The objective of this thesis is to study first the theory of fractional calculus and some of well known methods for evaluating derivatives of fractional orders for certain functions.The second objective is to study the G-spline interpolation functions and its construction using a new approach in formulating the Heremite-Birkhoff problem using fractional derivatives instead of integer order derivatives.
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 main objective of this work is to study the numerical solution of fractional ordinary differential equations using G-spline interpolation functions. Two numerical approaches are used, the first approach utilize the explicit linear multistep methods which can be applied easily for linear and nonlinear problems while the second approach is a modified approach by using the implicit linear multistep methods for solving nonlinear fractional ordinary differential equations which has so many difficulties in their solution.
