In my previous post I introduced the idea of a group, which is an abstract mathematical concept that is defined as a set with a binary operation fulfilling a few mathematical properties. Many cryptography primitives are based on the mathematics of groups such as asymmetric encryption, digital signatures and zero knowledge proofs, which we will cover later. In this post I introduce rings and fields and explain how they relate to groups.
Group theory is a branch of mathematics in the area of abstract algebra that studies the symmetries that recur throughout mathematics. A group is a very abstract concept and a generalisation of some other algebraic types such as rings and fields which I will explain more in a future post. It turns out that the patterns and laws described by group theory show up in number theory, algebraic equations and geometry. Most importantly cryptography relies on group theory and finite fields for primitives such as the Diffie-Hellman key exchange, RSA encryption and also for zero knowledge proofs.
So what is a group?