Thesis

 The NTRU [Number theory research unit] cryptosystem is a relatively new public key cryptographic algorithm that was first introduced in 1998, and that key runs are much faster than conventional public key algorithms such as RSA, ECC. The main advantage of this cryptosystem is its high speed generation keys, which is often the most important part of public key cryptography. The security of NTRU cryptosystem comes from the interaction of the polynomial mixing system with the independence of reduction modulo two relatively prime integers' p and q.

The Fractional Order Bounded Delay Differential Equations (FOBDDE’s) has been studied in this work. The Existence and Uniqueness theorems of such type of differential equation have been proved, by using the successive approximation techniques. Also, the analytic solution of (FOBDDE’s) are presented, using Laplace Transformation, and the numerical solutions are discussed, using general one-step methods and linear multi-step methods. The comparison, among these methods and the exact solutions are presented.

Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject. In this work, some properties and basic definitions of fractional integral and derivatives of Riemann-Liouvill are presented.

This thesis consider the normal distribution with its important appearance in many statistical fields of applications. Some mathematical and statistical properties of the distribution have been collected and illustrated with moments and higher moments. Six related theorems have been studied in the applications of this type of distribution.  The estimation manner and its properties have been illustrated throughout two methods (Moment and Maximum Likelihood methods) which are used to estimate the distribution parameters theoretically.