Computing A^X mod n or A^XB^Y mod n for large X, Y, and n is very important in many ElGamal-like public key cryptosystems. In this paper, we proposed using block method in sparse form to improve the performance of modular exponentiation and analyzing the computational cost by state transition diagram. We also extended the concept of Block Method and make it more general. This method is suitable for some devices with limited storage space, such as smart card.

The ability to extract accurately the stress intensity factor and the T-Stress for fractured engineering materials is very significant in the decision-making process for in-service engineering components, mainly for their functionality and operating limit. The subject of computational fracture mechanics in engineering make this possible without resulting to expensive experimental processes. In this thesis, the Finite Block Method (FBM) has been developed for the meshless study of interface stationary crack under both static and dynamic loading in bi-materials. The finite block method based on the Lagrangian interpolation is introduced and the various mathematical constructs are examined. This includes the use of the mapping technique. In a one-dimensional and a two-dimensional case, numerical studies were performed in order to determine the interpolation error. The finite block method in both the Cartesian coordinate and the polar coordinate systems is developed to evaluate the stress intensity factors and the T-stress for interface cracks between bi-materials. Using the William's series for bi-material, an expression for approximating the stress and displacement at the interface crack tip is established. In order to capture accurately the stress intensity factors and the T-stress at the crack tip, the asymptotic expansions of the stress and displacement around the crack tip are introduced with a singular core technique. The accuracy and capability of the finite block method in evaluating interface cracks is demonstrated by several numerical assessments. In all cases, comparisons have been made with numerical solutions by using the boundary collocation method, the finite element method and the boundary element method, etc.

In this work are discussed a block parameter estimation method for linear periodically time varying system. The whole work consists of two parts: theoretical and practical. The theoretical part is based on model’s description, its creation and structure. There it is shown estimation Markova, or an estimation of the least squares generalized method and the description of the generalized model. The practical part is devoted to fulfilling experiments and their describing. The conclusion about estimation of block parameter method’s achievement was also made. The experiments have been fulfilled using Matlab program. In addition count correctly Matlab (matrica, period) have been used. The results of experiments are given in the tables and schedules.

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