The application of group theory behind modern cryptography

Abstract

The importance of cryptography and securing data has become increasingly important, and the safety of previous cryptographic methods are also being questioned. In the past few decades, with the rise of modern mathematical tools, notably group theory, cryptography has quickly advanced to more complex and safer levels. This paper will begin by recalling the definition and some terminology about groups. Then, it will summarize two of the most used cryptography systems with underlying group theory: the Diffie-Hellman Key Exchange Protocol, together with one of its variants, the Ko-Lee-Cheon-Han-Kang-Park Key Agreement, and the RSA Protocol. Also, it examines what the safety of a cryptography system means, and methods to increase the security of these protocols. When talking about the two protocols, extensive group theory is used, both as a means of operation and as a method to prove the protocols validity. Finally, it mentions the possible directions of improvement in this field and whether these cryptographic methods are still reliable, even with the widespread use of quantum computers in the future.