Gram - Schmidt process and LLL algorithm

Práce se zabývá Gram - Schmidtovým ortogonalizačním procesem a uvádí pojmy s ním související. Dále se pak zabývá mřížkami a hledáním krátké báze dané mřížky v dímenzi 2, kde je dokonce možné najít nejkratší bázi. Více se věnuje LLL algoritmu, LLL redukované bázi mřížky a aplikaci LLL algoritmu. U každé kapitoly jsou vypočítané příklady a ukázky výpočtů v programu Mathematica.Katedra matematiky, fyziky a technické výchovyObhájenoThis master thesis will be concerned with LLL algorithm. The target of the thesis is to introduce LLL algorithm to Czech readers and demonstrate contribution of algorithm in mathematical science. My thesis is divided into 4 chapters. The first chapter deals with the Gram?Schmidt process. This is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space . In mathematics, the goal of lattice basis reduction is given an integer lattice basis as input, to find a basis with short, nearly orthogonal vectors. This is realized by using different algorithms, whose running time is usually at least exponential in the dimension of the lattice. The second chapter is just about lattices and their reduction. In the third chapter, I finally defined the LLL algorithm, which can be found in polynomial time quite short based on the lattice. The fourth chapter includes application of LLL algorithm. Each chapter involves amount of practical examples for better understanding, supplemented by calculations in the computer program Mathematica 8. Illustration images are created in the program GeoGebra