Number Theory for Beginners
“This blog represents a significant piece of my PhD thesis chapter, where I delved into mathematical algebraic structures within number systems. Among my fascinating discoveries, one stood out — the profound connection between prime fields and modern public key cryptography, especially elliptic curve cryptography. Through my research, I cracked prime fields' magical properties, including how they exhibit cyclic behaviour within their multiplicative groups. Furthermore, I found that prime fields possess only two subfields: the field itself and the field of integers. I learned the pivotal role of prime fields, mainly when they align with the realm of elliptic curves, in enabling the robust and secure elliptic curve cryptographic scheme.”
In this blog, you will visualise the fundamental building blocks of public key cryptography: groups, rings, integral domains, and fields. These mathematical structures form the foundation of secure cryptographic systems. I explored the properties and significance of each structure, highlighting their role in enabling secure communication. Come along with me on this journey as I break down and make understandable the complex mathematics utilised in cryptography.
After reading this blog, readers can learn about the building blocks of modern cryptography: Groups, Rings, and Fields. This course includes the following parts:
