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165
A New Efficient Algorithm for Computing Gröbner Bases Without Reduction to Zero (F5
 In: ISSAC ’02: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation
, 2002
"... This paper introduces a new efficient algorithm for computing Gröbner bases. To avoid as much as possible intermediate computation, the algorithm computes successive truncated Gröbner bases and it replaces the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduc ..."
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Cited by 253 (54 self)
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This paper introduces a new efficient algorithm for computing Gröbner bases. To avoid as much as possible intermediate computation, the algorithm computes successive truncated Gröbner bases and it replaces the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduction of several polynomials. This powerful reduction mechanism is achieved by means of a symbolic precomputation and by extensive use of sparse linear algebra methods. Current techniques in linear algebra used in Computer Algebra are reviewed together with other methods coming from the numerical field. Some previously untractable problems (Cyclic 9) are presented as well as an empirical comparison of a first implementation of this algorithm with other well known programs. This comparison pays careful attention to methodology issues. All the benchmarks and CPU times used in this paper are frequently updated and available on a Web page. Even though the new algorithm does not improve the worst case complexity it is several times faster than previous implementations both for integers and modulo computations. 1
REDLOG Computer Algebra Meets Computer Logic
 ACM SIGSAM Bulletin
, 1996
"... . redlog is a package that extends the computer algebra system reduce to a computer logic system, i.e., a system that provides algorithms for the symbolic manipulation of firstorder formulas over some temporarily fixed language and theory. In contrast to theorem provers, the methods applied know a ..."
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Cited by 105 (30 self)
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. redlog is a package that extends the computer algebra system reduce to a computer logic system, i.e., a system that provides algorithms for the symbolic manipulation of firstorder formulas over some temporarily fixed language and theory. In contrast to theorem provers, the methods applied know about the underlying algebraic theory and make use of it. Though the focus is on simplification, parametric linear optimization, and quantifier elimination, redlog is designed as a generalpurpose system. We describe the functionality of redlog as it appears to the user, and explain the design issues and implementation techniques. ? The second author was supported by the dfg (Schwerpunktprogramm: Algorithmische Zahlentheorie und Algebra) 1 Introduction redlog stands for reduce logic system. It provides an extension of the computer algebra system (cas) reduce to a computer logic system (cls) implementing symbolic algorithms on firstorder formulas w.r.t. temporarily fixed firstorder languag...
Algebraic Cryptanalysis of Hidden Field Equation (HFE) Cryptosystems Using Gröbner Bases
 In Advances in Cryptology — CRYPTO 2003
, 2003
"... Abstract. In this paper, we review and explain the existing algebraic cryptanalysis of multivariate cryptosystems from the hidden field equation (HFE) family. These cryptanalysis break cryptosystems in the HFE family by solving multivariate systems of equations. In this paper we present a new and ef ..."
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Cited by 104 (27 self)
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Abstract. In this paper, we review and explain the existing algebraic cryptanalysis of multivariate cryptosystems from the hidden field equation (HFE) family. These cryptanalysis break cryptosystems in the HFE family by solving multivariate systems of equations. In this paper we present a new and efficient attack of this cryptosystem based on fast algorithms for computing Gröbner basis. In particular it was was possible to break the first HFE challenge (80 bits) in only two days of CPU time by using the new algorithm F5 implemented in C. From a theoretical point of view we study the algebraic properties of the equations produced by instance of the HFE cryptosystems and show why they yield systems of equations easier to solve than random systems of quadratic equations of the same sizes. Moreover we are able to bound the maximal degree occuring in the Gröbner basis computation. As a consequence, we gain a deeper understanding of the algebraic cryptanalysis against these cryptosystems. We use this understanding to devise a specific algorithm based on sparse linear algebra. In general, we conclude that the cryptanalysis of HFE can be performed in polynomial time. We also revisit the security estimates for existing schemes in the HFE family. 1
Using the Groebner basis algorithm to find proofs of unsatisfiability
, 1996
"... A propositional proof system can be viewed as a nondeterministic algorithm for the (coNP complete) unsatisfiability problem. Many such proof systems, such as resolution, are also used as the basis for heuristics which deterministically search for short proofs in the system. We discuss a propositio ..."
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Cited by 97 (5 self)
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A propositional proof system can be viewed as a nondeterministic algorithm for the (coNP complete) unsatisfiability problem. Many such proof systems, such as resolution, are also used as the basis for heuristics which deterministically search for short proofs in the system. We discuss a propositional proof system based on algebraic reasoning, which we call the Groebner proof system because of a tight connection to the Groebner basis algorithm. For an appropriate measure of proof size, we show that (a degreelimited implementation of) the Groebner basis algorithm finds a Groebner proof of a tautology in time polynomial in the size of the smallest such proof, In other words, unlike most proof systems, the nondeterministic algorithm can be converted to a deterministic one without loss in power. We then compare the power of the Groebner proof system to more studied systems. We show that the Groebner system polynomially simulates Horn clause resolution, quasipolynomially simulates tree...
Noncommutative Elimination in Ore Algebras Proves Multivariate Identities
 J. SYMBOLIC COMPUT
, 1996
"... ... In this article, we develop a theory of @finite sequences and functions which provides a unified framework to express algorithms proving and discovering multivariate identities. This approach is vindicated by an implementation. ..."
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Cited by 90 (9 self)
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... In this article, we develop a theory of @finite sequences and functions which provides a unified framework to express algorithms proving and discovering multivariate identities. This approach is vindicated by an implementation.
On the complexity of Gröbner basis computation of semiregular overdetermined . . .
, 2004
"... ..."
An introduction to commutative and noncommutative Gröbner bases
 Theoretical Computer Science
, 1994
"... In 1965, Buchberger introduced the notion of Gröbner bases for a polynomial ideal and an algorithm (Buchberger Algorithm) for their computation ([B1],[B2]). Since the end of the Seventies, Gröbner bases have been an essential tool in the development of computational ..."
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Cited by 71 (3 self)
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In 1965, Buchberger introduced the notion of Gröbner bases for a polynomial ideal and an algorithm (Buchberger Algorithm) for their computation ([B1],[B2]). Since the end of the Seventies, Gröbner bases have been an essential tool in the development of computational
A Geometric Constraint Solver
, 1995
"... We report on the development of a twodimensional geometric constraint solver. The solver is a major component of a new generation of CAD systems that we are developing based on a highlevel geometry representation. The solver uses a graphreduction directed algebraic approach, and achieves interact ..."
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Cited by 60 (9 self)
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We report on the development of a twodimensional geometric constraint solver. The solver is a major component of a new generation of CAD systems that we are developing based on a highlevel geometry representation. The solver uses a graphreduction directed algebraic approach, and achieves interactive speed. We describe the architecture of the solver and its basic capabilities. Then, we discuss in detail how to extend the scope of the solver, with special emphasis placed on the theoretical and human factors involved in finding a solution  in an exponentially large search space  so that the solution is appropriate to the application and the way of finding it is intuitive to an untrained user. 1 Introduction Solving a system of geometric constraints is a problem that has been considered by several communities, and using different approaches. For example, the symbolic computation community has considered the general problem, in the Supported in part by ONR contract N0001490J...
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 42 (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