Elliptic Curve Cryptography: Making the Simple Complex

Abstract

The fast paced development of new technologies continues to advance the newest security measures and innovative cryptographic methods. As a result, the intention of this thesis is to perform an in-depth study of the most contemporary procedure available to the general public at the time of this study: ElGamal Elliptic Curve Cryptography. This modern form of encryption merges the notion of public key cryptography with some of the most powerful mathematical proceduresknown to this field. In order to adequately establish the mathematical foundation for this applied topic, the key topics for upcoming chapters will include Cryptography, Number Theory, Abstract Algebra and elliptic curves. Subsequently, this study will merge these four topics to properly establish the procedure of elliptic curve cryptography. As a supplement to the procedures and computations introduced during the body of this thesis, Appendix A steps through the mathematical processes of the entire ElGamal cryptosystem using code generated in Maple R . Despite increasing popularity, the high level of theoretical mathematics involved in elliptic curve cryptography prevents its fast adoption. Therefore, this thesis serves as an introductory level explanation of the mathematical procedures behind the implementation of ElGamal Elliptic Curve Cryptography