hash function H, we show that ElGamal encryption with an added Schnorr signature is secure against the adaptive chosen ciphertext attack, in which an attacker can freely use a decryption oracle except for the target c ..."
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Cited by 40 (3 self)
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. Assuming a cryptographically strong cyclic group G of prime order q and a random hash function H, we show that ElGamal encryption with an added Schnorr signature is secure against the adaptive chosen ciphertext attack, in which an attacker can freely use a decryption oracle except for the target ciphertext. We also prove security against the novel onemoredecyption attack. Our security proofs are in a new model, corresponding to a combination of two previously introduced models, the Random Oracle model and the Generic model. The security extends to the distributed threshold version of the scheme. Moreover, we propose a very practical scheme for private information retrieval that is based on blind decryption of ElGamal ciphertexts. 1 Introduction and Summary We analyse a very practical public key cryptosystem in terms of its security against the strong adaptive chosen ciphertext attack (CCA) of [RS92], in which an attacker can access a decryption oracle on arbitrary ciphertexts (ex...
Computing Parametric Geometric Resolutions
, 2001
"... Given a polynomial system of n equations in n unknowns that depends on some parameters, we de ne the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters. The coefficients of this resolution are rational functions of the parameters; we f ..."
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Cited by 20 (7 self)
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Given a polynomial system of n equations in n unknowns that depends on some parameters, we de ne the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters. The coefficients of this resolution are rational functions of the parameters; we first show that their degree is bounded by the Bézout number d n , where d is a bound on the degrees of the input system. We then present a probabilistic algorithm to compute such a resolution; in short, its complexity is polynomial in the size of the output and the probability of success is controlled by a quantity polynomial in the Bézout number. We present several applications of this process, to computations in the Jacobian of hyperelliptic curves and to questions of real geometry.
Factoring Rational Polynomials over the Complex Numbers
, 1989
"... eskeleton on the surface (P = 0) whose number of connected components is precisely the number of connected components of P =0minus its singular points. The connectivity of this curveskeleton is constructed symbolically using Sturm sequences associated with the various polynomials de ning these ..."
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Cited by 19 (2 self)
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eskeleton on the surface (P = 0) whose number of connected components is precisely the number of connected components of P =0minus its singular points. The connectivity of this curveskeleton is constructed symbolically using Sturm sequences associated with the various polynomials de ning these maps. Given the number of irreducible factors and their degree, the actual factors can be reconstructed using the recent result of Ne [22] on nding zeroes of one variable polynomials in NC. 1 Introduction Factoring polynomials is a basic problem in symbolic computation with applications as diverse as theorem proving and computeraided design. Our goal is to approximate the factors, irreducible over the complex numbers, of a multivariable polynomial with rational coecients in deterministic NC with respect to the polynomial's degree and coecient size, assuming that the number of variables is xed. Further if the number of variables is not xed, we will nd the number of irreducible facto
Algorithms on Compressed Strings and Arrays
 In Proc. 26th Ann. Conf. on Current Trends in Theory and Practice of Infomatics
, 1999
"... . We survey the complexity issues related to several algorithmic problems for compressed one and twodimensional texts without explicit decompression: patternmatching, equalitytesting, computation of regularities, subsegment extraction, language membership, and solvability of word equations. Our ..."
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Cited by 18 (0 self)
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. We survey the complexity issues related to several algorithmic problems for compressed one and twodimensional texts without explicit decompression: patternmatching, equalitytesting, computation of regularities, subsegment extraction, language membership, and solvability of word equations. Our basic problem is one and twodimensional patternmatching together with its variations. For some types of compression the patternmatching problems are infeasible (NPhard), for other types they are solvable in polynomial time and we discuss how to reduce the degree of corresponding polynomials. 1 Introduction In the last decade a new stream of research related to data compression has emerged: algorithms on compressed objects. It has been caused by the increase in the volume of data and the need to store and transmit masses of information in compressed form. The compressed information has to be quickly accessed and processed without explicit decompression. In this paper we consider severa...
Fast algorithms for zerodimensional polynomial systems using duality
 APPLICABLE ALGEBRA IN ENGINEERING, COMMUNICATION AND COMPUTING
, 2001
"... Many questions concerning a zerodimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A = k[X1,..., Xn]/I, where I is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the q ..."
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Cited by 16 (3 self)
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Many questions concerning a zerodimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A = k[X1,..., Xn]/I, where I is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the question of speeding up the linear algebra phase for the computation of minimal polynomials and rational parametrizations in A. We present new formulæ for the rational parametrizations, extending those of Rouillier, and algorithms extending ideas introduced by Shoup in the univariate case. Our approach is based on the Amodule structure of the dual space � A. An important feature of our algorithms is that we do not require � A to be free and of rank 1. The complexity of our algorithms for computing the minimal polynomial and the rational parametrizations are O(2 n D 5/2) and O(n2 n D 5/2) respectively, where D is the dimension of A. For fixed n, this is better than algorithms based on linear algebra except when the complexity of the available matrix product has exponent less than 5/2.
Probabilistic Algorithms for Geometric Elimination
 in Engineering, Communication and Computing
, 1999
"... We develop probabilistic algorithms that solve problems of geometric elimination theory using small memory resources. These algorithms are obtained by means of the adaptation of a general transformation due to A. Borodin which converts uniform boolean circuit depth into sequential (Turing machine) s ..."
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Cited by 12 (5 self)
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We develop probabilistic algorithms that solve problems of geometric elimination theory using small memory resources. These algorithms are obtained by means of the adaptation of a general transformation due to A. Borodin which converts uniform boolean circuit depth into sequential (Turing machine) space. The boolean circuits themselves are developed using techniques based on the computation of a primitive element of a suitable zerodimensional algebra and diophantine considerations. Our algorithms improve...
Computing the Frobenius Normal Form of a Sparse Matrix
 CASC 2000 Proc. the Third International Workshop on Computer Algebra in Scientific Computing
, 2000
"... . We probabilistically determine the Frobenius form and thus the characteristic polynomial of a matrix A 2 F nn by O(n log(n)) multiplications of A by vectors and O n 2 log 2 (n) log log(n) arithmetic operations in the eld F. The parameter is the number of distinct invariant factors of A, ..."
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Cited by 11 (2 self)
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. We probabilistically determine the Frobenius form and thus the characteristic polynomial of a matrix A 2 F nn by O(n log(n)) multiplications of A by vectors and O n 2 log 2 (n) log log(n) arithmetic operations in the eld F. The parameter is the number of distinct invariant factors of A, it is less than 3 p n=2 in the worst case. The method requires O(n) storage space in addition to that needed for the matrix A. 1 Introduction The known complexity estimates of the computation of the characteristic polynomial and a fortiori, of the Frobenius normal form of special { sparse or black box { square matrices A over a eld F, seem to not be satisfactory. We refer to Kaltofen [8, Open Problem 3] and to Pan et al. [16, 15] for discussions on this subject and survey of current solutions. We denote by M(n) the number of operations in F required for nn matrix multiplications. The characteristic polynomial of a general matrix A can be computed at cost of O(n 3 ) or O(M(n) log...
Decomposition Of Algebras Over Finite Fields And Number Fields
, 1991
"... We consider the boolean complexity of the decomposition of semisimple algebras over finite fields and number fields. ..."
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Cited by 8 (2 self)
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We consider the boolean complexity of the decomposition of semisimple algebras over finite fields and number fields.