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Showing 1 to 16 of 16 for “"Gröbner Basis"”.
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Weyl Gröbner Basis Cryptosystems
… certain class of algebraic cryptosystems called Gröbner Basis Cryptosystems. In 1994, Koblitz introduced the Polly Cracker cryptosystem that is based on the theory of Gröbner basis in commutative polynomials rings. The security of this cryptosystem relies on the fact that the computation of …
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Gröbner basis techniques for certain problems in coding and systems theory
… theory. Fitzpatrick has shown that a Göbner basis approach leads to efficient algorithms in the decoding of Reed-Solomon codes and in scalar interpolation and partial realization. This thesis simultaneously generalizes and simplifies that approach and presents applications to discrete-time …
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An F4-Style Involutive Basis Algorithm
… then give us the solution. Similarly, Gröbner Basis is the “nice form” of nonlinear equation systems that can span all the polynomials in the given ideal [4]. So we can use Gröbner Basis to analyze the solution of a nonlinear equation system.</p> <p>But how to compute a Gröbner Basis? …
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Non-commutative Gröbner Bases and Applications
Commutative Gröbner bases have a lot of applications in theory and practice, because they have many nice properties, they are computable, and there exist many efficient improvements of their computations. Non-commutative Gröbner bases also have many useful properties. However, applications of …
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Monomization of power Ideals and parking functions
… very efficient, in particular far faster than Gröbner basis techniques. We extend this monomization theory from the known case of the central zonotopal algebra to the other two main cases of the external and internal zonotopal algebras.
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Algorithms and Orders for Finding Noncummutative Gröbner Bases
… and good admissible orders for computing Gröbner bases in noncommutative algebras is considered. Gröbner bases are an important tool that make many problems in polynomial algebra computationally tractable. However, the computation of Gröbner bases is expensive, and in noncommutative …
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Gröbner Bases and Syzygy Modules
The theory of Gröbner bases has become a useful tool in computational commutative algebra. In this paper, we outline some basic results, as they are found in [1], including the concepts of terms ordering, multivariable polynomial division, Gröbner bases, Buchberger's algorithm, and syzygy modules. …
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On the Complexity of Computing Galois Groups of Differential Equations
… complexity, in the worst case, of computing a Gröbner basis is doubly exponential in the number of variables. Feng chose to represent the radical of the ideal generated by the defining equations of a proto-Galois group by its Gröbner basis. Hence, a double-exponential degree bound for computing …
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Exploiting chordal structure in systems of polynomial equations
… that relies in elimination theory and Gröbner bases. Chordal elimination can be seen as a generalization of sparse linear algebra. Unlike the linear case, the elimination process may not be exact. Nonetheless, we show that our methods are well-behaved for a large family of problems. We …
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Geometric Properties of Over-Determined Systems of Linear Partial Difference Equations
… the role of the kernel representation's Gröbner basis in the Cauchy problem, we extend notions of controllability from the classical behavioral setting to accommodate this generalization. We then address the question as to when an autonomous behavior admits a Livšic-system state-space …
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Mathematical Methods for Image Based Localization
… six papers. In the first paper, enhancements of Gröbner basis techniques to solve systems of polynomial equations are presented. The new strategies improve the numeric stability with several orders of magnitudes, compared to previous state of the art. This framework is then applied in the next …
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Die Tits-Alternative für eine relevante Klasse endlich präsentierter Gruppen unter besonderer Berücksichtigung computeralgebraischer Aspekte
… eine Methode angewandt, die auf Berechnungen von Gröbner-Basen in nicht-kommutativen Polynomringen basiert. Es wird dabei die Dimension der Gruppenringe betrachtet als Vektorräume berechnet. Für die betrachtete Klasse von Gruppen wird für Relationen der Blocklänge 1 die Tits-Alternative …
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New Techniques for Polynomial System Solving
… with two elements. Instead of using traditional Gröbner basis techniques we focus on highly developed methods from several other areas like linear algebra, discrete optimization, numerical analysis and number theory. We study some techniques from combinatorial optimization to transform a …
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Analysis and Implementation of Algorithms for Noncommutative Algebra
… a quotient of a path algebra, with a known Gröbner basis, is finite or infinite dimensional is described. This algorithm is based on the Aho-Corasick pattern matching automata. The resulting automata is used to efficiently determine the dimension of the algebra, enumerate a basis for the …
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SAT Solving Using XOR-OR-AND Normal Forms and Cryptographic Fault Attacks
… we can then find optimal GD attacks utilizing a Gröbner basis. As a case study, we apply this method to aid in the construction of novel fault attacks on the ciphers KCipher-2 and Enocoro-128v2. Using ad hoc methods combining algebraic and logical approaches, we show that both ciphers are …
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Algebraic theory for discrete models in systems biology
… which can be efficiently solved by computing Gröbner bases. Section 5 introduces a bit-wise implementation of a Gröbner basis algorithm for Boolean polynomials. This implementation has been incorporated into the core engine of Macaulay 2. Chapter 6 discusses bistability for Boolean models …