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296
A New Efficient Algorithm for Computing Gröbner Bases (F4)
 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 315 (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.
Quantum Schubert Polynomials
 J. AMER. MATH. SOC
, 1997
"... We compute GromovWitten invariants of the flag manifold using a new combinatorial construction for its quantum cohomology ring. Our construction provides quantum analogues of the BernsteinGelfandGelfand results on the cohomology of the flag manifold, and the LascouxSchutzenberger theory of S ..."
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Cited by 68 (6 self)
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We compute GromovWitten invariants of the flag manifold using a new combinatorial construction for its quantum cohomology ring. Our construction provides quantum analogues of the BernsteinGelfandGelfand results on the cohomology of the flag manifold, and the LascouxSchutzenberger theory of Schubert polynomials. We also derive the quantum Monk's formula.
Fast Hierarchical Clustering and Other Applications of Dynamic Closest Pairs
, 1999
"... We develop data structures for dynamic closest pair problems with arbitrary distance functions, that do not necessarily come from any geometric structure on the objects. Based on a technique previously used by the author for Euclidean closest pairs, we show how to insert and delete objects from an n ..."
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Cited by 60 (2 self)
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We develop data structures for dynamic closest pair problems with arbitrary distance functions, that do not necessarily come from any geometric structure on the objects. Based on a technique previously used by the author for Euclidean closest pairs, we show how to insert and delete objects from an nobject set, maintaining the closest pair, in O(nlog² n) time per update and O(n) space. With quadratic space, we can instead use a quadtreelike structure to achieve an optimal time bound, O(n) per update. We apply these data structures to hierarchical clustering, greedy matching, and TSP heuristics, and discuss other potential applications in machine learning, Gröbner bases, and local improvement algorithms for partition and placement problems. Experiments show our new methods to be faster in practice than previously used heuristics.
Effective Algorithms for Parametrizing Linear Control Systems over Ore Algebras
 APPLICABLE ALGEBRA IN ENGINEERING, COMMUNICATION AND COMPUTING
"... ..."
Variation of Cost Functions in Integer Programming
 MATHEMATICAL PROGRAMMING
, 1994
"... We study the problem of minimizing c \Delta x subject to A \Delta x = b, x 0 and x integral, for a fixed matrix A. Two cost functions c and c 0 are considered equivalent if they give the same optimal solutions for each b. We construct a polytope St(A) whose normal cones are the equivalence classe ..."
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Cited by 46 (8 self)
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We study the problem of minimizing c \Delta x subject to A \Delta x = b, x 0 and x integral, for a fixed matrix A. Two cost functions c and c 0 are considered equivalent if they give the same optimal solutions for each b. We construct a polytope St(A) whose normal cones are the equivalence classes. Explicit inequality presentations of these cones are given by the reduced Gröbner bases associated with A. The union of the reduced Gröbner bases as c varies (called the universal Gröbner basis) consists precisely of the edge directions of St(A). We present geometric algorithms for computing St(A), the Graver basis [Gra], and the universal Gröbner basis.
Algebraic Cryptanalysis of McEliece Variants with Compact Keys
 In Proceedings of Eurocrypt 2010
"... Abstract. In this paper we propose a new approach to investigate the security of the McEliece cryptosystem. We recall that this cryptosystem relies on the use of errorcorrecting codes. Since its invention thirty years ago, no efficient attack had been devised that managed to recover the private key ..."
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Cited by 38 (11 self)
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Abstract. In this paper we propose a new approach to investigate the security of the McEliece cryptosystem. We recall that this cryptosystem relies on the use of errorcorrecting codes. Since its invention thirty years ago, no efficient attack had been devised that managed to recover the private key. We prove that the private key of the cryptosystem satisfies a system of bihomogeneous polynomial equations. This property is due to the particular class of codes considered which are alternant codes. We have used these highly structured algebraic equations to mount an efficient keyrecovery attack against two recent variants of the McEliece cryptosystems that aim at reducing public key sizes. These two compact variants of McEliece managed to propose keys with less than 20,000 bits. To do so, they proposed to use quasicyclic or dyadic structures. An implementation of our algebraic attack in the computer algebra system MAGMA allows to find the secretkey in a negligible time (less than one second) for almost all the proposed challenges. For instance, a private key designed for a 256bit security has been found in 0.06 seconds with about 2 17.8 operations. 1
Hybrid approach for solving multivariate systems over finite fields
 JOURNAL OF MATHEMATICAL CRYPTOLOGY
, 2009
"... In this paper, we present an improved approach to solve multivariate systems over finite fields. Our approach is a tradeoff between exhaustive search and Gröbner bases techniques. We give theoretical evidences that our method brings a significant improvement in a very large context and we clearly d ..."
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Cited by 33 (9 self)
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In this paper, we present an improved approach to solve multivariate systems over finite fields. Our approach is a tradeoff between exhaustive search and Gröbner bases techniques. We give theoretical evidences that our method brings a significant improvement in a very large context and we clearly define its limitations. The efficiency depends on the choice of the tradeoff. Our analysis gives an explicit way to choose the best tradeoff as well as an approximation. From our analysis, we present a new general algorithm to solve multivariate polynomial systems. Our theoretical results are experimentally supported by successful cryptanalysis of several multivariate schemes (TRMS, UOV,...). As a proof of concept, we were able to break the proposed parameters assumed to be secure until now. Parameters that resists to our method are also explicitly given. Our work permits to refine the parameters to be chosen for multivariate schemes.
Algebraic Structure of Quasicyclic Codes
 DISCRETE APPL. MATH
"... We use Gröbner bases of modules as a tool in the construction and classification of quasiscyclic codes. Whereas previous studies have been mainly concerned with the 1generator case, our results elucidate the structure of arbitrary quasicyclic codes and their duals. As an application we provide a co ..."
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Cited by 29 (1 self)
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We use Gröbner bases of modules as a tool in the construction and classification of quasiscyclic codes. Whereas previous studies have been mainly concerned with the 1generator case, our results elucidate the structure of arbitrary quasicyclic codes and their duals. As an application we provide a complete characterisation of selfdual quasicyclic codes of index 2.
Gomory Integer Programs
, 2001
"... The set of all group relaxations of an integer program contains certain special members called Gomory relaxations. A family of integer programs with a fixed coefficient matrix and cost vector but varying right hand sides is a Gomory family if every program in the family can be solved by one of its G ..."
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Cited by 28 (4 self)
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The set of all group relaxations of an integer program contains certain special members called Gomory relaxations. A family of integer programs with a fixed coefficient matrix and cost vector but varying right hand sides is a Gomory family if every program in the family can be solved by one of its Gomory relaxations. In this paper, we characterize Gomory families. Every TDI system gives a Gomory family, and we construct Gomory families from matrices whose columns form a Hilbert basis for the cone they generate. The existence of Gomory families is related to the Hilbert covering problems that arose from the conjectures of Sebö. Connections to commutative algebra are outlined at the end.