is presented. It describes a systematic exploration of different choices for the irreducible polynomials that generate the extension fields. It also examines all possible transformation matrices that map one field representation to another. We evaluated the optimal candidates using Xilinx and Altera FPGA implementations. And finally in the last section of the thesis two public key cryptography protocols are introduced: Public key cryptography based on semi modules over quotient semi rings and also Public Key Cryptography using Semi-Group Actions and Semi-Rings. Student: Reza Ebrahimi Atani Supervisor: Dr. Sattar Mirzakuchaki Advisor: Dr. Shahabaddin Ebrahimi Atani Referees: Dr. Ahmadian; Dr. Raeesei; Dr. Sadr; Dr Shokouhi; Dr. Shahhosseini Defense date: 12th of May by 8 AM Auditorium of Electrical Engineering Department '> is presented. It describes a systematic exploration of different choices for the irreducible polynomials that generate the extension fields. It also examines all possible transformation matrices that map one field representation to another. We evaluated the optimal candidates using Xilinx and Altera FPGA implementations. And finally in the last section of the thesis two public key cryptography protocols are introduced: Public key cryptography based on semi modules over quotient semi rings and also Public Key Cryptography using Semi-Group Actions and Semi-Rings. Student: Reza Ebrahimi Atani Supervisor: Dr. Sattar Mirzakuchaki Advisor: Dr. Shahabaddin Ebrahimi Atani Referees: Dr. Ahmadian; Dr. Raeesei; Dr. Sadr; Dr Shokouhi; Dr. Shahhosseini Defense date: 12th of May by 8 AM Auditorium of Electrical Engineering Department '>