The objective of this research work is to find a better understanding of the Interior fixed point theory. We plan to expand the previous result on the Boundary Fixed Point in Index theorem, which only concerned the extension maps on the 2 dimensional balls, to the similar result in the Euclidean plane on the unit-sphere-preserving maps. We also will try to propose the extended result on the Interior fixed point property theorem that was done on the n dimensional balls to n-Euclidean space with the unit-sphere-preserving maps. We will try to show that if we specify the derivative condition of the extension map then the interior fixed points will be guaranteed.
Moreover, we investigate the possibility of the applications of algebraic topology in the area of Cryptography. In particular, we want to create a new cryptosystem based on the problem in fixed point theory.
