52 research outputs found
Computing a Gröbner basis of a polynomial ideal over a Euclidean domain
AbstractAn algorithm for computing a Gröbner basis of a polynomial ideal over a Euclidean domain is presented. The algorithm takes an ideal specified by a finite set of polynomials as its input; it produces another finite basis of the same ideal with the properties that using this basis, every polynomial in the ideal reduces to 0 and every polynomial in the polynomial ring reduces to a unique normal form. The algorithm is an extension of Buchberger's algorithms for computing Gröbner bases of polynomial ideals over an arbitrary field and over the integers as well as our algorithms for computing Gröbner bases of polynomial ideals over the integers and the Gaussian integers. The algorithm is simpler than other algorithms for polynomial ideals over a Euclidean domain reported in the literature; it is based on a natural way of simplifying polynomials by another polynomial using Euclid's division algorithm on the coefficients in polynomials. The algorithm is illustrated by showing how to compute Gröbner bases for polynomial ideals over the integers, the Gaussian integers as well as over algebraic integers in quadratic number fields admitting a division algorithm. A general theorem exhibiting the uniqueness of a reduced Gröbner basis of an ideal, determined by an admissible ordering on terms (power products) and other conditions, is discussed
On Factorized Gröbner Bases
We report on some experience with a new version of the well known Gröbner algorithm with factorization and constraint inequalities, implemented in our REDUCE package CALI, [12]. We discuss some of its details and present run time comparisons with other existing implementations on well splitting examples
Efficient computation of regular differential systems by change of rankings using Kähler differentials
We present two algorithms to compute a regular differential system for some ranking, given an equivalent regular differential system for another ranking. Both make use of Kähler differentials. One of them is a lifting for differential algebra of the FGLM algorithm and relies on normal forms computations of differential polynomials and of Kähler differentials modulo differential relations. Both are implemented in MAPLE V. A straightforward adaptation of FGLM for systems of linear PDE is presented too. Examples are treated
Efficient computation of regular differential systems by change of rankings using Kähler differentials
We present two algorithms to compute a regular differential system for some ranking, given an equivalent regular differential system for another ranking. Both make use of Kähler differentials. One of them is a lifting for differential algebra of the FGLM algorithm and relies on normal forms computations of differential polynomials and of Kähler differentials modulo differential relations. Both are implemented in MAPLE V. A straightforward adaptation of FGLM for systems of linear PDE is presented too. Examples are treated
Computations involving differential operators and their actions on functions
The algorithms derived by Grossmann and Larson (1989) are further developed for rewriting expressions involving differential operators. The differential operators involved arise in the local analysis of nonlinear dynamical systems. These algorithms are extended in two different directions: the algorithms are generalized so that they apply to differential operators on groups and the data structures and algorithms are developed to compute symbolically the action of differential operators on functions. Both of these generalizations are needed for applications
Gröbnerovy báze v kryptografii
PĹ™edloĹľenĂ© práce studuje vyuĹľitĂ GrobnerovĂ˝ch bázĂ v kryptografii, a to speciálnÄ› pĹ™i kryptoanalĂ˝ze blokovĂ˝ šifer. Nejprve seznamujeme se základnĂmi pojmy teorie GrobnerovĂ˝ch bázĂ a metodou pro jejich nalezenĂ, kterou je BuchbergerĹŻv algoritmus. Je vysvÄ›tlen princip Ĺ™ešenĂ soustav polynomiálnĂch rovnic pomocĂ vhodnĂ˝ch GrobrenovĂ˝ch bázĂ. NáslednÄ› je vÄ›nována pozornost modernĂm algoritmĹŻm pro nalezenĂ Grobnerovy báze, jeĹľ BuchbergerĹŻv algoritmus vylepšujĂ. V poslednà části jsou shrnuty dosavadnĂ vĂ˝sledky dosaĹľenĂ© v kryptografii pomocĂ metod zaloĹľenĂ˝ch na GrobnerovĂ˝ch bázĂch a je pĹ™edstaven pojem algebraickĂ© kryptoanalĂ˝zy. Ta pĹ™evádĂ problĂ©m prolomenĂ kryptosystĂ©mu na problĂ©m nalezenĂ Ĺ™ešenĂ soustavy polynomiálnĂch rovnic nad koneÄŤnĂ˝m tÄ›lesem. Na pĹ™Ăkladech je vysvÄ›tleno jak konstruovat soustavy polynomĹŻ vĂce promÄ›nnĂ˝ch popisujĂcĂ blokovĂ© šifry a jsou prezentovány vĂ˝sledky praktickĂ˝ch pokusĹŻ s takovĂ˝mi soustavami.The thesis focuses on the use of GrĂ„obner bases in cryptography and especially on applications in cryptanalysis of block ciphers. Some elementary concepts from the theory of GrĂ„obner bases are introduced together with Buchberger's algorithm, a method for constructing such bases. The principle of solving of poly nomial systems using suitable GrĂ„obner bases is explained. This is followed by pre sentation of modern algorithms that improve the Buchberger's algorithm. In the last part of the thesis present results achieved by GrĂ„obner bases are summarised and the notion of algebraic cryptanalysis is introduced. In algebraic cryptanalysis we transform breaking of given cryptosystem into a problem of solving polynomial equations over some nite eld. Examples of polynomial descriptions of block ciphers are provided together with some experimental result on arising polynomial systems.Department of AlgebraKatedra algebryFaculty of Mathematics and PhysicsMatematicko-fyzikálnĂ fakult
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