A mixture experiment is an experiment in which the k ingredients are nonnegative and subject to the simplex restriction £Ux_i=1 on the (k-1)-dimensional probability simplex S^{k-1}. This dissertation discusses optimal designs for linear and quadratic log contrast models for experiments with mixtures suggested by Aitchison and Bacon-Shone (1984), where the experimental domain is restricted further as in Chan (1992). In this study, firstly, an essentially complete class of designs under the Kiefer ordering for linear log contrast models with mixture experiments is presented. Based on the completeness result, £X_p-optimal designs for all p, -¡Û<p≤1 including D- and A-optimal are obtained, where the eigenvalues of the design moment matrix are used. By using the approach presented here, we gain insight on how these £X_p-optimal designs behave. Following that, the exact N-point D-optimal designs for linear log contrast models with three and four ingredients are further investigated. The results show that for k=3 and N=3p+q ,1 ≤q≤2, there is an exact N-point D-optimal design supported at the points of S_1 (S_2) with equal weight n/N, 0≤n≤p , and puts the remaining weight (N-3n)/N uniformly on the points of S_2 (S_1) as evenly as possible, where S_1 and S_2 are sets of the supports of the approximate D-optimal designs. When k=4 and N=6p+q , 1 ≤q≤5, an exact N-point design which distributes the weights as evenly as possible among the supports of the approximate D-optimal design is proved to be exact D-optimal. Thirdly, the approximate D_s-optimal designs for discriminating between linear and quadratic log contrast models for experiments with mixtures are derived. It is shown that for a symmetric subspace of the finite dimensional simplex, there is a D_s-optimal design with the nice structure that puts a weight 1/(2^{k-1}) on the centroid of this subspace and the remaining weight is uniformly distributed on the vertices of the experimental domain. Moreover, the D_s-efficiency of the D-optimal design for quadratic model and the design given by Aitchison and Bacon-Shone (1984) are also discussed Finally, we show that an essentially complete class of designs under the Kiefer ordering for the quadratic log contrast model is the set of all designs in the boundary of T or origin of T . Based on the completeness result, numerical £X_p -optimal designs for some p, -¡Û<p≤1 are obtained.

This diploma thesis is devoted to the multivariate cryptosystems. It includes an overview of commutative algebra with emphasis on Gröbner bases. Of all algorithms, especially the ones using Gröbner bases are studied, i.e. Buchberger's algorithm, which is already implemented in Wolfram Mathematica, and F4 algorithm, for which a program package has been created in the Wolfram Mathematica environment. Also Zhuang-Zi algorithm is described. To simplify its steps a program to compute the Lagrange interpolation polynomial has been created in Python.