Designing pair of nonlinear components of a block cipher over quaternion integers

In the field of cryptography, block ciphers are widely used to provide confidentiality and integrity of data. One of the key components of a block cipher is its nonlinear substitution function. In this paper, we propose a new design methodology for the nonlinear substitution function of a block cipher, based on the use of Quaternion integers (QI). Quaternions are an extension of complex numbers that allow for more complex arithmetic operations, which can enhance the security of the cipher. We demonstrate the effectiveness of our proposed design by implementing it in a block cipher and conducting extensive security analysis. Quaternion integers give pair of substitution boxes (S-boxes) after fixing parameters but other structures give only one S-box after fixing parameters. Our results show that the proposed design provides superior security compared to existing designs, two making on a promising approach for future cryptographic applications

Exploring the Efficiency of the <i>q</i>-Homotopy Analysis Transform Method for Solving a Fractional Initial Boundary Value Problem with a Nonlocal Condition

This article employs the q-homotopy analysis transformation method (q-HATM) to numerically solve, subject to an integral condition, a fractional IBVP. The resulting numerical scheme is applied to solve, in which the exact solution is obtained, several test examples in order to illustrate its efficiency

Approximation on the complex sphere

The aim of this thesis is to study approximation of multivariate functions on the complex sphere by spherical harmonic polynomials. Spherical harmonics arise naturally in many theoretical and practical applications. We consider different aspects of the approximation by spherical harmonic which play an important role in a wide range of topics. We study approximation on the spheres by spherical polynomials from the geometric point of view. In particular, we study and develop a generating function of Jacobi polynomials and its special cases which are of geometric nature and give a new representation for the left hand side of a well-known formulae for generating functions for Jacobi polynomials (of integer indices) in terms of associated Legendre functions. This representation arises as a consequence of the interpretation of projective spaces as quotient spaces of complex spheres. In addition, we develop new elements of harmonic analysis on the complex sphere, and use these to establish Jackson's and Kolmogorov's inequalities. We apply these results to get order sharp estimates for m-term approximation. The results obtained are a synthesis of new results on classical orthogonal polynomials geometric properties of Euclidean spaces. As another aspect of approximation, we consider interpolation by radial basis functions. In particular, we study interpolation on the spheres and its error estimate. We show that the improved error of convergence in n dimensional real sphere, given in [7], remain true in the case of the complex sphere

Approximation on the complex sphere

The aim of this thesis is to study approximation of multivariate functions on the complex sphere by spherical harmonic polynomials. Spherical harmonics arise naturally in many theoretical and practical applications. We consider different aspects of the approximation by spherical harmonic which play an important role in a wide range of topics. We study approximation on the spheres by spherical polynomials from the geometric point of view. In particular, we study and develop a generating function of Jacobi polynomials and its special cases which are of geometric nature and give a new representation for the left hand side of a well-known formulae for generating functions for Jacobi polynomials (of integer indices) in terms of associated Legendre functions. This representation arises as a consequence of the interpretation of projective spaces as quotient spaces of complex spheres. In addition, we develop new elements of harmonic analysis on the complex sphere, and use these to establish Jackson's and Kolmogorov's inequalities. We apply these results to get order sharp estimates for m-term approximation. The results obtained are a synthesis of new results on classical orthogonal polynomials geometric properties of Euclidean spaces. As another aspect of approximation, we consider interpolation by radial basis functions. In particular, we study interpolation on the spheres and its error estimate. We show that the improved error of convergence in n dimensional real sphere, given in [7], remain true in the case of the complex sphere.EThOS - Electronic Theses Online ServiceGBUnited Kingdo